3G Wireless Communications and Beyond

EURASIP Journal on Applied Signal Processing

3G Wireless Communicationsand Beyond

Guest Editors: Anand G. Dabak, Erik Dahlman,and Giridhar D. Mandyam

3G Wireless Communications and Beyond


3G Wireless Communications and Beyond

Guest Editors: Anand G. Dabak, Erik Dahlman,and Giridhar D. Mandyam


Copyright © 2002 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in volume 2002 of “EURASIP Journal on Applied Signal Processing.” All articles are open accessarticles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

Editor-in-ChiefK. J. Ray Liu, University of Maryland, College Park, USA

Associate EditorsKiyoharu Aizawa, Japan Jiri Jan, Czech Antonio Ortega, USAGonzalo Arce, USA Shigeru Katagiri, Japan Mukund Padmanabhan, USAJaakko Astola, Finland Mos Kaveh, USA Ioannis Pitas, GreeceMauro Barni, Italy Bastiaan Kleijn, Sweden Raja Rajasekaran, USASankar Basu, USA Ut Va Koc, USA Phillip Regalia, FranceShih-Fu Chang, USA Aggelos Katsaggelos, USA Hideaki Sakai, JapanJie Chen, USA C. C. Jay Kuo, USA William Sandham, UKTsuhan Chen, USA S. Y. Kung, USA Wan-Chi Siu, Hong KongM. Reha Civanlar, USA Chin-Hui Lee, USA Piet Sommen, The NetherlandsTony Constantinides, UK Kyoung Mu Lee, Korea John Sorensen, DenmarkLuciano Costa, Brazil Y. Geoffrey Li, USA Michael G. Strintzis, GreeceIrek Defee, Finland Heinrich Meyr, Germany Ming-Ting Sun, USAEd Deprettere, The Netherlands Ferran Marques, Spain Tomohiko Taniguchi, JapanZhi Ding, USA Jerry M. Mendel, USA Sergios Theodoridis, GreeceJean-Luc Dugelay, France Marc Moonen, Belgium Yuke Wang, USAPierre Duhamel, France José M. F.Moura, USA Andy Wu, TaiwanTariq Durrani, UK Ryohei Nakatsu, Japan Xiang-Gen Xia, USASadaoki Furui, Japan King N. Ngan, Singapore Zixiang Xiong, USAUlrich Heute, Germany Takao Nishitani, Japan Kung Yao, USAYu Hen Hu, USA Naohisa Ohta, Japan

Contents

Editorial, Anand G. Dabak, Erik Dahlman, and Giridhar D. MandyamVolume 2002 (2002), Issue 8, Pages 755-756

Chip-Level Channel Equalization in WCDMA Downlink, Kari Hooli, Markku Juntti,Markku J. Heikkilä, Petri Komulainen, Matti Latva-aho, and Jorma LillebergVolume 2002 (2002), Issue 8, Pages 757-770

Reduced-Rank Chip-Level MMSE Equalization for the 3G CDMA Forward Link withCode-Multiplexed Pilot, Samina Chowdhury, Michael D. Zoltowski, and J. Scott GoldsteinVolume 2002 (2002), Issue 8, Pages 771-786

EM-Based Multiuser Detection in Fast Fading Multipath Environments, Mohammad Jaber Borranand Behnaam AazhangVolume 2002 (2002), Issue 8, Pages 787-796

Performance of Reverse-Link Synchronous DS-CDMA System on a Frequency-Selective MultipathFading Channel with Imperfect Power Control, Seung-Hoon Hwang and Duk Kyung KimVolume 2002 (2002), Issue 8, Pages 797-806

Joint Transmitter-Receiver Optimization in the Downlink CDMA Systems, Mohammad Saquiband Habibul IslamVolume 2002 (2002), Issue 8, Pages 807-817

An Adaptive Channel Estimation Algorithm Using Time-Frequency Polynomial Model for OFDMwith Fading Multipath Channels, Xiaowen Wang and K. J. Ray LiuVolume 2002 (2002), Issue 8, Pages 818-830

On Bandwidth Efficient Modulation for High-Data-Rate Wireless LAN Systems, John D. Terry,Juha Heiskala, Victor Stolpman, and Majid FozunbalVolume 2002 (2002), Issue 8, Pages 831-843

Dual Switched Predictive DIR MLSD Receiver for Dynamic Channels, Michael Boyleand Anthony D. FaganVolume 2002 (2002), Issue 8, Pages 844-853

Spatial Block Codes Based on Unitary Transformations Derived from Orthonormal Polynomial Sets,Giridhar D. MandyamVolume 2002 (2002), Issue 8, Pages 854-864

EURASIP Journal on Applied Signal Processing 2002:8, 755–756c© 2002 Hindawi Publishing Corporation

Editorial

Anand G. DabakWireless Communications Branch, DSPS R&D Center, Texas Instruments,12500 TI Boulevard, MS 8649, Dallas, TX 75243, USAEmail: [email protected]

Erik DahlmanEricsson Research, 164 80 Stockholm, SwedenEmail: [email protected]

Giridhar D. MandyamNokia Research Center, 6000 Connection Drive, Irving, TX 75039, USAEmail: [email protected]

The past few years have seen worldwide standardizationactivity of third-generation (3G) wireless systems. Stan-dard bodies such as the Third-Generation PartnershipProject (3GPP) and Third-Generation Partnership Project 2(3GPP2) have been developing 3G standards. The 3G sys-tems are intended to deliver high data rates and are ex-pected to handle data and multimedia applications in addi-tion to voice. Large-scale deployment of 3G systems is ex-pected to begin during 2002. The first releases of the 3Gstandards can deliver data rates up to 384 kbits with wide-area coverage and 2 Mbits for indoor environments. Fur-thermore, evolution of the 3G systems, delivering peak-data rates of up to 10 Mbits in the downlink direction,are already being standardized. One of the enabling tech-nologies to make these high data rates available is the ad-vance in signal processing. The standards developed byboth 3GPP and 3GPP2 already include support for a num-ber of advanced signal processing techniques, such as mul-tiuser detection, smart antennas, and advanced coding tech-niques.

Concurrently, much recent work in standardization hasfocused on noncellular technologies such as Bluetooth andwireless local-area network technologies (e.g., HIPERLAN,IEEE 802.11). Such systems are intended to deliver data rateshigher than 3G cellular systems under the constraints of lim-ited mobility and indoor operation. These types of technolo-gies and their operating environments present a set of chal-lenges to which advanced signal processing methods are usedto address.

The first three papers are on CDMA receiver design forthird-generation wireless systems.The paper by K. Hooli,

M. Juntti, M. J. Heikkila, P. Komulainen, M. Latva-aho, andJ. Lilleberg provides an overview of different methods forchip-level equalization in the downlink of wideband CDMA(WCDMA) systems. The paper by S. Chowdhury, M. D.Zoltowski, and J. S. Goldstein concentrates on chip-levelequalization for CDMA reception in the downlink using asimplified version of the minimum mean-squared equal-izer. The paper by M. J. Borran and B. Aazhang, on theother hand, presents a method for uplink multiuser detectionin CDMA systems based on the expectation-maximization(EM) algorithm.

The next two papers cover system level aspects of CDMA.The paper by S.-H. Hwang and D. K. Kim provides an ana-lytical approach to determining the performance of a syn-chronous CDMA system in the reverse link under the as-sumption of imperfect power control. The paper by M.Saquib and M. H. Islam describes a method for optimizingCDMA systems that employ multiple transmit and receiveantennas in the downlink.

The next two papers describe new methods for improv-ing OFDM system performance. The paper by X. Wang andK. J. R. Liu describes a new method for channel estimationin OFDM systems. The paper by J. D. Terry, J. Heiskala, V.Stolpman, and M. Fozunbal examines the performance ben-efits from coded bandwidth-efficient modulation approachesin OFDM systems.

The next paper by M. Boyle and A. D. Fagan describesa new approach to maximum likelihood sequence detec-tion (based on the Viterbi algorithm) that addresses theproblem of finding the best-desired impulse response (DIR)given the actual channel conditions. Finally, the paper by

756 EURASIP Journal on Applied Signal Processing

G. D. Mandyam presents a family of block codes that can beused in multiple transmit antenna systems.

Anand G. DabakErik Dahlman

Giridhar D. Mandyam

Anand G. Dabak received his B.Tech. inelectrical engineering from the Indian Insti-tute of Technology, Bombay, India in 1987and the M.S. and Ph.D. degrees in electricalengineering in 1989 and 1992, respectively,from Rice University, Houston, Texas. Af-ter receiving his Ph.D., he worked for ViasatInc., Carlsbad, California in satellite com-munications. He joined Texas InstrumentsInc., Dallas, Texas in 1995 and has since then worked on the systemand algorithm issues related to wireless communications. Presently,he is a Senior Member of Technical Staff and Manager of mo-bile wireless research in the DSPS R&D center of Texas Instru-ments. He has been involved in the standardization activity inthird-generation wireless systems, Bluetooth and IEEE 802.15.3.His work in applying space-time block codes for wideband CDMAsystems resulted in the standardization of open loop transmit diver-sity techniques for third-generation cellular/wireless systems. Heholds eight patents in the area of digital communications.

Erik Dahlman received his M.S. in elec-trical engineering in 1987 and the Ph.D.in Telecommunication in 1992, both fromthe Royal Institute of Technology in Stock-holm. Since 1993, he has been at EricssonResearch, involved in research and develop-ment of radio-access technologies for cellu-lar communication. Erik Dahlman was ex-tensively involved in the development andstandardization work on wideband CDMA for third-generationmobile communication, both within ETSI and ARIB. Later on, hejoined the Third-Generation Partnership Project (3GPP) for thedevelopment of the global WCDMA standard. Recently, he hasbeen involved in the standardization of WCDMA release 5, includ-ing the support for High Speed Downlink Packet Access. He is alsoinvolved in research on radio-access for future-generation cellularsystems. Erik Dahlman holds more than twenty patents within thearea of wireless communication. In 1998, he was awarded the In-ventor of the Year Award of Ericsson, and in 1999, he was awardedthe IEEE Vehicular technology Society Jack Neubauer Award for thebest system paper.

Giridhar D. Mandyam is the ResearchManager of the Wireless Data Access Groupat Nokia Research Center, Irving, Texas.He received his B.S. degree Magna CumLaude in electrical engineering from South-ern Methodist University (Dallas, Texas) in1989, the M.S. degree in electrical engineer-ing from the University of Southern Cal-ifornia (Los Angeles, California) in 1993,and the Ph.D. degree in the same major from the University ofNew Mexico (Albuquerque, New Mexico) in 1996. He has worked

for several companies on wireless communications equipment, in-cluding Qualcomm and Texas Instruments. In 1998, he joinedNokia, where he has worked on standardization and implemen-tation concepts for cdma2000, 1X-EV, and WCDMA. He has au-thored and coauthored over fourty journal and conference publi-cations and four book chapters. He also holds four US patents inthe area of wireless communications technology.


Chip-Level Channel Equalization in WCDMA Downlink

Kari HooliCentre for Wireless Communications, P.O. Box 4500 FIN-90014, University of Oulu, FinlandEmail: [email protected]

Markku JunttiCentre for Wireless Communications, P.O. Box 4500 FIN-90014, University of Oulu, FinlandEmail: [email protected]

Markku J. HeikkilaNokia Mobile Phones, P.O. Box 50, 90571 Oulu, FinlandEmail: [email protected]

Petri KomulainenNokia Mobile Phones, P.O. Box 50, 90571 Oulu, FinlandEmail: [email protected]

Matti Latva-ahoCentre for Wireless Communications, P.O. Box 4500 FIN-90014, University of Oulu, FinlandEmail: [email protected]

Jorma LillebergNokia Mobile Phones, P.O. Box 50, 90571 Oulu, FinlandEmail: [email protected]

Received 14 August 2001 and in revised form 7 March 2002

The most important third generation (3G) cellular communications standard is based on wideband CDMA (WCDMA). Receiversbased on TDMA style channel equalization at the chip level have been proposed for a WCDMA downlink employing long spread-ing sequences to ensure adequate performance even with a high number of active users. These receivers equalize the channelprior to despreading, thus restoring the orthogonality of users and resulting in multiple-access interference (MAI) suppression.In this paper, an overview of chip-level channel equalizers is delivered with special attention to adaptation methods suitable forthe WCDMA downlink. Numerical examples on the equalizers’ performance are given in Rayleigh fading frequency-selectivechannels.

Keywords and phrases: WCDMA, multiple-access interference, channel equalization.

1. INTRODUCTION

The air interface of universal terrestrial radio access (UTRA),the most important third generation (3G) cellular mobilecommunications standard, is based on wideband code-division multiple-access (WCDMA). In 3G, cellular net-works downlink capacity is expected to be more crucialthan uplink capacity due to asymmetric capacity require-ments, that is, the downlink should offer higher capacitythan the uplink [1]. Therefore, the use of efficient down-link receivers is important. In order to avoid performancedegradation, near-far resistant (or multiuser) receivers [2]

can be used. Several suboptimal receivers feasible for prac-tical implementations have been proposed, including linearminimum mean-squared error (LMMSE) receivers [3, 4].The adaptive versions of LMMSE receivers are the mostpromising for single-user terminals. However, the adap-tive symbol-level LMMSE receivers rely on cyclostation-arity of multiple access interference (MAI), and thus re-quire periodic spreading sequences with a very short pe-riod. Hence, they cannot be applied to the frequency divi-sion duplex (FDD) mode of the WCDMA downlink, whichuses spreading sequences with a one radio frame (10 ms)period.


In a synchronously transmitted downlink employing or-thogonal spreading codes, MAI is mainly caused by mul-tipath propagation (neighboring cells form another sourceof MAI). Due to the nonzero cross-correlations betweenspreading sequences with arbitrary time shifts, there is in-terference between propagation paths (or Rake fingers) afterdespreading, causing MAI. With a moderate or high numberof active users, the performance of a Rake receiver becomeslimited by interpath MAI. If the received chip waveform, dis-torted by the multipath channel, is equalized prior to correla-tion by the spreading code or matched filtering, there is onlya single path in the despreading. With orthogonal spreadingsequences the equalization effectively retains, to some extent,the orthogonality of users lost due to multipath propagation,thus suppressing MAI. Since the signal is equalized at thechip level, not on the symbol level, they can also be applied insystems using long spreading sequences. Such a receiver con-sists of a linear equalizer followed by a single correlator anda decision device.

The chip-level channel equalizer has proven to be one ofthe most promising terminal receivers for a WCDMA/FDDdownlink. It has drawn attention and inspired numerouspublications in recent years. Chip-level equalization has beentreated in [5, 6, 7, 8, 9, 10]. It has been addressed in mis-cellaneous contexts also earlier, as in [11]. Although chip-level equalization resembles at first glance the classical linearTDMA-type equalization (time-division multiple access), ithas several aspects characteristic for CDMA or 3G networks,thus making it an interesting research topic. These aspectsinclude efficient adaptation of the equalizer due to differ-ent training signals in CDMA, soft hand-over, and trans-mit diversity. Chip-level equalization was considered withsoft hand-over in [12, 13], and with different transmit diver-sity schemes in [14, 15]. Performance evaluations of chip-level equalizers with channel coding have been presentedin [16], and with adaptive equalizers in [17, 18]. A chipequalizer based receiver for a CDMA downlink employ-ing a long scrambling sequence and nonorthogonal chan-nelization sequences was proposed in [19]. A large vari-ety of adaptive chip-level equalizers have also been pre-sented and studied in a number of publications, includ-ing, for example, [20, 21, 22, 23, 24, 25, 26, 27]. A differ-ent approach for improving the performance of downlinkreceiver, that is, generalized Rake receiver, is discussed in[28].

The purpose of this paper is to provide an overview ofchip-level equalization and on the adaptation methods pro-posed for chip-level equalizers. The possible performancegains offered by the chip-level channel equalizers are ad-dressed, and the performance of six different adaptive chipequalizers are compared in a frequency-selective fading chan-nel. The paper is organized as follows. The system modelis defined in Section 2, and the zero-forcing and LMMSEequalizers with perfect knowledge of the channel are ad-dressed in Section 3. The adaptive versions of chip-levelequalizers are discussed in Section 4. Numerical examples forthe receivers are presented in Section 5, followed by conclud-ing remarks in Section 6.

2. SYSTEM MODEL

In this section, we present the system model used in definingthe receivers in the sequel. Only a single-base station and asingle-receive antenna are included into the system model.1

With a single-receive antenna, signal processing is restrictedto the time domain. Due to the lack of spatial characteristicsin the signal structure, moderate level other cell interferencecan be considered to be included in the white Gaussian noise.

Since the downlink is considered, all signals are syn-chronously transmitted through the same multipath chan-nel. The complex envelope of the received signal at the userterminal can be written as

r(t) =K∑k=1

rk(t) + n(t)

=K∑k=1

Mk−1∑m=0

Akb(m)k

L∑l=1

cl(t)s(m)k

(t −mTk − τl

)+ n(t),

(1)

where K is the number of users, Mk is the number of kthuser’s symbols in the observation window,2 L is the number

of paths, Ak is the average received amplitude of kth user, b(m)k

is the mth symbol of kth user, cl(t) is the time-variant com-

plex channel coefficient of lth path, s(m)k (t) is the spreading

waveform of mth symbol of kth user given by convolutionof spreading sequence and chip waveform, Tk is the symbolinterval for kth user, τl is the delay of lth path, and n(t) iscomplex white Gaussian noise process.

The discrete-time received signal after appropriate down-conversion and filtering can be written as

r =K∑k=1

DCSkAkbk + n ∈ CNcNs , (2)

where Ns is the number of samples per chip and Nc is thenumber of chips in the observation window.3 In (2), D =[d(1)

1 , . . . ,d(1)L ,d(2)

1 , . . . ,d(Nc)L ] ∈ RNcNs×LNc is a path delay and

chip waveform matrix where the column vector d(n)l contains

samples from appropriately delayed chip waveform for thelth path of nth chip, C = diag(c(1), . . . , c(Nc)) ∈ CLNc×Nc is ablock diagonal channel matrix with column vector c(n) ∈ CL

containing the time-variant channel coefficients for L paths.The term DC models the combination of chip waveformand multipath channel and is common for all users; Sk =diag(s(1)

k , . . . , s(Mk)k ) ∈ CNc×Mk is a block diagonal spread-

ing sequence matrix where column vector s(m)k ∈ ΞGk

s , Ξs isthe chip alphabet, contains the spreading sequence for thekth user’s mth symbol with a spreading factor Gk. The cellspecific scrambling sequence is included in the spreadingsequences, that is, Sk = SSc(t)Sk,Ch, where the scrambling

1It is straightforward to extend the model to include multiple-base sta-tions or receive antennas.

2Thus the model allows multiple transmission rates.3Nc = GkMk , where Gk is the spreading factor for kth user. The product

GkMk is constant for all users.

Chip-Level Channel Equalization in WCDMA Downlink 759

sequence is in the diagonal matrix SSc(t) and the block di-agonal matrix Sk,Ch contains the user specific channeliza-tion sequence for the kth user. The sequences are normal-ized so that SH

k Sk = I, and they are also orthogonal, thatis, SH

k S j = 0 if k = j. The average received amplitude forthe kth user is contained in a diagonal matrix Ak = AkIMk ,

4

vector bk = [b(1)k , . . . , b(Mk)

k ]T ∈ ΞMk

b , Ξb is the symbol alpha-bet, contains the transmitted symbols of kth user, assumed tobe i.i.d. with unit variance, and n ∈ CNcNs contains samplesfrom a white complex Gaussian noise process with covari-ance Cnn = σ2

nINcNs .

3. RECEIVERS

The chip-level version of zero-forcing and LMMSE equaliz-ers as well as the conventional Rake receiver are presented inthis section. As mentioned earlier, the multipath channel isequalized prior to despreading in the chip equalizer receivers,thus restoring to some extent the orthogonality of channel-ization codes and suppressing MAI. This means that the termDC in (2) is suppressed. The equalizer is followed by a singlecorrelator and decision device.

The equalizers treated in this paper are all linear, due tothe difficulties faced when nonlinear equalizers are consid-ered for chip-level equalization. The detection of the desireduser at chip-level is highly counter-intuitive, since it wouldignore the processing gain associated with spreading. Thus,the desired output signal of a chip-level equalizer is the to-tal transmitted signal from the base station, that is, the sumof all users’ signals. The nonlinear equalizers rely on priorknowledge of constellation of the desired signal. This infor-mation is not easily available at the terminal, since the con-stellation of all users’ signal sum is a high-order QAM withuneven spacing. The constellation order changes at framerate (100 Hz) with the changing number of active users, andthe spacing in the constellation changes at power control rate(1.5 kHz). However, for example, in [29] an interesting non-linear equalizer based on soft decision cancelling was pre-sented for a WCDMA downlink.

With the introduced system model, the decision variableof the Rake receiver for arbitrary selected user 1 is given by

yR = SH1 CHDHr, (3)

that is, the received signal is filtered by the chip waveform,appropriately delayed and weighted with channel coefficientsin the Rake fingers, coherently combined and finally de-spread. The abnormal order of the maximal ratio combin-ing (MRC) and despreading has no effect on receiver perfor-mance due to the linearity of operations.

When a zero-forcing equalizer is applied, the tap coeffi-cients of the equalizer are given by

WZ = DC(

CHDHDC)−1

. (4)

4The use of Ak allows power differences between different symbols of anindividual user. In here, constant power A2

k is assumed.

The first term on the right-hand side, DC, performs chipwaveform matched filtering, and is followed by the conven-tional zero-forcing equalizer [30, 31]. The decision variableafter despreading can be written as

yZ = SH1 WH

Z r = SH1

(CHDHDC

)−1CHDHr. (5)

In (4), the zero-forcing equalizer is given in block form(WZ ∈ CNcNs×Nc). The filter form of the zero-forcing equal-izer is obtained by taking a middle column from WZ .5

The second considered equalizer, an LMMSE chip-levelequalizer, can be obtained by solving

WL = arg minW

E

∣∣∣∣∣WHr−K∑k=1

SkAkbk

∣∣∣∣∣2, (6)

where E[·] denotes expectations, and the minimization iscarried out elementwise. By estimating the sum of all users’chips instead of the chips of a single user, the signal-to-noiseratio (SNR) faced in the estimation problem is significantlyimproved. It should also be noted that the chip equalizer doesnot try to suppress other users’ signals at the chip level, butjust restores the orthogonality of users. It can be shown thatfor (6) WL is given by [32]

WL =(

DC

( K∑k=1

A2kSkSH

k

)CHDH + σ2

nI

)−1

DC, (7)

resulting in the decision variable

yL = SH1 WH

L r

= SH1 CHDH

(DC

( K∑k=1

A2kSkSH

k

)CHDH + σ2

nI

)−1

r.(8)

The standard symbol-level LMMSE equalizer [3, 4], fea-sible for DS-CDMA systems employing short spreading se-quences, is defined for user 1 by

WLs = arg minW

E[∣∣WHr− b1

∣∣2], (9)

which results in the decision variable

yLs = AH1 SH

1 CHDH

(DC

( K∑k=1

A2kSkSH

k

)CHDH + σ2

nI

)−1

r.

(10)

Comparing (8) and (10), we can see that, for a given ob-servation window, the chip-level and symbol-level LMMSEreceivers are equal up to a scalar. This is an expected re-sult, since the LMMSE estimator commutes over linear (oraffine) transformations [32], like despreading. The equal per-formance of LMMSE receivers was verified numerically in[33].

5In the sequel, filter forms are denoted with lower case w.


As seen from (7), the LMMSE solution of a chip-levelequalizer depends on the spreading sequences of all userswith the period of a long scrambling code. This followsfrom the dependency between consecutive chips to be es-timated. The optimal solution changes from chip to chip,and an adaptive chip-level equalizer will not reach the exactoptimal tap coefficients. The adaptive versions of chip-levelLMMSE equalizer are built on simplifications that (i) spread-ing sequences are random, and (ii) the random spreading se-quences are white and independent from those of other users.The simplifications are unavoidable due to the nonstationar-ity of the LMMSE solution in (7).6 The LMMSE chip-levelequalizer derived according to the simplifications (or false as-sumptions) is given by

WL =(s2

K∑k=1

A2kDCCHDH + σ2

nI

)−1

DC, (11)

where s2 denotes the square value of chip. The decision vari-able after correlation with the spreading sequence is given by

yL = SH1 WH

L r

= SH1 CHDH

(s2

K∑k=1

A2kDCCHDH + σ2

nI

)−1

r.(12)

The performance losses caused by the aforementioned sim-plifications can be assessed by comparing the performancesof the equalizers given in (7) and (11) [33, 34]. Again, thechip-level equalizer in (11) is in a block form, and the equal-izer is obtained in filter form by taking a middle column fromWL.

4. ADAPTATION METHODS

In this section, several methods for adapting chip-levelequalizer are discussed. WCDMA terminal receivers haveaccess to two-pilot signals, one carried on the continu-ous common pilot channel (CPICH) and another carriedon the time-multiplexed dedicated physical control chan-nel (DPCCH). The high content of pilot signals combinedwith the DS-CDMA signal structure allows accurate chan-nel response estimation in the terminal. On the other hand,pilot signals form only a small portion of the received sig-nal. Thus, for the adaptation of the equalizer, a good es-timate of the channel response and a relatively weak buttime-continuous reference signal are available. As a result,several adaptation methods that are not typical with linearTDMA equalizers have been proposed for chip-level equal-ization. In [22] equalizing was proposed to be carried out byan adaptive chip separation filter, which is based on blindlydecorrelating the multipath combined chip estimates. Themethod applies to systems employing long random scram-bling codes, in which the original transmitted multiuser chip

6The LMMSE solution cannot be considered to be cyclostationary, sincethe channel is likely to change during the spreading sequence period of10 ms.

sequence is also uncorrelated. Kalman filtering was appliedfor chip-level equalization in [25]. Constrained minimumoutput energy equalizers, minimizing the energy in the sub-space of unused spreading sequences, are proposed for IS-95and CDMA2000 terminals in [21, 35, 36, 37]. However, theyrequire knowledge of unused spreading sequences, thus im-posing restrictions too strict for radio resource managementin WCDMA/FDD systems.

The most straightforward adaptation, that is, the leastmean square (LMS) adaptation based on the common pi-lot channel (CPICH), is studied in Section 4.1. The use of apilot channel as a reference signal was suggested in [7], butthe adaptation was not studied any further. In [19], a pilotchannel was proposed to be used with the stabilized fast a-posteriori error sequential technique (SFAEST) and in [38]with the multistage nested Wiener filter (MSNWF). In thefollowing, the adaptation is done at chip rate. The adapta-tion can be performed also at symbol rate, as shown in [39].7

The use of Griffiths’ algorithm [40] with a chip-level equal-izer is presented in Section 4.2, followed by the introductionof a minimum output energy equalizer constrained with thechannel response in Section 4.3. Equalizers consisting of aseparate filter and Rake receiver are referred to as prefilter-Rake equalizers in the sequel [20, 27]. In Section 4.4, adap-tive versions of a prefilter-Rake based on square-root RLS,the Levinson algorithm and Griffiths algorithm are studied.Finally, the properties of adaptive equalizers are compared inSection 4.5.

4.1. Common pilot channel based adaptation

In the adaptive equalizers discussed in Sections 4.1, 4.2, and4.3, the appropriately sampled received signal is filtered bythe chip waveform, equalized and correlated with the spread-ing sequence. The decision variable for user 1 after correla-tion with the spreading sequence is given by y = SH

1 z, wherevector z contains equalizer outputs for corresponding chipintervals. The nth element of z is w(n)Hr(n), where w(n) ∈C2D+1 contains the equalizer taps and r(n) = [r(nNs − D),. . . , r(nNs), . . . , r(nNs + D)]T contains output samples fromthe chip waveform matched filter within the equalizer at nthchip interval. The number of samples per chip is given by Ns

and 2D + 1 is the number of equalizer taps.The most straightforward solution to the adaptation of

a chip-level equalizer is to use the normalized LMS (NLMS)algorithm with a common (or dedicated) pilot channel as areference signal. The receiver is depicted in Figure 1, and re-ferred to as a CPICH trained equalizer in the following. TheNLMS adaptation step for the equalizer is [41, Chapter 9]

wC(n + 1) = wC(n) + µr(n)e∗(n)rH(n)r(n)

, (13)

where 0 < µ < 2 is the adaptation step size, and e∗(n) =(s(n)−wH

C (n)r(n))∗ is the complex conjugate of error signalbetween the equalizer output and reference signal s(n).

7When comparing results, one should pay attention to the channelmodel assumptions made for the fractionally spaced equalizer.


r r z yChip waveform

matched filterEqualizer Correlator

LMS + ×−

Pilot signal

Spreadingsequence

Figure 1: CPICH trained LMS equalizer.

Since the adaptation is carried out at chip level, the chipsof CPICH are used as reference samples. It should be notedthat although the error e(n) contains the signals of all activeusers, also the desired one, the equalizer does not suppressthem due to the pseudorandomness of spreading codes. Therelatively large error signal values also mean that the SNRin the adaptation is low, and small adaptation step sizes arerequired to provide sufficient averaging. The small values ofµ are partially compensated by a high adaptation rate.

4.2. Griffiths’ algorithm adapted equalizer

Several adaptation algorithms are obtained through differentapproximations of the gradient vector of the mean square er-ror cost function

J = E[∣∣d −wHr

∣∣2], (14)

where d is the desired output of the equalizer. The gradientvector is given by

∇J = −2E[d∗r

]+ 2E

[rrH]w, (15)

where E[d∗r] is the cross-correlation vector between the in-put signal r and the desired output of the equalizer, andE[rrH] = R is the covariance matrix of the input signal [41].For example, the aforementioned standard LMS algorithm isobtained by replacing expectations with instantaneous esti-mates, that is, signal vectors r(n).

In [23], the Griffiths algorithm is used for the adaptationof a chip-level channel equalizer. The algorithm is obtainedfrom (15) by replacing the cross-correlation vector E[d∗r]with p, the channel response estimate. Instantaneous esti-mates are used for the covariance matrix R, as in the LMSalgorithm. The resulting adaptation is

wG(n + 1) = wG(n)− µ(z∗(n)r(n)− p

), (16)

where µ is the adaptation step size and z(n) is the equalizeroutput at nth chip interval.

4.3. CR-MOE equalizer

In the channel-response constrained minimum-output-energy (CR-MOE) equalizer [26], the equalizer is decom-posed into a constraint (or nonadaptive) component and anadaptive component. This is the well-known idea of the gen-

eralized side-lobe canceler, described, for example, in [41,Chapter 5]. The same approach has been applied in blindMOE multiuser receivers, in which the spreading sequenceof a desired user is used as the constraint [2, 42]. As men-tioned, the equalizer is decomposed into two parts, that is,wM = p + x. The channel response estimate p is used as thenonadaptive part, and the adaptive part x is constrained ontoa subspace orthogonal to p to avoid suppression of the de-sired signal. The mean square error J = E|d − wH

M r|2 can bewritten as J = E[d2] − 2pHp + (p + x)HE[rrH](p + x). Fora given p, the mean square error is minimized by minimiz-ing the last term of J , that is, the equalizer output energy—other terms of J are not affected by vector x. To obtain anadaptive algorithm for x, stochastic approximation is ap-plied to the gradient ∇J = E[rrH](p + x) taken with re-spect to x. The orthogonality condition is maintained at eachiteration by projecting the gradient onto the subspace or-thogonal to p. The orthogonal component of the gradientis

∇J⊥p =(

r− ppHrpHp

)rHwM, (17)

and the resulting adaptation algorithm becomes

x(n + 1) = x(n)− µz∗(n)(

r(n)− zp(n)p), (18)

where zp(n) = pHr(n)/(pHp) is the output of the channelresponse filter normalized with the energy of the channel re-sponse estimate, and z(n) = wM(n)Hr = (p(n) + x(n))Hr isthe output of the CR-MOE equalizer. The structure of thisequalizer is depicted in Figure 2.

The CR-MOE has the typical weaknesses of the MOEadaptation [2]. The orthogonality between x and the channelresponse estimate p is lost when p is updated. Thus periodicre-orthogonalization of x is required, given by

x⊥p = x − pHxpHp

p. (19)

The second problem of the MOE adaptation is the un-avoidable estimation error in p. Due to the estimation error,x has small projection on true p while maintaining orthog-onality with p. Since x is adapted to minimize output en-ergy, the projection on p translates to partial suppression ofthe desired signal component. The surplus energy χ = ‖x‖2,where ‖x‖2 = xHx, required for total suppression of the de-sired signal component is given by

χ ≥ ‖p‖2∣∣pHp

∣∣2

‖p‖2‖p‖2 − ∣∣pHp∣∣2 . (20)

Since the channel estimation error is usually relatively small,(|pHp|2 ∼ ‖p‖2‖p‖2), suppression of the desired signalmeans large ‖x‖2 values and significant noise enhancement.Therefore, in noisy environments the suppression remains atacceptable levels. However, to avoid the desired signal sup-pression at high SNR, ‖x‖2 values must be restricted. Onesolution is to introduce tap leakage [2], that is,


r Chip waveformmatched filter

Channelestimation

Filter p

Filter x

Adaptationalgorithm

+ Correlatory

Figure 2: Channel-response constrained minimum-output-energy (CR-MOE) equalizer.

r Chip waveformmatched filter

Channelestimation

Adaptiveprefilter

QR-RLS

Rakey

Figure 3: Prefilter-Rake receiver with inverse QR-RLS adaptation.

x(n + 1) = (1− µα)x(n)− µz∗(n)(

r(n)− zp(n)p), (21)

where α, a small positive constant, controls the tap leakage.On the other hand, large values of α cause too large tap leak-age, thus preventing efficient channel equalization. However,a predefined constant α has proven to be adequate, if p isnormalized with its energy.

4.4. Prefilter-Rake equalizer

When (3) and (12) are compared, it can be seen that theequalizer consists of a received signal’s covariance matrix in-verse, and of the part corresponding to the conventional Rakereceiver. In [20] it was suggested to actually divide the equal-izer into these parts, and to use the matrix inversion lemma(or appropriate parts of the RLS-algorithm) for the estima-tion of covariance matrix inverse R−1 (R = E[rrH]). If thechip waveform matched filter is sampled at chip rate, thecovariance matrix has a Toeplitz structure. Also the matrixinverse R−1 approaches a Toeplitz matrix with increasing ma-trix dimension and finite effective length of the autocorrela-tion function. Thus the multiplication with R−1 can be effec-tively replaced by filtering r with a middle row of R−1.

The structure of prefilter-Rake with channel estimation isdepicted in Figure 3. A filter matched to the chip waveform ispreceding the prefilter, and the output of the matched filteris sampled at chip rate. The output of the prefilter is fed tothe Rake receiver performing despreading and maximal ratiocombining. There exist several possibilities for adapting the

prefilter. In the following, three methods are presented. First,a blind approach utilizing square-root RLS algorithms [27] isdiscussed, followed by a description of adaptation using theLevinson algorithm. The use of the Levinson algorithm wasproposed also in the simultaneous and independent work ofMailaender [43]. Finally, the application of Griffiths’ algo-rithm to the prefilter adaptation [18] is discussed.

Square-root RLS

The coefficients of the prefilter v(n) are given by the mid-dle column of R−1(n),8 estimate of covariance matrix in-verse R−1. The square-root matrix of R−1(n) can be up-dated using appropriate parts of the (inverse) QR-RLS al-gorithm [41, Chapter 14], or as in this paper, HouseholderRLS (HRLS) [44]. Both algorithms operate on the square-root matrix R1/2(n) of R−1, that is, R−1(n) = R1/2(n)RH/2(n).Thus the prefilter coefficients are obtained by v(n) =R1/2(n)[RH/2(n)]:,D+1, where [RH/2(n)]:,D+1 denotes the mid-dle column of RH/2(n). The square-root matrix is restrictedto a triangular matrix in the QR-RLS algorithm, whereas inthe HRLS algorithm there is no such restriction. This allowsan efficient use of block annihilation properties of House-holder reflection. The HRLS update of RH/2(n) can be writ-ten as [44]

Θ(n)

[β−1/2RH/2(n) β−1/2RH/2(n)r(n)

0T 1

]

=[

RH/2(n + 1) 0

kT(n) −δ(n)

],

(22)

where 0 < β < 1 is a weighting factor and RH/2(n)r(n)is a preprocessed input vector; Θ(n) is a Householder trans-formation matrix annihilating RH/2(n)r(n) to a zero vec-tor in the postarray. The resulting adaptation algorithm forthe prefilter coefficients is tabulated in Table 1. It should benoted that prefilter-Rake does not require computation ofδ(n) or k(n) in the postarray. Although the square-root RLS

8The filtering with a prefilter is defined as vH(n)r(n).


Table 1: Prefilter adaptation with Householder RLS algorithm.

At every prefilter update

A = β−1/2RH/2(n)

a = Ar(n)

B = AHa

γ = √aHa + 1

γ = [γ(γ + 1)]−1

B = γB

RH/2(n + 1) = A− aBH

v(n + 1) = R1/2(n + 1)[RH/2(n + 1)]:,D+1 Prefilter length is 2D + 1

algorithms are computationally intensive, the filter lengthsin the chip-level equalizers are relatively short, roughly twicethe effective channel delay spread.

Levinson algorithmSince the ideal prefilter coefficients v(n) are given by a middlecolumn of matrix inverse R−1, the prefilter coefficients can beobtained by solving periodically

R(n)v(n) = [I]:,D+1, (23)

where [I]:,D+1 is the middle column of the identity matrix,where I is the identity matrix, and 2D + 1 is the prefilterlength. Due to the Toeplitz structure of R, (23) is efficientlysolved with the standard Levinson algorithm, tabulated, forexample, in [45]. Since the covariance matrix R is Toeplitzand Hermitean, it is defined by its first row. Thus, it is suffi-cient to estimate the autocorrelation function ρ of r for non-negative delays.

Estimation of autocorrelation function can be done ei-ther directly from the received signal [32], or the estimate ρcan be calculated from the channel response estimate p andnoise power estimate σ2

n by

ρ[k] = pH1:2D+1−kp1+k:2D+1 + δkσ

2n (24)

for lags k = 0, . . . , 2D. Calculation of the autocorrelation esti-mate from the channel response estimate has lower complex-ity than the aforementioned possibilities, since the channelresponse estimate is required in other parts of receiver andthe autocorrelation estimate can be updated at the prefilterupdate rate. However, it is less robust against any unknowninterfering signals.

To maintain good performance, the prefilter coefficientsshould be recalculated several times during the coherencetime of the channel. Thus the update frequency can be fixed,for example, to the WCDMA slot rate (1.5 kHz), or can bemade to adapt to the velocity of the terminal. To avoid anyperformance degradations due to a compromise with theprefilter update rate, a 15 kHz update rate is used in the sim-ulation cases presented in Section 5.

Modified Griffiths’ algorithmThe ideas of Griffiths’ algorithm can be used to deriving anLMS algorithm variant for the prefilter adaptation [18]. The

gradient of mean square error cost function for the prefilterv is

∇J = −2E[d∗r

]+ 2E

[rrH]v. (25)

The desired response of the prefilter is the received signal fil-tered with the desired prefilter, that is, d = [R−1]H

:,D+1r. Thisallows a further development of the cost function gradient

∇J = −2E[

rrH][R−1]:,D+1 + 2E

[rrH]v

= −2[

I]

:,D+1 + 2E[

rrH]v.(26)

The adaptive algorithm is obtained by replacing the remain-ing expectation with an instantaneous estimate of expecta-tion, that is, with rrH. The resulting adaptation algorithm forthe prefilter can be written as

v(n + 1) = v(n) + µ([I]:,D+1 − g∗(n)r(n)

), (27)

where g(n) = v(n)Hr(n) is the output of the prefilter.Since the inverse of a Hermitean Toeplitz matrix is

Hermitean and persymmetric [46], the optimal prefilter[R−1]:,D+1 is conjugate symmetric with respect to its middleelement. This can be utilized to speed up the adaptation byforcing

vD−m(n) = 12

(vD−m(n) + v∗D+m(n)

),

vD+m(n) = v∗D−m(n), m = 0, 1, . . . , D(28)

at each adaptation step and by using filter v(n) instead of v(n)to generate the new prefilter output.

4.5. Observations on adaptive equalizers

In this section, the similarities and differences between thestudied equalizers are briefly discussed. Firstly, it should benoted that the computational requirements of the presentedequalizers are quite similar, with the exception of prefilter-Rake with square-root RLS adaptation. The complexity ofthe prefilter-Rake with RLS-type adaptation increases as aquadratic function of the number of taps, whereas the com-plexities of the other equalizers depend roughly linearly onthe number of taps. Thus, the complexity of the square-root RLS adapted prefilter-Rake is considerably higher thanthe complexities of the other presented equalizers. Also theLevinson algorithm has quadratic complexity, but the highercomplexity is effectively compensated for by the significantlylower activation rate of the Levinson algorithm.

A CPICH trained equalizer treats other received signalsthan CPICH signal as noise, whereas all other equalizers uti-lize the whole received signal from the desired base stationin the adaptation. The use of the whole received signal sig-nificantly enhances the SNR available in the equalizer adap-tation, thus providing an advantage for the other equalizersover the CPICH trained equalizer.

On the other hand, a CPICH trained equalizer does notrequire channel response estimate, whereas the other equal-izers rely on channel estimation. In prefilter-Rake receivers


Table 2: Channel parameters.

Vehicular A Channel II

Path # Power [dB] Delay [ns] Power [dB] Delay [ns]

1 0 0 0 0

2 −1 310 −3 521

3 −9 710 −6 1042

4 −10 1090 — —

5 −15 1730 — —

6 −20 2510 — —

channel estimates are needed for maximal ratio combining,but the adaptation of the prefilter can be carried out blindly.In the CR-MOE and Griffiths’ equalizers, MAI suppressionis based on channel response estimate. With the CR-MOEequalizer an inaccurate channel response estimate also resultsin self-cancellation. Thus the CR-MOE receiver can be con-sidered to be the most sensitive to channel estimation errorsof the considered equalizers.

It can be easily noted that the CR-MOE and Griffiths’equalizers have a distinctive resemblance. For example, thepart of the adaptation step orthogonal to p in (16) is equal tothe adaptation step in (18), assuming the same equalizer tapsw(n). However, the estimated channel response is directly in-serted into the equalizer in CR-MOE, whereas in Griffiths’algorithm it is gradually introduced through the adaptation.This provides a small advantage in the tracking of the chang-ing channel.

Finally, it should be noticed that the prefilter-Rake struc-ture offers an additional advantage in the soft hand-over sit-uation. Since the prefilter depends only on the properties ofreceived signal, the same prefilter can be used in the detec-tion of signals coming from different base stations. In the softhand-over situation, the prefilter is followed by more thanone Rake receiver, each of them assigned to a different basestation.

5. NUMERICAL EXAMPLES

To obtain a good understanding and comparison of the po-tential performance that the presented receivers can offer, biterror probabilities (BEPs) and signal-to-interference-plus-noise ratios (SINRs) were evaluated for the Rake receiver,zero-forcing equalizer and LMMSE equalizers with ideal co-efficients. Bit error rates (BERs) were simulated for the pre-sented adaptive chip-level equalizers.

The performance evaluations were carried out inRayleigh fading frequency-selective channels described inTable 2. The ITUs vehicular A channel model [47] was usedwith BEP and SINR evaluations. The BER simulations werecarried out in Channel II, which is a 3-path channel with anexponentially decaying average power profile. QPSK modu-lation was used employing root raised cosine pulses with aroll-off factor of 0.22. Random cell specific scrambling andWalsh channelization codes were used. The chip rate was setto 3.84 Mchip/s corresponding to a 260 ns chip interval. The

received signal was modeled with two samples per chip, andchannel coding was excluded from the study.

5.1. Bit error probability andsignal-to-interference-plus-noise evaluations

BEPs were evaluated by applying the semianalytical methoddiscussed, for example, in [48]. For a linear receiver w,9

the conditional bit error probability can be expressed in theform10

Pb[E | C,b1, . . . ,bK

]= Q

(Re

(∑Kk=1 wHDCSkAkbk

)√σ2nwHw

),

(29)

given that the arbitrary selected desired user 1 transmittedQPSK symbol 1. The bit error probability is conditional onchannel realization, on the symbols of other users as well ason the previous and following symbols of the desired user.The BEPs were obtained by taking the average of the condi-tional bit error probabilities over a sample set of bit patternand channel realizations. A sample set size of 8000 realiza-tions was used in the evaluations.

In the same manner SINR was also evaluated. The con-ditional SINR depending on channel realization and bit pat-terns is given by

γ= Re(

wHDCS1A1b1)2

σ2nwHw+

∣∣∑Kk=2 wHDCSkAkbk

∣∣2+Im

(wHDCS1A1b1

)2 ,

(30)and a Gaussian approximation for the conditional bit errorprobability is obtained by

Pb[E | C,b1, . . . ,bK

] = Q(√

γ). (31)

Both the conditional SINR and the Gaussian approximationfor conditional bit error probability were sampled over asample set of bit pattern and channel realizations.

BEPs given by (29) and (31) were evaluated for 8 usersemploying a spreading factor of 16 and equal transmissionpowers in the vehicular A channel. The length of zero-forcingand LMMSE equalizers was set to 98 taps (49 chips), and theobservation window of the received signal was 80 chips. BEPsare presented in Figure 4 for the conventional Rake receiver,as well as for the ZF and LMMSE chip-level equalizers WL

and WL given by (7) and (11), respectively. Also the theo-retical single-user bound [49] for the considered channel isgiven in the figure.

From the results in Figure 4 it can be immediately notedthat the Gaussian approximation for BEP gives a good matchwith the BEP values obtained with (29). Also it is easily seenthat as Eb/N0 increases, the BEP of the Rake receiver saturatesdue to the MAI. The LMMSE equalizers show a significantBEP improvement when compared to conventional Rake

9Correlation with a spreading code is included in w.10Gray code mapping of symbols to bits is assumed, and the term Q2(·)

is ignored.


0 2 4 6 8 10 12 14 16 18 20

Eb/N0[dB]

10−4

10−3

10−2

10−1

100

Bit

erro

rpr

obab

ility

Figure 4: BEP versus Eb/N0 for 8 QPSK users with a spreading fac-tor of 16 in the Vehicular A channel. BEP (solid line) are presentedwith Gaussian approximation (dashed line) for Rake receiver (),zero-forcing equalizer (∗), as well as for the LMMSE receivers givenby (7) () and by (11) (×). Also the performance bound of channel() is included.

receiver, whereas the ZF equalizer attains the performance ofa Rake receiver only at high Eb/N0. The performance differ-ence between ZF and LMMSE chip-level equalizer receiversis caused by the noise enhancement typical to the ZF equal-izers [50]. The performance difference between the LMMSEequalizers WL and WL is relatively small, less than 2 dB at thepractical Eb/N0 range. Hence, it can be noted that the chang-ing chip correlations do not have a significant effect on theperformance of the LMMSE equalizer in a WCDMA down-link. However, the LMMSE equalizer defined by (11) showsmoderate saturation in performance at high Eb/N0 values.

In Figure 5, the sample distribution functions are pre-sented for the SINR γ with the Rake receiver, the ZF equalizerand LMMSE equalizers. The sample distribution functionsare given for ES/N0 values of 6 dB and 18 dB. The SINR for azero-forcing equalizer presents a strongly skewed sample dis-tribution function with a heavily weighted lower tail. Com-parison of Figures 4 and 5 reveals that the BEP performanceis dominated by the lower tail characteristics of an SINR dis-tribution, as expected. At Eb/N0 = 15 dB, the BEP of the ZFequalizer is close to that of the Rake receiver. At the corre-sponding ES/N0 (18 dB), the median SINR of the ZF equal-izer is closer to the median SINR of WL, whereas the 10thpercentiles11 of SINR are almost equal for the ZF equalizerand Rake receiver. From Figure 5 we can also notice that withincreasing ES/N0 the variance of SINR increases significantlymore for the Rake receiver than for LMMSE equalizers. ThusLMMSE equalizers provide a more stable SINR performancethan the Rake receiver, which is an important advantage forradio resource management processes.

1110th percentile of a sample set is such a value x that at least 10% ofsamples are smaller than or equal to x.

−10 −5 0 5 10 15 20 25

Signal to interefence-and-noise ratio (SINR)[dB]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sam

ple

dist

ribu

tion

fun

ctio

n

Figure 5: Sample distribution functions of SINR are presentedat 6 dB (dashed line) and 18 dB (solid line) signal-to-noise ratiosfor Rake receiver (), zero-forcing equalizer (∗), as well as for theLMMSE receivers given by (7) () and by (11) (×).

In Figure 6, the average and the 10th percentile of SINRare presented with respect to ES/N0. As we could expect fromFigure 5, the average SINR is not an adequate figure of merit.According to average SINR, the performance of the ZF equal-izer approaches the performance of LMMSE equalizers withhigh ES/N0. However, the 10th percentile of SINR appearsto be able to capture several performance characteristics. Itshows that the performance of the Rake receiver saturatesafter ES/N0 = 13 dB (Eb/N0 = 10 dB), and that the per-formances of the ZF equalizer and Rake receiver cross atES/N0 = 18 dB. It also shows that the performance of theLMMSE equalizer WL starts to saturate at ES/N0 = 23 dB.

The ratio of symbol energy Es to the total power PT ofbase station signal was varied by scaling the powers of in-terfering users. The ratio Es/PT describes the load of a basestation, with low Es/PT values indicating a heavily loadedsystem. At the same time, the Es/N0 values that maintainedthe 10th percentile of SINR at a predefined target value weresearched. The target values were set to−3 dB and 0 dB, whichresulted in all cases 13–15% and 6–8% BEPs, respectively.The results are presented in Figure 7. The horizontal differ-ences between the curves indicate how much the transmis-sion powers of interfering users can be increased while main-taining the desired user’s transmission power constant. Theamount that the desired user’s transmission power can be de-creased while maintaining the transmission powers of inter-fering users at a constant level is also indicated by the differ-ences between the curves, as depicted in the figure.

5.2. Bit error rate simulations

The performance of adaptive chip-level equalizers were eval-uated by simulations in Rayleigh fading Channel II definedin Table 2. In the simulations, the terminal speed was as-sumed to be 60 km/h, which results in a 56 Hz maximum


0 5 10 15 20 25

Signal to noise ratio [dB]

−15

−10

−5

0

5

10

15

20

25

Ave

rage

sign

alto

inte

refe

nce

-an

d-n

oise

rati

o[d

B]

(a)

0 5 10 15 20 25

SNR [dB]

−15

−10

−5

0

5

10

15

20

25

10th

per

cen

tile

ofsi

gnal

toin

tere

fen

ce-a

nd-

noi

sera

tio

[dB

]

(b)

Figure 6: Average (a) and 10th percentile (b) of SINR versus signal-to-noise ratio are presented for Rake receiver (), zero-forcingequalizer (∗), as well as for the LMMSE receivers given by (7) ()and by (11) (×).

Doppler frequency shift at the 2 GHz carrier frequency. Aspreading factor of 64 was used on the common pilot chan-nel, and the power of the pilot channel was scaled to 10%of the total transmitted power PT . Filter lengths of 19 tapswere used with the equalizers, except for the 16-tap CPICHtrained equalizer.

BERs for different Eb/N0 values are presented in Figure 8for the considered adaptive receivers, excluding the CPICHtrained equalizer. Also the performance of the LMMSEequalizer WL and the theoretical single-user bound [49] forthe considered channel are included in the figure. In Figure 8,all receivers had a perfect knowledge of the channel. The per-formance of the Rake receiver is degraded by MAI even at low

−3 −2 −1 0 1 2 3 4 5 6

ES/PT [dB]

0

2

4

6

8

10

12

ES/N

0[d

B]

SNR gain of 1.4 dB

ES/PT gain of 3 dB

Figure 7: Required Es/N0 versus Es/PT is presented for −3 dB (solidline) and 0 dB (dashed line) target values of 10th percentile of SINRwith Rake receiver () and LMMSE receivers given by (7) () andby (11) (×).

0 2 4 6 8 10 12 14 16 18 20

Eb/N0 (dB)

10−3

10−2

10−1

100

Bit

erro

rra

te

Rake receiverGriffiths equalizerCR-MOE equalizer

Prefilter-Rake with GriffithsPrefilter-Rake with HRLSPrefilter-Rake with Levinson

LMM

SE

Performance bound

Figure 8: Bit error rates versus Eb/N0 for 4QPSK users (spread-ing factor 8) and common pilot channel (spreading factor 64) withknown channel response in Channel II.

Eb/N0 values, whereas the adaptive equalizers attain almostthe performance of the LMMSE receiver at low Eb/N0 values.At high Eb/N0 values the prefilter-Rake receivers provide bet-ter performance than CR-MOE or Griffiths’ equalizers. How-ever, the performance differences remain at a moderate level.

To see the effect of channel estimation, the channel co-efficients were estimated with a common pilot channel anda moving average smoother spanning over two slots. BERsare presented in Figure 9 for different Eb/N0 values and in


0 2 4 6 8 10 12 14 16 18 20

Eb/N0 (dB)

10−3

10−2

10−1

100

Bit

erro

rra

te

Rake receiverCPICH-trained equalizerGriffiths equalizerCR-MOE equalizer


LMM

SE

Performance bound

Figure 9: Bit error rates versus Eb/N0 for 4 QPSK users (spread-ing factor 8) and common pilot channel (spreading factor 64) withestimated channel response in Channel II.

−2 0 2 4 6 8 10 12 14

Es/PT (dB)

10−3

10−2

10−1

Bit

erro

rra

te

Rake receiverCPICH-trained equalizerGriffiths equalizerCR-MOE equalizer


LMMSE

Figure 10: Bit error rates versus Es/PT at Eb/N0 = 12 dB (Es/N0 =15 dB) with estimated channel response in Channel II.

Figure 10 with respect to the ratio of the desired user’s sym-bol energy Es to the total transmitted power PT . DifferentEs/PT values were obtained by simulating different numbersof equal power users employing a spreading factor 64.

Comparing Figures 8 and 9, it can be noted that the per-

formance of a CR-MOE equalizer is affected by channel es-timation errors. However, the performance degradation dueto channel estimation is not severe. It is slightly surprisingto see that the prefilter-Rake and Rake receivers are not sig-nificantly affected by channel estimation. This indicates thattheir performances are limited by other factors, like MAI inthe case of the Rake receiver.

From the results it can be seen that both prefilter-Rake receivers as well as CR-MOE and Griffiths’ equaliz-ers provide performance improvements when compared tothe conventional Rake receiver. The performance gain alsoincreases with increasing Eb/N0 (Figure 9) and decreasingEs/PT (Figure 10), that is, when MAI becomes more domi-nant. However, at high Es/PT values (low number of users)the Rake receiver provides equal or better performance thanthe CR-MOE or Griffiths’ equalizer. From Figure 10 it canbe noted that with decreasing numbers of users and, thus,MAI, the performance of prefilter-Rake and Rake receiversapproach the performance of the LMMSE receiver.

The CPICH trained equalizer provides performance im-provement over the Rake receiver at a relatively high Eb/N0

range (Figure 9) or in severe MAI situations (Figure 10). TheCPICH trained equalizer suffers from insufficient adaptationcaused by low SNR in the adaptation. The SNR is especiallylow at high Es/PT values, indicating low CPICH power. Atthe same range also the BER of the equalizer saturates as seenin Figure 10.

6. CONCLUSIONS

One approach to improve the performance of WCDMAdownlink receivers was studied in this paper, namely, channelequalization prior to despreading. The presented receivers,consisting of a channel equalizer, a correlator, and a decisiondevice, restore to some extent the orthogonality of users, and,thus, suppress MAI when orthogonal spreading sequencesare employed.

The zero-forcing and LMMSE solutions for chip-levelchannel equalizers were defined and the effects of WCDMAdownlink signal structure to the equalizers were addressed.The performance of chip-level channel equalizers was stud-ied with respect to bit error probability and signal-to-interference-plus-noise ratio in a frequency-selective fadingchannel.

An overview of several adaptation methods proposed forthe chip-level channel equalizers was given. Adaptive chip-level channel equalizers based on training with a common pi-lot channel, on MOE equalization constrained with channelresponse, on the Griffiths’ algorithm, as well as on the combi-nation of a blind prefilter and Rake were studied. The perfor-mance of the equalizers was numerically evaluated and com-pared to the performance of the Rake receiver in a Rayleighfading multipath channel. The results show significant per-formance improvements when chip-level channel equalizersare employed instead of the conventional Rake receiver in aWCDMA downlink. The prefilter-Rake receivers appear asthe most promising adaptive solution for the equalizers.


ACKNOWLEDGMENTS

The research at the University of Oulu has been supported bythe Academy of Finland, Nokia, and Texas Instruments.

REFERENCES

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Kari Hooli received his M.S. degree inelectrical engineering from the Universityof Oulu, Finland in 1998. Since 1998, hehas been a Research Scientist at Centrefor Wireless Communications, University ofOulu. Currently, he is pursuing Dr.Tech. de-gree in electronic engineering with reseachinterests in statistical signal processing, in-terference suppression, and CDMA systems.

Markku Juntti was born in Kemi, Fin-land, in 1969. He received his M.S. (Tech.)and Dr.Tech. degrees in electrical engineer-ing from University of Oulu, Oulu, Fin-land in 1993 and 1997, respectively. Dr.Juntti has been a Research Scientist andResearch Project Manager at Telecommu-nication Laboratory and Centre for Wire-less Communications, University of Oulu in1992–1997. In academic year 1994–1995 hewas a Visiting Research Scientist at Rice University, Houston, Tex.In 1998 he was an Acting Professor at the University of Oulu. In1999–2000 he was with Nokia Networks, Radio Access Systems inOulu as a Senior Specialist of W-CDMA Research and Solutions.Dr. Juntti has been a Professor of Telecommunications at Universityof Oulu since 2000. He is also Research Manager of UMTS Researchat Centre for Wireless Communications, University of Oulu. Dr.Juntti consults the telecommunication industry, for example, bytraining its personnel. Dr. Juntti’s research interests include com-munication theory and signal processing for wireless communica-tion systems as well as their application in wireless communicationsystem design. He is an author in book “W-CDMA for UMTS”published by Wiley. Dr. Juntti is a member of IEEE. He was Sec-retary of IEEE Communication Society Finland Chapter in 1996–1997 and the Chairman for years 2000–2001. He has been Secretaryof the Technical Program Committee of the 2001 IEEE Interna-tional Conference on Communications (ICC ’01), and Chairmanof the Technical Program Committees of 1999 Finnish Signal Pro-cessing Symposium (FINSIG ’99) and the 2000 Finnish WirelessCommunications Workshop (FWCW ’00).

Markku J. Heikkila received his M.S. degreein electrical engineering from the Universityof Oulu, Oulu, Finland, in 1997. He is cur-rently a Senior Research Engineer in Nokia,Research and Technology Access, Finland.From 1996 to 1997, he was employed as aResearch Assistant at the Telecommunica-tion Laboratory of the University of Oulu.Since 1997, he has been with Nokia, Fin-land. His research interests include multi-antenna transmission and reception techniques, space-time codingand baseband signal processing algorithms.


Petri Komulainen was born in Kiuruvesi,Finland, in 1971. He received the M.S.(Dipl. Eng.) degree in 1996 from the pro-gramme of Information Engineering of theUniversity of Oulu, Finland, with a thesison turbo codes. In 1997, he worked as aVisiting Research Scholar at the New Jer-sey Institute of Technology (NJIT), Newark.Since then he has worked for Nokia MobilePhones in Oulu, Finland, specialising in theresearch and development of various WCDMA terminal receiveralgorithms.

Matti Latva-aho received the M.S. in elec-trical engineering, Lic.Tech., and Dr.Tech.degrees from the University of Oulu, Fin-land in 1992, 1996, and 1998, respectively.From 1992 to 1993, he was a Research En-gineer at Nokia Mobile Phones, Oulu, Fin-land. During the years 1994–1998 he wasa Research Scientist at TelecommunicationLaboratory and Centre for Wireless Com-munications at the University of Oulu. Cur-rently, Professor Latva-aho is Director of Centre for Wireless Com-munications at the University of Oulu. His research interests in-clude future broadband wireless communication systems and re-lated transceiver algorithms. Professor Latva-aho has publishedmore than 50 conference or journal papers in the field of CDMAcommunications.

Jorma Lilleberg was born in Rovaniemi,Finland, in 1953. He received the DiplomaEngineer and Licentiate of Technology de-grees in electrical engineering at the Uni-versity of Oulu, Oulu, Finland in 1979 and1984, respectively and the Doctor of Tech-nology degree at the Tampere Universityof Technology, Tampere, Finland in 1992.From 1992 to 1993 he worked at the Tech-nical Research Center of Finland, in Oulu,Finland as an acting Research Professor and Chief Scientist. SinceAugust 1993 he has been working at Nokia Mobile Phones, Oulu,Finland as a Principal Scientist and NMP Fellow. Dr. Lilleberg’s re-search interest is in digital communications theory and the appli-cation of statistical signal processing methods to digital radio re-ceivers. He has published several tens of research papers and holds9 patents. He is also a Docent to the University of Oulu, Oulu, Fin-land where he has given lectures on estimation theory and guideddoctoral students.


Reduced-Rank Chip-Level MMSE Equalization for the 3GCDMA Forward Link with Code-Multiplexed Pilot

Samina ChowdhurySchool of Electrical Engineering, Purdue University, West Lafayette, IN 47907-1285, USAEmail: [email protected]

Michael D. ZoltowskiSchool of Electrical Engineering, Purdue University, West Lafayette, IN 47907-1285, USAEmail: [email protected]

J. Scott GoldsteinSAIC 4001 N. Fairfax Drive, Suite 400, Arlington, VA 22203, USAEmail: [email protected]

Received 31 July 2001 and in revised form 15 March 2002

This paper deals with synchronous direct-sequence code-division multiple access (CDMA) transmission using orthogonal channelcodes in frequency selective multipath, motivated by the forward link in 3G CDMA systems. The chip-level minimum mean squareerror (MMSE) estimate of the (multiuser) synchronous sum signal transmitted by the base, followed by a correlate and sum, hasbeen shown to perform very well in saturated systems compared to a Rake receiver. In this paper, we present the reduced-rank,chip-level MMSE estimation based on the multistage nested Wiener filter (MSNWF). We show that, for the case of a knownchannel, only a small number of stages of the MSNWF is needed to achieve near full-rank MSE performance over a practicalsingle-to-noise ratio (SNR) range. This holds true even for an edge-of-cell scenario, where two base stations are contributingnear equal-power signals, as well as for the single base station case. We then utilize the code-multiplexed pilot channel to train theMSNWF coefficients and show that adaptive MSNWF operating in a very low rank subspace performs slightly better than full-rankrecursive least square (RLS) and significantly better than least mean square (LMS). An important advantage of the MSNWF is thatit can be implemented in a lattice structure, which involves significantly less computation than RLS. We also present structuredMMSE equalizers that exploit the estimate of the multipath arrival times and the underlying channel structure to project the datavector onto a much lower dimensional subspace. Specifically, due to the sparseness of high-speed CDMA multipath channels, thechannel vector lies in the subspace spanned by a small number of columns of the pulse shaping filter convolution matrix. Wedemonstrate that the performance of these structured low-rank equalizers is much superior to unstructured equalizers in terms ofconvergence speed and error rates.

Keywords and phrases: CDMA forward link, minimum mean square error equalization, pilot code.

1. INTRODUCTION

Mobile units in current code-division multiple access(CDMA) cellular systems use a Rake receiver, which is amaximal-ratio combiner and can be interpreted as a bankof filters matched to the channel that combine the energyfrom multiple paths [1]. The Rake filter is the optimum(maximum likelihood) demodulator when there is no in-terference from other users [1]. In IS-95 and the proposedthird-generation (3G) systems, orthogonal Walsh-Hadamardcodes are used to spread the different users’ data symbols onthe forward link. At the downlink receiver, after removing thecoherent carrier, the signal is multiplied by the synchronized

base station long code and then decorrelated with the desireduser’s spreading code. In a flat fading environment, this willensure that any interference due to other users in the samecell is eliminated.

However, in urban wireless systems, the fading is of-ten not flat and the orthogonality of the underlying Walsh-Hadamard codes is destroyed at the receiver, resulting inmultiple-access interference (MAI) at the receiver. Further-more, if the multipath delay spread is a significant portionof the symbol period, there will be considerable intersym-bol interference (ISI) in addition to the MAI. There are alsomajor interference issues if the mobile unit is near the edgeof a cell and is receiving significant out-of-cell transmission,


regardless of whether the fading is flat or not. In such en-vironments, the Rake receiver is suboptimal, because it in-herently treats MAI as uncorrelated noise. The multipath in-duced MAI also necessitates very tight power control. Whenmany or all users are active in the cell, the bit error rate (BER)curve of the standard Rake receiver flattens out at highersignal-to-noise ratio (SNR) [2]. Thus, in situations where thenumber of active users approaches the spreading gain, theRake receiver does not provide adequate performance.

The maximum likelihood multiuser detector was derivedin [3] for the general case, and was shown to have com-putational complexity that increases exponentially with thenumber of users. This makes the optimal receiver practi-cally infeasible. Recently, chip-level downlink equalizers havebeen proposed to significantly increase the capacity of 3GCDMA based high-speed wireless communication links. Inthe CDMA downlink, for a given user, the signals from allusers in the same cell propagate through the same multi-path channel, so the multiple-access and interchip interfer-ence can be suppressed by linear channel equalization atthe chip-level [4]. One advantage of chip-level equalizationis that the equalizer coefficients depend only on the down-link multipath channel. In contrast, due to the base stationdependent pseudorandom scrambling code, the optimumsymbol-level equalizer varies from symbol to symbol, regard-less of whether the multipath channel changes or not. Also,the chip-level equalizer is valid for all users, as the coeffi-cients do not depend on the channel short code [5]. Down-link equalization prior to despreading to restore the orthog-onality of the different users’ signals and hence suppress MAIhas been suggested in [2, 4, 5, 6, 7, 8, 9]. Of these, linearzero-forcing (ZF) and minimum mean square error (MMSE)equalizers (proposed by Ghauri and Slock [7] and indepen-dently by Krauss et al. [2] and Zoltowski and Krauss [9]) em-phasized the multichannel aspect by means of oversamplingand/or multiple antennas at the mobile station.

Given a perfect estimate of the common downlink chan-nel, zero-forcing equalization is possible in the noiseless case,regardless of the number of users, provided that sufficientspatio-temporal diversity is available [7]. But in the practicalsituation of noise in the received signal, the ZF equalizer maysuffer from significant noise enhancement. In [2], the sum ofthe chip sequences from all the users is modeled as an i.i.d.random sequence, resulting in a “simple” chip-level equal-izer that does not depend on the (Walsh-Hadamard) channelcode, or the base station dependent long code. The equalizeris followed by correlation with the desired user’s spreadingcode and the output, downsampled by the spreading factor,gives the symbol estimate. The resulting chip-level MMSE es-timators with perfect channel estimation were shown to out-perform both ZF and Rake [2].

References [2, 4, 5, 6, 7, 8, 9] all assumed that the chan-nel statistics and noise power are known at the receiver,hence the performances presented therein represent “upper-bounds” achievable with perfect channel estimation. A blindlinear equalization algorithm which equalizes for the com-mon downlink channel was proposed in [10], based on max-imizing signal-to-interference plus noise ratio (SINR). Sev-

eral adaptive versions of chip-level equalizers have been pro-posed, such as in [11, 12, 13]. In [11], the matched filter forthe chip sequence was followed by an adaptive chip decorre-lator based on blind linear decorrelation. In [12], the Grif-fith algorithm was used to update the multiuser chip esti-mator. However, both receivers were assumed to have perfectchannel knowledge. A channel-response minimum-outputenergy equalizer was proposed in [14], which outperformsRake for large number of users, provided that the channel es-timation error is not significant. Adaptive equalization whichrequires no knowledge of the channel or the other users’codes was proposed by Frank and Visotsky in [5], where theyassumed synchronization with the base station long code andsuggested training at chip-level using the code-multiplexedcommon pilot channel as reference, but no simulation re-sults were presented. This straightforward adaptation doesnot exploit the orthogonality of the channel codes, and can-not suppress the intracell users due to pseudorandomness ofthe long code. Hence the SNR in the adaptation is low andthe pilot-chip trained equalizer performs as poorly as Rakeexcept at high SNR [13].

For high data rate applications, the multipath delayspread may span numerous chips so that the MMSE equalizerrequires computation of a large number of coefficients, andmay thus take an unacceptably long time to converge in adap-tive implementations. As a result, application of reduced-rank filtering methods in the context of equalization inCDMA has been an active topic of research in recent years.Goldstein et al. [15, 16, 17] first formulated the multistagenested reduced-rank technique for approximating the classi-cal Wiener filter. Their approach uses information from boththe covariance matrix and the cross-correlation vector to de-termine the basis of a lower-dimension subspace that theWiener filter is constrained to lie within. This method doesnot require any knowledge of the eigenvectors of the chan-nel covariance matrix, and so involves much less computa-tion than either the principal components (PC) [18] or thecross-spectral components (CS) [19] methods, the two mostwidely known reduced-rank techniques. A similar approachwas developed in [20, 21, 22] where they used an orthonor-mal auxiliary vector (AV) in conjunction with the matchedfilter (Rake) to maximize the SINR subject to constraints.

The authors of [23, 24], applied adaptive multistagenested Wiener filter (MSNWF) to the reverse link with asyn-chronous users, flat fading, and no long code. An importantresult from their analysis is that the rank D needed to achievea desired SINR does not scale with system size. Indeed,through extensive supporting simulations, the MSNWF wasshown to achieve near optimal SINR performance with asubspace of dimension roughly equal to D = 8 or less wherethe full-rank space was of dimension 128. Another knownapplication of the MSNWF is the cancellation of narrowbandand wideband jammers for global positioning system (GPS)employing a power minimization based space-time prepro-cessor [25, 26].

In this paper, we present reduced-rank, chip-level MMSEestimators based on the MSNWF algorithm. The main goalof this paper is to demonstrate the superior performance

Reduced-Rank Chip-Level MMSE Equalization for the 3G CDMA Forward Link with Code-Multiplexed Pilot 773

of reduced-rank MMSE equalization for the CDMA down-link. We show that, with perfect knowledge of the chan-nel statistics, the MSNWF requires only a small number ofstages to achieve near full-rank MMSE performance overa wide range of SNR. We then use a training-based blockadaptive algorithm whereby the filter coefficients are adaptedon the symbol-level using the code-multiplexed pilot channelthat carries known symbols, to analyze the performance ofthe MSNWF when the channel is unknown. The SINR plotshows a convergence speed comparable to full-rank recursivelease square (RLS) and much faster than full-rank least meansquare (LMS). Simulated bit error rate curves further illus-trate the excellent performance of the MSNWF.

We also present a reduced-rank MMSE equalizer that ex-ploits the structure and sparseness of the multipath chan-nel under high-speed, wideband conditions. In this case, thechannel coefficients lie approximately in a subspace associ-ated with only a few columns of the pulse shaping filter con-volution matrix. We project the full-rank chip-level MMSEequalizer onto this much lower-rank subspace and illustratea much better convergence rate than that achieved with anunstructured MMSE equalizer of similar rank.

The results in this paper are for the CDMA forward linkwith synchronous users, saturated loading, frequency selec-tive fading, long code scrambling and employing two an-tennas at the mobile receiver. The channel is assumed to beunvarying with time, which is generally true over a shorttime interval. We also assume that synchronization with thebase station long code has been achieved. The rest of thepaper is organized as follows. After describing the channeland data models in Section 2, Section 3 derives the chip-level MMSE estimator. Reduced-rank filtering based on theMSNWF is discussed in Section 4, and simulation results ob-tained for the CDMA downlink is presented in Section 5.Adaptive equalization based on the MSNWF is described inSection 6 along with simulation results. We present the newstructured equalizer based on sparse multipath channel inSection 7 and corresponding simulation results in Section 8.Our conclusions are drawn in Section 9.

2. DATA AND CHANNEL MODEL

The channel model is shown in Figure 1. If the transmissionis from only one base station, the impulse response for the ithantenna channel between the transmitter and receiver (mo-bile station) is given by

hi(t) =Nm−1∑k=0

hci[k]prc(t − τk

), i = 1, 2, (1)

where hci[k] is the time-invariant complex gain associatedwith the kth multipath at the ith antenna; prc(t) is the com-posite chip waveform (including the matched low-pass filterson the transmit and receive end). The chip waveform is as-sumed to have a raised cosine spectrum. And Nm is the totalnumber of delayed paths, that is, multipath arrivals, some ofwhich may have zero or negligible power, so that the channelimpulse response is sparse.

The transmitted “sum” signal may be described as

s[n] = cbs[n]Nu∑j=1

Ns−1∑m=0

bj[m]cj[n−Ncm

], (2)

where cbs[n] is the base station dependent long code, bj[m] isthe bit/symbol sequence of the jth user, cj[n] is the jth user’schannel (short) code of length Nc, Nu is the total numberof active users, Ns is the number of bit/symbols transmittedduring a given time window.

The signal received at the ith antenna (after convolvingwith a matched filter having a square-root raised cosine im-pulse response) is given by

xi(t) =∑n

s[n]hi(t − nTc

)+ ηi(t), i = 1, 2, (3)

where Tc is the duration of one chip and ηi(t) is a noise pro-cess assumed white and Gaussian prior to coloration by thereceiver chip-pulse matched filter.

2.1. Edge of cell/soft handoff

We consider the interference problem when the desired useris at the edge of a cell so that the total received signal at themobile station is the sum of the contributions from two basestations, plus noise

xi(t) = x(1)i (t) +x(2)

i (t) +ηi(t), i = 1, 2 antennas at receiver,(4)

where x(l)i (t) denotes the signal received at the ith antenna

due to transmission from the lth base station, and xi(t) de-notes the total received signal at the ith antenna.

At each antenna, we oversample the signal xi(t) to obtainr1i[n] = xi(nTc) and r2i[n] = xi(Tc/2 + nTc).

In the soft handoff mode, the desired user’s data is trans-mitted simultaneously from the two base stations. At the re-ceiver, two equalizers are designed, one for each base station.The output of each of the two chip-level equalizers is cor-related with the desired user’s channel code times the cor-responding base station’s long code. These two symbol es-timates are averaged to get the symbol estimate for the softhandoff mode. The block-diagram for the chip-level MMSEequalizer employing soft handoff is shown in Figure 2. In thenormal mode, the second base station is treated as interfer-ence.

3. CHIP-LEVEL MMSE ESTIMATOR

The chip-level MMSE equalizer is designed to minimize theMSE between the multiuser synchronous sum signal, s[n],and the sum of the equalizer outputs, as depicted in Figures1 and 2. Given the orthogonality of the channel codes, anestimate of the symbol, b j[m], can be obtained via a correlateand sum with cj and cbs at the output of the chip-level MMSEequalizer.

Equation (4) can be written more compactly in vectorform as


Symbolestimate

b j[m]

Ncc∗bs

[−n]c∗j [−n]+

g1[n]

g2[n]

+

+

η1

η2

Antenna 1

Antenna 2

h1[n]

h2[n]

s[n]

Figure 1: Chip-level MMSE equalization for CDMA downlink with one base station.

Symbolestimate

b j[m]+

b(1)j [m]

b(2)j [m]

Nc

Ncc(2)∗bs

[−n]c∗j [−n]

c(1)∗bs

[−n]c∗j [−n]+

+

g(1)1 [n]

g(1)2 [n]

g(2)1 [n]

g(2)2 [n]

+

+

η1

η2

+

+

Antenna 2Antenna 1

h(1)1 [n]

h(1)2 [n]

s(1)[n]

Base-station 1

h(2)1 [n]

h(2)2 [n]

s(2)[n]Base-station 2

Figure 2: Chip-level MMSE equalization for CDMA downlink, two base stations, soft handoff.

x[n] = H(1)s(1)[n] + H(2)s(2)[n] + η[n]

=Hs[n] + η[n],(5)

where H(l) is the 2Ng × (L + Ng − 1) channel convolutionmatrix (Ng is the length of the equalizer, and L is the lengthof the channel in chips.), comprised of

H(l) =H(l)

1

H(l)2

, (6)

where

H(l)i =

h(l)i [0] 0 · · · 0

h(l)i [1] h(l)

i [0] · · · 0

.... . .

. . .. . .

h(l)i [L− 1] h(l)

i [L− 2] · · · · · ·...

. . .. . .

. . .

0 0 h(l)i [L− 1] · · ·

T

. (7)

H is the composite channel convolution matrix H =[H(1) H(2)]. And s(l)[n] is an (L+Ng −1) vector of the trans-mitted signal, with the superscript denoting the correspond-ing base station.

The corresponding formulae for only one transmitting

base station can be found by simply removing the term in-volving H(2) from (5).

Krauss et al. [2] made some simplifying assumptions toderive a chip-level MMSE equalizer that can be easily im-plemented. The sequence values for the multiuser sum sig-nal are assumed to be i.i.d. random variables. Otherwise,the covariance matrix of the sum signal s[n] is a compli-cated expression involving the Walsh-Hadamard spreadingcodes that vary from index to index. The i.i.d. assumptionis valid if the (long) scrambling code is viewed as a randomi.i.d. sequence and/or all users are active with equal power.With this assumption, the covariance matrix of the signal isEs[n]Hs[n] = σ2

s I, and the MMSE equalizer was shownto be

g(l)c =

σ2s HHH + Rηη

−1H(l)δDc , (8)

where δDc is a column vector of all zeros except 1 in the (Dc +1)th position, Dc is the combined delay of the equalizer andchannel, σ2

s is the signal power, and Rηη = E[ηH[n]η[n]].In [2] and Section 5 of this paper, the delay Dc, 0 ≤ Dc ≤(Ng + L− 2), that yields the smallest MMSE is calculated us-ing the actual channel statistics and that Dc is used in thesimulations.

Equation (8) has the form of the well-known Wiener-Hopf solution,

w = R−1xx rdx, (9)


ε0∑

w1ε1∑

w2ε2∑

wN−2

εN−2∑

wN−1

dN−2

BN−2

pN−2xN−3

B2x2

p2x1B1

p1x0

Analysis filter bankSynthesis filter bank

d0

d1

d2

xN−2 = dN−1 = εN−1

Figure 3: Structure of successive stages of the multistage nested Weiner filter.

where Rxx is the channel covariance matrix and rdx is thecross-correlation vector.

In [2, 27], the authors showed that the MMSE signifi-cantly outperforms the Rake receiver, especially when a largenumber of channel codes are active relative to the spreadingfactor. The difference is more pronounced when soft handoffis unavailable.

4. REDUCED RANK FILTERING

In general, the length of an MMSE equalizer should be atleast equal to the channel length to achieve the desired per-formance, and longer equalizers yield better error rates [28].Hence equalizers in the high-speed CDMA downlink will bynecessity span many chips in length with a correspondinglarge number of degrees of freedom. In order to reduce thenumber of filter coefficients to be estimated, the received sig-nal vector may be projected onto a lower dimensional sub-space, and the Wiener filter given by (9) constrained to liein this subspace. This increases the speed of convergencedramatically for adaptive methods, if the subspace is cho-sen properly. But the overall MMSE for the reduced-rankfilter may be higher than the MMSE for the full-rank fil-ter. The most widely known reduced rank techniques in sig-nal processing are the principal components method [18]and the cross-spectral methods [19], both based on eigen-decomposition of the channel covariance matrix.

4.1. Multistage nested Wiener filter

Goldstein et al. [15, 16, 17] first formulated the MSNWF,which uses information from the channel covariance matrix,Rxx, and cross-correlation vector, rdx, to determine the basesof the lower-dimension that w is constrained to lie within.The structure of the MSNWF is depicted in Figure 3. At eachstage, a rank-one basis is selected based on maximal correla-tion between the desired signal, d0, and the observed signalx0[n]. The observed vector process is decomposed by a se-quence of nested filters p1,p2, . . . ,pD, where D is the order ofthe filter,

pk = E[

xk−1 d∗k−1

]∥∥E[xk−1 d∗k−1

]∥∥ , (10)

and dk = pHk xk−1.

• Initialization:c1 = rdx = E[x0d

∗0 ], δ1 =

√rHdxrdx,

• Forward recursion:For k = 1, 2, . . . N − 1 Do

(1) pk = ck/δk ,(2) Bk = null(pk),(3) dk = pH

k xk−1,(4) xk = Bkxk−1,(5) ck+1 = E[xkd

∗k ],

(6) δk+1 =√

cHk+1ck+1,• End

xN−1 = dN = εN ,

• Backward recursion:ξN = E[|xN−1|2], δN = E[xN−1 d

∗N−1], wN = δN/ξN ,

• For k = N − 1, . . . , 2, 1 Do(1) εk = dk −wk+1εk+1,(2) ξk = σ2

dk− δ2

k+1/ξk+1,(3) wk = δk/ξk ,

• End

Algorithm 1: Basic MSNWF algorithm.

The input process to the (k + 1)th stage is xk[n] =Bkxk−1[n], where Bk is a blocking matrix such that Bkpk = 0.

The outputs of the various stages are linearly combinedvia the scalar weights, w1, w2, . . . , wD−1, chosen so that themean square error at each stage is minimized. If the decom-position is carried out for the full N stages, then the multi-stage nested filter is exactly equivalent to the full-rank clas-sical Wiener filter [15]. The filter-bank structure whitens theerror residue at each stage, and compresses the colored por-tion of the observed data subspace and hence, it is optimalin terms of reducing the MSE for a given rank. The MSNWFdoes not require any eigen decomposition or inversion of thecovariance matrix, and so represents a significant reductionin complexity over the full-rank Wiener solution and otherreduced-rank techniques. This is very important for practicalimplementations, particularly if the rank one decompositioncan be stopped after a few stages. The basic algorithm, basedon [15] is listed in Algorithm 1.

It is straightforward to see that the “desired” signal at


The filter can betruncated at anystage

D ≤ N − 1.

∑+−

p2d2(k)

∑ +−

ε2w2

pH2

x1(k)∑+−

p1d1(k)

∑ +−

ε1w1

pH1

x0(k)

∑ +−

d0(k)

ε0

Modular structure

Figure 4: MSNWF as a lattice filter.

each stage, dk[n] is the output of a length N filter,

ck =(k−1∏

i=1

BHi

)pk. (11)

A notable feature is that the first filter is orthogonal tofilters of all the following stages, that is,

cHk c1 = δk1, k = 1, 2, . . . , N, (12)

δki denotes the Kronecker delta function, which is 1 for k =i, and 0 otherwise. However, the filters hk, k = 2, 3, . . . , N,are not mutually orthogonal in general. The operation of theanalysis filter-bank can be combined into a D × N transfermatrix, given by

TD =

pH1

pH2 B1

...

pHD

1∏k=D−1

Bk

=

cH1

cH2...

cHD

. (13)

The orthogonal decomposition ensures that the reduced di-mension D × D correlation matrix TDRxxTH

D is tridiagonal[15].

Honig and Xiao [24] first proposed choosing a projec-tion matrix on to the subspace orthogonal to pk as the block-ing matrix at each stage, hence retaining the length N of thefilter and the “observed signal” xk[n]. With this choice forthe blocking matrix, TD = [p1 p2 · · · pD] forms an or-thonormal basis for the reduced dimension subspace. More-over, w is constrained to lie in the Krylov subspace spannedby rdx,Rxxrdx,R2

xxrdx, . . .RD−1xx rdx [24]. In this case,

pk+1 =(

I− pkpHk

)Rk−1pk∥∥(I− pkpH

k

)Rk−1pk

∥∥ , (14)

where

Rk =(

I−pkpHk

)Rk−1

(I−pkpH

k

), for k = 1, 2, . . . , D, (15)

p1 = rdx, R0 = Rxx.Joham and Zoltowski [29] proved that this choice of

the blocking matrix, that is, Bk = I − pkpHk , is optimum

in terms of maximizing the correlation between the scalarsignals dk[n] and dk−1[n] at each stage. They developed acovariance-level order recursive form of the MSNWF workingwithin the Krylov subspace, in which the backwards recur-sion coefficients and hence the weight vector and the meansquare error, may be updated at each stage via a simple re-cursion.

4.2. Lattice structure of the MSNWF

The blocking matrix is a very useful concept to develop andanalyze the performance of the MSNWF, but in practice,there is no need to calculate or store these N ×N matrices. Anew reduced-complexity implementation was presented byRicks and Goldstein in [30] based on the following substitu-tion: at the kth stage

dk[n] = pHk xk−1[n],

xk[n] = Bkxk−1[n]

= [I− pH

k pk]

xk−1[n]

= xk−1[n]− dk[n]pk.

(16)

This leads to the “lattice-type” structure for a D-stageMSNWF, as shown in Figure 4. This architecture has the ben-efit of being modular and scalable for hardware implementa-tion, as well as being computationally more efficient than thestructure depicted in Figure 3.

5. APPLICATION OF MSNWF TO CDMA DOWNLINK

Our first set of results solves for chip-level MMSE equaliza-tion based on (8) when only one base station is transmitting


50403020100−10Time in chips

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Nor

mal

ized

mag

nit

ude

Antenna 1Antenna 2

Figure 5: Typical channel impulse response for simulated model.

and finds the “ideal” MSNWF solution after various stages,assuming that Rxx and rdx are known (perfect channel esti-mation). The same simulations are also done for the edge ofcell situation, when two base stations of equal power are re-ceived at the mobile-station and the receiver uses soft hand-off.

A wideband CDMA forward link was simulated similarto one of the options in the US CDMA2000 proposal. Thechip rate was 3.6864 MHz (Tc = 0.27 µs), 3 times that of IS-95. Simulations were performed for a saturated cell, that is, allpossible 64-channel codes are active, and all users have equalpower. The spreading factor was Nc = 64 chips per bit. Thedata symbols were BPSK, and spread with one of 64 Walsh-Hadamard codes. The signals were summed synchronouslyand then multiplied with a QPSK scrambling code of length32678, similar to the IS-95 standard.

The channels were modeled to have four equal-powermultipaths, uniformly distributed within a delay spread of10 microseconds (corresponding to about 37 chips). Themultipath coefficients are complex normal, independent ran-dom variables. The arrival times at antenna 1 and 2 arethe same, but the multipath coefficients are independent.Figure 5 shows a typical channel’s impulse response withtails, sampled at the chip rate.

In the two-base station case, the maximum delay spreadof the downlink channel from the second base station isalso 10 microseconds, with 4 dominant multipaths arrivalsat random. The channels are scaled so that the total energyfrom each of the two base stations is equal at the receiver.Specifically,

M∑m=1

E∣∣x(1)

m [n]∣∣2=

M∑m=1

E∣∣x(2)

m [n]∣∣2. (17)

SNR is defined to be the ratio of the sum of the average powerof the received signals over all the channels, to the average

noise power, after chip-matched filtering. Since the spread-ing factor (number of chips per symbol) is equal to the num-ber of users, and each user contributes the same amount ofpower, this chip signal SNR is equal to the postcorrelation (ordespread) SNR per user per symbol. The curves were gener-ated by averaging 100 or more different channels. Note thatthe abscissa in the graphs is the postcorrelation SNR for eachuser, which includes a processing gain of 10 log(64) ≈ 18 dB.

Figure 6 plots the mean square error for the differentreduced-rank methods as a function of the subspace dimen-sion, D. The channel statistics and noise power are assumedto be known. In the single base station case, Figure 6a, the di-mension of the full space is 114 (the equalizer length is 57 ateach of the 2 antennas, as multipath delay spread is 37 chipsand the chip pulse waveform is cut off after 5 chips at bothends). The MSE for MSNWF is seen to drop dramaticallywithD, and achieves the performance of the full-rank Wienerfilter at dimension approximately 7! In contrast, the prin-cipal components method takes longer than twice the de-lay spread, and the cross-spectral method does only slightlybetter.

Figure 7a displays the BER curves obtained with theMSNWF for different sizes of the reduced-dimension sub-space. For comparison, the BER for a conventional Rake andfull-rank chip-level MMSE equalizer are also shown. Thechannel statistics are assumed to be known perfectly, so thesecurves serve as an informative upper bound on the perfor-mance. It is observed that even a 2-stage reduced-rank filteroutperforms the Rake at all SNRs and only a small numberof stages of the MSNWF are needed in order to achieve nearfull-rank MMSE performance over a practical range of SNRs.

Figures 6b and 7b display similar plots, but for the edgeof cell scenario. In this case, there are 4 effective channels atthe receiver, because we sampled the received signal at twicethe chip-rate at each antenna. It can be shown that the two-polyphase channels created from either antenna are nearlylinearly dependent in the case of a sparse multipath chan-nel as in our simulations. The dimension of the full spaceis 228, which makes the full rank calculations very cumber-some. Amazingly, the MSE for MSNWF still goes down verysteeply with rank and achieves the full-rank value for sub-space dimension of only 8 or so. Compared to the PC andCS methods, this is a huge difference in effective rank reduc-tion. In the BER plots of Figure 7b, the bit error is calculatedfor the soft handoff mode. With perfect channel estimation,the MSNWF can achieve error rates similar to the full-rankMMSE over practical SNR range after stopping at stage as lowas 5!

6. ADAPTIVE EQUALIZATION

Next, we use the class of training-based adaptive algorithmspresented by Honig and Goldstein in [23] to simulate theperformance of MSNWF when the channel is unknown. Al-though the MMSE equalizer described in this paper estimatesthe chip-rate multiuser synchronous sum signal, it is not pos-sible to train the equalizer on this signal as that would require


120100806040200Dimension of reduced-rank subspace for gc

−8

−7

−6

−5

−4

−3

−2

−1

0

Mea

nsq

uar

eder

ror

indB

Multistage nested Wiener filterPrincipal componentsCross-spectral components

(a) One base station.

250200150100500Dimension of reduced rank subspace for gc

−6

−5

−4

−3

−2

−1

0

Mea

nsq

uar

eder

ror

indB

Multistage nested Wiener filterPrincipal componentsCross-spectral components

Normal handoff

(b) Two base stations.

Figure 6: MSE versus rank of reduced dimension subspace, knownchannels, SNR = 10 dB.

the knowledge of number of active users, all of the activechannel codes and the transmitted symbols. Instead, we usethe pilot channel of CDMA downlink, which has a knowncode and known symbols. For DS-CDMA with orthogonalspreading codes, the chip-level MMSE equalizer restores thesynchronous sum signal transmitted by the base station, sothe MMSE equalizer is identical for all channels within amultiplicative constant and the common pilot code can beused to train for any other channel code [5]. Our approach isto train off the pilot symbols by using directly the following

20151050SNR in dB

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Ave

rage

BE

R

RAKEMSNWF, stage 2MSNWF, stage 5

MSNWF, stage 10Exact MMSE, rank 114


20151050SNR in dB

10−6

10−5

10−4

10−3

10−2

10−1

100

Ave

rage

BE

R

Soft handoff

RAKEMSNWF, stage 5MSNWF, stage 10

MSNWF, stage 20MMSE, rank 4 ∗ 57


Figure 7: BER for different chip-level equalizers for CDMA down-link, known channels.

relations (cf. Figure 1):

(i) Chip-level equalization

z[n] = gHc x[n], x[n] =

x[n]

x[n− 1]...

x[n−Ng + 1

]

. (18)


(ii) Despreading

b1[m] =Nc−1∑i=0

z[n + i]c∗bs[n + i]c∗1 [i], n = mNc + Dc

=Nc−1∑i=0

gHc x[n + i]

c∗bs[n + i]

= gHc

Nc∑i=1

x[n + i]c∗bs[n + i]

≡ gHc CH

1 [m]x[m] = gHc y[m],

(19)

where

x[m] =

x[n + Nc − 1

]...

x[n]...

x[n−Ng + 1

]

,

C1[m]

=

cbs[mNc+Nc−1

]0 · · · · · ·

.... . .

. . ....

cbs[mNc

]cbs

[mNc+1

] · · · cbs[mNc+Ng−1

]. . .

. . .. . .

...0 0 · · · cbs

[mNc

]

(20)

assuming that Ng ≤ Nc.Thus we convolve the received chip-sequence with the

pilot channel code (which is all 1’s) times the appropriateportion of the base station scrambling code, and then trainthe equalizer on the pilot symbols. This is equivalent to firstequalizing and then despreading due to the commutativityof convolution. We use a block-adaptive training-based algo-rithm that implements the lattice-type MSNWF of Figure 4.The algorithm is given in Algorithm 2. We also find the full-rank Wiener solutions, using least mean squares and recur-sive least squares algorithms for purposes of comparison inthe simulations.

First, we simulate the single base station scenario, wherethe dimension of the full-rank solution is 114 as describedbefore. Figure 8 plots the output SINR for different chip-levelequalizers versus time in symbols, at a fixed SNR. The out-put SINR is calculated using the formula derived by Kraussin [27]. The MSNWF after stages 5 and 10 yields very goodperformance with low sample-support. The convergence rateis similar to that of a full-rank RLS, which even asymptoti-cally, does not beat the MSNWF of rank only 10! The LMSalgorithm converges much slower and to a lower SINR. Forthe two-base station case, we implement soft handoff. Theasymptotic SINR is almost 3 dB lower for all the equalizersdue to the added interference from the MAI of the secondbase station. But the convergence speed of low-rank MSNWFis still impressive.

Block size = Nt symbols.• Initialization:

d0 = b where b is the length Nt vector of the pilotsymbols, and Y0 = [y[1], . . . , y[Nt]], y[m] = CH

1 [m]x[m].

• Forward recursion:For k = 1, 2, . . . , D,

ck = Yk−1dHk−1,

δk = ‖ck‖,pk = ck/δk,

dk = pHk Yk−1,

Yk = Yk−1 − pkdk,

• Backward recursion:εD = dD ,

For k = D,D − 1, . . . , 1 Do

wk = (εkdHk−1)/‖εk‖2 = δk/‖εk‖2,

εk−1 = dk−1 −wkεk

Algorithm 2: Structured lattice MSNWF algorithm for the CDMAforward link.

The output SINR is plotted versus the rank of the reduceddimension subspace, D, in Figure 9 at two different SNRs.For comparison, the SINR output for an “ideal” MSNWF,that is, with perfect channel estimation, is included. At a lowSNR of 0 dB, the SINR after 200 symbols for the adaptiveMSNWF shows a distinct peak at a dimension of only 3! At10 dB SNR, the peak is less prominent, but the SINR outputgoes down after stage 8 or so. This can be explained as the“penalty” for learning the channels, that is, in the presenceof significant noise, the lower rank MSNWF trades off a biasin the symbol estimate for a lower variance by working in alower dimensional space so as to pass less noise. As the signalpower increases, the higher-dimensional filters yield betterapproximation to the full-rank solution, as the lower rankfilters do not have adequate degrees of freedom to suppressMAI and ISI.

The BER curves in Figure 10 illustrate the performanceof these equalizers after training with 200 symbols in the sin-gle base station case, and 300 symbols in the multiple basestations case. At low SNRs, the BER for MSNWF stage 5 isactually slightly lower than the BER for stage 10 or 15, as ex-pected from the SINR graphs of Figure 9.

It is noteworthy that over a practical SNR range, in thisadaptive implementation, the stage 5 MSNWF does better oralmost as good as full-rank RLS! Another remarkable fact isthat the pilot channel had the same power as all the trafficchannels, implying that the MSNWF reduced-rank adapta-tion does not require a strong pilot signal for fast conver-gence. Obviously, the rate of convergence would increase ifthe amplitude of the pilot channel was made higher than thetraffic channels, and the BER floor would improve for theadaptive equalizers.


+ +++

++

+

+

+

+

+

5004003002001000Number of training symbols

−10

−5

0

5

10

15

SIN

R(d

B)

MSNWF, stage 5MSNWF, stage 10

Full-rank LMSFull-rank RLS

(a) One base station, SNR = 10 dB.

+ +++

++

+

+

+

+

+

+

+

+o

o

o

oo

o

o

oo o

+o


−10

−6

−2

2

6

10

SIN

R(d

B)



(b) Two base stations, SNR = 10 dB, soft handoff.

Figure 8: Output SINR versus time for adaptive chip-level equaliz-ers for CDMA downlink.

7. STRUCTURED EQUALIZER IN SPARSE MULTIPATH

For high-speed CDMA, typically the vector of multipathchannel coefficients is sparse, but the chip-rate linear equal-izers will not be sparse in general. Under certain conditions,the overall channel coefficients lie in a subspace associatedwith the pulse shaping filter convolution matrix. A semi-blind direct equalization approach that utilizes this structureto impose a penalty on nonblind equalizers was presentedin [31], and compared to nonblind MMSE equalizers forindoor channels. For wideband CDMA, additional benefits

302520151050Dimension of reduced-rank subspace

−1

0

1

2

3

4

5

6

SIN

R(d

B)

Known channelsTraining with 200 symbols

(a) SNR = 10 dB.

302520151050Dimension of reduced-rank subspace

2

4

6

8

10

12

14SI

NR

(dB

)

Known channelsTraining with 200 symbols

(b) SNR = 10 dB.

Figure 9: SINR of MSNWF versus dimension of reduced-rank sub-space, 1 base station.

may be obtained by realizing that, due to the sparse nature ofthe multipath arrivals, the total channel vector lies in a sub-space spanned by only a few columns of the pulse shapingconvolution matrix [10]. If we project the full-rank chip-levelMMSE equalizer onto this much lower-rank subspace, theequalizer should converge much faster.

It was shown in Section 3 that the channel cross-correlation vector is given by

rdx =HδDc . (21)


20151050SNR in dB

10−6

10−5

10−4

10−3

10−2

10−1

100

Ave

rage

BE

R

After 200 symbols training

MSNWF, stage 5MSNWF, stage 10MSNWF, stage 15



20151050SNR in dB

10−4

10−3

10−2

10−1

100After 300 symbols training

MSNWF rank 5MSNWF rank 10MSNWF rank 15


Soft handoff


Figure 10: BER for adaptive chip-level equalizers for CDMA down-link.

If we restrict ourselves to the following two conditions:

(1) Ng ≥ L, and(2) Ng − 1 ≥ Dc ≥ L− 1,

then it can be easily seen that rdx now contains all the ele-ments of the channel impulse response. In particular, if wechoose Ng = L and Dc = L− 1, we get

rdx =[

II

][h1

h2

], I =

0 · · · 1

..

.1 · · · 0

. (22)

The impulse response for the channel between the trans-mitter and the ith antenna at the receiver is given by (cf. (1))

hi(t) =Nm−1∑k=0

hci[k]prc(t − τk

), i = 1, 2. (23)

If we sample at chip rate and assume a high enough chiprate so that the multipath delays are integer multiples of thechip period Tc (we will relax this assumption shortly), we canwrite the channel vectors in (22) as

hi = Ghci , i = 1, 2, (24)

where G is the convolution matrix corresponding to the chippulse waveform, and hci is a sparse vector containing themultipath coefficients. In this case, the vector channel hi liesin a subspace spanned by only a few columns of G and wecan simplify (21) as

rdx =[

Gp

Gp

][hp1

hp2

]= Ghp, (25)

where Gp contains only the Lc columns of IG correspondingto the Lc dominant multipath arrivals, and hp is the vectorof corresponding complex gains, that is, hp consists of onlynonzero elements of hc.

Thus the chip-level MMSE equalizer has the form

gc = R−1xx rdx = R−1

xx Ghp. (26)

This structure due to the sparse multipath channel can beutilized to increase the convergence speed of adaptive MMSEequalizers, if an estimate of the multipath arrival times τk,k = 1, . . . , Lc is available at the receiver. We can then projectthe observed data vector onto a rank 2Lc 2L subspaceby taking

yr[m] = GHR−1xx CH

j [m]x[m]. (27)

This would also require estimation of the chip-level covari-ance matrix. We refer to the MMSE equalizer residing in thislow-rank subspace as the “structured projected” equalizer,denoted by gr . Then the estimate of the desired user’s symbolis given by

b j[m] = gHr yr[m] ≡ hH

p yr[m]. (28)

7.1. Chip-level whitening with multistage nestedWiener filter

Direct inversion of Rxx in (27) can be avoided using theMSNWF to obtain a reduced-rank solution to



−2

2

6

10

14

SIN

R(d

B)

Structured projected equalizer, rank 8

MSNWF solves Rxxγ[m] = y[m]Rxx is 114 × 114

MSNWF, stage 5MSNWF, stage 10Exact inversion

Figure 11: Application of MSNWF to covariance matrix inversion,SNR = 10 dB.

Rxxγ[m] ≈ y[m]. (29)

Then yr[m] = GHγ[m] is an approximation to the projecteddata-vector.

The efficacy of the MSNWF is illustrated in Figure 11,which plots the output SINRs of 3 structured equalizers, theonly difference is the use of 5 or 10 stages of the MSNWFto get approximate solution yr[m] or direct inversion of Rxx

to obtain exact solution yr[m]. The dimension of Rxx is114×114, so its inversion is prohibitively expensive, but only10 stages of MSNWF is sufficient to give the same SINR over-all.

7.2. Generalized arrival times

When the multipath arrival times are not exact multiples ofthe chip period, (24) is only an approximate relation. In thiscase we form the basis vector matrix G(ν) by sampling thechip-pulse shaping filter at rate Tc/ν. The approximation er-ror can be made arbitrarily small by increasing ν.

The basis matrix Gp is now formed by taking (ν+ 1) con-secutive columns of IG(ν) for each multipath arrival, corre-sponding to ντk/Tc, . . . , ντk/Tc. Maximum rank of struc-tured projected equalizer is now 2Lc(ν + 1). Note that ourscheme does not require any increase in the sampling rate ofthe received signal.

7.3. Delay estimation

The structured projected equalizer requires estimate of themultipath delays, but no knowledge of the multipath coeffi-cients is needed. The multipath delays will change relativelyslowly as compared to the complex gains even in a time-varying situation. The use of multiple antennas at the re-ceiver enhances the quality of the delay estimates since thearrival times are the same at both antennas (the propagation

5004003002001000Time in symbols

−10

−5

0

5

10

15

SIN

R(d

B)

MSNWF, stage 8MSNWF, stage 5MSNWF, stage 2

(a) Structured projected equalizers.


−6

−2

2

6

10

14SI

NR

(dB

)

Structured projected

Unstructured, reduced-rank


(b) Comparison of structured projected and unstructuredequalizers.

Figure 12: SINR convergence for chip-level equalizers usingMSNWF, arrivals at exact chip-periods, SNR = 10 dB.

delay between two antennas at the mobile is negligible fora given multipath). Typically in CDMA mobile receivers, aserial or block serial search is performed over a very shortinterval where the channels are assumed to be unchanging(perhaps over 512 chips), where the received sequence is cor-related with the base station long code. These short coherentcorrelations are combined in energy to obtain the delay es-timates. This approach yields fast, accurate delay estimates.If the estimate is deemed to be not enough reliable, 2 or



−6

−2

2

6

10

14

SIN

R(d

B)


Unstr. MSNWF, stage 8Structured projected

Figure 13: SINR versus time for different equalizers, arrivals at ex-act chip-periods, SNR = 10 dB.

3 columns of IG centered on the doubtful estimate may betaken to form Gp.

8. RESULTS

Figure 12a shows the output SINR vs. time (in symbols) ata fixed SNR of 10 dB for “structured projected” chip-levelequalizers, for channels simulated as described in Section 5,that is, the arrival times are at exact multiples of the chip-period Tc. We assume that the receiver has already formedestimates of the multipath arrival times. After the projectionvia (27), we use different stages of the MSNWF algorithm.The stage 2 MSNWF does not perform very well, but the con-vergence rate for stages 5 and 8 (maximum) is very good, asexpected.

Figure 12b compares the SINR convergence speed oftraining-based unstructured chip-level equalizers and thestructured projected equalizers, all of which use a MSNWFsolution. It is clear that the structured projected equaliz-ers exploit the structure in the sparse channels to yield amuch superior performance. The structured projected equal-izer is compared to full-rank RLS and LMS convergence inFigure 13. After training with 100 symbols there is approxi-mately 5 dB difference in output SINRs of the structured pro-jected MMSE equalizer and an unstructured MSNWF solu-tion of rank 10 and full-rank RLS.

Next, we simulate frequency-selective channels where thefirst multipath arrival is at 0, and the other three are uni-formly distributed within 10 µs, with the only constraint be-ing that the multipath delays are spaced at least one chip-period apart. Figure 14a shows the SINR plot after variousstages of the MSNWF where the basis vectors are formedby sampling the pulse-shaping filter at rate Tc. In this casethe structured projected equalizers are of dimension 14, as


−15

−10

−5

0

5

10

15

SIN

R(d

B)

Stage 14Stage 8

Stage 5Stage 2

(a) Basis vectors formed by sampling at Tc .


−15

−10

−5

0

5

10

15SI

NR

(dB

)

v = 1, stage 8v = 2, stage 8v = 2, stage 20

v = 4, stage 8v = 4, stage 32

(b) Basis vectors formed by sampling at Tc/ν.

Figure 14: SINR convergence for chip-level equalizers usingMSNWF, random arrivals, SNR = 10 dB.

we form Gp by taking two consecutive columns of IG cor-responding to each multipath arrival that is in between twochip-periods. We see that 8 stages of the MSNWF are suf-ficient but there is a loss of about 2 dB in asymptotic SINRcompared to Figure 12a due to the arrival times not being atexact chip periods.

Figure 14b shows the SINR plot where the basis vec-tors are formed by sampling the pulse-shaping filter at rateTc, Tc/2, and Tc/4. The SINR loss compared to Figure 12a is


20151050SNR in dB

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Ave

rage

BE

R

RAKEFull-rank LMSFull-rank RLS

Unstr. MSNWF, stage 10Str. proj., rank 20

Figure 15: Bit error rates graph, random arrival times.

recouped if we oversample by a factor of 2 and perform full-rank equalization (which is now rank 20). At Tc/4 the conver-gence goes down due to the increased dimension of the filter(which is now 32) but there is no improvement in asymptoticSINR.

We plot the bit error rate graphs of the structured pro-jected equalizer where the pulse-shaping filter is sampled atrate Tc/2, compare it with the full-rank RLS, and reduced-rank unstructured equalizer curves in Figure 15. The multi-path delays are randomly distributed within 10 µs. The struc-tured projected equalizer (of rank 20) exhibits significantlylower bit errors for all SNRs.

9. CONCLUSIONS

We presented reduced-rank chip-level MMSE equalizers forthe CDMA downlink with frequency-selective multipathbased on the multistage nested Wiener filter, for knownchannel case and also for training-based adaptation. The per-formance for the single base station case, and for the edge-of-cell scenario with soft handoff are very satisfactory. The con-vergence rate for MSNWF operating in a very low-rank sub-space was significantly better than LMS, and somewhat betterthan RLS. The BER performance showed improvement overthe full-rank methods for practical SNR range. This excel-lent performance is achieved at a computational complexityin between LMS and RLS due to lattice-type structure thatallows block-adaptive implementation through simple filter-ing operations.

We also developed a structured MMSE equalizer that uti-lizes the estimate of the multipath arrival times and sparsenature of the multipath channel to substantially reduce thenumber of parameters that need to be adapted. The con-vergence rate for this projected MMSE equalizer was sig-

nificantly better than unstructured MSNWF operating in asubspace of similar rank. The bit error rate performance ofthis structured MMSE equalizer was shown to be superior tofull-rank RLS and reduced-rank unstructured MMSE equal-izer over a wide SNR range.

ACKNOWLEDGMENT

This research was supported by the Air Force Office of Sci-entific Research under grant No. F49620-00-1-0127 and byTexas Instruments’ DSP University Research Program.

REFERENCES


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[3] W. van Etten, “Maximum likelihood receiver for multiplechannel transmission systems,” IEEE Trans. Communications,vol. 24, no. 2, pp. 276–283, 1976.

[4] S. Werner and J. Lilleberg, “Downlink channel decorrelationin CDMA systems with long codes,” in Proc. IEEE Int. Conf.on Communications, Vancouver, Canada, June 1999.

[5] C. D. Frank and E. Visotsky, “Adaptive interference suppres-sion for direct-sequence CDMA systems with long spread-ing codes,” in Proc. 36th Allerton Conf. on Communica-tion, Control, and Computing, Monticello, Ill, USA, September1998.

[6] A. Klein, “Data detection algorithms specially designed forthe downlink of CDMA mobile radio systems,” in Proc. IEEEVehicular Technology Conference, vol. 1, pp. 203–207, Phoenix,Ariz, USA, May 1997.

[7] I. Ghauri and D. T. M. Slock, “Linear receivers for the DS-CDMA downlink exploiting orthogonality of spreading se-quences,” in Proc. 32nd Asilomar Conf. on Signals, Systems,and Computers, Pacific Grove, Calif, USA, November 1998.

[8] K. Hooli, M. Latva-aho, and M. Juntti, “Multiple access inter-ference suppression with linear chip equalizers in WCDMAdownlink receivers,” in Proc. IEEE Global TelecommunicationsConf., vol. 1, pp. 467–471, Rio de Janero, Brazil, December1999.

[9] M. D. Zoltowski and T. P. Krauss, “Two-channel zero forc-ing equalization on CDMA forward link: Trade-offs betweenmulti-user access interference and diversity gains,” in Proc.33rd Asilomar Conf. on Signals, Systems and Computing, Pa-cific Grove, Calif, USA, October 1998.

[10] I. Ghauri and D. T. M. Slock, “Structured estimation of sparsechannels in quasi-synchronous DS-CDMA,” in Proc. IEEEInt. Conf. Acoustics, Speech, Signal Processing, Istanbul, Turkey,June 2000.

[11] M. J. Heikkila, P. Komulainen, and J. Lilleberg, “Interfer-ence suppression in CDMA downlink through adaptive chan-nel equalization,” in Proc. Vehicular Technology Conference,vol. 2 of Gateway to 21st Century Communications, pp. 978–982, Amsterdam, Netherlands, September 1999.

[12] P. Komulainen and M. J. Heikkila, “Adaptive channel equal-ization based on chip separation for CDMA downlink,” in


Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Com-munications, September 1999.

[13] K. Hooli, M. Latva-aho, and M. Juntti, “Performance eval-uation of adaptive chip-level channel equalizers in WCDMAdownlink,” in IEEE Int. Conf. on Communications, Helsinki,Finland, June 2001.

[14] K. Hooli, M. Latva-aho, and M. Juntti, “Novel adaptive chan-nel equalizer for WCDMA downlink,” in Proc. COST 262Workshop, pp. 7–11, Ulm, Germany, January 2001.

[15] J. S. Goldstein, I. S. Reed, and L. L. Scharf, “A multistage rep-resentation of the Wiener filter based on orthogonal projec-tions,” IEEE Transactions on Information Theory, vol. 44, no.7, pp. 2943–2959, 1998.

[16] J. S. Goldstein and I. S. Reed, “A new method of Wiener filter-ing and its application to interference mitigation for commu-nications,” in Proc. IEEE Military Communications Conference,Monterey, Calif, USA, November 1997.

[17] J. S. Goldstein, J. R. Guerci, and I. S. Reed, “An optimalgeneralized theory of signal representation,” in Proc. IEEEInt. Conf. Acoustics, Speech, Signal Processing, Phoenix, Ariz,USA, March 1999.

[18] X. Wang and H. V. Poor, “Blind multiuser detection: A sub-space approach,” IEEE Transactions on Information Theory,vol. 44, no. 2, pp. 677–690, 1998.

[19] J. S. Goldstein and I. S. Reed, “Reduced-rank adaptive filter-ing,” IEEE Trans. Signal Processing, vol. 45, no. 2, pp. 492–496,1997.

[20] D. A. Pados and S. N. Batalama, “Low-complexity blind detec-tion of DS/CDMA signals: Auxilliary-vector receivers,” IEEETrans. Communications, vol. 45, no. 12, pp. 1586–1594, 1997.

[21] D. A. Pados, F. J. Lombardo, and S. N. Batalama, “Auxiliary-vector filters and adaptive steering for DS/CDMA single-userdetection,” IEEE Trans. Vehicular Technology, vol. 48, no. 6,pp. 1831–1839, 1999.

[22] A. Kansal, S. N. Batalama, and D. A. Pados, “Adaptive max-imum SINR rake filtering for DS-CDMA multipath fadingchannels,” IEEE Journal on Selected Areas in Communications,vol. 16, no. 9, pp. 1831–1839, 1998.

[23] M. L. Honig and J. S. Goldstein, “Adaptive reduced-rankresidual correlation algorithms for DS-CDMA interferencesuppression,” in Proc. 32nd Asilomar Conf. on Signals, Systems,and Computing, Pacific Grove, Calif, USA, November 1998.

[24] M. L. Honig and W. Xiao, “Large system performance ofreduced-rank linear interference supression for DS-CDMA,”in Proc. Allerton Conf. on Communication, Control, and Com-puting, UIUC, October 1999.

[25] W. L. Myrick, M. D. Zoltowski, and J. S. Goldstein, “Anti-jamspace-time preprocessor for GPS based on multistage nestedWiener filter,” in Proc. IEEE Military Communications Confer-ence, Atlantic City, NJ, USA, October 1999.

[26] W. L. Myrick, M. D. Zoltowski, and J. S. Goldstein, “Low-sample performance of reduced-rank power minimizationbased jammer suppression for GPS,” in IEEE Sixth Interna-tional Symposium on Spread Spectrum Techniques & Applica-tions, pp. 93–97, Parsippany, NJ, USA, September 2000.

[27] T. P. Krauss and M. D. Zoltowski, “MMSE equalization un-der conditions of soft hand-off,” in IEEE Sixth InternationalSymposium on Spread Spectrum Techniques & Applications, pp.540–544, NJIT, Parsippany, NJ, USA, September 2000.

[28] T. P. Krauss, W. J. Hillery, and M. D. Zoltowski, “MMSEequalization for forward link in 3G CDMA: symbol-level ver-sus chip-level,” in Proc. 10th IEEE Workshop on Statistical Sig-

nal and Array Processing, pp. 18–22, Pocono Manor, Pa, USA,August 2000.

[29] M. Joham and M. D. Zoltowski, “Interpretation of the multi-stage nested Wiener filter in the Krylov subspace framework,”Tech. Rep. TR-ECE-00-51, Purdue University, West Lafayette,Ind, USA, November 2000.

[30] D. Ricks and J. S. Goldstein, “Efficient architectures for im-plementing adaptive algorithms,” in Allerton Antenna ArraysSymp., UIUC, September 2000.

[31] B. C. Ng, D. Gesbert, and A. Paulraj, “A semi-blind ap-proach to structured channel equalization,” in Proc. IEEEInt. Conf. Acoustics, Speech, Signal Processing, Seattle, Wash,USA, May 1998.

Samina Chowdhury was born in Dhaka,Bangladesh in December 1971. She receivedher B.S. degree in electrical and electronicengineering from Bangladesh Universityof Engineering and Technology, Dhaka,Bangladesh in June 1996 with the top po-sition in her class. She was employed as aLecturer in the Electrical Engineering De-partment of BUET from October 1996 toJuly 1997. She enrolled in the Ph.D. pro-gram in electrical and computer engineering at Purdue Univer-sity, West Lafayette, Indiana, in August 1997. She completed herPh.D. in December 2001 under supervision of Professor Michael D.Zoltowski. Her research interests include mobile cellular commu-nications, space-time processing, reduced-rank linear processing,and physical layer system design for wireless communications.

Michael D. Zoltowski was born in Philadel-phia, PA, on August 12, 1960. He receivedboth the B.S. and M.S. degrees in electri-cal engineering with highest honors fromDrexel University in 1983 and the Ph.D. insystems engineering from the University ofPennsylvania in 1986. In fall 1986, he joinedthe faculty of Purdue University where hecurrently holds the position of Professor ofelectrical and computer engineering. In thiscapacity, he was the recipient of the IEEE Outstanding BranchCounselor Award for 1989–1990 and the Ruth and Joel SpiraOutstanding Teacher Award for 1990–1991. In August 2001, hewas named a University Faculty Scholar by Purdue University. Dr.Zoltowski was the recipient of the IEEE Signal Processing Soci-ety’s 1991 Paper Award (Statistical Signal and Array ProcessingArea), “The Fred Ellersick MILCOM Award for Best Paper in theUnclassified Technical Program” at the IEEE Military Communi-cations (MILCOM ’98) Conference, and a Best Paper Award atthe IEEE International Symposium on Spread Spectrum Techniquesand Applications (ISSSTA 2000). He is also a corecipient of theIEEE Communications Society 2001 Leonard G. Abraham PrizePaper Award in the Field of Communications Systems. He is acontributing author to Adaptive Radar Detection and Estimation,Wiley, 1991, Advances in Spectrum Analysis and Array Processing,Vol. III, Prentice-Hall, 1994, and CRC Handbook on Digital SignalProcessing, CRC Press, 1996. He has served as a consultant to theGeneral Electric Company, and currently serves as a consultant toSAIC, Zenith, and ATLINKS. He has served as an associate editor


for both the IEEE Transactions on Signal Processing and the IEEECommunications Letters. Within the IEEE Signal Processing Society,he has been a member of the Technical Committee for the Statisti-cal Signal and Array Processing Area and the Technical Committeeon DSP Education. Currently, he is a member of both the TechnicalCommittee on Signal Processing for Communications (SPC) andthe Technical Committee on Sensor and Multichannel (SAM) Pro-cessing. From 1998 to 2001, he was a member-at-large of the Boardof Governors, and Secretary of the IEEE Signal Processing Society.He is a fellow of IEEE. His present research interests include space-time adaptive processing for all areas of mobile and wireless com-munications, GPS, and radar.

J. Scott Goldstein is an Assistant Vice-President at SAIC where he manages theTargeted Information Processing SolutionsDivision. He has over 17 years of experi-ence in the fields of radar, sonar, commu-nications, navigation, and imaging sensors.He is also an Adjunct Professor of electri-cal engineering at Virginia Tech, where heteaches courses on radar systems and signalprocessing. Dr. Goldstein is a Fellow of theIEEE, an Associate Vice-President of the IEEE Aerospace and Elec-tronic Systems Society, a member of the IEEE Radar Systems Panel,and a member of the IEEE Fellow Selection Committee. His recentawards include the 2002 IEEE Fred Nathanson Radar Engineer ofthe Year Award and the 2001 IEE Clarke-Griffith Memorial PaperAward and Premium for the best paper in the IEE Proceedings—Radar, Sonar, and Navigation. Dr. Goldstein has authored or co-authored over 100 technical publications and is a member of EtaKappa Nu, Tau Beta Pi, and Sigma Xi.


EM-Based Multiuser Detection in Fast FadingMultipath Environments

Mohammad Jaber BorranDepartment of Electrical and Computer Engineering, Rice University, P.O. Box 1892, Houston, TX 77251-1892, USAEmail: [email protected]

Behnaam AazhangDepartment of Electrical and Computer Engineering, Rice University, P.O. Box 1892, Houston, TX 77251-1892, USAEmail: [email protected]

Received 1 October 2001 and in revised form 8 March 2002

We address the problem of multiuser detection in fast fading multipath environments for DS-CDMA systems. In fast fadingscenarios, temporal variations of the channel cause significant performance degradation even with the Rake receiver. We use apreviously introduced time-frequency (TF) Rake receiver based on a canonical formulation of the channel and signals to simul-taneously combat fading and multipath effects. This receiver uses the Doppler spread caused by rapid time-varying channel asanother means of diversity. In dealing with multiaccess interference and as an attempt to avoid the prohibitive computationalcomplexity of the optimum maximum-likelihood (ML) detector, we use the expectation maximization (EM) algorithm to derivean approximate ML detector. The new detector turns out to have an iterative structure very similar to the well-known multistagedetector with some extra parameters. At the two extreme values of these parameters, the EM detector reduces to either one-shotTF Rake or generalized multistage detector. For the intermediate values of the parameters, it combines the two estimates to obtaina better decision for the bits of the users. Because of using the EM algorithm, this detector has better convergence properties thanthe multistage detector; the bit estimates always converge, and if an appropriate initial vector is used, they converge to the globalmaximizer of the likelihood function. As a result, the new detector provides significantly improved performance while maintainingthe low complexity of the multistage detector. Our simulation results confirm the expected performance improvements comparedto the base case of the TF Rake as well as the multistage detector used with the TF Rake.

Keywords and phrases: CDMA systems, multiuser detection, EM algorithm, multipath-Doppler diversity, time-frequency Rake.

1. INTRODUCTION

Multipath, fading, and multiple-access interference are themajor factors that limit the performance of the existing mo-bile wireless communication systems. Fading of the receivedsignal caused by wireless channels, coupled with the interfer-ence from other transmitters using the same channel, signif-icantly degrades the performance of the receiver.

Wideband code-division multiple access (WCDMA), theaccepted technology for the next generation cellular net-works, provides intrinsic protection against the multipatheffects of the channel. A Rake receiver structure is used toexploit the large time-resolution of the wideband signal andcapture the information in its multipath components.

In fast-fading scenarios, temporal variations of the chan-nel cause significant performance degradation even with theRake receiver. The Doppler spread caused by rapid time-varying channel can be used as another means of diver-sity in such environments. Joint multipath-Doppler diver-sity schemes [1, 2, 3] use a canonical representation of the

channel and signals to capture the multipath-Doppler com-ponents of the signal.

In multiple-access environments, the minimum prob-ability of error reception can be achieved by a maximumlikelihood (ML) receiver [4]. Although this optimal receivershows significant performance gains over the conventionaldetector, its computational complexity, which grows expo-nentially with the number of users, prohibits its practical im-plementation. Therefore, some practical suboptimum detec-tors have been introduced for multiuser detection [5, 6, 7, 8,9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

Lupas and Verdu [5] describe a family of linear detectorscalled decorrelator. These detectors eliminate multiuser in-terference at the expense of increased noise power. Further-more, the linear decorrelating detectors require the correla-tion matrix inversion, which may be difficult to perform inreal time, especially for asynchronous systems. Some subop-timal approaches have been taken to implement the decor-relating detector for asynchronous systems [6, 7, 8, 9]. The


most important advantage of the decorrelating detector isthat it does not require the estimation of the received am-plitudes.

Madhow and Honig [10] and Xie et al. [6] describea minimum mean-squared error (MMSE) linear detector,which minimizes the mean-squared error between the actualdata and the conventional detector soft outputs. Because oftaking the background noise into account, the MMSE detec-tor generally performs better than the decorrelating detec-tor, and converges to the decorrelating detector as the back-ground noise goes to zero.

Duel-Hallen in [11] presents a nonlinear multiuserdetector called decorrelating decision-feedback detector(DDFD) in which the users are ranked according to theirsignal strengths from the strongest one to the weakest one.This detector is based on a white noise channel model whosenoise-whitening filter is obtained by the Cholesky decom-position of the cross-correlation matrix. The detector per-forms successive interference cancellation at the output of thenoise-whitening filter using past decisions. For the strongestuser, this detector performs similarly to the decorrelator, butas the user’s power decreases compared to the power of inter-ferers, the detector outperforms the decorrelator and its per-formance approaches the single user bound. However, its im-portant difficulty is the need for computing the Cholesky de-composition. Other successive interference cancellation de-tectors are described in [12, 13].

In [14, 15], Varanasi and Aazhang describe a parallel in-terference cancellation detector called multistage detector inwhich the tentative decisions obtained from the previousstage are used to estimate and subtract the multiuser inter-ference. The first stage decisions are usually obtained fromthe conventional detector. This detector, like the DDFD of[11], outperforms the decorrelator when interfering users arestronger than the user under consideration, but its perfor-mance degrades as the energies of the interfering users de-crease.

The expectation maximization (EM) algorithm has alsobeen applied for multiple-access interference suppression inCDMA systems [16, 17, 18, 19], as well as for channel esti-mation [20, 21, 22]. In [16, 19], an iterative interference can-cellation method in additive white Gaussian noise (AWGN)channels based on the EM algorithm is proposed. Since thelikelihood function is bounded above, and since the EM es-timates monotonically increase in likelihood, the suggestedreceiver is convergent. Also, because of taking into accountthe previous decision about the data symbol of each user inmaking new decision for that user, this detector outperformsthe parallel interference cancellation detector of [15] forstrong users, while having similar performance for the otherusers.

In [17], Nelson and Poor propose some other iterativemultiuser receivers for CDMA systems, based on the EM al-gorithm and its generalized versions, such as space alternat-ing generalized EM (SAGE), and missing-parameter space-alternating algorithm. The suggested multiuser detectorshave structures similar to the parallel interference cancella-tion method of [14], except that updates of the estimates are

made sequentially, rather than in parallel. For the same rea-son mentioned above, these algorithms are also convergent.The MPEM receiver suggested in this paper has a computa-tional complexity which is proportional to the square of thenumber of the users, whereas the computational complex-ity of the original parallel interference cancellation methodgrows only linearly with the number of users.

In [18], the EM algorithm is applied to maximize thelikelihood function over a nondiscrete set. The discrete se-quence is obtained by quantizing the unquantized estimatedsequence at convergence. Since the nondiscrete maximiza-tion problem has a closed form solution, namely, the decor-relator, the performance of this scheme is expected to beupper bounded by the performance of the decorrelating re-ceiver. However, depending on the number of the iterationsused, the computational complexity of this scheme might belower. The proposed receiver also iterates between path com-ponent estimation and maximal-ratio combining to refinethe nondiscrete sequence estimate.

In this paper, we first review the canonical representationof the signal and channel in fast fading multipath environ-ments [1, 3]. Then, in Section 3, we review some of the mul-tiuser detection techniques in fast fading channels using thisrepresentation. These include the optimal (minimum proba-bility of error) and the linear suboptimal decorrelating andMMSE receivers, rederived in [23] for the time-frequency(TF) Rake, as well as a generalization of the multistage de-tector of [15].

As mentioned earlier, we intend to use the EM algorithmto find an iterative approximate ML solution for the mul-tiuser detection problem. For this, we first, in Section 4, re-view the EM algorithm, and then, in Section 5, in a similarway to [16], derive the new detection scheme for fast fad-ing multipath environments with canonical representation.The proposed detector uses the two-dimensional TF Rake re-ceiver [1, 3] to combat the fading and multipath effects. Thesimulation results are reported in Section 6, and show the su-perior performance of the proposed detector compared tothe original TF Rake, as well as the generalized multistagedetector. Finally, Section 7 contains the conclusions.

2. CANONICAL TIME-FREQUENCY REPRESENTATIONOF THE SIGNALS AND CHANNEL

The TF canonical representation [1, 3] exploits the multi-path and Doppler effects for obtaining diversity and resultsin a two-dimensional Rake receiver, which extracts Dopplercomponents in addition to multipath components. This rep-resentation reduces the channel to a set of independent chan-nels for the different time-delayed frequency-shifted versionsof the signal for each user. Figure 1 illustrates the locations ofcanonical coordinates in the time delay-Doppler shift plane,used for TF representation of the channel.

In a multiuser system, the received signal is a superposi-tion of the signals of different users and noise. In this work,we consider a synchronous CDMA system in which the sig-nature sequences of different users are aligned in time. With

EM-Based Multiuser Detection in Fast Fading Multipath Environments 789

Multipath

−M

M

Dop

pler

θTc

1/Ts

L τ

Figure 1: Canonical coordinates.

this assumption, if the delay spread of the channel is muchsmaller than the symbol interval, we can ignore the correla-tion terms between the symbols of different users in adjacenttime intervals, and use a one-shot detector for estimating thedata bits of different users, as in [23]. Therefore, we can re-strict ourselves to only the first time interval and assume thatthe received signal is as follows:

r(t) =K∑k=1

bkxk(t) + n(t), for 0 ≤ t ≤ Ts, (1)

where K is the number of users, bk denotes the data bit of thekth user, n(t) is a white Gaussian noise with zero mean andvariance σ2, Ts is the symbol interval, and

xk(t) =∫ Tm

0hk(t, τ)sk(t − τ)dτ, for k = 1, 2, . . . , K. (2)

In this equation, sk(t) and hk(t, τ) are, respectively, the sig-nature signal and the time-varying channel impulse responsefor the kth user, and Tm denotes the multipath (delay) spreadof the channel.

An equivalent representation for the signal xk(t) in termsof the channel spreading function Hk(θ, τ) [24] (i.e., theFourier transform of hk(t, τ) with respect to t), is

xk(t) =∫ Tm

0

∫ Bd

−Bd

Hk(θ, τ)e j2πθtsk(t − τ)dθ dτ, (3)

where θ corresponds to Doppler shifts introduced by thechannel and Bd denotes the Doppler spread of the channel.We use the wide-sense stationary uncorrelated scatterer (WS-SUS) [24] model for the channel, which assumes that H(θ, τ)is a two-dimensional uncorrelated Gaussian process.

For a spread spectrum signal s(t) of duration Ts and chipinterval Tc, and with the WSSUS assumption for the chan-nel, using the canonical coordinates [1, 3], we can rewritethe signal xk(t) as

xk(t) ≈L∑l=0

M∑m=−M

Hmlk sml

k (t), for 0 ≤ t ≤ Ts, (4)

where

smlk (t) = sk(t−lTc)e j(2πmt/Ts)s, Hml

k = Tc

TsHk

(m

Ts, lTc

), (5)

for l = 0, 1, . . . , L, m = −M,−M + 1, . . . ,M, k = 1, 2, . . . , Kwith the number of multipath components L = Tm/Tc,and the number of Doppler components M = BdTs. Here,Hk(θ, τ) is the time-frequency smoothed version of H(θ, τ)[23] given by the following expression:

Hk(θ, t) = Ts

Tc

∫ Tm

0

∫ Bd

−Bd

Hk(θ, τ)e− jπ(θ−θ′)Ts sinc((θ − θ′

)Ts)

× sinc(τ − τ′

Tc

)dθ′ dτ′.

(6)

In order to simplify the mathematical expressions, weuse the following vector notation for the time-delayed andfrequency-shifted versions of the signature waveforms of theusers,

s(t) =[

s1(t)T s2(t)T · · · sK (t)T]T

, (7)

where

sk(t) =[s−M0k (t) · · · s−ML

k (t)s(−M+1)0k (t) · · · s(−M+1)L

k (t)

· · · sM0k (t) · · · sML

k (t)]T(8)

for k = 1, 2, . . . , K . Using this representation, the K(L +1)(2M+1)×K(L+1)(2M+1) cross-correlation matrix of thecomponents of the signature waveforms of different users is

R =∫ Ts

0s∗(t)sT(t)dt =

R11 R12 · · · R1K

R21 R22 · · · R2K

......

. . ....

RK1 RK2 · · · RKK

, (9)

where

Rkl =∫ Ts

0s∗k (t)sTl (t)dt, for k, l = 1, 2, . . . , K. (10)

We also define the channel matrix H as

H =

h1 0 · · · 00 h2 · · · 0...

.... . .

...

0 0 · · · hK

, (11)


where

hk =[H−M0

k · · ·H−MLk H(−M+1)0

k · · ·H(−M+1)Lk · · ·HM0

k · · ·HMLk

]T(12)

for k = 1, 2, . . . , K .Using the above notations, (4) and (1) can be rewritten

as

xk(t) ≈ sk(t)Thk, for 0 ≤ t ≤ Ts, (13)

r(t) ≈ s(t)THb + n(t), for 0 ≤ t ≤ Ts, (14)

where b = [b1 b2 · · · bK ]T . In Section 3, we will see thatthe outputs of the time-frequency Rake receiver, given as

zk =∫ Ts

0s∗k (t)r(t)dt, for k = 1, 2, . . . , K, (15)

form a set of sufficient statistics for ML multiuser detec-tion. We collect all of these vectors in one vector z =[zT1 zT2 · · · zTK ]T . Using (14) and (9), it can be easily shownthat

z =∫ Ts

0s∗(t)r(t)dt ≈ RHb + w, (16)

where

w =∫ Ts

0s∗(t)n(t)dt (17)

is a zero mean complex Gaussian noise vector withE[wwH] = σ2R.

3. REVIEW OF SOME MULTIUSER DETECTIONSCHEMES

In this section, we review the optimal and linear suboptimalmultiuser detectors rederived in [23] for fast fading channels.We also consider the generalization of the well-known mul-tistage detector to fast fading channels using the TF Rake.

3.1. Conventional single user receiver

The single user receiver assumes that there is no multiaccessinterference, that is, either there are no interfering users, orthe signature codes of all of the users and their shifted ver-sions are orthogonal. It can be easily shown [1, 2] that, inthis case, the TF Rake receiver with maximal ratio combin-ing (MRC), given by the following expression, is the optimal(i.e., minimum probability of error) receiver:

bk = sgn[

hHk zk

], for k = 1, 2, . . . , K. (18)

This receiver coherently combines the different multipath-Doppler shifted components of the signal to achieve a diver-sity of order (L + 1)(2M + 1). Of course, it is assumed thatthe receiver has complete channel state information (CSI). In

practice, channel coefficients,Hmlk , may be estimated through

a pilot signal transmission.In the presence of multiaccess interference, that is, when

the signature codes of the interfering users are not completelyorthogonal, the above receiver is no longer optimal, and doesnot show acceptable performance. The optimal multiuser de-tector is discussed in Section 3.2, and has a much more com-putational complexity.

3.2. Minimum probability of error receiver

Initially introduced by Verdu [4], the ML multiuser receiverachieves the minimum probability of error and is optimalin this sense. For the problem under consideration, the log-likelihood function of the received signal (1) can be writtenas

log fR(r; b) = A− 12σ2

∫ Ts

0

∣∣∣∣∣r(t)−K∑k=1

bkxk(t)

∣∣∣∣∣2

dt, (19)

where A is a constant. The ML receiver finds the vectorbopt = [b1 b2 · · · bK ]T , such that the above log-likelihoodfunction is maximized for b = bopt.

Ignoring the constant terms and the terms which do notdepend on the unknown bits of the users, and using (9),(10), (13), (14), (15), and (16), we define the simplified log-likelihood function as

Λ(r; b) =K∑k=1

2hHk zk

bk −

K∑k=1

K∑l=1

bkhHk Rklhlbl

= 2[bTHHz

]− bTHHRHb.

(20)

Therefore, the decision rule for the ML receiver can be writ-ten as

bopt = arg maxb∈−1,1K

Λ(r; b). (21)

We observe that the outputs of the TF Rake, zk for k =1, 2, . . . , K , form a set of sufficient statistics for the detectionproblem. We also observe that, still, maximal ratio combin-ing of the outputs of the TF Rake is necessary, though notsufficient.

The above maximization is a K-dimensional discrete op-timization problem and requires a search over 2K possi-bilities. As a result, the computational complexity of thereceiver increases exponentially with the number of users,which makes its real-time implementation prohibitive forlarge number of users. Therefore, several suboptimal ap-proaches have been proposed. In the next subsections, we re-view some of these suboptimal receivers. Later, in Section 5,we introduce a new detection scheme, which iteratively solvesthe above optimization problem, and even with a few num-ber of stages, shows better performance compared to the ex-isting schemes with similar complexity.

3.3. Linear suboptimal multiuser receivers

Having established that z = [z1 z2 · · · zK ] is a sufficientstatistic for the detection problem, we can try other low com-plexity processings of this vector to obtain some suboptimal


receivers. The approach is motivated by the fact that, in theabsence of multiaccess interference, that is, when the noisefree output of the correlators for the kth user is equal to hkbk,the MRC is optimal. Therefore, we first try to find a reliableestimate for the vectors hkbk for k = 1, 2, . . . , K , given theobservation z, and then, to coherently combine them to ob-tain the bit estimate for each user. In [23], based on the aboveidea, the well-known decorrelating and MMSE receivers arerederived for the TF Rake. Since the noise vector at the out-puts of these linear processings is correlated, a whiteningoperation is performed before maximal ratio combining ofthese outputs.

If the linear operation involved in the linear detector isperformed using a matrix F, the general form of the overalllinear multiuser TF Rake receiver will be

b = sgn[

HHDFz], (22)

where D is a block diagonal whitening matrix. The entriesof this matrix depend on the type of the linear processing,that is, the matrix F, as well as the correlation matrix of thesignature codes, R,

D =

Q−1

11 0 · · · 0

0 Q−122 · · · 0

......

. . ....

0 0 · · · Q−1KK

, (23)

where

Q = E[

FwwHFH] = σ2FRFH

=

Q11 Q12 · · · Q1K

Q21 Q22 · · · Q2K

......

. . ....

QK1 QK2 · · · QKK

.(24)

In Sections 3.3.1 and 3.3.2, we will consider two specialcases of the above generic linear detector, called decorrelatingand linear MMSE receivers.

3.3.1 Decorrelating receiver

From the likelihood function (20), it is easy to show that theML estimate for u = Hb is given by

uML = arg maxu

2[

uHz]− uHRu

= R−1z. (25)

Therefore, from (22) by letting F = R−1, a generalization ofthe decorrelating receiver of [5] can be obtained,

bdec = sgn[

HHDdecR−1z], (26)

where Ddec is defined as in (23), with Q = Qdec = σ2R−1.This detector eliminates multiuser interference at the ex-

pense of increasing the noise power. It also requires the cor-relation matrix inversion, which may be difficult to performin real time.

3.3.2 Linear MMSE receiver

A generalization of the linear MMSE multiuser detector of[6, 10] results from employing a linear MMSE estimate foru = Hb. It is shown in [23] that the corresponding linearoperation, F, for this detector is given by

FMMSE = arg minF

E‖Hb− Fz‖2 = (R + σ2Ψ−1)−1

, (27)

where Ψ = E[HHH]. The resulting linear MMSE TF Rakereceiver is given by

bMMSE = sgn[

HHDMMSE(

R + σ2Ψ−1)−1z]

, (28)

where DMMSE is defined as in (23), with Q = QMMSE = σ2(R+σ2Ψ−1)−1R(R + σ2Ψ−1)−1 (for a WSSUS channel, Ψ is a realdiagonal matrix [23]).

Because of taking the background noise into account, thisdetector generally performs better than the decorrelating de-tector. However, like the decorrelating detector, it requires acorrelation matrix inversion, which may be difficult to per-form in real time.

3.4. Generalized multistage receiver

In [14], Varanasi and Aazhang describe a parallel interfer-ence cancellation detector called multistage detector, whichattempts to iteratively maximize the likelihood function. Ateach stage, the bit estimate for each user is obtained by maxi-mizing the likelihood function over the possible values of thedata bit of that user, and by using the bit estimates from theprevious stage for all other users. From the likelihood func-tion (20), it is easy to show that for the system with TF Rake,the (n + 1)st-stage estimate of the data bit of the kth user,using this multistage detector will be given by the followingexpression:

b(n+1)k = arg max

bk∈−1,1bl=b(n)

l ,l =k

Λ(r; b)

= sgn

[

hHk zk −

K∑j=1, j =k

b(n)j hH

k Rk jh j

].

(29)

As it can be seen from the above expression, the tentativedecisions obtained from the previous stage are used to esti-mate and subtract the multiuser interference. The first stagedecisions are usually obtained from the conventional detec-tor, which will be given by

b(0)k = sgn

[hHk zk

], (30)

if the TF Rake is used. This detector outperforms the decor-relating detector when the interfering users are stronger thanthe user under consideration, but its performance degradesas the energies of the interfering users decrease. In this case,that is, when the interfering users are not much strongerthan the user under consideration, because of the enor-mous errors in the estimate of the interference, the perfor-mance of the multistage detector can be even worse than the


conventional detector, and using more stages may only resultin even more degraded performance. Examples of this situa-tion are given in Figures 3 and 5 and discussed in Section 6.

In general, there is no guarantee that the multistage de-tector will converge, or in convergence, if at all, will producethe global maximizer of the likelihood function. However, itslower computational complexity, which is a result of its iter-ative nature, is a motivation to look for other iterative meth-ods for maximizing the likelihood function, which have bet-ter convergence properties. The EM algorithm is one of thesemethods, and will be reviewed in the next section.

4. EM ALGORITHM

Expectation maximization algorithm is an iterative methodfor maximizing log-likelihood functions. The original prob-lem is formulated as the following optimization problem:

maximizeb

log fR(r; b), (31)

where r is the observed data. The vector b can be any setof parameters. In the problem under consideration, it is thevector of unknown data bits of different users. This is a K-dimensional discrete optimization problem whose real-timeimplementation is prohibitive because of exponential com-plexity in K (number of users). To construct an iterative sub-optimal solution for this problem, a set of complete data, y,is defined such that

r = g(y1, y2, . . . , yK

) = g(y), (32)

where g is some many-to-one transformation relating thecomplete data set, y, to the observation r. Then, instead ofsolving the problem given in (31), we solve the followingmaximization problem:

maximizeb

log fY(y; b). (33)

However, as mentioned above, y is related to r by a many-to-one transformation and there is no unique y for each valueof r. Therefore, we replace the log-likelihood function in (33)with its expected value with respect to y given r, and maxi-mize the following expression:

EY

log fY(y; b) | R = r; b = ∫

log fY(y; b) fY|R(y | r; b)dy.

(34)Since b is also unknown, we cannot calculate fY|R(y | r; b) in(34), therefore we replace b in fY|R(y | r; b) with the current

estimate of b, that is, b, and maximize the following functionwith respect to its first argument, b,

U(

b, b) = ∫

log fY(y; b) fY|R(y | r; b)dy. (35)

Using Jensen’s inequality, it can be shown that

U(

b, b)> U

(b, b

) =⇒ fR(r; b) > fR(r; b

). (36)

This provides the following iterative method for max-imizing likelihood function and guarantees that the likeli-hood function does not decrease along the iterations:

• E-step (Expectation calculation step): computeU(b, b(n)),

U(

b, b(n)) = ∫log fY(y; b) fY|R

(y | r; b(n))dy, (37)

where b(n) is the estimate of b in the nth iteration.• M-step (Maximization step): maximize U(b, b(n)),

b(n+1) = arg maxb

U(

b, b(n)). (38)

Since the likelihood function is bounded above, and sincethe above estimates monotonically increase in likelihood, weexpect the algorithm to converge to at least a local maximizer.

By an appropriate choice of the initial estimates, b(0), the al-gorithm can produce the global maximizer of the likelihoodfunction.

In most cases, if the complete data is chosen properly,the maximization step of the above algorithm can be decom-posed into K one-dimensional maximization, which has lin-ear complexity in K and can be easily implemented for real-time processing.

5. EM-BASED MULTIUSER DETECTOR

In order to apply the EM algorithm to the problem in hand,we define the complete data, y(t) = [y1(t) · · · yK (t)]T ,where

yk(t) = bkxk(t) + nk(t), for k = 1, . . . , K, (39)

and nk(t), k = 1, . . . , K are independent additive white Gaus-sian noise with variance σ2

k . Then we have r(t) =∑Kk=1 yk(t),

and the log-likelihood function of the complete data is

log fY(y; b) = B −K∑k=1

12σ2

k

∫ Ts

0

∣∣yk(t)− bkxk(t)∣∣2

dt, (40)

where B is a constant.In the appendix, we will show that with this choice of

complete data, the result of the E-step, that is, U(b, b(n)), isgiven by the following equality:

U(

b, b(n))= K∑k=1

bkσ2k

[b(n)k hH

k Rkkhk

+σ2k

σ2

(hHk zk−

K∑j=1

b(n)j hH

k Rk jh j

)].

(41)

Since the data bit of each user appears only in one of theterms in the summation in (41), we can maximize each termseparately in the M-step. Therefore, defining βk = σ2

k /σ2, the


MAI estimation &cancellation · · · MAI estimation &

cancellation

β β· · ·

HHz

TF RAKE+

MRC

sgnb(0)

I-β + sgnb(1)

· · ·b(n−1)

I-β + sgnb(n)

Figure 2: Multiuser receiver structure.

iterative equation for updating the (n+ 1)st-stage estimate ofthe data bit of the kth user will be

b(n+1)k =sgn

[(

1−βk)b(n)k

+βk

hHk Rkkhk

(hHk zk−

K∑j=1, j =k

b(n)j hH

k Rk jh j

)].

(42)

As mentioned in Section 4, by an appropriate choice ofthe initial values for the unknown parameters, the algorithmconvergence to the global maximizer of the log-likelihoodfunction. As in the well-known multistage detector, a good

choice for b(0)k can be the output of the filter matched to the

signature signal of the kth user, or if, as in our case, multipathand Doppler diversities are available, the maximal ratio com-bined outputs of the time-frequency Rake receiver for the kthuser,

b(0)k = sgn

[hHk zk

]. (43)

The block diagram of this multiuser detection scheme isshown in Figure 2.

With the above assumption for the initial value for b, wecan consider two extreme special cases of the new detectionscheme as follows:

• if βk = 1, then the new detector for user k will be thesame as the multistage one;

• if βk = 0, then the new detector for user k will lose itsiterative nature, and will reduce to the time-frequencyRake receiver with maximal ratio combining.

With a suitable choice of parameter β for different users,we hope to achieve better performance than both TF Rakeand multistage receivers. According to the discussions ofSection 3.4, we expect that large (close to one) values of β willresult in good performance for weak (in terms of the signal-to-interference ratio) users, whereas for strong users, smallervalues of β will provide better performance. This parameteralso determines the speed of convergence of the iterative al-gorithm. In our simulations discussed in Section 6, the valueof this parameter for each user is chosen by simulation for the

best performance. However, further simulations show thatthe performance of the detector is not very sensitive to theexact values of these parameters, and the values from the fol-lowing heuristic expression:

βk = 11 + SIRk

, (44)

where SIRk is a measure of the signal-to-interference ratio,calculated as

SIRk =∣∣hH

k Rkkhk

∣∣∑l =k

∣∣hHk Rklhl

∣∣ , (45)

provide similar performance.

6. SIMULATION RESULTS

We implemented the EM-based multiuser detector and com-pared its performance with the base case of the time-frequency Rake as well as the multiuser detector. The simu-lations are done for a system with five users with Gold se-quences of spreading length 7. In the EM and multistagedetectors we obtained the performance curves for two- andthree-stage cases. The channel was modeled as a three-pathchannel, with independent Jakes’ models for each path.

Figures 3 and 4 show the plots of bit error rate (BER) ver-sus the signal-to-noise Ratio (SNR) for a case with Dopplerfrequency of 100 Hz. We observe that the performance ofthe EM-based detector is better for both users than the basecase of the TF Rake as well as the multistage detector. No-tice that for the multistage detector, the performance of thethree-stage detector is worse than the two-stage detector foruser 2, and does not show much improvement in the perfor-mance for user 5. As a result, the performance of the two-stage EM detector is better than the three-stage multistagedetector with higher computational complexity. It should benoted that the computational complexities of these two de-tectors with the same number of stages are similar. Finally,we observe that the three-stage EM provides significant gainswith respect to the multistage case.

Similarly, Figures 5 and 6 show that the performance isconsistent with other values of the Doppler (200 Hz). EM de-tector also shows similar performance for other users.


151050−5−10SNR in dB

10−3

10−2

10−1

100

BE

R

User 2, β2 = 0.6

TF RAKEMultistage 2-stageEM 2-stage

Multistage 3-stageEM 3-stage

Figure 3: BER versus SNR plot for Doppler = 100 Hz.

151050−5−10SNR in dB

10−4

10−3

10−2

10−1

100

BE

R

User 5, β5 = 0.8




Note that the different users have different β’s in the dif-ferent plots. The appropriate value for parameter β can resultin a rapid convergence of the EM algorithm. In our simu-lations, these parameters are chosen by simulation for thebest performance within two or three stages. As mentioned inSection 5, however, even the values obtained from the heuris-tic expression (44) provide satisfactory performance.

151050−5−10SNR in dB

10−4

10−3

10−2

10−1

100

BE

R

User 3, β3 = 0.8




151050−5−10SNR in dB

10−3

10−2

10−1

100

BE

R

User 5, β5 = 0.8




7. CONCLUSIONSWe have presented a new multiuser detector for CDMA sys-tems in fast fading multipath channels. The detector uses thetime-frequency Rake receiver at the front end to exploit mul-tipath and Doppler spreads as two sources of diversity. Themultiaccess interference cancellation part of the detector isbased on the EM algorithm. It has an iterative structure very


similar to the generalized multistage detector but with betterconvergence properties. As a result, unlike the multistage de-tector whose performance could become very poor for strongusers because of the errors in the decisions of the weak users,this detector shows good performance for all users. Our sim-ulation results show that the new EM-based detector can pro-vide a substantial improvement in performance compared tothe generalized multistage detector as well as the TF Rake.

The improvement in the performance comes at the ex-pense of introducing a set of new parameters, which haveto be chosen appropriately. In this paper, the optimum val-ues for these parameters were found by simulation and ex-haustive search. Finding an analytical expression for the op-timum values of these parameters is not addressed in this pa-per and requires more investigation, but we have providedan ad hoc expression which is shown to provide satisfactoryperformance, very close to that of optimum values found bysimulation.

APPENDIX

In this appendix, we apply the E-step of the EM algorithm to(40) to obtain (41). Expanding the squared absolute value in(40) and noting that b2

k = 1, we have

log fY(y; b) = g(y) +K∑k=1

1σ2k

[bk

∫ Ts

0

yk(t)x∗k (t)dt],

(A.1)

where g(y) is a function of y and does not depend on b.

According to the definition of U(b, b(n)), we haveto compute the conditional expected value of thelog-likehood function in (A.1) given the observed sig-

nal r(t), at a parameter value b(n). Defining C(t) =[(b1/σ

21 )x∗1 (t) · · · (bK/σ2

K )x∗K (t)]T and ignoring the firstterm g(y), which has no effect on the maximization process,we have

U(

b, b(n))=

∫ Ts

0CT(t)E

y(t) | r(t); b(n)

dt

. (A.2)

Since both y(t) and r(t) given b(n) are Gaussian, we canwrite

E

y(t) | r(t); b(n)

= E

y(t) | b(n)

+ CYrC

−1rr

[r(t)−E

r(t) | b(n)

],

(A.3)

where

CYr

= E

(y(t)−E

y(t) | b(n)

)∗(r(t)−E

r(t) | b(n)

)| b(n)

,

Crr = E

(r(t)−E

r(t) | b(n)

)2 | b(n).

(A.4)

It can be easily shown that

E

y(t) | b(n)

=[b(n)

1 x1(t) · · · b(n)K xK (t)

]T,

E

r(t) | b(n)

=

K∑k=1

b(n)k xk(t),

CYr =[σ2

1 · · · σ2K

]T,

Crr = σ2.

(A.5)

Substituting (A.5) in (A.3), we have

E

y(t) | r(t); b(n)

=

b(n)

1 x1(t)...

b(n)K xK (t)

+

σ2

1...σ2K

1σ2

r(t)−

K∑k=1

b(n)k xk(t)

,

(A.6)

and (41) can be obtained by substituting (A.6) in (A.2) andusing (10), (13), and (15).

ACKNOWLEDGMENTS

The first author wishes to thank Srikrishna Bhashyam forhelpful discussions, and Zeljko Cakareski, Ahmad Khosh-nevis, and Vishwas Sundaramurthy for providing some of thesimulation programs.

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[21] H. V. Poor, “On parameter estimation in DS/SSMA formats,”in Advances in Communications and Signal Processing, vol. 129of Lecture Notes in Control and Information Sciences, pp. 59–70, Springer-Verlag, Heidelberg, Germany, 1989.

[22] U. Fawer and B. Aazhang, “A multiuser receiver for code di-vision multiple access communications over multipath chan-nels,” IEEE Trans. Communications, vol. 43, no. 2–4, pp. 1556–1565, 1995.

[23] A. M. Sayeed, A. Sendonaris, and B. Aazhang, “Multiuser de-tection in fast-fading multipath environments,” IEEE Journalon Selected Areas in Communications, vol. 16, no. 9, pp. 1691–1701, 1998.

[24] J. G. Proakis, Digital Communications, McGraw-Hill, BurrRidge, Ill, USA, 1995.

Mohammad Jaber Borran received his B.S.degree in electronics and M.S. degree incommunication systems from Sharif Uni-versity of Technology, Tehran, Iran, in 1993and 1996, respectively. He is currently pur-suing the Ph.D. degree at the Electrical andComputer Engineering Department, RiceUniversity, Houston, Texas. His research in-terests are in communications, informationtheory, and coding.

Behnaam Aazhang received his B.S. (withhighest honors), M.S., and Ph.D. degrees inelectrical and computer engineering fromUniversity of Illinois at Urbana-Champaignin 1981, 1983, and 1986, respectively. From1981 to 1985, he was a Research Assis-tant in the Coordinated Science Laboratory,University of Illinois. In August 1985, hejoined the faculty of Rice University, Hous-ton, Texas, where he is now the J. S. Aber-crombie Professor in the Department of Electrical and ComputerEngineering and the Director of Center for Multimedia Commu-nications. His research interests are in the areas of communicationtheory, information theory, and their applications with emphasison multiple access communications, cellular mobile radio commu-nications, and optical communication networks. Dr. Aazhang is aFellow of IEEE, a recipient of the Alcoa Foundation Award 1993,the NSF Engineering Initiation Award 1987–1989, and the IBMGraduate Fellowship 1984–1985, and is a member of Tau Beta Piand Eta Kappa Nu. He is currently serving on Houston Mayor’sCommission on Cellular Towers.


Performance of Reverse-Link Synchronous DS-CDMASystem on a Frequency-Selective Multipath FadingChannel with Imperfect Power Control

Seung-Hoon HwangStandardization & System Research Group, UMTS System Research Laboratory, LG Electronics,533 Hogye-dong, Dongan-gu, Anyang-shi, Kyungki-do, KoreaEmail: [email protected]

Duk Kyung KimInformation & Communication Engineering, Inha University, 253 Yonghyun-dong, Nam-gu, Inchon, KoreaEmail: [email protected]

Received 28 July 2001 and in revised form 12 March 2002

We analyze the performance for reverse-link synchronous DS-CDMA system in a frequency-selective Rayleigh fading channelwith an imperfect power control scheme. The performance degradation due to power control error (PCE), which is approximatedby a log-normally distributed random variable, is estimated as a function of the standard deviation of the PCE. In addition, weinvestigate the impacts of the multipath intensity profile (MIP) shape and the number of resolvable paths on the performance.Finally, the coded bit error performance is evaluated in order to estimate the system capacity. Comparing with the conventionalCDMA system, we show an achievable gain of from 59% to 23% for reverse-link synchronous transmission technique (RLSTT) inthe presence of imperfect power control over asynchronous transmission for BER = 10−6. As well, the effect of tradeoff betweenorthogonality and diversity can be seen according to the number of multipaths, and the tendency is kept even in the presence ofPCE. We conclude that the capacity can be further improved via the RLSTT, because the DS-CDMA system is very sensitive topower control imperfections.

Keywords and phrases: reverse link synchronous transmission technique, frequency-selective multipath fading, imperfect powercontrol.

1. INTRODUCTION

Direct-sequence code-division multiple-access (DS-CDMA)has been considered as the most promising multiple-accessscheme for the next generation mobile communications, be-cause of its high flexibility in offering various services withvariety of rates and its possibility of achieving greater capac-ity [1, 2]. The capacity of DS-CDMA system is mainly limitedby multiple-access interference (MAI), and thus techniquesto reduce the MAI, such as multiuser detection or interfer-ence cancellation, are currently of great interest [3, 4, 5]. Inparticular, techniques for reverse links have attracted muchattention, as the capacity of a reverse voice cellular networklink is smaller than that of the forward link. One reason forthis is that the code orthogonality is not maintained, becausein the reverse link the arrival times of signals from mobilestations (MSs) at a cell site (CS) are different, given the ran-dom geographical distribution of MSs within the cell sector.

For terrestrial mobile systems, the reverse link syn-chronous transmission technique (hereafter, we denote it by

RLSTT) has been proposed to reduce the interchannel in-terferences over a reverse link [6]. In RLSTT, a closed-formtiming control based on a new parameter called the timingcontrol bit is introduced. The DS-CDMA system considereduses an orthogonal reverse-link spreading sequence and tim-ing control algorithm that allows the main paths to be syn-chronized. Analyses for a single-cell system have shown goodperformance, especially for an exponentially decaying multi-path intensity profile with a large decay factor [6, 7]. How-ever, the previous analyses have assumed perfect power con-trol, that is, all the users’ transmissions arrive with the samepower at the CS receiver. In a practical mobile radio environ-ment, an adaptive power control (APC) scheme is always es-sential to compensate for the distance losses, shadowing, andfading effects. Such a scheme attempts to maintain a con-stant average performance among the users, and reduce theMAI effect. This results in a randomly varying power controlerror (PCE), which may be caused by the dynamic range ofthe APC, the spatial user distributions, and the propagationstatistics [8, 9, 10, 11].


Evaluation of the system capacity degradation due toPCE is the main focus of this paper. To investigate the over-all effect of imperfect power control, PCE is considered interms of the standard deviation of the lognormal distribu-tion. We consider the capacity of a reverse-link synchronousDS-CDMA system over frequency selective Rayleigh fadingchannels in the presence of imperfect power control scheme.Using the results of [6], which are described in more detail inthe appendices of this paper, the system performance degra-dation as a function of the standard deviation of the PCEis evaluated. We also investigate the impact of the multipathintensity profile (MIP) shape and the number of resolvablepaths on the performance of RLSTT, because the extent oforthogonality destruction depends on the multipath channelpower delay profile shape and number of resolvable paths. Toestimate the system capacity, the coded bit error probabilityis evaluated and compared with conventional asynchronousCDMA.

The remainder of the paper is organized as follows. InSection 2, channel and system model are described. The per-formance is analytically derived and evaluated, assuming co-herent binary phase shift keying (BPSK) data modulationand a Rake combiner using maximal ratio combining (MRC)in Section 3. Numerical results and conclusions are providedin Sections 4 and 5, respectively.

2. CHANNEL AND SYSTEM MODEL

2.1. Transmitted signal representation

We assume that the narrowband-modulated signal of eachuser is first spread by a short orthogonal sequence, and thenrandomized by a pseudonoise (PN) sequence. Assuming Kactive users (k = 1, 2, . . . , K), the kth transmitted signal isgiven by

S(k)(t) =√

2Pka(t)W (k)(t)b(k)(t) cos[ωct + φ(k)

], (1)

where Pk is the average transmitted power, ωc is the commoncarrier frequency, φ(k) is the phase angle of the kth modula-tor, which is assumed to be uniformly distributed in [0, 2π),and a(t) is a PN randomization sequence, which is commonto all the channels in a cell to maintain the CDMA orthogo-nality and is expressed as

a(t) =∞∑

j=−∞aj pTc

(t − jTc

), a j ∈ −1, 1. (2)

The orthogonal channelization sequence, W (k)(t), is given by

W (k)(t) =∞∑

j=−∞w(k)

j pTw

(t − jTw

), w(k)

j ∈ −1, 1 (3)

and user k’s data waveform, b(k)(t) is expressed as

b(k)(t) =∞∑

j=−∞b(k)j pT(t − jT), b(k)

j ∈ −1, 1, (4)

where Tw is the chip duration in the orthogonal sequenceand pT(t) is a rectangular pulse of unit height and durationT . The PN chip interval Tc is related to the data bit interval Tby the processing gain N = T/Tc. We assume, for simplicity,

that Tw equals to Tc. As well, w(k)j represents the sign of the

jth chip for the kth user’s orthogonal sequence, aj represents

the sign of the jth chip for the PN sequence, and b(k)j is the

sign of the jth transmitted symbol for the kth user.

2.2. Channel model

The low-pass impulse response of the band-pass channel forthe kth user may be written as [12]

hk(τ) =L(k)−1∑l=0

β(k)l e jθ

(k)l δ

[τ − τ(k)

l

]. (5)

Each path is characterized by three variables: its strength

β(k)l , its phase shift θ(k)

l , and its propagation delay τ(k)l . A

tapped delay line model describes the frequency selectivechannel with the lth multipath delay of the kth user given

by τ(k)l = τ(k)

0 + lTc [13]. Assuming Rayleigh fading, the re-ceived signal strength of the kth user on the lth propagationpath, l = 0, 1, . . . , L(k) − 1, has a probability density function(pdf) given by

p(β(k)l

)= 2β(k)

l

Ω(k)l

exp

−(β(k)l

)2

Ω(k)l

. (6)

The parameter Ω(k)l is the second moment of β(k)

l , that is,

Ω(k)l = E

[(β(k)

l )2]

, with∑∞

l=0 Ω1 = 1, and we assume it tobe related to the second moment of the initial path strength

Ω(k)0 for the MIP by

Ω(k)l = Ω(k)

0 e−lδ , δ ≥ 0. (7)

The parameter δ reflects the decay rate of the average pathstrength as a function of path delay. Note that a more realisticprofile model may be given by the exponential MIP, in whichthe main path occupies more than half of the total receivedsignal power [12, 14, 15].

2.3. Rake combiner output

The receiver is a coherent Rake receiver, where the number oftaps Lr is a variable that is less than or equal to L(k). The tapweights and phases are assumed to be perfect estimates of thechannel parameters. The received signal is represented as

r(t) = n(t) +√

2PK∑k=1

√λk

L(k)−1∑l=0

β(k)l a

[t − τ(k)

l

]W (k)

[t − τ(k)

l

]· b(k)

[t − τ(k)

l

]cos

[ωct + ϕ(k)

l

],

(8)

where λk corresponds to the PCE of the kth user, which is

Performance of Reverse-Link Synchronous DS-CDMA System 799

a random variable due to imperfect power control. We con-sider λk to be log-normally distributed with standard devia-tion σλk dB. In other words, λk = 10(x/10) where the variable

x follows a normal distribution. As well, φ(k)l is the phase of

the lth path of the carrier of the kth user, and n(t) is the addi-tive white Gaussian noise (AWGN) with a two-sided spectraldensity η0/2. For the user of interest (k = 1), the output ofthe receiver is given by

U =Lr−1∑n=0

∫ T+nTc

nTc

r(t)β(1)n a

(t − nTc

)W (1)(t − nTw

)· cos

[ωct + ϕ(1)

n

]dt

=Lr−1∑n=0

S(n) + I(n)

MAI + I(n)SI + I(n)

NI

,

(9)

where

S(n) =√

Pλ1

2b(1)

0 Tβ(1)n

2, (10a)

I(n)MAI =

K∑k=2

L(k)−1∑l=0

f (k, l), (10b)

where

f (k, l) =

√P

2

√λkβ

(1)n β(k)

l

·b(k)−1RWk1

[τ(k)nl

]+b(k)

0 RWk1

[τ(k)nl

]· cos

[ϕ(k)nl

], if τ(k)

l ≥ τ(1)l ;√

P

2

√λkβ

(1)n β(k)

l

·b(k)

0 RWk1

[τ(k)nl

]+b(k)

1 RWk1

[τ(k)nl

]· cos

[ϕ(k)nl

], if τ(k)

l ≥ τ(1)l ,

I(n)SI =

L(1)−1∑l=0l =n

g(l),

(10c)

where

g(l) =

√Pλ1

2β(1)n β(1)

l

b(1)−1RW11

[τ(1)nl

]+ b(1)

0 RW11

[τ(1)nl

]· cos

[ϕ(1)nl

], if n < l,√

Pλ1

2β(1)n β(1)

l

b(1)

0 RW11

[τ(1)nl

]+ b(1)

l RW11

[τ(1)nl

]· cos

[ϕ(1)nl

], if n > l,

I(n)NI =

∫ T+nTc

nTc

n(t)β(1)n a

(t − nTc

)W (1)(t − nTw

)· cos

[ωct + ϕ(1)

n

]dt

(10d)

with b(1)0 being the information bit to be detected, b(1)

−1 is the

preceding bit, τ(k)nl = τ(k)

l − τ(1)n , ϕ(k)

nl = ϕ(k)l − ϕ(1)

n , and RWand RW are Walsh-PN continuous partial cross-correlationfunctions defined by

RWk1(τ) =∫ τ

0a(t − τ)W (k)(t − τ)a(t)W (1)(t)dt,

RWk1(τ) =∫ T

τa(t − τ)W (k)(t − τ)a(t)W (1)(t)dt.

(11)

From (9), we see that the output of the nth branch,n = 0, 1, . . . , Lr − 1, consists of four terms. The first termrepresents the desired signal component to be detected. Thesecond term represents the MAI from the (K − 1) othersimultaneous users. The third term is the self-interference(SI) for the reference user. Finally, the last term is the noise-interference (NI) caused by AWGN.

3. PERFORMANCE ANALYSIS

Suppose we have K transmitters able to adjust their timingclock of the main paths to be aligned at the CS by the timingcontrol algorithm [6]. Therefore, in our analysis, the eval-uation is carried out for the case in which the arrival timeof paths is modeled as asynchronous in every branch (i.e.,for multipaths) but as synchronous in the first branch (i.e.,for main paths) exceptionally. We first estimate the uncodedbit error performance at different system parameter settings.Assuming perfect interleaving, we then evaluate an upperbound on the coded bit error performance of the system us-ing convolutional codes with hard decision Viterbi decoding.

3.1. Uncoded BER performance

Using the Gaussian approximation method, the MAI termsof the first branch and the rest of the branches are modeled asGaussian processes with variances equal to the MAI variancesfor n = 0 and for n ≥ 1, respectively [6]. Using the randomchip model for the Walsh-PN sequences [16] and performingsome mathematical operations described in more detail inthe appendices, we obtain the following results. The variance

of MAI for n = 0, conditioned on β(1)n , is

σ2MAI,0 =

EbT(2N − 3)12N(N − 1)

β(1)

0

2 K∑k=2

λk

L(k)−1∑l=1

Ω(k)l . (12)

Similarly, the variance of MAI for n ≥ 1 is

σ2MAI,n =

EbT(N − 1)6N2

β(1)n

2 K∑k=2

λk

L(k)−1∑l=0

Ω(k)l , (13)

where Eb = PT is the signal energy per bit. The conditionalvariance of σ2

SI,n is approximated by [13]

σ2SI,n ≈

EbT

4Nλ1

β(1)n

2 L(k)−1∑l=1

Ω(1)l . (14)


The variance of the AWGN term, conditioned on β(1)n , is given

as

σ2NI,n =

Tη0

4

β(1)n

2. (15)

Therefore, the output of the receiver U , conditioned onβ(1)n , is a Gaussian random process with a mean given by

Us =√

Ebλ1T

2

Lr−1∑n=0

β(1)n

2(16)

and the variance equal to the sum of the variances of all theinterference terms. From (12), (13), (14), and (15), we have

σ2T =

Lr−1∑n=0

(σ2

MAI,n + σ2SI,n + σ2

NI,n

)

= σ2MAI,0 +

Lr−1∑n=1

σ2MAI,n +

Lr−1∑n=0

(σ2

SI,n + σ2NI,n

)

= (EbT

)(

2− (1/(N − 1)

))∑Kk=2 λk

∑L(k)−1l=1 Ω(k)

l

12N

·β(1)

0

2

β(1)

0

2+∑Lr−1

n=1

β(1)n

2

+(N − 1)

∑Kk=2 λk

∑L(k)−1l=0 Ω(k)

l

6N2

·∑Lr−1

n=1

β(1)n

2

β(1)

0

2+∑Lr−1

n=1

β(1)n

2

+λ1

∑L(k)−1l=1 Ω(1)

l

4N+

η0

4Eb

Lr−1∑n=0

β(1)n

2 .

(17)

The variance of the total interference in (17) becomes

σ2T =

(EbTΩ0

) (2N − 3)[q(L(k), δ

)− 1]λI

12N(N − 1)

·β(1)

0

2

β(1)

0

2+∑Lr−1

n=1

β(1)n

2 +(N−1)q

(L(k), δ

)λI

6N2

·∑Lr−1

n=1

β(1)n

2

β(1)

0

2+∑Lr−1

n=1

β(1)n

2

+

[q(L(k), δ

)− 1]λ1

4N+

η0

4EbΩ0

Lr−1∑n=0

β(1)n

2,

(18)

where q(L(k), δ) = ∑L(k)−1l=0 e−lδ = (1− e−L(k)δ)/(1− e−δ). Fur-

thermore, if we define

S = 1Ω0

Lr−1∑n=0

β(1)n

2(19)

then the received signal-to-noise ratio (SNR) at the output ofthe receiver may be written as σ0S,

σ0S = λ1

[(2N − 3)

[q(L(k), δ

)− 1]λI

3N(N − 1)

·β(1)

0

2

β(1)

0

2+∑Lr−1

n=1

β(1)n

2

+2(N − 1)q

(L(k), δ

)λI

3N2

∑Lr−1n=1

β(1)n

2

β(1)

0

2+∑Lr−1

n=1

β(1)n

2

+

[q(L(k), δ

)− 1]λ1

N+

η0

EbΩ0

]−11Ω0

Lr−1∑n=0

β(1)n

2.

(20)

The pdf of Y = β(1)0 2 is

pY (y) = 1Ω0

e−y/Ω0 , y ≥ 0 (21)

and the pdf of X =∑Lr−1n=1 β(1)

n 2 for exponential MIP is

pX(x) =Lr−1∑k=1

πkΩk

e−x/Ωk , (22)

where

πk =L−1∏i=1i =k

xkxk − xi

=L−1∏i=1i =k

Ωk

Ωk −Ωi. (23)

In addition to fast signal fluctuations caused by multi-path reflections, slow signal fluctuations exist due to obstruc-tion of the signal by hills and buildings. We can express thereceived signal power as Pk = Pλk, and λk is the PCE, whichis a log-normally distributed random variable [8, 9, 10, 11]

p(λ1) = 1√

2πσλ1λ1exp

[−

(ln λ1 −mλ1

)2

2σ2λ1

]. (24)

The CS receives the desired power of the reference user andthe joint interference power. Fenton [17] showed that the pdfof λI =

∑Kk=2 λk for K − 1 users is approximately log-normal

with the following logarithmic mean and logarithmic vari-ance, which is valid for a logarithmic standard deviation σless than 4 dB:

p(λI) = 1√

2πσλI λIexp

[−

(ln λI −mλI

)2

2σ2λI

], (25)


where

σ2I = ln

(1

K − 1eσ

2λ +

K − 2K − 1

),

mI = ln(K − 1) + m +σ2λ

2− 1

2ln(K − 2K − 1

+1

K − 1eσ

2λ

).

(26)

The average bit error probability is then

Ple =

∫∫∫∫Ple|X,Y,λ1 ,λI

p(x, y, λ1, λI

)dx dy dλ1 dλI

=∫∫∫∫

Ple|X,Y,λ1 ,λI

p(x)p(y)p(λ1)p(λI)dx dy dλ1 dλI ,

(27)

where

Ple|X,Y,λ1 ,λI

= Q(√

2σ0S),

Q(x) = 1√2π

∫∞x

exp(− u2

2

)du.

(28)

From (20), (21), and (22),

Ple

(λ1, λI

)=∫ ∫∞0Q(√

2σ0S)·L−1∑k=1

πkΩk

e−x/Ωk · 1Ω0

e−y/Ω0 dx dy.

(29)

Using (24) and (29), we find that

Ple

(λI)=∫∞

0Ple

(λ1, λI

) 1√2πσλ1λ1

exp

[−(

ln λ1−mλ1

)2

2σ2λ1

]dλ1.

(30)

Assuming that z1 = (ln λ1 −mλ1 )/√

2σλ1 , we can rewrite(30) as

ple(λI)= 1√

π

∫∞−∞

ple(

exp(√

2σλ1z1 +mλ1

), λI

)exp

[−z21

]dz1.

(31)

Therefore, the bit error probability is now given by

ple =1√π

∫∞−∞

1√π

·∫∞−∞

ple(

exp(√

2σλ1z1 + mλ1

), exp

(√2σλI zI + mλI

))· exp

[− z21

]dz1 exp

[− z2I

]dzI .

(32)

Note that (32) can be calculated using the Hermite poly-nomial approach, which requires only summation and no in-tegration [18].

3.2. Coded BER performance

For convolutional codes with hard decision Viterbi decod-ing, the bit error rate (BER) transfer characteristic can be

302520151050Eb/N0 [dB]

1.00E-03

1.00E-02

1.00E-01

1.00E+00

BE

R

PCE = 0 [dB]PCE = 1 [dB]PCE = 2 [dB]PCE = 3 [dB]PCE = 4 [dB]

Sync.Async.

Figure 1: Uncoded BER versus Eb/N0 for different values of PCEstandard deviation (δ = 0.2).

upper-bounded by the well-known transfer function bound[12]

p0 <∞∑

x=d f

γxP(x), (33)

where d f is the free distance of the code, and γx are thecoefficients in the expansion of the derivative of T(D,N), thetransfer function of the code evaluated at N = 1 [19]. As well,P(x) is the probability of selecting an incorrect path, whichcan be bounded by the expression

P(x) <[4p(1− p)

]d/2, (34)

where p is the uncoded BER.

4. NUMERICAL RESULTS

In this section, we consider the capacity of a reverse-link synchronous DS-CDMA system over frequency selec-tive Rayleigh fading channels in the presence of imperfectpower control scheme. The system performance degrada-tion as a function of the standard deviation of the PCEis estimated. We also investigate the effects of the selec-tion of system parameters on the performance of a coher-ent BPSK Rake receiver with RLSTT in terms of the averageBER and the supportable number of users for exponentialMIPs. The effect of different values of δ on the effective totalcapacity is given. The BER analysis of a conventional asyn-chronous CDMA system with a diversity technique utiliz-ing maximal ratio combining (MRC) reception can be foundin [12, 13].

Figures 1 and 2 show the difference in the average un-coded BER performance as a function of Eb/N0, when δ = 0.2and δ = 1.0 are assumed for the different MIPs. Measure-ments made by Turin et al. [20] in an urban environment in-dicate that the MIP is exponential. For illustration, we have


302520151050Eb/N0 [dB]

1.00E-03

1.00E-02

1.00E-01

1.00E+00

BE

R


Sync.Async.

Figure 2: Uncoded BER versus Eb/N0 for different values of PCEstandard deviation (δ = 1.0).

chosen the arbitrary parameters K = 24, N = 128, andL(k) = Lr = 3, but we vary the values of σλ from 0 to4 dB. From these figures, we observe that the bit error per-formance degrades with increased σλ values, as expected. Inparticular, the degradation in performance is marginal whenσλ = 1 dB. However, for σλ > 2 dB, significant degradationis observed. The analytical results derived for asynchronouswith the PCE are found to closely match the simulation re-sults obtained by Chockalingam and Milstein in [8]. It isnoted that the average BER in RLSTT, when σλ = 1 dB,is better than that in the asynchronous transmission whenσλ = 0 dB (i.e., perfect power control). It means that the RL-STT with imperfect power control has better performance,compared to non-RLSTT with perfect power control. In ad-dition, as the decay constant δ of MIP increases, RLSTTin the DS-CDMA reverse link results in a significant BERimprovement over the asynchronous transmission even inthe presence of imperfect power control. For example, forEb/N0 > 15 dB, the average BER of RLSTT when σλ = 2 dBis better than that of the asynchronous transmission whenσλ = 0 dB.

Figures 3 and 4 show the effect of having a large num-ber of resolvable paths and coherently combining all of themat the Rake receiver, L(k)(= Lr) on the average BER. The av-erage BER is plotted as a function of L(k)(= Lr) for variousvalues of σλ, when K = 24, N = 128, Eb/N0 = 20 dB, and ex-ponential MIP are assumed. The figures illustrate the perfor-mance difference between RLSTT and the asynchronous caseis smaller as L(k)(= Lr) increases due to the increased MAIresulting from multipath interference. Meanwhile, for largeL(k)(= Lr), the potential improvement in performance existsdue to the diversity gain from increased frequency selectiv-ity. Therefore, we observe that there are tradeoff between or-thogonality and diversity as a function of L(k)(= Lr), and thetendency is kept even in the presence of PCE. However, the

54321L(k) = Lr

1.00E-03

1.00E-02

1.00E-01

1.00E+00

BE

R


Sync.Async.

Figure 3: Uncoded BER versus L(k) = Lr for different values of PCEstandard deviation (δ = 0.2).

54321L(k) = Lr

1.00E-03

1.00E-02

1.00E-01

1.00E+00B

ER


Sync.Async.

Figure 4: Uncoded BER versus L(k) = Lr for different values of PCEstandard deviation (δ = 1.0).

RLSTT still offers better performance over the asynchronoustransmission.

In Figures 5 and 6, the BER is plotted for the situationwith perfect and imperfect power control. We see from Fig-ures 5 and 6 that the BER increases with the increase in theimperfection of the power control scheme. This is due tothe random nature of the received power in the case of im-perfect power control, whereas in the case of perfect powercontrol the received power can be assumed to be determinis-tic.

We consider the use of a rate 1/3 convolutional code ofconstraint length 9 on the reverse link [21]. The γx coef-ficients for the corresponding code are taken from [19]. Theupper bound on the coded BER performance of the system as


43210PCE [dB]

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

BE

R

USER = 12USER = 24USER = 36USER = 48

Sync.Async.

Figure 5: Uncoded BER versus PCE standard deviation for differentnumber of users (δ = 0.2).

43210PCE [dB]

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

BE

R

USER = 12USER = 24USER = 36USER = 48

Sync.Async.

Figure 6: Uncoded BER versus PCE standard deviation for differentnumber of users (δ = 1.0).

a function of the number of users, when Eb/N0 = 20 dB andL(k) = Lr = 3 can be achieved for different σλ values. Basedon the results, the system capacity values as a function of σλin dB for different bit error rates (10−3 for voice and 10−6 or10−10 for data) are tabulated in Tables 1 and 2. The systemcapacity is defined as the number of simultaneous users thatcan be supported while maintaining an acceptable BER per-formance needed by the specific application. When RLSTTis employed, a channel with exponential MIP of δ = 1.0 hasmore capacity than a channel with δ = 0.2 for the same BERvalue. For BER=10−6 and δ = 1.0, when σλ = 0 dB the ca-pacity improvements are around 50%, while when σλ = 2 dBthe improvements increase to 59%. The results have shownthat the capacity of DS-CDMA system can be improved byemploying RLSTT even in the presence of imperfect powercontrol.

Table 1: CDMA system capacity with PCE (δ = 0.2, L(k) = Lr = 3,rate 1/3 convolutional code (K = 9) with perfect interleaving).

System capacity

BER Standard derivation of PCE, σλ

0 dB 1 dB 2 dB 3 dB 4 dB

10−3 Sync. 67 64 56 46 36

Voice Async. 54 51 46 39 30

10−6 Sync. 39 36 32 25 18

Data Async. 32 30 26 20 15

10−10Sync. 21 20 16 < 12 < 12

Async. 17 16 13 < 12 < 12

Table 2: CDMA system capacity with PCE (δ = 1.0, L(k) = Lr = 3,rate 1/3 convolutional code (K = 9) with perfect interleaving).

System capacity

BER Standard derivation of PCE, σλ

0 dB 1 dB 2 dB 3 dB 4 dB

10−3 Sync. 81 78 69 57 44

Voice Async. 49 47 42 35 27

10−6 Sync. 42 40 35 29 20

Data Async. 28 26 22 18 13

10−10Sync. 21 20 17 12 < 12

Async. 15 14 < 12 < 12 < 12

5. CONCLUSIONS

In this paper, we have considered the effect of imper-fect power control in the performance of reverse-link syn-chronous DS-CDMA system in Rayleigh multipath fadingchannel. Multiple-access and self-noise interference weremodeled as additional Gaussian noise. Under these assump-tions, the performance of the coherent system was derived interms of the uncoded BER and the capacity from the codedBER. The results indicate that in Rayleigh fading with perfectpower control, RLSTT shows capacity improvements from50% to 22% for BER = 10−6. When RLSTT is employed inimperfect power control, it shows more capacity gains from59% to 23%. It means that RLSTT with imperfect powercontrol has better performance, compared to non-RLSTTwith perfect power control. As well, the effect of tradeoff be-tween orthogonality and diversity can be seen according tothe number of multipaths, and the tendency is kept even inthe presence of PCE. Finally, we conclude that the capacitycan be somewhat further increased via the RLSTT, becausethe DS-CDMA system is very sensitive to power control im-perfections.


APPENDICES

A. THE AVERAGE CROSS-CORRELATION PARAMETERIN (12)

Let the discrete aperiodic cross-correlation CWk,1(i) be

CWk,1(i) =

N−1−i∑j=0

ajw(k)j a j+iw

(1)j+i, 0 < i ≤ N − 1,

N−1+i∑j=0

aj−iw(k)j−ia jw

(1)j , −(N − 1) ≤ i < 0,

0, i = 0 or otherwise.(A.1)

Let Ik be the correlator output resulting from the kth inter-ferer. Then the variance of Ik is given by

Var(Ik) = P

2E[

b(k)−1RWk1

(τ(k)nl

)+ b(k)

0 RWk1

(τ(k)nl

)2]E[

cos2(ϕ(k)nl

)]= PT2

c

12(N − 1)

[ N−1∑i=1

CW2k,1(1 + i−N)

+ CWk,1(1 + i−N)

· CWk,1(i−N) + CW2k,1(i−N)

+ CW2k,1(i + 1) + CW2

k,1(i)

+ CWk,1(i)CWk,1(i + 1)

].

(A.2)

The factor 1/(N −1) and the summation from i = 1 to N −1in (A.2) arise from the existence of N − 1 chip intervals in

time [Tc, T], and τ(k)0l can fall into any one of them with equal

probability. Using the random chip model for the Walsh-PNsequences [17], making Var(Ik) a random variable, and tak-ing the expectation of (A.2)

E[

Var(Ik)] = PT2

12N2(N − 1)

· E[ N−1∑

i=1

CW2k,1(1 + i−N) + CWk,1(1 + i−N)

· CWk,1(i−N) + CW2k,1(i−N)

+ CW2k,1(i + 1) + CW2

k,1(i)

+ CWk,1(i)CWk,1(i + 1)

]

= PT2

12N2(N − 1)

·[

3× N(N − 1)2

+(N − 2)(N − 1)

2− 1

]= PT2

(2N2 − 3N

)12N2(N − 1)

= PT2

12N3

(2N − 3)N2

(N − 1),

(A.3)

where CWk,1(0) = 0,

N−2∑i=1

E(CW2

k,1(1 + i−N)) = N−2∑

i=1

i∑m=0

E(1)

=N−2∑i=1

(i + 1) = (N − 1)N2

− 1,

N−1∑i=1

E(CW2

k,1(i)) = N−1∑

i=1

N−1−i∑m=0

E(1)

=N−1∑i=1

(N − i) = (N − 1)N2

,

N−1∑i=1

E(CW2

k,1(i−N)) = N−1∑

i=1

i−1∑m=0

E(1)

=N−1∑i=1

i = (N − 1)N2

,

N−1∑i=1

E(CW2

k,1(i + 1)) = N−2∑

i=1

N−i−2∑m=0

E(1) =N−2∑i=1

(N − i− 1)

= (N − 2)(N − 1)2

,

N−2∑i=1

E(CWk,1(1 + i−N)CWk,1(i−N)

)=

N−1∑i=1

E(CWk,1(i)CWk,1(i + 1)

)=0.

(A.4)

Thus, we find that

rk1(N) ≈ 2N − 3N − 1

N2 =(

2− 1N − 1

)N2. (A.5)

B. THE AVERAGE CROSS-CORRELATION PARAMETERIN (13)

Similarly, we can show that

rk1(N) = 2N(N − 1), (B.1)

Var(Ik) = P

2E[

b(k)−1RWk1

(τ(k)nl

)+ b(k)

0 RWk1

(τ(k)nl

)2]E[

cos2(ϕ(k)nl

)]= PT2

12N3

[ N−1∑i=0

CW2k,1(1 + i−N)

+ CWk,1(1 + i−N)CWk,1(i−N)

+ CW2k,1(i−N) + CW2

k,1(i + 1)

+ CW2k,1(i) + CWk,1(i)CWk,1(i + 1)

].

(B.2)

The factor 1/N and the summation from i = 0 to N − 1 in


(B.2) are required because there are N chip intervals in time

[0, T], and τ(k)nl can fall into any one of them with equal prob-

ability

E[

Var(Ik)] = PT2

12N3

· E[ N−2∑

i=0

CW2

k,1(1 + i−N)

+ CWk,1(1 + i−N)CWk,1(i−N)

+N−1∑i=0

CW2

k,1(i−N) + CW2k,1(i + 1)

+N−1∑i=1

CW2

k,1(i)+CWk,1(i)CWk,1(i+1)]

= PT2

12N3

[4× N(N − 1)

2

]= PT2

6N2(N − 1)

= PT2

12N32N(N − 1),

(B.3)

where CWk,1(0) = 0,

N−2∑i=0

E(CW2

k,1(1 + i−N)) = N−2∑

i=0

i∑m=0

E(1)

=N−2∑i=0

(i + 1) = (N − 1)N2

,

N−1∑i=1

E(CW2

k,1(i)) = N−1∑

i=1

N−1−i∑m=0

E(1)

=N−1∑i=1

(N − i) = (N − 1)N2

,

N−1∑i=0

E(CW2

k,1(i−N)) = N−1∑

i=0

E(CW2

k,1(i + 1))

= (N − 1)N2

,

N−2∑i=0

E(CWk,1(1 + i−N)CWk,1(i−N)

)=

N−1∑i=1

E(CWk,1(i)CWk,1(i + 1)

).

(B.4)

REFERENCES

[1] E. Dahlman, P. Beming, J. Knutsson, F. Ovesjo, M. Persson,and C. Roobol, “WCDMA—the radio interface for futuremobile multimedia communications,” IEEE Trans. VehicularTechnology, vol. 47, no. 4, pp. 1105–1118, 1998.

[2] F. Adachi, K. Ohno, A. Higashi, T. Dohi, and Y. Okumura,“Coherent multicode DS-CDMA mobile radio access,” IEICETrans. Communications, vol. E-79-B, pp. 1316–1325, Septem-ber 1996.

[3] R. A. Iltis, “Performance of constrained and unconstrainedadaptive multiuser detectors for quasi-synchronous CDMA,”IEEE Trans. Communications, vol. 46, no. 1, pp. 135–143,1998.

[4] H. Sari, F. Vanhaverbeke, and M. Moeneclaey, “Extending thecapacity of multiple access channels,” IEEE CommunicationMagazine, vol. 38, pp. 74–82, January 2000.

[5] H. Sari, F. Vanhaverbeke, and M. Moeneclaey, “Multiple ac-cess using two sets of orthogonal signal waveforms,” IEEECommunications Letters, vol. 4, no. 1, pp. 4–6, 2000.

[6] E. K. Hong, S. H. Hwang, K. J. Kim, and K. C. Whang, “Syn-chronous transmission technique for the reverse link in DS-CDMA terrestrial mobile systems,” IEEE Trans. Communica-tions, vol. 47, no. 11, pp. 1632–1635, 1999.

[7] D. K. Kim, S. H. Hwang, E. K. Hong, and S. Y. Lee, “Capac-ity estimation of uplink synchronized CDMA system with fastTPC and two antenna diversity reception,” IEICE Trans. Com-munications, vol. E84-B, no. 8, pp. 2309–2312, 2001.

[8] A. Chockalingam and L. B. Milstein, “Capacity of DS-CDMAnetworks on frequency selective fading channels with open-loop power control,” in Proc. IEEE Int. Conf. Communications,vol. 2, pp. 703–707, Seattle, Wash, USA, June 1995.

[9] N. Kong and L. B. Milstein, “Error probability of multicellCDMA over frequency selective fading channel with powercontrol error,” IEEE Trans. Communications, vol. 47, pp. 608–617, April 1999.

[10] A. Abrardo and D. Sennati, “On the analytical evaluation ofclosed loop power control error statistics in DS-CDMA cellu-lar systems,” IEEE Trans. Vehicular Technology, vol. 49, no. 6,pp. 2071–2080, 2000.

[11] A. Chockalingam, P. Dietrich, and R. R. Milstein, L. B. Rao,“Performance of closed loop power control in DS-CDMA cel-lular systems,” IEEE Trans. Vehicular Technology, vol. 47, no.3, pp. 774–789, 1998.

[12] J. G. Proakis, Digital Communications, McGraw-Hill, NewYork, NY, USA, 1983.

[13] T. Eng and L. B. Milstein, “Coherent DS-CDMA performancein Nakagami multipath fading,” IEEE Trans. Communications,vol. 44, no. 9, pp. 1117–1129, 1996.

[14] D. Parsons, The Mobile Radio Propagation Channel, Addison-Wesley, New York, NY, USA, 1992.

[15] Rec. ITU-R M. 1225, “Guideline for evaluation of radio trans-mission technologies for IMT-2000,” 1997.

[16] M. B. Pursley, “Performance evaluation for phase-codedspread-spectrum multiple-access communication—Part I:System analysis,” IEEE Trans. Communications, vol. 25, no.8, pp. 795–799, 1977.

[17] R. Prasad, CDMA for Wireless Personal Communications,Artech House Publishers, Boston-London, 1996.

[18] M. Abramowitz and I. A. Stegun, Handbook of MathematicalFunctions, National bureau of standards applied mathematicsseries. Dover, New York, NY, USA, 1965.

[19] J. Conan, “The weight spectra of some short low rate convo-lutional codes,” IEEE Trans. Communications, vol. 32, no. 9,pp. 1050–1053, 1984.

[20] G. Turin, F. Clapp, T. Johnston, S. Fine, and D. Lavry, “A sta-tistical model of urban multipath propagation,” IEEE Trans.Vehicular Technology, vol. 21, no. 1, pp. 1–9, 1972.

[21] R. Padovani, “Reverse link performance of IS-95 based cellu-lar system,” IEEE Personal Communications, vol. 1, no. 3, pp.28–34, 1994, Third quarter.


Seung-Hoon Hwang was born in Seoul,Korea on February 26, 1969. He received hisB.S., M.S., and Ph.D. degrees in electricalengineering from Yonsei University in 1992,1994, and 1999, respectively. His Ph.D. the-sis is entitled Performance Evaluation of aSynchronous DS-CDMA System in a MobileRadio Channel. Since 1999, he has workedfor LG Electronics where he is now a Re-search Engineer in UMTS System Labora-tory, LG R&D Center, participating in IMT-2000 physical layerstandardization activities. His research interests include interfer-ence cancellation techniques for DS-CDMA and various aspects ofwideband/broadband CDMA.

Duk Kyung Kim received his B.S. degree inelectrical engineering from Yonsei Univer-sity, Seoul, Korea, in 1992, and the M.S. andPh.D. degrees from the Korea Advanced In-stitute of Science and Technology (KAIST),in 1994 and 1999, respectively. From 1999to 2000, he was a Postdoctoral Researcher atthe Wireless Laboratories, NTT DoCoMo,Japan. From 2000 to 2002, he worked atR&D center, SK Telecom, Korea and in-volved in the standardization in 3GPP and also in 4G systemdevelopment in SK Telecom. In 2002, he joined in the facultyof Inha University. He was interested in asynchronous transfermode (ATM) network and ATM-based personal communicationservice (PCS) network. His research interests now include sys-tem performance evaluation at link/system level, handoff model-ing/management, power control, and multimedia provision in thenext generation wireless systems.


Joint Transmitter-Receiver Optimizationin the Downlink CDMA Systems

Mohammad SaquibWireless Communications Research Lab. (WiCoRe), Department of Electrical Engineering, University of Texas at Dallas,Richardson, TX 75083-0688, USAEmail: [email protected]

Md Habibul IslamWireless Communications Research Lab. (WiCoRe), Department of Electrical Engineering, University of Texas at Dallas,Richardson, TX 75083-0688, USAEmail: [email protected]


To maximize the downlink code-division multiple access (CDMA) system capacity, we propose to minimize the total transmittedpower of the system subject to users’ signal-to-interference ratio (SIR) requirements via designing optimum transmitter sequencesand utilizing linear optimum receivers (minimum mean square error (MMSE) receiver). In our work on joint transmitter-receiverdesign for the downlink CDMA systems with multiple antennas and multipath channels, we develop several optimization algo-rithms by considering various system constraints and prove their convergence. We empirically observed that under the optimiza-tion algorithm with no constraint on the system, the optimum receiver structure matches the received transmitter sequences. Asimulation study is performed to see how the different practical system constraints penalize the system with respect to the opti-mum algorithm with no constraint on the system.

Keywords and phrases: CDMA system, joint transmitter-receiver optimization, MMSE receiver, power control, downlink, multi-path and multiple antennas.

1. INTRODUCTION

Code-division multiple access (CDMA) systems are be-ing considered to support multimedia traffic in the nextgeneration mobile radio systems, such as CDMA2000 andWCDMA. Voice-based CDMA systems are generally equal intheir uplink and downlink traffic, whereas, in future CDMAsystems, which will support various types of high data rateimage and video traffic with voice messages, the downlinkwill carry the significant portion of the total system traffic.Therefore, an important area of research is to maximize thedownlink capacity via fully utilizing the limited system re-sources.

The capacity of a CDMA system is interference limited.Techniques that control or avoid interference improve theCDMA system capacity. There are three means of control-ling interference in a CDMA system: power control, mul-tiuser detection, and beamforming. Power control balancesreceived powers of all users so that no user suffers from exces-sive interference due to other users in the system. Multiuserdetection suppresses interference by exploiting the temporal

structure of the interference, whereas beamforming uses thespatial structure of the interference to cancel it.

Recently, several studies [1, 2, 3, 4] have been performedin order to integrate power control with multiuser detection.The motivation of these works was to achieve a performancegain over multiuser detection by providing power control formultiuser detection. In [1, 3], the problem of finding thejointly optimum powers and linear receivers in synchronousCDMA systems are addressed. It is shown that a distributedand iterative power control algorithm, where each user em-ploys linear minimum mean square error (MMSE) receiverfilter before each power control update, converges to thepoint where all users spend minimum transmit powers anduse corresponding MMSE receivers. The empirical results of[3] indicate that the linear MMSE receiver with optimumpower allocation to users can significantly improve the sys-tem capacity.

In [5] three basic interference management approachesare combined, transmit power control, multiuser detection,and beamforming to increase the uplink capacity of a syn-chronous CDMA system. Due to the fact that in a CDMA


system, a transmitter is a combination of power, spreadingsequences, and beamforming weights, joint power-receiveroptimization is suboptimum with respect to the joint trans-mitter and receiver optimization. Since it is most desirable tofully maximize the system capacity, in this work, we developseveral joint transmitter-receiver optimization algorithms forCDMA systems with multipaths and multiple antennas un-der different practical system constraints.

Joint transmitter-receiver optimization is proposed in [6]for multiuser systems over multipath channels. Jang et al. [6]consider a single antenna at the base station and the mean-squared error between the true bit value, and its estimate istaken as the cost function subject to average and peak powerconstraints. The capacity of a wireless system is interference-limited and the interference in the system will be mini-mized if the transmitter powers are minimized. Therefore, wewould like to develop joint transmitter-receiver optimizationalgorithm that minimizes the sum of the total transmittedpower by all the transmitters subject to the SIR requirementof each user. Thus, the problem we are interested in solvingis different from the problem addressed in [6].

The novel idea of multiple-access interference (MAI)elimination for CDMA systems by means of signature se-quence adaptation has been considered in [7, 8, 9, 10]. In[7], we see that it is possible to obtain the maximum in-formation theoretic capacity [11] as well as user capacity [9]by developing optimal set of transmitter spreading sequencesfor synchronous CDMA systems. The user capacity is definedas the maximum number of supportable users at a commonSIR target of a fixed processing gain CDMA system. For thesynchronous CDMA system, signature sequence sets havingthe least total squared correlation (TSC) are the optimumin all cases [7]. In [8] an algorithm similar to [7] is devel-oped, which yields the optimum sequences for asynchronousCDMA systems that achieve a lower bound on the totalsquared asynchronous correlation (TSAC) among the users.Here it is found that under the optimum signature sequencesthe user capacity of a single-cell asynchronous CDMA sys-tem is the same as that of a single-cell synchronous CDMAsystem. Recently, in [12] an algorithm analogous to [7, 8]is proposed to design joint transmitter power and spread-ing sequences for the uplink CDMA systems with multipathchannels. Here it is assumed that all the users have the com-mon SIR requirement and the proposed algorithm in [12] issuboptimum in multipath channels.

It should be mentioned that the transmitter beamform-ing with multielement transmit antenna arrays [13, 14]is also related to the signature sequence design. However,the key difference between transmit beamforming in mul-tielement antenna arrays and signature sequence design inCDMA systems is that the spreading (temporal) sequencesof the users in CDMA systems can be fully controlled bythe transmitter, whereas, since the beamforming weights inmultielement antenna systems are created by the wirelesscommunications channel, they cannot be directly controlledby the transmitters. Recently, for CDMA systems with mul-timedia services, a downlink beamforming technique thatconverts the downlink beamforming problem into a virtual

uplink one by taking into account the data rate informationof all users is proposed [15].

In this work, we formulate the transmitter and receiverdesign problem as a joint optimization problem. We de-velop the optimum transmitter sequences by minimizing thecost function of the optimization problem, which is the totaltransmitted power by the system. The maximization prob-lem in the constraint of the minimization problem yields thelinear optimum receivers of users subject to their SIR con-straints. We consider multipath CDMA systems with multi-ple antennas at the transmitter and a single antenna elementat each receiver. As a consequence, our designed transmit-ter sequences are a combination of spreading sequences andoptimum beamforming weights (spatial sequences). Practi-cal constraints in the systems lead us to develop two con-strained joint transmitter-receiver optimization algorithmsfor the downlink CDMA systems. Finally, it should be men-tioned that the joint transmitter-receiver optimization prob-lem in the uplink is not much different from that in thedownlink.

In [14], Visotsky and Madhow has studied joint transmit-receive beamforming with the objective of minimizing thetotal transmitter power by the system subject to SIR require-ments of users in space division multiple access (SDMA)systems. In SDMA systems, transmitters are modeled byantenna (beamforming) weights only. The differences be-tween this work and [14] are that we concentrate on mul-tipath CDMA systems, where a transmitter is a combina-tion of power, temporal, and spatial sequences. We formulatea joint transmitter-receiver optimization problem with noconstraint on the system and provide an iterative algorithmto solve it. Considering different practical system constraints,we also develop two other constrained joint transmitter-receiver optimization algorithms and prove their conver-gence.

The rest of this paper is organized as follows. Section 2derives a vector model for a multipath single-cell CDMA sys-tem, where a transmitter broadcasts signals to many usersby using multiple antennas. In Section 3, we propose theoptimization algorithms that yield the transmitter and re-ceiver structures of users under various system constraints.Section 4 shows the convergence of the proposed algorithms.Some empirical results and final remarks are presented inSections 5 and 6, respectively.

2. SYSTEM MODEL

Consider a CDMA system that has one transmitter and manyreceivers, where each receiver is equipped with a single an-tenna element and the transmitter broadcasts signals to re-ceivers using an M element antenna array. This CDMA sys-tem model could be applied to the downlink of WCDMA[16] or that of Infostations, which is an array of isolated wire-less ports proposed in [17] to provide convenient and fre-quent access to a wide range of useful and economical In-ternet type services. There are K receivers in the system and,without loss of generality, receiver q is the receiver of inter-est. We assume that each receiver corresponds to a unique

Joint Transmitter-Receiver Optimization in the Downlink CDMA Systems 809

user and thus receiver k and user k are one in the same. It isassumed that a single transmitter employs M antennas trans-mitting asynchronously over M different multipath channelsto each receiver. Here, the transmitter has accurate feedbackfrom receivers regarding transmitter-receiver channels seenby the receivers and it executes the proposed algorithms. Itis also the responsibility of the transmitter to inform the re-ceivers of their coefficients.

At each transmit antenna, each bit results in the trans-mission of a sequence of pulses, or chips, p[t], where eachpulse has a duration of one chip period Tc. The bit trans-mission time is T and the processing gain is L = T/Tc. Slowfading is assumed on each channel. We also assume that themaximum delay spread is very small with respect to T andthus we ignore intersymbol-interference (ISI). The length ofthe observation window is DTc, where D ≥ L is an integer. Inthis work, XT denotes the transpose of the matrix X and XH

the Hermitian of the matrix X.Let a(m,i)

k be the mth chip for user k at antenna i; we usea(m,i)

k to denote the set of temporal or spreading sequencesfor user k. The temporal sequence vector for user k at an-tenna i is a(i)

k = [a(1,i)k , . . . , a(L,i)

k ]. Using O(n) to denote a zerorow vector of size n, we can simplify the description of thereceived signal by defining the ML× 1 vector

A(i)k =

[O((i− 1)L

), a(i)

k ,O((M − i)L

)]T. (1)

Using A(i)k , we define

Ak =[

A(1)k , . . . ,A(M)

k

], (2)

where Ak is an ML×M matrix. For each receiver k, the trans-mitter employs an antenna weight vector

wk =[w(1)k , . . . , w(M)

k

]T, (3)

where w(i)k is the weight used at antenna i for user k. We call

w(i)k the set of spatial sequences for user k. Using (2) with

(3), we define Sk as

Sk = Akwk. (4)

The term Sk denotes a column vector of size ML and its el-ements from (i− 1)L + 1 to iL are the joint temporal-spatialsequences that have been employed at antenna i for receiverk. The total transmitter power for receiver k is ‖Sk‖2 = Pk.In [18], we show that at the end of a bit interval, the chipmatched filter outputs of user q yield the received signal vec-tor

Rq =K∑k=1

bkHqAkwk + N =K∑k=1

bkHqSk + N, (5)

where bk ∈ −1, 1 is the transmitted bit for receiver k, Hq

is a D ×ML matrix whose components are functions of pa-rameters of M multipath channels and the structure of thetransmitter pulse p[t]. If the contribution of the ith antenna

to Hq is H(i)q , then

Hq =[

H(1)q , . . . ,H(M)

q

], (6)

where the size of H(i)q is D×L. The term N is a complex white

Gaussian noise vector with zero mean and covariance

E[

NNH] = σ2ID, (7)

where ID is an identity matrix of size D ×D.

3. JOINT TRANSMITTER-RECEIVER OPTIMIZATION

To maximize the system capacity, we will minimize the to-tal power transmitted by the system subject to quality of ser-vice (QoS) requirements of users, where QoS is defined interms of the SIR. Here, our main assumption for developingjointly optimum transmitters and receivers for CDMA sys-tems is that the transmitters and receivers have the accurateinformation that they need to operate.

The detection of the information bit of the desired re-ceiver q is done by taking the sign of the decision statistics,which is to be found, using the observation vector Rq. Ob-servations from the spatial and temporal domains are to beprocessed intelligently in making the bit decisions at the de-sired receiver. Our goal is to minimize the total transmittedpower by the system subject to the SIR constraint at eachmobile that employs the linear receiver to maximize the SIR.The problem we are formulating here is the joint transmitter-receiver optimization problem over transmitter signatures Sq

and receiver filters Cq given by

minSq

K∑q=1

∥∥Sq

∥∥2(8)

subject to

maxCq

∣∣∣CHq HqSq

∣∣∣2

∑Kk =q

∣∣∣CHq HqSk

∣∣∣2+ σ2

∥∥Cq

∥∥2≥ γq, ∀q, (9)

where γq is the minimum SIR requirement of user q.The MMSE receiver [19] minimizes mean-squared error

(MSE) and maximizes the SIR [5]. Thus, the solution to themaximization problem addressed in the constraint of (8), (9)is the MMSE receiver filter coefficients for given Sq, whichis

Cq =[ K∑k=1

(HqSkSH

k HHq

)+ σ2ID

]−1

HqSq, ∀q. (10)

Assuming Cq as given in the above equation, we rewritethe optimization problem of (8), (9) as

minSq

K∑q=1

∥∥Sq

∥∥2(11)


subject to ∣∣∣CHq HqSq

∣∣∣2

∑Kk =q

∣∣∣CHq HqSk

∣∣∣2+ σ2

∥∥Cq

∥∥2≥ γq, ∀q. (12)

Now we will establish some elementary properties of theoptimization problem just proposed in (11), (12). Since|XHY|2 = |YHX|2 for any two vectors X and Y of the samesize, denoting CH

q = CHq Hq and σ2

q = σ2‖Cq‖2, the optimiza-tion problem in (11), (12) can be formulated as

minSq

K∑q=1

∥∥Sq

∥∥2(13)

subject to

γq

( K∑k =q

∣∣∣SHk Cq

∣∣∣2+ σ2

q

)−∣∣∣SH

q Cq

∣∣∣2 ≤ 0, ∀q. (14)

For given Cq or Cq, we recognize that the above optimizationproblem is similar to the optimization problem proposedin [13] to find the optimum (may not be global optimum)transmit beamforming weights for time division multiple ac-cess (TDMA) type of systems. The optimum solution is ob-tained with equality in (13), (14). In [14] it is shown that ifthe optimization problem proposed in (13), (14) is feasible,then the algorithm in [13], applied to an appropriately scaledversion of the problem, will converge to the global minimumof the original optimization problem.

The algorithm that converges to a fixed point of the de-sired minimization problem proposed in (8), (9) is called op-timization algorithm or OA. We find that the convergence ofthe OA is a special case of the convergence of the constrainedoptimization algorithm, which will be proposed in the fol-lowing subsection.

3.1. Fixed transmitter temporal spreading sequences

In the downlink of the existing IS-95, users’ spreading se-quences are generated by multiplying the Walsh codes withunique spreading sequences of the base station. Thus, CDMAsystems, similar to IS-95, may not allow us to change thetransmitter temporal sequences. For this case, we proposeto maximize the system capacity by designing the optimumspatial sequences (or antenna weights) and the MMSE re-ceiver for each user. We call the algorithm that solves thisconstrained optimization problem as optimization algorithmwith fixed spreading sequence or OAS. The OAS is presented inScheme 1 and its proof is given in Section 4. Of course, due tothe constraint on temporal sequences, the solution providedby the OAS will be suboptimum with respect to the solutiongiven by the OA. Without loss of generality, we can assumethat under the OAS, the temporal sequences at the transmit-ters are normalized to unity.

Letting Aq as an identity matrix of size ML ×ML andchanging wq fromM×1 toML×1 and then denoting Sq = wq

in Step 4 of the OAS, we will obtain the OA. In Section 4, we

Step 1. Set n=1 and choose nonzero initialtemporal-spatial sequences Sq(n).

Step 2. Derive MMSE receiver

Cq(n) =[ K∑k=1

(HqSk(n)SH

k (n)HHq

)+σ2ID

]−1

HqSq(n), ∀q.

Step 3. Equate

CHq (n) = CH

q (n)Hq,

σ2q (n) = σ2

∥∥Cq(n)∥∥2, ∀q.

Step 4. Set n = n + 1 and select joint spatial sequencesand powers at the transmitter

wq(n)

= arg minwq

K∑q=1

∥∥wq

∥∥2

such that

γq

( K∑k =q

∣∣∣wHk AH

k Cq(n− 1)∣∣∣2

+ σ2q (n− 1)

)

−∣∣∣wH

q AHq Cq(n− 1)

∣∣∣2≤ 0, ∀q.

Step 5. Derive the joint temporal-spatial sequences

Sq(n) = Aqwq(n), ∀q

and Go to Step 2.

Scheme 1: Optimization algorithm with fixed spreading sequence(OAS).

present the algorithm that solves the optimization problemof Step 4 as well as prove its convergence.

3.2. Optimum transmit and receive beamforming

In this paper, our assumption is that the transmitter willinform receivers of their filter coefficients. In practice, thenumber of filter coefficients or the length of Cq will be rea-sonably large and moreover, those coefficients are complex.As a result, this feedback transmission will reduce the ef-fective transmitter-receiver throughput, which may not bedesirable. In addition, complexities in constructing linearMMSE receivers at each iteration at Step 2 may not be man-ageable. Thus, in practice, we may come across constraintsnot only on the temporal sequences but also on feeding backto or constructing receiver coefficients for users. To handlethis problem, we propose the following receiver structure foruser q:

Cq = HqAqZq, (15)

where the matrix Aq contains the normalized temporal se-quences for user q as described in (2). The term Zq, a vectorof size M × 1, is the receive beamforming weights (or spatial


receiver filter) of receiver q and is the solution to

Zq = arg minZq

E[∣∣∣ZH

q AHq HH

q Rq − bq∣∣∣2]. (16)

Denoting Bq,k = AHq HH

q HqAk as an M ×M matrix, we canwrite the desired receiver structure as

Cq = HqAqZq

= HqAq

[ K∑k=1

(Bq,kwkwH

k BHq,k

)+ σ2Bq,q

]−1

Bq,qwq.(17)

The receiver proposed in (15) has been developed by em-ploying optimal weights on the received fixed temporal se-quences. This receiver is a constrained linear MMSE receiver.We prove in the following section that the MMSE receivermaximizes the SIR in the unconstrained as well as in the con-strained filter spaces as shown in (15). This result will per-mit us to claim that if we develop an algorithm similar tothe OAS, it will converge to a fixed point where the antennaweights and the structure of the linear receiver will be jointlyoptimized. Since this algorithm will optimally develop thebeamforming weights at both the transmitter and receiverends, we call this optimization algorithm the optimum beam-forming algorithm (OBA). The difference between the OBAand OAS, is that the OBA uses constrained MMSE receiver inStep 2, whereas the OAS employs the unconstrained MMSEreceiver.

4. CONVERGENCE OF ALGORITHMS

In this section, we prove the convergence of the proposedalgorithms. To do so, we first prove the following lemma,which provides the property of the receiver structures inStep 2.

Lemma 1. The unconstrained MMSE receiver of (10) satisfies

Cq =[ K∑k=1

(HqSkSH

k HHq

)+ σ2ID

]−1

,

HqSq = αq

[ K∑k =q

(HqSkSH

k HHq

)+ σ2ID

]−1

HqSq,

(18)

where

αq = 1

1 + SHq HH

q

[∑Kk =q

(HqSkSH

k HHq

)+ σ2ID

]−1HqSq

(19)

and similarly, the constrained MMSE receiver of (17) satisfies

Cq = HqAq

[ K∑k=1

(Bq,kwkwH

k BHq,k

)+ σ2Bq,q

]−1

Bq,qwq

= βqHqAq

[ K∑k =q

(Bq,kwkwH

k BHq,k

)+ σ2Bq,q

]−1

Bq,qwq,

(20)

where

βq = 1

1 + wHq BH

q,q

[∑Kk =q

(Bq,kwkwH

k BHq,k

)+ σ2Bq,q

]−1Bq,qwq

.

(21)

In order to establish the convergence of the proposed al-gorithms we will need the following proposition.

Proposition 1. A linear receiver maximizes the SIR in the un-constrained or constrained filter space if and only if that receiveris the MMSE receiver or its scaled version in the correspondingfilter space.

Now we concentrate on Step 4, where we are seeking op-timum transmitters for users given that users’ receiver struc-tures are fixed. Recall that Step 4 of the OAS is the same asthat of the OBA and is a generalization to that of the OA.Thus, we will concentrate on solving the optimization prob-lem proposed in Step 4 of the OAS. To solve this optimizationproblem, we extend the algorithm of [14], which yields opti-mum downlink beamforming weights (i.e, spatial sequences)for SDMA systems to CDMA systems. Note that at each iter-ation of the OA, Step 4 will yield optimum joint temporal-spatial sequences for given linear receivers derived in Step 2,whereas, the OAS and the OBA generate optimum spatial se-quences for the corresponding linear receivers in the down-link.

Normalizing Cq = Cq/σq for all q, we formulate the de-sired downlink constrained optimization problem as

minwq

K∑q=1

∥∥wq

∥∥2(22)

subject to

γq

( K∑k =q

∣∣∣wHk AH

k Cq

∣∣∣2+ 1

)−∣∣∣wH

q AHq Cq

∣∣∣2 ≤ 0, ∀q. (23)

In problem (22), (23), it can be shown that there exists aglobal minimum, if a feasible solution exists. Moreover, bycontradiction we find that at the global minimum, the con-straints in (22), (23) are satisfied with equality. Normalizingthe downlink spatial sequences wq such that wq = √

PqWq

where Pq is a nonzero scalar and WHq AH

q Cq = 1, we obtainthe following lemma.

Lemma 2. At the global minimum of the downlink problem,the following set of equations will be satisfied for all q:

Wq − λqAHq Cq +

K∑k =q

λkγkAHq CkCH

k AqWq = 0,

WHq AH

q Cq = 1,

(24)

γq

( K∑k =q

Pk∣∣∣WH

k AHk Cq

∣∣∣2+ 1

)= Pq, (25)

where λq are Lagrangian coefficients.


The proof of the above lemma is given in the appendix. Nowwe show that the optimal transmit spatial sequences of thedownlink are a scaled version of the receive filter coefficientsfor the virtual uplink problem. One iteration for the down-link optimization algorithm consists of one iteration for vir-tual uplink optimization algorithm. In fact, the feasibility ofthe virtual uplink problem and that of the downlink problemare equivalent [13]. In the virtual uplink problem, the mo-biles seek to minimize the total transmitter powers subject totheir SIR requirements. Here, user q uses Cq as its spread-ing sequences and the base station employs a linear receiver,which maximizes the SIR in the constrained filter spaces. Theproblem, which we would like to solve now, is as follows:

minPq

K∑q=1

Pq∣∣Cq

∣∣2(26)

subject to

maxwq

Pq∣∣∣wH

q AHq Cq

∣∣∣2

∑Kk =qPk

∣∣∣wHq AH

q Ck

∣∣∣2+∣∣wq

∣∣2≥ γq (27)

subject to ∣∣∣wHq AH

q Cq

∣∣∣2 = 1, ∀q. (28)

Recall that Cq is derived from the downlink receiver ofuser q after processing as HH

q Cq/σq. We can view Aqwq as atemporal-spatial filter of user q, where Aq is the temporal fil-ter and wq is the spatial filter. Also note that γq is the same inboth the uplink and the downlink problems.

We can show by contradiction that at the global mini-mum of the above optimization problem, the constraint willbe satisfied with equality and the optimum power vector iscomponent-wise less than or equal to any feasible power vec-tor of the optimization problem. Thus, the power vector,which will be optimum to problem (26), (27), and (28), willalso be the optimum solution if we change the cost functionof the above problem as

∑Kq=1 Pq. Now we state a proposition,

which yields an algorithm to solve the optimization problemof the virtual uplink with the assumption that it is feasible.

Proposition 2. The following iterative algorithm has a uniquefixed point, thus, it is the global minimum of the virtual uplinkproblem:

Pq(n + 1) = γq minwq

( K∑k =q

Pk(n)∣∣∣wH

q AHq Ck

∣∣∣2+∣∣wq

∣∣2)

(29)

subject to ∣∣∣wHq AH

q Cq

∣∣∣2 = 1. (30)

The proof of Proposition 2 follows from the framework ofstandard interference functions developed in [20]. The con-vergence of the above type of algorithms has been provenusing the properties of standard interference functions in

[3, 5, 14]. Now we present some properties of the transmitterpowers Pq and the spatial filter wq of user q at the globalminimum of the uplink problem.

Lemma 3. The transmitter powers Pq and the spatial filterswq at the global minimum of the uplink problem will satisfythe following set of equations for all q:

wq −Pqγq

AHq Cq +

∑k =q

PkAHq CkCH

k Aqwq = 0, (31)

wHq AH

q Cq = 1. (32)

The proof of the above lemma will be found in the ap-pendix. Employing a similar technique used in [14], we ob-tain the following lemma.

Lemma 4. If the transmitter powers pq and the spatial filtersωq are the solutions to (31) and (32), then they are unique.

The above lemma implies that (31) and (32) will be satis-fied only by the global minimum of the virtual uplink prob-lem, if that is feasible. If we denote λqγq = Pq for all q in (31)and compare (31) and (32) with (24), then from Lemma 4,we get the following two lemmas.

Lemma 5. Any normalized spatial sequences that satisfy (24)will be unique.

Lemma 6. The global optimum spatial filters wo,q of the up-link problem are the global optimum normalized spatial se-quences Wo,q in the downlink problem.

Using Lemma 6, we claim that the iterative algorithm inProposition 2 yields the global optimum normalized spatialsequences not only for the uplink problem but also for thedownlink problem. In Scheme 2, we present the algorithmthat solves the optimization problem in Step 4 of the OASfor all q. Now we conclude the convergence of the algorithmin Step 4 by the following remarks:

(1) Proposition 2 implies that Step 4d of Scheme 2 willconverge to the global optimum transmitter powersand the spatial filter coefficients wo,q of the virtualuplink;

(2) Lemma 6 states that the optimum spatial filters wo,qof the uplink problem are the global optimum normal-ized spatial sequences Wo,q in the downlink prob-lem;

(3) once Step 4d converges, the convergence of Step 4 tothe global minimum will be ensured by the conver-gence of Step 4f, which is guaranteed by the followingproposition.

Proposition 3. If the downlink problem is feasible, then

Pq(m + 1) = γq

( K∑k =q

Pk(m)∣∣∣WH

o,kAHk cq

∣∣∣2+ 1

), ∀q (33)

will converge to a unique fixed point.


Step 4a. Normalize Cq(n) by σq as Cq(n) = Cq(n)/σq forall q

Step 4b. Set m = 1 and denote cq = Cq(n) for all q

Step 4c. Initialize Pq(m) = 0 and Pq(m) = 0 for all q

Step 4d. Update the spatial filters and transmitter powersof the virtual uplink problem

wq(m) = arg minwq

( K∑k =q

Pk(m)∣∣∣wH

q AHq ck

∣∣∣2+∣∣wq

∣∣2

)

such thatwH

q AHq cq = 1 ∀q

Pq(m + 1)

= γq

( K∑k =q

Pk(m)∣∣∣wH

q (m)AHq ck

∣∣∣2+∣∣wq(m)

∣∣2

)∀q

Step 4e. Derive spatial filters Wq(m) = wq(m) for all q

Step 4f. Scale the spatial filters of the downlink problem

Pq(m + 1) = γq

( K∑k =q

Pk(m)∣∣∣WH

k (m)AHk cq

∣∣∣2+ 1

)∀q

Step 4g. Set m = m + 1 and Go to Step 4d.

Scheme 2: This algorithm solves the optimization problem of Step 4of the OAS algorithm.

The proof of Proposition 3 simply follows from theframework of standard interference functions developed in[20].

Now we are about to conclude the convergence of theproposed algorithms. To do so, we will need the followinglemma. The proof of this lemma straightforwardly followsfrom the fact that the constraint in the optimization prob-lem of Step 4 does not alter if we multiply Cq by a complexconstant α.

Lemma 7. The solution to the optimization problem in Step 4will be the same if the receiver Cq is scaled by a complex factor α.

The MMSE receivers are unique and maximize the SIR ofall users in Step 2. This fact with Proposition 1 and Lemma 7help us to prove, in the appendix, the following proposition.

Proposition 4. Each of the three proposed algorithms will con-verge to a fixed point, which is not necessarily the global opti-mum of the desired minimization problem.

5. EMPIRICAL RESULTS

To observe the performance of the OA, OAS, and OBA, anempirical study was performed with a single circular cell DS-

161412108642Number of users

10−0.04

10−0.03

10−0.02

10−0.01

100

100.01

100.02

100.03

100.04

Cro

ss-c

orre

lati

on

Figure 1: The cross-correlations between the normalized receiverfilter of user 1 and its normalized received temporal-spreading se-quences is shown.

CDMA system. The radius of the cell was r0 = 1000 meters.Mobiles were uniformly distributed within the cell. This as-sumption yielded a probability density function f (r) = 2r/r2

0

for the distance of a user from the base station. The path lossexponent was 4. The height of the base station was 30 metersso that the downlink channel gain to a user from the base sta-tion was h = 1/(r2 + 302)2. The number of transmitting an-tennas at the base station was 2. Transmitted pulses were rect-angular. Transmitted signal of each antenna was received bya user over a multipath channel, where the number of pathswas two and the path delays were 0 and Tc. We used Tc = 1.Channel coefficients were modeled as independent identi-cally distributed complex Gaussian random variables withmean zero and mean square value h/2. The processing gainof the system was 16. To implement the OAS and OBA, weused the same Walsh code for each user at both antennas.Thespreading sequences of different users were derived by mul-tiplying a unique spreading sequence with different orthog-onal Walsh codes. We assumed that each of the three algo-rithms converged, when minq(|γq,2(n) − γq|/γq) < 0.00001,where γq,2(n) is the SIR of user q at iteration n immedi-ately after Step 2. The algorithm of Step 4 was assumed tobe converged when |TP(n) − TP(n + 1)|/ TP(n) < 0.00001,where TP(n) was the total transmitter power by the down-link system at iteration n. The back-ground noise variancewas σ2 = 6× 10−14. The target SIR of all users were 10 dB.

Figure 1 shows (CH1 /‖C1‖)(H1S1/‖H1S1‖), which is the

cross-correlations between the normalized receiver filter ofuser 1 and its normalized received temporal-spreading se-quences as a function of the number of users. We observedthat under the OA, the linear MMSE receiver matches the re-ceived transmitter sequences. Recently [9] identifies that ina synchronous CDMA system, the linear MMSE receiver isthe matched filter under the optimum transmitter sequences[9, 11] and powers. In the synchronous CDMA system, theoptimum signature sequences minimize the total squaredcorrelation (TSC); they form a set of orthogonal sequences,if the number of users is less than or equal to the processing



0

2

4

6

8O

BA

/OA

S(i

ndB

)


25

30

35

OA

S/O

A(i

ndB

)

Figure 2: The relative performance of the algorithms is shown.

gain, and a set of WBE sequences otherwise. The results of [9]motivate [21] to design a linear receiver for multi-inputs andmulti-outputs (MIMO) dispersive communication channels,whose coefficients matches the efficient transmitted trans-mitter sequences.

In the second experiment, we performed a simulationstudy to see how the OA, OAS, and OBA perform with re-spect to each other. Figure 2 shows the relative performanceof the algorithms where the total transmitter power subjectto SIR requirement by the system was taken as the perfor-mance metric. In this experiment, we observed that the OBAbecame infeasible when the number of users was greater than9. Our experimental result suggested that the OA is far supe-rior to the OAS, whereas the OAS significantly outperformedthe OBA. This result is well expected and can be explainedas follows. The OA takes advantage of both the temporal andspatial diversities at both the transmitter and receiver ends.However, unlike the OA, OAS is constrained by the fixedtransmitter temporal sequences. Although both the OAS andOBA use the same temporal spreading sequences, the betterperformance of the OAS than that of the OBA is due to thefact that the OBA uses constrained MMSE receiver, whereasthe OAS employs the unconstrained MMSE receiver.

6. CONCLUSION

Next generation wireless systems are being designed to sup-port both voice and high capacity flexible data servicesthrough available limited bandwidths. Interference and mul-tipath fading inherent to the wireless link make this a diffi-cult task. However, future wireless systems must adapt to thisadverse radio environment efficiently. This situation leads usto develop the joint transmitter-receiver optimization algo-rithms for the next generation CDMA systems under vari-ous system circumstances where we minimize the total powertransmitted by the system subject to SIR requirements ofusers. First, we develop the optimization algorithm con-

sidering no constraint on the system and referred to it asOA. Afterwards, taking practical system constraints on thetransmitter-receiver structures into account, we propose twojoint transmitter-receiver optimization algorithms and callthem as OAS and OBA. Both the OAS and OBA are con-strained by the fixed temporal spreading sequences, however,the OAS employs the unconstrained MMSE receiver, whereasthe OBA employs the constrained MMSE receiver. We provethe convergence of all the proposed three algorithms. In ourexperiments, we observed that under the OA, the optimumreceiver structure matches the received transmitter sequences.Our empirical results also indicated that the OA is far supe-rior to the OAS in terms of performance, whereas the OASoutperformed the OBA significantly.

APPENDIX

Proof of Lemma 1. Matrix inversion lemma [5] states that aninvertible matrix M and vectors u and v satisfy the followingequality:

(M + uvH)−1 = M−1 − M−1uvHM−1

1 + vHM−1u. (A.1)

In (10), if we assume that

M =K∑k =q

(HqSkSH

k HHq

)+ σ2ID,

u = v = HqSq,

(A.2)

then (A.1) yields

Cq = M−1u− M−1uuHM−1u1 + uHM−1u

= M−1u(

I− uHM−1u1 + uHM−1u

I)= 1

1 + uHM−1uM−1u.

(A.3)

Using a similar method, we can show that the constrainedMMSE receiver of (17) satisfies (20).

Proof of Proposition 1. In order to prove this proposition,first we show that the MMSE receiver and its scaled ver-sion maximize the SIR in the unconstrained filter spaces aswell as in the constrained spatial-temporal filter spaces. TheMMSE receiver [19] minimizes mean-squared error (MSE)and maximizes the SIR [5, 22] in the unconstrained filterspaces. Now, we prove that the MMSE receiver maximizesthe SIR in the constrained spatial filter spaces as proposed in(15). For the receiver filter Cq = HqAqZq, the SIR of user qis

γq(

Zq) =

∣∣∣ZHq Bq,qwq

∣∣∣2

∑Kk =q

∣∣∣ZHq Bq,kwk

∣∣∣2+ σ2ZH

q Bq,qZq

. (A.4)

Our goal is to find Zq = Zq that maximizes γq(Zq), that


is,

Zq = arg maxZq

(γq(

Zq)). (A.5)

However, the solution to the above optimization problem isnot unique. To see that, simply observe that Zq and αZq willproduce the same SIR, where α is a scalar. Thus if a linearreceiver maximizes the SIR, then its scaled versions will alsomaximize the SIR in the constrained as well as unconstrainedfilter spaces.

An optimum solution to (A.5) will be obtained if we solvethe optimization problem with the constraint ZH

q Bq,qwq =g, where g is an arbitrary complex-valued gain. Under theconstraint ZH

q Bq,qwq = g, the maximization problem in (A.5)will be equivalent to the following minimization problem:

Zq = arg minZq

( K∑k =q

∣∣∣ZqBq,kwk

∣∣∣2+ σ2ZH

q Bq,qZq

)(A.6)

subject to

ZHq Bq,qwq = g. (A.7)

We solve the above optimization problem explicitly as

Zq = g∗

wHq BH

q,qVqVq, (A.8)

where g∗ is the conjugate of g and

Vq =[ K∑k =q

(Bq,kwkwH

k BHq,k

)+ σ2Bq,q

]−1

Bq,qwq. (A.9)

Letting

g = wHq BH

q,qVq

1 + wHq BH

q,q

[∑Kk =q

(Bq,kwkwH

k BHq,k

)+ σ2Bq,q

]−1Bq,qwq

,

(A.10)

we find that the MMSE receiver in (20) maximizes the SIR inthe constrained filter space.

Now, we prove that any linear receiver that maximizesthe SIR is either the MMSE receiver or a scaled version ofthe MMSE receiver. From (A.8) we see that a linear receiver,which maximizes the SIR of user q and satisfies the constraintCHq HqSq = g, can be explicitly obtained by solving the follow-

ing minimization problem:

Cq = arg minCq

( K∑k =q

∣∣∣CHq HqSk

∣∣∣2+ σ2

∥∥Cq

∥∥2)

(A.11)

subject to

CHq HqSq = g. (A.12)

Since the above cost function is convex, the solution to

the above minimization problem is unique. We have alreadyproved that Cq will be either the MMSE receiver or its scaledversion depending on the value of g. Note that when Cq isconstrained to be in the form of HqAqZq, Problem (A.11),(A.12) is equivalent to (A.6), (A.7).

Let Cq be a linear receiver, which maximizes user q’sSIR. Without loss of generality, we can assume the projec-tion of this receiver on the received transmitter sequences asCHq HqSq = g. Note that Cq is a feasible solution to the above

optimization problem and provides the same SIR as the op-timum receiver Cq. Thus, Cq is also an optimum solution.Since the optimum solution to the above minimization prob-lem is unique, we get Cq = Cq.

Proof of Lemma 2. The Lagrangian for the optimizationproblem of (22), (23) is given by

L(W, λ)

=K∑q=1

∥∥wq

∥∥2

+K∑q=1

λq

(γq

( K∑k =q

∣∣∣wHk AH

k Cq

∣∣∣2+ 1

)−∣∣∣wH

q AHq Cq

∣∣∣2).

(A.13)

Differentiating the above equation with respect to wq, the fol-lowing K equations are obtained as the necessary conditionsfor optimality:

wq − λqAHq CqCH

q Aqwq +K∑k =q

λkγkAHq CkCH

k Aqwq = 0, ∀q.

(A.14)In addition, the optimal solution must satisfy the followingK constraints, which simply follows from the optimizationproblem of (22), (23):

γq

( K∑k =q

∣∣∣wHk AH

k Cq

∣∣∣2+ 1

)=∣∣∣wH

q AHq Cq

∣∣∣2, ∀q. (A.15)

Using wq = √PqWq where WH

q AHq Cq = 1 with (A.14) and

(A.15), we prove the lemma.

Proof of Lemma 3. Using Lagrangian with the following op-timization problem:

minwq

( K∑k =q

Pk∣∣∣wH

q AHq Ck

∣∣∣2+∣∣wq

∣∣2)

(A.16)

subject to

∣∣∣wHq AH

q Cq

∣∣∣2 = 1 (A.17)

and then differentiating, we get the necessary condition foroptimality of the optimization problem (26), (27), and (28)


as

wq − λqAHq CqCH

q Aqwq +K∑k =q

PkAHq CkCH

k Aqwq = 0, (A.18)

wHq AH

q Cq = 1. (A.19)

Multiplying both sides of (A.18) by wHq and using (A.19), we

get

λq =K∑k =q

Pk∣∣∣wH

q AHq Ck

∣∣∣2+∣∣wq

∣∣2. (A.20)

Now we use the fact that at the global minimum of the vir-tual uplink optimization problem, the constraint of the op-timization problem (26), (27), and (28) will be satisfied withequality. Coupling this fact with (A.20), we get at the globalminimum, λq = Pq/γq. Substituting λq back to (A.18), weprove the lemma.

Proof of Proposition 4. To prove this proposition, we will startfrom Step 2. Let γq,2(n) denote the SIR of user q at the nth it-eration immediately after Step 2, where n > 1. Proposition 1implies that Cq(n) will maximize the SIR of user q for giventransmitter sequences of all users.

Recall that Steps 4 and 5 yield the optimum trans-mitter sequences Sq(n) for given Cq(n− 1), which arethe MMSE receivers of users for the transmitter sequencesSq(n− 1). The global optimum solution of the minimiza-tion problem in Step 4 satisfies the constraint with equality,and thus, Sq(n) and Cq(n− 1) will provide all users SIRsγq. Since the MMSE receiver maximizes the SIR, after exe-cuting Step 2, we will get

γq,2(n) ≥ γq, ∀q. (A.21)

If γq,2(n) > γq for user q, then it is easy to show that afeasible solution can be obtained for the minimization prob-lem of Step 4 by reducing the power of that user q. Thuswhen γq,2(n) > γq, the cost function of Step 4 will strictlydecrease at its next execution. When γq,2(n) = γq for allq, Proposition 1 yields that Cq(n− 1) are either MMSEreceivers or their scaled versions for transmitter sequencesSq(n). Coupling this fact with Lemma 7, we get

Sq(n) = Sq(n− 1), ∀q. (A.22)

Since the MMSE receivers are unique for given Sq(n), weobtain

Cq(n) = Cq(n− 1), ∀q. (A.23)

Therefore, γq,2(n) = γq for all q implies that the proposedalgorithms have converged to a fixed point, which is not nec-essarily the global optimum.

ACKNOWLEDGMENT

This work has been presented in part in the Conference onInformation Science and Systems, Johns Hopkins University,March 21–23, 2001.

REFERENCES

[1] P. S. Kumar and J. Holtzman, “Power control for a spreadspectrum system with multiuser receivers,” in Proc. IEEEInternational Symposium on Personal, Indoor, and Mobile Ra-dio Communications, vol. 3, pp. 955–959, Toronto, Canada,September 1995.

[2] M. Saquib, R. Yates, and A. Ganti, “Power control for an asyn-chronous multi-rate decorrelator,” IEEE Trans. Communica-tions, vol. 48, no. 5, pp. 804–812, 2000.

[3] S. Ulukus and R. Yates, “Adaptive Power Control with MMSEMultiuser Detectors,” in Proc. IEEE International Conferenceon Communications, Montreal, Quebec, Canada, June 1997.

[4] M. Varanasi, “Power control for multiuser detection,” inProc. 30th Annual Conference on Information Sciences and Sys-tems, pp. 866–873, Princeton University, Princeton, NJ, USA,March 1996.

[5] A. Yener, R. Yates, and S. Ulukus, “Interference managementfor CDMA systems through power control, multiuser detec-tion, and beamforming,” IEEE Trans. Communications, vol.49, no. 7, pp. 1227–1239, 2001.

[6] W. M. Jang, B. R. Vojcic, and R. L. Picholtz, “Joint transmitter-receiver optimization in synchronous multiuser communica-tions over multipath channels,” IEEE Trans. Communications,vol. 46, no. 2, pp. 269–278, 1998.

[7] S. Ulukus and R. Yates, “Iterative construction of optimumsignature sequence sets in synchronous CDMA systems,” IEEETransactions on Information Theory, vol. 47, no. 5, pp. 1989–1998, 2001.

[8] S. Ulukus and R. Yates, “User capacity of asynchronousCDMA systems with optimum signature sequences,” sub-mitted to IEEE Transactions on Information Theory, April2001.

[9] P. Viswanath, V. Anantharam, and D. N. C. Tse, “Optimalsequences, power control, and user capacity of synchronousCDMA systems with linear MMSE multiuser receivers,” IEEETransactions on Information Theory, vol. 45, no. 6, pp. 1968–1983, 1999.

[10] T. F. Wong and T. M. Lok, “Transmitter adaptation in mul-ticode DS-CDMA systems,” IEEE Journal on Selected Areas inCommunications, vol. 19, no. 1, pp. 69–82, 2001.

[11] M. Rupf and J. L. Massey, “Optimum sequences multisets forsynchronous code-division multiple-access channels,” IEEETransactions on Information Theory, vol. 40, no. 4, pp. 1261–1266, 1994.

[12] J. I. Concha and S. Ulukus, “Optimization of CDMA signa-ture sequences in multipath channels,” in Proc. IEEE VehicularTechnology Conference, Rhodes, Greece, May 2001.

[13] F. Rashid-Farrokhi, R. K. J. Liu, and L. Tassiulas, “Transmitbeamforming and power control for cellular wireless systems,”IEEE Journal on Selected Areas in Communications, vol. 16, no.8, pp. 1437–1450, 1998.

[14] E. Visotsky and U. Madhow, “Optimal multiuser space-timetransmit filtering,” Coordinated Science Laboratory, Univer-sity of Illinois, Urbana, Ill, USA, 1999.

[15] Y. Liang, F. P. S. Chin, and K. J. R. Liu, “Downlink beamform-ing for DS-CDMA mobile radio with multimedia services,”IEEE Trans. Communications, vol. 49, no. 7, pp. 1288–1298,2001.


[16] F. Adachi, M. Sawahashi, and H. Suda, “Wideband DS-CDMAfor next-generation mobile communication systems,” IEEECommunication Magazine, vol. 36, no. 9, pp. 56–69, 1998.

[17] R. H. Frenkiel and T. Imielinski, “Infostations: the joy of‘many-time, many-where’ communications,” WINLAB Tech-nical Report (WINLAB-TR-119), Rutgers University, April1996.

[18] M. Saquib, Md. H. Islam, and S. Kumar, “Power control andtransmit diversity in multipath CDMA systems,” in Proc.IEEE Wireless Communications and Networking Conference,Chicago, Ill, USA, September 2000.

[19] Z. Xie, R. T. Short, and C. K. Rushforth, “A family of sub-optimum detectors for coherent multiuser communications,”IEEE Journal on Selected Areas in Communications, vol. 8, no.4, pp. 683–690, 1990.

[20] R. D. Yates, “A framework for uplink power control in cellularradio systems,” IEEE Journal on Selected Areas in Communica-tions, vol. 13, no. 7, pp. 1341–1347, 1995.

[21] P. Dimitrie and C. Rose, “New approach to multiple antennasystems,” in Proc. Conference on Information Science and Sys-tems, Baltimore, Md, USA, March 2001.


Mohammad Saquib received his B.S. de-gree (1991) in electrical and electronicsengineering from Bangladesh Universityof Engineering & Technology, Bangladesh(BUET). After his baccalaureate, he workedas a System Analyst (1991–92) at the EnergyResearch Corporation, Danbury, CT. He re-ceived the M.S. (1995) and the Ph.D. (1998)degrees in electrical engineering from Rut-gers University, New Brunswick, NJ, wherehe was a Graduate Research Assistant in the Wireless InformationNetworks Laboratory (WINLAB). From 1998 to 1999, he was withthe MIT Lincoln Laboratory, Lexington, Mass, as a member of theTechnical Staff. In January 1999, he joined the Electrical and Com-puter Engineering Department at Louisiana State University (LSU),where he was the Donald Ceil & Elaine T. Delaune Endowed Assis-tant Professor. Since July 2000, he has been with the Electrical Engi-neering Department at the University of Texas at Dallas (UTDallas)as Assistant Professor. His research interests include power control,interference suppression, and media access protocols for wirelesscommunications systems.

Md Habibul Islam received his M.S. degree(1995) in electrical engineering and Mas-ters of Business Administration (1999) fromTajik Technical University, USSR and theInstitute of Business Administration, TheUniversity of Dhaka, Bangladesh, respec-tively. Currently, he is pursuing the Ph.D.degree in electrical engineering at The Uni-versity of Texas at Dallas, Richardson, Texas.He is a Graduate Research Assistant in theWireless Communications Research Laboratory (WiCoRe). His re-search interests include power control, multiuser detection, inter-ference management, and adaptive modulation for wireless com-munications systems.


An Adaptive Channel Estimation Algorithm UsingTime-Frequency Polynomial Model for OFDM withFading Multipath Channels

Xiaowen WangWireless Systems Research Department, Agere Systems, Murray Hill, NJ 07974, USAEmail: [email protected]

K. J. Ray LiuElectrical and Computer Engineering Department, University of Maryland, College Park, MD 20742, USAEmail: [email protected]


Orthogonal frequency division multiplexing (OFDM) is an effective technique for the future 3G communications because of itsgreat immunity to impulse noise and intersymbol interference. The channel estimation is a crucial aspect in the design of OFDMsystems. In this work, we propose a channel estimation algorithm based on a time-frequency polynomial model of the fadingmultipath channels. The algorithm exploits the correlation of the channel responses in both time and frequency domains andhence reduce more noise than the methods using only time or frequency polynomial model. The estimator is also more robustcompared to the existing methods based on Fourier transform. The simulation shows that it has more than 5 dB improvementin terms of mean-squared estimation error under some practical channel conditions. The algorithm needs little prior knowledgeabout the delay and fading properties of the channel. The algorithm can be implemented recursively and can adjust itself to followthe variation of the channel statistics.

Keywords and phrases: channel estimation, OFDM, polynomial approximation.

1. INTRODUCTION

The 3G wireless communication system is the next genera-tion mobile cellular system that aims to provide high ratedata communications of a bit rate up to 2 Mbit/s. Amongmany technical challenges in this broadband system, the se-vere intersymbol interference (ISI) caused by multipath ef-fect of wireless channels is an essential one. One effectivetechnique to deal with this problem is the orthogonal fre-quency division multiplexing (OFDM) [1, 2]. In OFDM sys-tems, the entire bandwidth is partitioned into parallel sub-channels by dividing the transmit data into several paral-lel low bit rate data streams to modulate the carriers cor-responding to those subchannels. By doing so, the OFDMsystem has a relatively longer symbol duration, thus pro-vides a great resistance to ISI and impulse noise. When thenumber of subchannels is large enough, the subchannels canbe treated as independent of each other and only a one-tapequalizer is needed for each subchannel. Because of theseadvantages, OFDM has become a promising technique forbroadband wireless communications.

Channel estimation is a key issue in a communication

system, as is the case for the OFDM system. Without theknowledge of channel information, noncoherent detection,such as differential modulation, has to be used and resultsin some performance loss compared to the coherent detec-tion. The channel estimation problem becomes more im-portant for the 3G systems because many sophisticated sig-nal processing techniques that require the knowledge of thechannel information are expected to be used to meet thechallenge of throughput and performance. For example, theindependence of the subchannels in OFDM systems pro-vides an easy way to optimize the transmitter design byadjusting the bit rate and transmit power across subchan-nels according to their channel conditions [3], which im-plies that the channel information has to be known at thetransmitter.

The channel estimation problem is also more challeng-ing in the 3G system because both the multipath effect andthe fading effect have to be considered in this mobile broad-band system. The important observation to solve the channelestimation problem in the OFDM systems is that the fadingmultipath channel in the 3G system is correlated in both timeand frequency domain, even though subchannels are treated

An Adaptive Channel Estimation Algorithm Using Time-Frequency Polynomial Model for OFDM 819

independently when performing the signal detection. Thechannel estimation algorithms should exploit such correla-tion to improve the accuracy of the estimation. Van de Beeket al. [4] tried to exploit the correlation of the channel pa-rameters in frequency domain while Mignone and Morello[5] used the correlation in time domain. Li et al. [6] con-sidered the correlation in both time and frequency domains.The estimators designed in these literatures are all Fourier-transform-based approaches, which implicitly assumed thatthe channel power spectrum can be viewed as band lim-ited. The assumption is true when we consider the ensemblestatistics. However, in practice, we can only get finite discretesamples of the channel response of the time varying channel.The leakage can be very severe and then degrade the perfor-mance dramatically.

In this work, we consider the problem from anotherpoint of view. Because of the correlation of the fading multi-path channel, it can be viewed as a smoothly varying functionof both time and frequency. It has been stated in the approx-imation theory that such a smoothly varying function canbe approximated by a set of basis functions [7], for example,the polynomial basis [8]. Borah and Hart [9, 10] used thetime domain polynomial approximation while Luise et al. in[11] used the frequency domain polynomial approximation.However, the channel responses used in coherent detectionof OFDM are located in the time-frequency plane. Therefore,it is naturally to exploit the channel correlation in both timeand frequency domains using a time-frequency polynomialmodel. The noise can then be greatly suppressed by estimat-ing a smaller number of coefficients of the basis functionsover a large number of observations. Moreover, it also makethe estimator design more flexible and robust to the variationof channel statistics. We can also view Fourier transform as atype of model basis function and hence Fourier-transform-based method is the same type of method as the polynomial-model-based channel estimation scheme but with differ-ent model accuracy and different noise reduction capability.These two methods compared, the model error of the Fourierbasis is very sensitive to the channel statistics and works onlyfor some very specific system parameters and channel statis-tics. On the contrary, the polynomial-model-based methodperforms more consistently and robustly for variety ofchannels.

A key problem in using the polynomial model to estimatethe channel responses is to decide the model order and time-frequency window dimensions of observations. The modelapproximation error of polynomial model decreases whenincreasing model order or decreasing the window dimen-sions. On the other hand, the noise is reduced more whendecreasing the model order or increasing the window dimen-sions. It is important to reach a tradeoff between the modelerror and noise reduction. In this paper, we propose an adap-tive algorithm that adjusts the window dimensions to balancethe tradeoff. With this adaptive algorithm, the channel corre-lation function or the fading and delay characteristics are nolonger that essential in the design of the channel estimator.The estimator can adapt its settings to the variation of thechannel statistics.

S/PLoading

&mapping

X0,k

X1,k

...

Xm−1,k

IFFT

X0,k

X1,k...

Xm−v,k

Xm−1,k

...

Xm−1,k

P/S

(a)

S/P

y−v,k...

y−1,k

y0,k

y1,k

...

ym−1,k

FFT

Y0,k

Y1,k...

Ym−1,k

One-tapequalizer

w

X0,k

X1,k...

Xm−1,k

X0,k

X1,k...

Xm−1,k

(b)

Figure 1: OFDM transmitter and receiver. (a) Transmitter, (b) re-ceiver.

The rest of the paper is organized as follows. First,we introduce the OFDM system in Section 2 and the fad-ing multipath channel in Section 3.1. Then, we discussthe time-frequency polynomial model Section 3.2 and de-rive the corresponding recursive channel estimation algo-rithm in Section 4. The performance analysis is discussed onthe general-model-based estimation approach in Section 5.Then the window dimension adaptive algorithm is derivedin Section 6 based on the performance analysis. Finally, thesimulation results are presented to demonstrate the perfor-mance in Section 7 and the conclusion is drawn in Section 8.

2. OFDM SYSTEMS

Figures 1a and 1b show the transmitter and receiver of anOFDM system, respectively. The OFDM system divides thewhole bandwidth Bd into m subchannels by buffering theinput data to blocks, and then partitions the block into mlower rate bit streams. In most of OFDM systems, the sub-channels are divided evenly, the bandwidth of the subchan-nels or the rate of the bit streams is ∆ f = Bd/m. The bitstreams may contain different amount of bits and use dif-ferent transmit energy according to the channel condition.The bit and energy allocation is done by a loading algorithm.Then the bit streams are mapped to some complex constella-tion points Xi,k, i = 0, . . . ,m−1 at the kth block. The modula-tion is then implemented by m-point inverse discrete Fouriertransform (IDFT). Then the modulated data go through P/Sconverter to form the serial data xi,k. A cyclic prefix which isconstructed using the last v samples of xi,k’s is inserted beforesending the xi,k’s to the channel. Now it follows that the sym-bol duration is m/Bd, however, the actual block duration is


−1 0 1 2 3 4 5 6

t (µs)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pow

er

(a)

−2 0 2 4 6 8 10 12 14 16 18

t (µs)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pow

er

(b)

Figure 2: Two typical delay profiles (a) TU and (b) HT.

Tf = (m + v)/Bd. For a system with Bd = 800 kHz, m = 128,and v = 16, the block duration is Tf = 180 microseconds.Such a system is used in the rest of this paper.

At the receiver, the prefix part is discarded. The demodu-lation is performed by the discrete Fourier transform (DFT)operation. If the cyclic prefix is long enough, then the in-terference between two OFDM blocks is eliminated and thesubchannels can be viewed as independent of each other, thatis, the demodulated data Yi,k can be expressed as

Yi,k = Hi,kXi,k + Ni,k, (1)

where Hi,k is the channel frequency response at i∆ f of kthblock and Ni,k is the corresponding channel noise that is as-sumed to be white Gaussian process with zero mean and vari-ance σ2.

Because of the simple relation of (1), only a one-tapequalizer is needed for each subchannel at the receiver, thatis,

Xi,k = Yi,kWi,k, (2)

where the equalizer coefficient Wi,k is some function of Hi,k.For example, the zero-forcing equalizer is constructed asWi,k = 1/Hi,k. Then the decision or decoding is made uponXi,k.

3. POLYNOMIAL CHANNEL MODEL

3.1. Fading multipath channel

In a mobile broadband wireless communication system suchas 3G, the transmission is impaired by both fading that is dueto the mobility, and multipath that is due to the wide band-width. This fading multipath channel has long been knownto be modeled as a time-varying linear filter [12],

h(t, τ) =∑i

γi(t)δ(τ − τi

), (3)

where γi(t)’s are independent complex Gaussian processeswith zero mean and variance pi’s. For OFDM systems, wecan assume that the channel is time varying for differentblocks but time-invariant within one block. The channel fre-quency response Hi,k’s are samples of the continuous channelresponse H(t, f ) = ∫

h(t, τ)e− j2π f τ dτ, that is,

Hi,k = H(kT f , i∆ f

). (4)

The correlation function of H(t, f ) is defined as rH(t, f )=

E[H(t1, f1)H∗(t1 − t, f1 − f )]. Assume that the correlationfunction of γi(t) follows E[γi(t1)γ∗i (t1− t)] = pir(t), then wehave

rH(t, f ) = rt(t)r f ( f ). (5)

For the Rayleigh fading channel [12], rt(t) = J0(2π fDt) andr f ( f ) = ∑

i pie− j2π∆ f τi with J0(·) denoting the zero-order

Bessel function, fD being the Doppler frequency describingthe channel variation along t, and pi’s together with τi’s be-ing delay profiles describing the channel dispersion whichis also often characterized by the maximum delay spread

Td= maxi τi. Three types of delay profiles are used in this

work, TU, HT, and 2-ray [13]. The TU and HT delay profilesare shown in Figure 2. The 2-ray delay profile has two equalpower paths and the delay between two paths is Td. We alsoassume that the channel is normalized in our simulation, thatis,

∑i pi = 1.

From (5), the power spectrum of the channel response is

SH(ξ, ν) =∫ ∫

rH(t, f )e− j(tξ+ f ν) dξ dν = St(ξ)S f (ν), (6)

where St(ξ) = ∫rt(t)e− jtξ dξ and S f (ν) = ∫

r f ( f )e− j f ν dν.Because of the physical mechanism of the propagation, thechannel varies smoothly and most of the energy is concen-


trated in a finite bandwidth in both time and frequency do-mains. The bandwidths are fD for St(ξ) and Td for S f (ν), re-spectively.

3.2. Time-frequency polynomial channel modelof OFDM systems

We know from the approximation theory [7, 8] that thesmoothly varying channel responses can be approximated byprojecting to a finite set of basis functions. In [14], it wasshown that the channel responses in a small time domainwindow around a center point k0 of dimension 2K + 1 canbe closely approximated by a small set of polynomial basisfunctions, that is,

Hi,k=M−1∑m=0

Hi,k0 (m)(k−k0

)m+RM, for k0−K ≤ k ≤ k0+K,

(7)where

Hi,k0 (m) =Tm

f

m!∂mH(t, f )

∂tm

∣∣∣∣t=k0Tf

,

RM =((k − k0

)Tf

)MM!

∂MH(t, f )∂tM

∣∣∣∣t=t′

(8)

with k0Tf ≤ t′ ≤ kT f .For such an approximation, it can be proved that the

mean-squared model error is bounded by

E[∥∥RM

∥∥2] ≤ (((k − k0

)Tf

)MM!

)2 ∫ fD

0(2πξ)2MSt(ξ)dξ

≤ fD

(2π

(k − k0

)fDT f

M!

)2M

.

(9)

Here we assume that∫ fD

0 St(ξ)dξ = 1. It can be seen from(10) that the sufficient condition for this error to con-verge to zero is fDT f 1, that is, if fDT f 1 thenlimM→∞ E[‖RM‖2] = 0.

Similarly, if the channel delay spread Td satisfies Td∆ f 1, which means that the frequency variation of Hi,k’s issmooth enough along frequencies, then Hi,k’s in a frequencydomain window of dimension 2I +1 around i0, [i0− I, i0 + I],can be approximated by the polynomial bases, that is,

Hi,k =N−1∑n=0

Hi0 ,k(n)(i−i0

)n+RN, for i0−I ≤ i ≤ i0+I, (10)

where

Hi0 ,k(n) = ∆ f n

n!∂nH(t, f )

∂ f n

∣∣∣∣f=i0∆ f

,

RN =((i− i0

)∆ f

)NN !

∂NH(t, f )∂ f N

∣∣∣∣f= f ′

(11)

with i0∆ f ≤ f ′ ≤ i∆ f .The mean-squared model error of this approximation is

bounded by

E[∥∥RN

∥∥2] ≤ (((i− i0

)∆ f

)NN !

)2 ∫ Td

0(2πν)NS f (ν)dν. (12)

The time domain expansion (7) is used for channel esti-mation in [10], while the frequency domain expansion (10)is adopted in [11]. For a given channel, the selection of theabove two types of expansions depends on the channel statis-tics, fD and Td, and the system parameters, Tf and ∆ f .Moreover, it is naturally to expand the channel responses inboth time and frequency domains [15] for the OFDM sys-tem, since its signal is distributed in a time frequency plane.The expansion in the time-frequency window of dimensions(2I + 1)× (2K + 1) around i0 and k0 is

Hi,k =M−1∑m=0

N−1∑n=0

Hi0,k0 (nm)(k − k0

)m(i− i0

)n+ RMN, (13)

for k0 − K ≤ k ≤ k0 + K and i0 − I ≤ i ≤ i0 + I , where

Hi0 ,k0 (nm) =Tm

f ∆ f n

m!n!∂m∂nH(t, f )∂tm∂ f n

∣∣∣∣t=k0Tf , f=i0∆ f

,

RMN

= RM + RN

−((k − k0

)Tf

)M((i− i0

)∆ f

)NM!N !

∂M∂NH(t, f )∂tM∂ f N

∣∣∣∣t=t′ , f= f ′

(14)

with k0Tf ≤ t′ ≤ kT f and i0∆ f ≤ f ′ ≤ i∆ f .The mean-squared model error is then bounded by

E[∥∥RMN

∥∥2] ≤ ((KTf

)MM!

)2 ∫ fD

0(2πν)MSt(ξ)dξ

+

((I∆ f )N

N !

)2 ∫ Td

0(2πν)NS f (ν)dν

+

((KTf )M

M!

)2((I∆ f )N

N !

)2

×∫ fD

0

∫ Td

0(2πξ)M(2πν)NSt(ξ)S f (ν)dξ dν.

(15)

Without loss of generality, assuming M = N and us-ing the multipath Rayleigh fading channel described inSection 3.1, we can show that [15]

E[∥∥RMM

∥∥2] ≤ 2M!(2πKTf fD

)2M

22M(M!)4+

(2πI∆ f Td

)2M

(M!)2

+2M!

(4π2KITf∆ f fDTd

)2M

22M(M!)6.

(16)

Again the sufficient conditions for convergence of the aboveexpansion are fDT f 1 and Td∆ f 1. It is noticedthat these two conditions are usually satisfied in a practical


1 2 3 4 5

Model order

−100

−80

−60

−40

−20

0

MSE

(dB

)

Bound on model errorResidual noise (10 dB)Residual noise (20 dB)

Figure 3: Bound on mean-squared model error.

OFDM systems. When fDT f > 1, the block duration is solong or the channel changes are so fast that the channel can-not be viewed as invariant during one block and the systemsuffers large interchannel interference (ICI). On the otherhand, when Td∆ f > 1, the block duration is so short or thechannel dispersion is so large that the subchannels can nolonger be treated independently and the system would sufferboth ISI and ICI. The OFDM system cannot work in eithercase. Hence, it is reasonable to assume that both conditionsare satisfied in a well designed OFDM system.

Now we take a close look at the upper bound of the modelerror. Suppose that the length of the cyclic prefix v can be ig-nored compared to the number of the subchannels. Then thefirst term in (16) is determined by fDT f = m fD/Bd, while thesecond term is determined by ∆ f Td = BdTd/m. The thirdterm is actually determined by fDTd and is much smallerthan the first two terms, since they both are smaller thanone. For the first two terms, when m is large, the first termis dominating, then we should choose smaller K or larger M.While m is small, then the second term is dominating andI should be smaller or the model order N should be larger.If the Doppler frequency fD, maximum delay Td, and band-width Bd are fixed, we can adjust the window dimensions ac-cording to m to keep the time-frequency model error to cer-tain level but we still have a small MN/IK . However, if onlytime or frequency domain expansion is used, the model errorcannot be adjusted to maintain a small level with the sameM/K or N/I when the number of subchannels m varies.

Figure 3 shows the upper bound of mean the squaredmodel error with fDT f = Td∆ f = 10−2 and I = K = 5according to model order M. It shows that the model erroris under −40 dB as the model order is 3. This means thatwe only need to estimate 9 model coefficients to get the 121channel responses. In this figure, we also show the residualnoise for SNR of 10 dB and 20 dB. It shows that the noise

can be greatly reduced with very small penalty on model er-ror. Moreover, such a model approximation does not need toknow the actual channel correlation function.

4. CHANNEL ESTIMATION ALGORITHMWITH POLYNOMIAL MODEL

4.1. Estimator structure

The channel estimation problem in OFDM systems is to es-timate the channel response Hi,k based on the transmittedsignal Xi,k and the received signal Yi,k. The information ofthe transmitted signal Xi,k’s is obtained either from trainingor from detected feedback. In OFDM systems, an instanta-neous estimate can be easily constructed as Hi,k = Yi,k/Xi,k.Then suppose that we have chosen the model order and win-dow dimensions such that the model error is small and canbe ignored, we can approximate (13) in a matrix form

Hi0 ,k0 QM,N (I, K)bi0,k0 , (17)

where

Hi0 ,k0 =[H−I+i0 ,−K+k0· · ·HI+i0 ,−K+k0· · ·H−I+i0 ,K+k0· · ·HI+i0 ,K+k0

]T,

bi0,k0 =[Hi0 ,k0 (0, 0) · · ·Hi0 ,k0 (N−1, 0) · · ·Hi0 ,k0 (0,M−1)

· · ·Hi0 ,k0 (N−1,M−1)]T

,

QM,N (I, K) =

q−I,−K0,0 · · · q−I,−K0,N−1 · · · q−I,−KM−1,0· · · q−I,−KM−1,N−1

......

......

qI,−K0,0 · · · qI,−K0,N−1· · · qI,−KM−1,0· · · qI,−KM−1,N−1

......

......

qI,K00 · · · qI,K0,N−1· · · qI,KM−1,0· · · qI,KM−1,N−1

,

(18)

with qi,km,n = inkm, for i = −I, . . . , 0, . . . , I , k = −K, . . . , 0,. . . , K , m = 0, . . . ,M − 1, and n = 0, . . . , N − 1.

Define

Hi0 ,k0 =[H−I+i0 ,−K+k0· · · HI+i0 ,−K+k0· · · H−I+i0 ,K+k0· · · HI+i0 ,K+k0

]T,

(19)

then

Hi0 ,k0 = Hi0 ,k0 + Ni0 ,k0

QM,N (I, K)bi0,k0 + Ni0 ,k0 ,(20)

where

Ni0 ,k0 =[N−I+i0 ,−K+k0

X−I+i0,−K+k0

· · ·NI+i0 ,−K+k0

XI+i0,−K+k0

· · ·N−I+i0 ,K+k0

X−I+i0,K+k0

· · ·NI+i0 ,K+k0

XI+i0,K+k0

]T.

(21)


Using least square (LS) methods, we can get the estimationof the coefficients of the polynomial basis from the instanta-neous estimates

bi0,k0 = Q†M,N (I, K)Hi0,k0 , (22)

where Q†M,N (I, K) is the pseudoinverse of QM,N (I, K). The

channel estimation then can be constructed as

Hi,k = qM,N(i− i0, k − k0

)Tbi0,k0

= qM,N(i− i0, k − k0

)TQ†(I, K)Hi0,k0 ,

(23)

where

qM,N(i− i0, k − k0

)=[qi−i0,k−k0

0,0 · · · qi−i0,k−k00,N−1 · · · qi−i0,k−k0

M−1,0 · · · qi−i0,k−k0M−1,N−1

]T.

(24)

Usually, we fix the value of i− i0 and k− k0, that is, we fix thepoint of estimation inside the window and slide the windowto get all the estimations. Then the estimator can be viewedas a two-dimensional filtering process. Arranging the instan-taneous estimation inside the window into a matrix form,

H =

H−I+i0 ,−K+k0 · · · H−I+i,K+k0

......

HI+i0,−K+k0 · · · HI+i0 ,K+k0

. (25)

Then the estimation is

Hi,k = qTN

(i− i0

)Q†T

N (I)HQ†M(K)qM

(k − k0

), (26)

where

qN (i) =[i0 i1 · · · iN−1

]T,

QN (I) =[

qN (−I) · · · qN (0) · · · qN (I)]T

.(27)

The estimator structure is shown in Figure 4. The coefficientsof the frequency domain filter are Q†

N (I)qN (i − i0) and thecoefficients of the time domain filter are Q†

M(K)qM(k − k0).

4.2. Recursive algorithms

The two-dimensional filter can actually be implemented re-cursively in time and frequency domains, respectively.

Define the basis functions Q′N (I) as

Q′N (I)=

[qN (−I + 1) · · · qN (0) · · · qN (I) qN (I + 1)

]T.

(28)The basis Q′

N (I) and QN (I) are actually homomorphic toeach other, that is, there is an invertible matrix R such that

Q′N (I) = QN (I)R. (29)

This means that instead of using QN (I) as basis, we can useQ′

N (I) as the basis to construct the estimator, that is,

Hi,k

Hi,k−1

Hi,k−2K Frequencydomain filter

...

Frequencydomain filter

Frequencydomain filter

P/STime

domain filter

Reset every2K + 1 samples

Hi,k

Figure 4: The estimator structure.

qTN

(i + 1− i0

)Q′†

N (I) = qTN

(i− i0

)Q†

N (I). (30)

Substitute (30) into (26), we can estimate the channel usingQ′†

n(I). Then the core of the recursive algorithm is to calcu-late Q′†

N (I) from Q†N (I) iteratively.

Let P f = (QTN (I)QN (I))−1 and Pt = (QT

M(K)QM(K))−1.

At initialization, we estimate model coefficients b f (k) or

bt(i) regarding to P f or Pt over a window of dimension 2I+1or 2K + 1. Similar to the recursive least square (RLS) algo-rithm, using the matrix inverse lemma [16], we can calculateP+f = (Q′T

N (I + 1)Q′N (I + 1))−1 or P+

t = (Q′TM(K + 1)Q′

M(K +

1))−1 and then the corresponding model coefficients b+f or

b+t recursively over the window of dimension 2I + 2 or 2K + 2

during the updating process. After that apply the matrix in-verse lemma again, we can calculate P−f = (Q′T

N (I)Q′N (I))−1

or P−t = (Q′TM(K)Q′

M(K))−1 and the corresponding modelcoefficients b−f or b−t from P+

f or P+t over the window of di-

mension 2I + 1 or 2K + 1 during the downdating process.Then according to (30), the channel can be estimated as

Hi+1,k = qTN

(i + 1− i0

)b−f ,

Hi,k+1 = qTM

(k + 1− k0

)b−t .

(31)

As this recursive process going on, the basis function be-comes qT

M(k+ l−k0) where l is the index of the iteration. Thedynamic range of such a basis function may become so largethat it will affect the numerical stability of the algorithm.Therefore, regularization using R f or Rt should be used pe-riodically to scale the basis back to QN (I) or QM(K). Thefrequency domain and time domain recursive algorithms aresummarized in Algorithms 1 and 2, respectively. The matri-ces K+

f and K−f or K+

t and K−t are the corresponding gain ma-

trices in updating and downdating. The two-dimensional fil-ter in the tables is implemented first by frequency domainfiltering then by time domain filtering. The order can beswitched. In that way, the input in Algorithm 2 are instanta-neous estimates while the inputs in Algorithm 1 are the out-puts of the time domain filters of Algorithm 2. It is also notedthat the order of downdating and updating can be switched,too.

In both tables, K+f , K−

f , P+f , P−f and K+

t , K−t , P+

t , P−t can be


Initialization:with temporary estimation Hk =

[H−I+i0 ,k · · · HI+i0 ,k

],

calculateb f (k) = Q†

N (I)Hk,

P f =(

QTN (I)QN (I)

)−1.

Updating:with the new input HI+i0+1,k , calculate

K+f = I− P f qN (I + 1)qT

N (I + 1)

1 + qTN (I + 1)PqN (I + 1)

,

P+f = K+

f P f .

∆b+f = qN (I + 1)HI+i0+1,k ,

b+f = K+

f

(b f (k) + ∆b+

f

).

Downdating:

K−f = I +

P+f qN (−I)qT

N (−I)1− qT

N (−I)P+f qN (−I) ,

P−f = K−f P+

f .

∆b−f = qN (−I)H−I+i0 ,k ,

b−f = K−f

(b+f − ∆b−f

).

Regularization:b f (k) = R f b−f ,

Hi+1,k = qTN

(i− i0

)b f (k).

Algorithm 1: Frequency domain recursive algorithm.

calculated off-line and do not change if the model order andwindow dimensions do not change. However, we still put thecalculations inside the updating and downdating process incase that the window dimensions may change as what hap-pened in the adaptive algorithm described in Section 6.

The recursive algorithm needs less calculation comparedto direct computation of the product of pseudoinverse whenthe window dimensions are much larger than the model or-der. Many fast algorithms of recursive least square (RLS) canbe used for the practical implementation of such a recursivealgorithm [16]. It also provides an easy way to adjust the win-dow dimensions for the implementation of the adaptive algo-rithm.

5. PERFORMANCE ANALYSIS

Suppose that the channel can be modeled by some basis func-tion, that is, a set of channel responses H can be projected toa set of basis function Q and the coefficients of the basis func-tions are b, that is,

H = Qb. (32)

The length of H is L and the length of b is l. In order to getan accurate channel estimation, we expect that l L. Thisis true if the channel parameters in H is highly correlated.

Initialization:with frequency domain filter resultsH f =

[H(i + 1,−K + k0) · · · H(i + 1, K + k0)

], calculate

bt(i + 1) = Q†M(K)H f ,

Pt =(

QTM(K)QM(K)

)−1.

Updating:with the new input H(i + 1, K + k0 + 1), calculate

K+t = I− PtqM(K + 1)qT

M(K + 1)1 + qT

M(K + 1)PtqM(K + 1),

P+t = K+

t Pt .

∆b+t = qM(K + 1)H

(i + 1, K + k0 + 1

),

b+t = K+

t

(bt(i + 1) + ∆b+

t

).

Downdating:

K−t = I +

P+t qM(−K)qT

M(−K)1− qT

M(−K)P+t qM(−K)

,

P−t = K−t P+

t .

∆b−t = qM(−K)H(i + 1,−K + k0

).

b−t = K−t

(b+t − ∆b−t

).

Regularization:

bt = Rtb−t ,

Hi+1,k+1 = qTM

(k − k0

)bt(i + 1).

Algorithm 2: Time domain recursive algorithm.

Given a set of noisy observations,

H = H + N, (33)

the LS estimation of the coefficients is

b = Q†H. (34)

The channel estimation is then

H = QQ†H. (35)

Define the mean-squared estimation error matrix as

ε = E[(

H−H)(

H−H)H]

. (36)

We can show that

ε = (I−QQ†)RH

(I−QQ†) + QQ†RNQQ†, (37)

where RH = E[HHH] and RN = E[NNH] = σ2I if the trans-mitted signals of all subchannels are all using the same con-stant envelop modulation and transmit energy of 1.

The estimation error consists of two parts; one is relatedto the model inaccuracy, that is,

εH =(

I−QQ†)RH(

I−QQ†), (38)


and the other is related to the residual noise, that is,

εN = σ2QQ†. (39)

Since RH is a Toeplitz matrix, it can be decomposed as

RH =[

U1 U2

]Λ 0

0 0

UH1

UH2

, (40)

where Λ is a diagonal matrix with eigenvalues of RH on its di-agonal. If Q = U1, then the model error is zero. This meansthat the optimal function basis, which we can find in termsof model accuracy, is the eigenbasis U1. However, it requiresthe knowledge about the statistics of the channel responses.In some special cases, we can easily find some specific func-tion bases that can diagonalize RH without actually knowingRH . For example, if H is the channel response for one OFDMblock with all the delay paths, τi’s, at the sampling instance ofthe OFDM system, then such an optimal function basis is theDFT matrix [6]. However, in most of practical situations, thechannel delay profiles do not satisfy this condition. There-fore, using DFT matrix may cause severe leakage problemand incur a large model error.

The average energy of the residual noise over the entireestimation window can be calculated as follows:

eN = σ2

Ltr[

QQ†] = lσ2

L. (41)

The average mean-squared error over the whole estimationwindow is actually lower bounded by (41). The lower boundis achieved when Q = U1.

Although the average energy of the residual noise main-tains the same once the data length and model order is fixed,the estimation error inside the window is often distributedunevenly and differently for different basis functions. For thepolynomial model, the estimation error is the least at the cen-ter point of the window and larger at the edge. Therefore,we prefer to choose the center of the window to get a betterperformance. However, along the time domain, we can onlychoose the end point to get a causal filter.

6. OPTIMAL MODEL PARAMETERS ADAPTATION

With estimation point chosen at the center of the frequencydomain window and end point at the time domain window,the estimation error from (23) becomes

εI,K = E[∥∥Hi0 ,k0 − Hi0 ,k0

∥∥2] = εh + εn, (42)

where the model error is

εh = E[∥∥Hi0 ,k0 − qM,N (0, K)TQ†

M,N (I, K)Hi0,k0

∥∥2]= rH(0, 0)− E

[Hi0 ,k0 HT

i0 ,k0

]Q†T

M,N (I, K)qM,N (0, K)

− qM,N (0, K)TQ†M,N (I, K) E

[Hi0,k0H

∗i0 ,k0

]+ qM,N (0, K)TQ†

M,N (I, K) E[

Hi0 ,k0 HTi0 ,k0

]×Q†T

M,N (I, K)qM,N (0, K),

(43)

(1) Initialization: use I0 × K0 calculate estimation andε0 = εI0 ,K0 .

(2) Use window dimensions I × K to estimate the kthblock and compute the estimated estimation errorεI,K , εI+1,K and εI,K+1.

(3) If εI,K < ε0, then I0 = I , K0 = K , ε0 = εI,K , and

(a) if |εI,K − εI+1,K | < ε fth, then I remains un-

changed. Otherwise, if εI,K > εI+1,K , then I =I + 1, if εI,K < εI+1,K , then I = I − 1.

(b) If |εI,K − εI,K+1| < εtth, then K remains un-changed. Otherwise, if εI,K > εI,K+1, then K =K + 1, if εI,K < εI,K+1, then K = K − 1.

Otherwise, I = I0 and K = K0.(4) Go to step 2 for block k + 1.

Algorithm 3: Window dimension adaptive algorithm.

and the residual noise is

εn = σ2qM,N (0, K)TQ†M,N (I, K)Q†T

M,N (I, K)qM,N (0, K). (44)

The residual noise is reduced more when the model or-der M × N becomes small or the window dimension I × Kbecomes large. However, the model error will increase in thiscase. With fixed polynomial model order M and N , the opti-mal window dimension is obtained by

minI,K

εI,K = εh + εn. (45)

Usually, there are several local minima in this optimiza-tion problem. Considering the computational complexity, wewould prefer the one with small I × K .

In order to adaptively adjust the window dimensions weneed to know the estimation error. Since the actual chan-nel responses are not known, we have to estimate the esti-mation error using the instantaneous estimates and the finalestimates. Suppose that the noise statistics is known, we cancalculate the estimated estimation error as

εI,K =∑i

∑k

∥∥Hi0,k0 − Hi0 ,k

∥∥2 − σ2

+ E[Ni0 ,k0 NH

i0,k0

]Q†

M,N (I, K)qM,N (0, K)

+ qM,N (0, K)TQ†M,N (I, K) E

[Ni0,k0N

∗i0 ,k0

].

(46)

Using this approximation, the window dimension adap-tive algorithm for the optimization of (45) is given inAlgorithm 3.

If the recursive algorithm in Algorithms 1 and 2 isadopted, the window adaptation can be implemented easily.We just eliminate one downdating when increasing the win-dow dimension, or eliminate one updating when decreasingwindow dimension.

One important problem in the adaptive algorithm is todetermine the threshold ε f

th and εtth. With large threshold,the algorithm converges faster, but with larger deviation.Especially when the local minima are located closely, the


0 5 10 15 20 25 30

SNR (dB)

−40

−35

−30

−25

−20

−15

−10

−5

0

MSE

(dB

)

Polynomial approximation in time-frequencyPolynomial approximation in time onlyPolynomial approximation in frequency only

(a)

10 12 14 16 18 20 22 24 26 28 30

SNR (dB)

10−3

10−2

10−1

SER


(b)

0 5 10 15 20 25 30

SNR (dB)

−40

−35

−30

−25

−20

−15

−10

−5

MSE

(dB

)


(c)

10 12 14 16 18 20 22 24 26 28 30

SNR (dB)

10−3

10−2

10−1

SER


(d)

Figure 5: Estimation error and symbol error rate versus SNR (2-ray, M ×N = 3× 3) (a) MSE, (b) SER ( fD = 40 Hz, Td = 5 microseconds,I × K = 7× 10), (c) MSE, and (d) SER ( fD = 20 Hz, Td = 10 microseconds, I × K = 4× 30).

large threshold may result in unstable convergence. Hence,it would be preferred to use smaller thresholds here.


The OFDM system used in the simulations is the system in-troduced in Section 2. QPSK modulation is used throughoutall subchannels. Figure 5 shows the mean-squared estimationerror and the symbol error rate (SER) comparison of the al-

gorithm based on the approximations in both time and fre-quency domains with those based on approximation eitherin time or frequency domain. Figures 5a and 5b show thecase of a 2-ray channel with delay spread of 5 microsecondsand Doppler frequency of 40 Hz, while Figures 5c and 5dshow the case of another 2-ray channel with delay spread of10 microseconds and Doppler frequency of 20 Hz. In bothcases, fDTd remains the same. We can see that the perfor-mance of using both time and frequency domain expansions


0 5 10 15 20 25 30

SNR (dB)

−40

−35

−30

−25

−20

−15

−10

MSE

(dB

)

Polynomial model (TU)Polynomial model (2-ray)Fourier transform (TU)Fourier transform (2-ray)

(a)

0 5 10 15 20 25 30

SNR (dB)

−40

−35

−30

−25

−20

−15

−10

−5

MSE

(dB

)

Polynomial model (HT)Polynomial model (2-ray)Fourier transform (HT)Fourier transform (2-ray)

(b)

Figure 6: Estimation error versus SNR (M ×N = 3× 3, fD = 40 Hz), (a) I × K = 5× 15 and (b) I × K = 2× 15.

is better than that of using only frequency domain expansionor using only time domain expansion in both cases. However,in the first case, the delay spread is smaller while the Doppleris larger, then the channel responses have more correlationin the frequency domain than in the time domain. There-fore, we use larger frequency domain window to exploit thefrequency domain correlation. In the second case, the delayspread is larger while the Doppler frequency is smaller, thenthe channel responses have more correlation in the time do-main and we use a larger time domain window to exploit it.It is shown that we have to use different time and frequencyestimator to best exploit the channel correlations for differ-ent channels. Using only time or frequency domain schemeis not enough.

Figure 6 shows the estimation error under different de-lay profiles with Doppler frequency of 40 Hz. Figure 6ashows the estimation error with TU delay profile and 2-raydelay profile of Td = 5 microseconds, which is the max-imal delay spread of TU while Figure 6b shows the esti-mation error with HT delay profile and 2-ray delay pro-file of Td = 17.2 microseconds which is the maximal de-lay spread of HT. We also compared the results using theFourier-transform-based method of [6]. We can see thatfor TU or HT, the proposed algorithm performs muchbetter than the Fourier-transform-based method. However,for 2-ray channel with Td = 5 microseconds, the Fourier-transform-based method performs the best. The reason isthat Td = 5 microseconds is an integer multiplication ofthe sampling period of the OFDM system, which is ts =1/800 KHz = 1.25 microseconds. The impulse response ofthis 2-ray channel has energy only at the sampling instance

1 2 3 4 5 6 7 8 9 10

Td (µs)

−40

−35

−30

−25

−20

−15

−10

MSE

(dB

)

Polynomial modelFourier transform

Figure 7: Estimation error versus delay spread (SNR = 20 dB, fD =40 Hz, 2-ray, M ×N = 3× 3, I × K = 5× 15).

of the OFDM system, hence there is no leakage or model er-ror using Fourier transform, which is used as frequency do-main estimator in [6]. In this case, the Fourier-transform-based method actually provides a minimum mean-squarederror estimator. Unfortunately, in the practice, such a caseis quite unlikely especially for the time-varying channel. Itis show that for TU or HT delay profiles and 2-ray withTd = 17.2 microseconds, there is great amount of leakage


0 25 50 75 100 125 150

Iteration k

5

8

11

14

17

20

Win

dow

dim

ensi

on

Time domain window K

Frequency domain window I

AdaptiveOptimal

(a)

0 25 50 75 100 125 150

Iteration k

10

12

14

16

18

20

Win

dow

dim

ensi

on

Time domain window K

Frequency domain window I

AdaptiveOptimal

(b)

0 25 50 75 100 125 150

Iteration k

−27

−26

−25

−24

−23

−22

−21

−20

−19

MSE

(dB

)

AdaptiveOptimal

(c)

0 25 50 75 100 125 150

Iteration k

−27

−26

−25

−24

−23

−22

−21

−20

−19M

SE(d

B)

AdaptiveOptimal

(d)

Figure 8: Window dimensions adaptation (TU, SNR = 10 dB, fD = 40 Hz) (a) Window dimensions (starting from 5 × 5), (b) Windowdimensions (starting from 20× 20), (c) Estimation error (starting from 5× 5), and (d) Estimation error (starting from 20× 20).

using Fourier-transform-based on the sampling frequencyof the OFDM system. The leakage greatly degrades the per-formance of the Fourier-transform-based method. In con-trast, the polynomial-model-based method performs consis-tently for the channels with same maximal delay spread andhence is more robust to the channel statistics. This is becausethe model errors are bounded by the same bound for thechannels with the same Td and fD as stated in Section 3.2.Therefore, the performance of the polynomial-model-basedchannel estimation is not sensitive to the specific correlation

functions of the channels with the same Doppler frequencyand maximum delay spread.

Figure 7 shows the mean-squared estimation error atSNR of 20 dB with different delay spread of a 2-ray channel.It further demonstrates the robustness of the polynomial-model-based method compared to the Fourier-transform-based method. The Fourier-transform-based method per-forms better only when the delay spread is at the samplinginstance of the system. For most of the cases, it performspoorly. However, for the polynomial-model-based method,


it performs consistently and outperforms the Fourier-transform-based method most of the time.

Figure 8 shows the window dimension adaptation. Thewindow dimension variation is shown in Figures 8a and 8b.The estimation error is shown in Figures 8c and 8d. Twocases with different initial conditions are simulated, whichare shown in Figures 8a and 8c and Figures 8b and 8d, re-spectively. In Figures 8a and 8c, the window dimension is5 × 5 at the beginning, while in Figures 8b and 8d, it is20 × 20. In both cases, after about 100 iterations, the algo-rithm converges to a window dimension of 12 × 10 and anestimation error under −26 dB. However, as mentioned inSection 6, smaller window dimensions are preferred for thesake of the computation complexity. With this adaptationalgorithm, the polynomial-model-based method is not onlyrobust to the specific correlation of the channel variation anddispersion, but also robust to Td and fD and can follow thevariation of the statistics of the channel. Moreover, in theprevious simulation, fixed window dimensions are used, byapplying this window dimension adaptation algorithm, theperformance in Figure 6 can be further improved.

8. CONCLUSIONS

In this work, we proposed a channel estimation algorithm forthe OFDM system with fading multipath channels, which issuitable for the applications in 3G wireless communications.The algorithm is based on the time-frequency polynomialmodel that exploits the correlation of the channel responsesin both time and frequency domains. The channel responseis approximated by a small number of time-frequency poly-nomial basis functions and estimated by first estimating thecoefficients of the bases. The residual noise is significantly re-duced in this way, compared to the results when approxima-tion is only done either in time or frequency domain, and theestimator design is more flexible. Therefore, the approach ismore robust to the channel statistics and system parametersthan the existing Fourier-transform-based method. It doesnot require the delay profiles to be integer multiples of thesystem sampling period. Moreover, the algorithm can be im-plemented recursively and can adjust the model parametersadaptively to the delay and fading characteristics.

REFERENCES

[1] J. A. C. Bingham, “Multicarrier modulation for data trans-mission: An idea whose time has come,” IEEE Communica-tions Magazine, vol. 28, no. 5, pp. 5–14, 1990.

[2] L. J. Cimini Jr., “Analysis and simulation of a digital mo-bile channel using orthogonal frequency division multiplex-ing,” IEEE Trans. Communications, vol. 33, no. 7, pp. 665–675,1985.

[3] P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical dis-crete multitone transceiver loading algorithm for data trans-mission over spectrally shaped channels,” IEEE Trans. Com-munications, vol. 43, no. 2, pp. 773–775, 1995.

[4] J.-J. van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O.Baorjesson, “OFDM channel estimation by singular value de-

composition,” IEEE Trans. Communications, vol. 46, no. 7, pp.931–939, 1998.

[5] V. Mignone and A. Morello, “CD3-OFDM: a novel de-modulation scheme for fixed and mobile receivers,” IEEETrans. Communications, vol. 44, no. 9, pp. 1141–1151, 1996.

[6] Y. (G.) Li, L. J. Cimini Jr., and N. R. Sollenberger, “Robustchannel estimation for OFDM systems with rapid dispersivefading channels,” IEEE Trans. Communications, vol. 46, no. 7,pp. 902–915, 1998.

[7] E. W. Cheney, Introduction to Approximation Theory,McGraw-Hill, New York, NY, USA, 1966.

[8] H. N. Mhaskar, Introduction to the Theory of Weighted Polyno-mial Approximation, World Scientific Publishing, Singapore,1996.

[9] D. K. Borah and B. D. Hart, “A robust receiver structure fortime-varying, frequency-flat Rayleigh fading channels,” IEEETrans. Communications, vol. 47, no. 3, pp. 360–364, 1999.

[10] D. K. Borah and B. D. Hart, “Frequency-selective fadingchannel estimation with a polynomial time-varying channelmodel,” IEEE Trans. Communications, vol. 47, no. 6, pp. 862–873, 1999.

[11] M. Luise, R. Reggiannini, and G. M. Vietta, “Blind equal-ization/detection for OFDM signals over frequency-selectivechannels,” IEEE Journal on Selected Areas in Communications,vol. 16, no. 8, pp. 1568–1578, 1998.

[12] W. C. Jakes, Microwave Mobile Communications, Wiley, NewYork, NY, USA, 1974.

[13] Y. (G.) Li, N. Seshadri, and S. Ariyavisitakul, “Channel esti-mation for OFDM systems with transmitter diversity in mo-bile wireless channels,” IEEE Journal on Selected Areas in Com-munications, vol. 17, no. 3, pp. 461–471, 1999.

[14] P. A. Bello, “Characterization of randomly time-variant linearchannels,” IEEE Trans. Communications Systems, vol. 11, no.4, pp. 360–393, 1963.

[15] X. Wang and K. J. R. Liu, “OFDM channel estimation basedon time-frequency polynomial model of fading multipathchannel,” in VTC, Fall 2001.

[16] S. Haykin, Adaptive Filter Theory, Prentice Hall, EnglewoodCliffs, NJ, USA, 1996.

Xiaowen Wang received her B.S. degreefrom the Department of Electronics En-gineering, Tsinghua University, Beijing,China in 1993, and the M.S. and Ph.D.degrees from the Department of Electri-cal and Computer Engineering, Universityof Maryland, College Park, Md, in 1999and 2000, respectively. From 1993 to 1996,Dr. Wang was a Teaching Assistant withTsinghua University, Beijing, China. From1996 to 2000, she was a Research Assistant with University of Mary-land, College Park, Md. Since 2000, she has been with the WirelessSystems Research Department, Agere Systems (formerly Bell Labs,Lucent Technologies, Microelectronics). Her research interests in-clude adaptive digital signal processing, wireless communicationsand networking. Dr. Wang was ranked first among the class of de-partment of Electronics Engineering for her B.S. degree from Ts-inghua University in 1993, and was the recipient of the GraduateSchool Fellowship from University of Maryland.


K. J. Ray Liu received his B.S. degree fromthe National Taiwan University, and thePh.D. degree from UCLA, both in electri-cal engineering. He is Professor of Electri-cal and Computer Engineering Departmentof University of Maryland, College Park.His research interests span broad aspects ofsignal processing architectures; multimediasignal processing, wireless communicationsand networking, information security, andbioinformatics in which he has published over 230 refereed papers,of which over 70 are in archival journals. Dr. Liu is the recipient ofnumerous awards including the 1994 National Science FoundationYoung Investigator, the IEEE Signal Processing Society’s 1993 Se-nior Award, IEEE 50th Vehicular Technology Conference Best Pa-per Award, Amsterdam, 1999. He also received the George Corco-ran Award in 1994 for outstanding contributions to electrical engi-neering education and the Outstanding Systems Engineering Fac-ulty Award in 1996 in recognition of outstanding contributions ininterdisciplinary research, both from the University of Maryland.Dr. Liu is the Editor-in-Chief of EURASIP Journal on Applied Sig-nal Processing, and has been an Associate Editor of IEEE Trans-actions on Signal Processing, a Guest Editor of special issues onMultimedia Signal Processing of Proceedings of the IEEE, a GuestEditor of special issue on Signal Processing for Wireless Commu-nications of IEEE Journal of Selected Areas in Communications, aGuest Editor of special issue on Multimedia Communications overNetworks of IEEE Signal Processing Magazine, a Guest Editor ofspecial issue on Multimedia over IP of IEEE Trans. on Multimedia,and an editor of Journal of VLSI Signal Processing Systems.


On Bandwidth Efficient Modulation for High-Data-RateWireless LAN Systems

John D. TerryNokia Research Center, 6000 Connection Drive, Irving, TX 75039, USAEmail: [email protected]

Juha HeiskalaNokia Research Center, 6000 Connection Drive, Irving, TX 75039, USAEmail: [email protected]

Victor StolpmanSouthern Methodist University, 3145 Dyer Street, Dallas, TX 75275-0338, USAEmail: [email protected]

Majid FozunbalGeorgia Institute of Technology, Atlanta, GA 30332, USAEmail: [email protected]

Received 3 October 2001 and in revised form 25 March 2002

We address the problem of high-data-rate orthogonal frequency division multiplexed (OFDM) systems under restrictive band-width constraints. Based on recent theoretic results, multiple-input multiple-output (MIMO) configurations are best suited forthis problem. In this paper, we examine several MIMO configurations suitable for high rate transmission. In all scenarios con-sidered, perfect channel state information (CSI) is assumed at the receiver. In constrast, availability of CSI at the transmitter isaddressed separately. We show that powerful space-time codes can be developed by combining some simple well-known tech-niques. In fact, we show that for certain configurations, these space-time MIMO configurations are near optimum in terms ofoutage capacity as compared to previously published codes. Performance evaluation of these techniques is demonstrated withinthe IEEE 802.11a framework via Monte Carlo simulations.

Keywords and phrases: OFDM, WLAN, MIMO, antenna diversity, space-time block codes, TCM.

1. INTRODUCTION

Currently, the IEEE 802.11a standard offers data rates rang-ing from 6 Mbit/s to 54 Mbit/s. However, there is a grow-ing interest for a 100 Mbit/s mode of operation for the IEEEstandard. Unfortunately, to achieve such a rate within theIEEE 802.11a framework requires the system to operate ata spectral efficiency better than 6 bit/s/Hz. The spectral effi-ciency problem is further complicated by the fact that 20%of the available bandwidth is used for the cyclic prefix to mit-igate the effects of frequency selective fading. From a sys-tem design perspective, the complexity associated with thismode of operation should not be much greater than that forthe 54 Mbit/s mode of the standard. These stringent require-ments constitute a very interesting research problem.

Recent information theoretic results [1] suggest thatthere is a tremendous capacity potential for wireless com-munication systems using antenna diversity. Foschini and

Gans [1] and others [2, 3] noted that orthogonal frequencydivision multiplexed (OFDM) systems are particularly wellsuited for antenna diversity techniques. Hence, it is expectedthat multiple element antenna arrays will play an increasinglyimportant role in emerging wireless LAN networks. Indeed,when used in conjunction with appropriately designed signalprocessing algorithms, these arrays can dramatically enhanceperformance.

In systems where channel state information (CSI) is notknown to the transmitter, space-time coding (STC) is a band-width and power efficient solution for communication overwireless Rayleigh or Rician fading channels. STC guaranteestransmit diversity and, optionally, receive diversity. Further-more, the code construction is done such that the diversityadvantage is achieved without any sacrifice in the transmis-sion rate. In [4], Tarokh constructed space-time trellis codes(STTC) using design criteria derived for the Rayleigh fadingchannels, where ideal channel state information is available


at the receiver. It was shown that for a quasi-static Rayleighor Rician channels, performance is determined by the diver-sity advantage quantified by the rank of certain matrices andby the coding advantage, that is, quantified by the determi-nants of these matrices.

In contrast, when the temporal and spatial channel gainis available in transmitter, there is no need to use space-time codes. Because of the availability of channel gains inboth transmitter and receiver, spatial processing can be per-formed at both transmitter and receiver to change the statis-tic of multiple-input multiple-output (MIMO) channel atthe receiver into a parallel bank of single-input single-output(SISO) channels [2]. When the channel has severe nulls in itsfrequency response or there is a powerful narrowband noise,the signal frequency components are completely canceled outin those frequencies. It is obvious that transmitting energyin these frequencies is a waste of power. Therefore, to ap-proach the capacity of these channels, a kind of spectral shap-ing should be applied to the transmitted signal. Correspond-ingly, new coding schemes, which are adapted to the fre-quency response of the channel, are required. In other words,the power and rate should be optimally distributed over fre-quency components of the transmitted signal by an algo-rithm called water-filling. It has long been known that multi-carrier modulation [5] could in principle be used to achievethe power and rate allocations prescribed by water-filling.However, practical realizations of multicarrier modulation incombination with powerful codes have been achieved onlyin recent years. Goldsmith in [6, 7] was one of the first re-searchers to develop powerful codes for the wireless channel.

The remainder of this paper is organized as follows.Section 2 describes the basic channel model used for MIMOsystems, which is the basis for later development in the paper.Section 3 is dedicated to the design criteria for STC. It brieflysummarizes the popular work by Tarokh in [4], but also in-cludes the work of Ionescu [8], who developed a new crite-rion for the coding gain for STC. Section 4 leverages heav-ily on the work presented in Section 3 to derive simple con-struction techniques for near-optimum STC. Section 5 re-views the previous work in adaptive modulation and intro-duces novel extensions to this work. Finally, we summarizethe work presented and salient points for future research.

2. CHANNEL MODEL FOR MIMO SYSTEMS

We provide sufficient details to generally characterize thechannel for any multiple-input multiple-output (MIMO)systems employing an OFDM modulation. The final form forthe channel model is denoted in matrix/vector for notationalconvenience. Now, we consider a communication link com-prising N transmitter antennas and M receiver antennas thatoperates in an OFDM MIMO channel. Each receiver antennaresponds to each transmitter antenna through a statisticallyindependent fading path. The received signals are corruptedby additive noise, that is, statistically independent among theM receiver antennas and the transmission periods. For easeof presentation, discrete baseband notation is used, that is,at sample time index l, the complex symbols, si(l), sent by

the N transmit antennas and, subsequently, detected by thekth receive antenna, is denoted by yk(l). Then yk(l) can beexpressed as

yk(l) =√

P

N

N∑i=1

hki(l)si(l) + vk(l), 1 ≤ k ≤M, (1)

where hik(l) is the complex scalar associated with path be-tween the ith transmitter antenna and the kth receive an-tenna at time index l. The noise samples at time index i, vk(l),are complex zero-mean spatially and temporally white Gaus-sian random variable with variance N0/2 per dimension. Itis further assumed that the transmitted signals si(l) are nor-malized such that the average energy for the signal constel-lation is unity. Recall that, for OFDM signal, the channel ismade circulant by prepending a cyclic prefix (CP) to the datasequence prior to transmission. This simple mechanism al-lows us to replace the linear convolution in (1) with a circu-lar one. Hence, the frequency selective channel given in (1)is transformed into a parallel bank of flat fading channels viaan L-point FFT, that is,

Yk(m) =√

P

N

N∑i=1

Hki(m)Si(m) + Vk(m), for 1 ≤ m ≤ L,

(2)

where Yk(m), Hki(m), Si(m), and Vk(m) denote the fre-quency domain representations of the mth subcarrier of thereceived signal for the kth antenna, complex channel gainsbetween the ith transmitter antenna and the kth receive an-tenna, transmitted signal for the ith antenna, and noise sig-nal for the kth receive antenna, respectively. If the channelgains are slowly fading, then it can be assumed that, during aperiod of T time indexes, the gains are constant and we canapproximate the channel as a block fading channel. Thus, thereceived signal vector for themth subcarrier using matrix no-tation is given by

Y(m) =√

P

NH(m)S(m) + V(m), (3)

where

Y(m) = [Y1(m), Y2(m), . . . , YM(m)

]T,

H(m) =

H1(m)H2(m)

...HM(m)

,

S(m) = [S1(m), S2(m), . . . , SN (m)

]T.

(4)

For ease of presentation, the MIMO analysis herein willbe developed for the mth subcarrier, knowing that analysisapplies equally to the remaining subcarriers. Hence, the in-dex for the subcarriers will be dropped here for notationalsimplicity and re-introduced when channel coding acrosssubcarriers are discussed later in this paper.

On Bandwidth Efficient Modulation for High-Data-Rate Wireless LAN Systems 833

3. REVIEW OF STC DESIGN CRITERIA

We briefly review STC design criteria for quasi-static chan-nels. In general, a wireless communication system is com-prised of N transmit antennas and M receive antennas. Theinformation data is encoded using a channel code to protectit against imperfections caused by the channel. The encodeddata stream is split into N parallel streams each of whichis modulated and then transmitted using separate antennas.Each path for the separate antennas is assumed i.i.d. andquasi-static, that is, the complex gains of the paths are con-stant over each data frame but change from frame to frame.

To develop codes that perform well over fading channels,we choose to minimize the pairwise error probability. Thatis, the probability that the code word c is transmitted overthe channel and a maximum-likelihood (ML) receiver de-cides in favor of a different code word e. The ML receivermakes decoding decisions based on a performance metricm(Y, c,H) provided that estimates of the fading amplitudes,H, are available at the receiver. Formally, the maximum like-lihood criterion for the optimum decoder requires the con-ditional probability of receiving Y given that the code word cwas transmitted to be greater than the probability of receiv-ing Y assuming any other code word e was transmitted, thatis,

Pr(c −→ e | H) = Pr[m(Y, e; H) ≥ m(Y, c; H) | H

], (5)

where

m(Yk, ek;Hk

) = −∣∣Yk −Hkek∣∣2

; (6)

Yk is again the received signal for the kth antenna; Hk andek are the complex path gains and transmitted symbols, re-spectively, from the N transmit antennas to the kth receivedantenna, and | · |2 represents the squared Euclidean norm.The pairwise error probability is found by taking the sta-tistical expectation of (5). Rather than solving for the ex-act pairwise error probability, which can only be evaluatednumerically, an upper bound can be found for (5) usingthe Chernoff bound techniques. Evaluation of the Chernoffbound for Rician and Rayleigh channels leads to the follow-ing two design criteria.

First, we define the quantity referred to the code word dif-ference matrix D(c, e) defined as

D(c, e) =

c1(0)− e1(0) · · · c1(l)− e1(l)

.... . .

...

cN (0)− eN (0) · · · cN (l)− eN (l)

T

, (7)

where l represents the length of an error event path. In (7),the columns of D(c, e) index the transmit antennas and therows index the symbol epochs1 of the codes c and e. The rank

1For this case, the symbol epochs corresponds to subcarriers of theOFDM symbol.

of D(c, e) determines the diversity advantage of the code.That is, in order to achieve a diversity of pM in a rapid fad-ing environment for any two code words c and e the stringsc1(l)c2(l) · · · cN (l) and e1(l)e2(l) · · · eN (l) must be differentat least for p values of 1 ≤ l ≤ N . The coding advantage d2

P

is determined from the geometric mean of the eigenvalues λiof the matrix of D(e, c)†D(e, c), that is,

d2P(r) =

( r∏i=1

λi

)1/r

. (8)

Typically, the minimum coding advantage d2P(r) amongst all

code word pairs is the dominate factor in the performance ofSTC. Further, the minimum Euclidean distance dmin betweenany two code words c and e determines the minimum codingadvantage. Hence, codes with larger dmin have better codingadvantages.

3.1. Improvement of design criterion

Ionescu [8] demonstrated that the determinant crite-rion can be strengthened by requiring the eigenvalues ofD(e, c)†D(e, c) to be as close as possible, for any pair of codewords c, e. Formally, the criterion was expressed as follows.

Theorem 1 (new determinant criterion). The equal eigen-value criterion: for N-transmit antenna system operating ini.i.d. Rayleigh fading with perfect channel state information(CSI), an upper bound to the pairwise error probability ismade as small as possible if and only if, for all pair of codewords c, e, the squared Euclidean distance tr[D(e, c)†D(e, c)]is made as large as possible. Further, the nonsquare ma-trices D(e, c) are semiunitary—up to a scale factor—, thatis, D(e, c)†D(e, c) = (tr[D(e, c)†D(e, c)]/N)∗IN . Essentially,maximizing minc,e det[D(e, c)†D(e, c)], as specified by the de-terminant criterion, requires maximizing the minimum eigen-value product over all D(e, c)†D(e, c).

Proof. By Hadamard’s theorem, the eigenvalue product fora square, positive definite matrix A, with elements ai, j as-sumes its maximum value Πiai,i, if and only if A is diagonal.Once D(e, c)†D(e, c) is diagonalized, the product of its diag-onal elements is maximized if and only if they are renderedequal and their sum, tr[D(e, c)†D(e, c)], is maximized. Con-sider the arithmetric-mean geometric mean inequality, thatis,

n

√√√√√ N∏i=1

λi ≤(

1N

N∑i=1

λi

). (9)

Hence, the product distance d2P(r) given in (8) is up-

per bounded by the arithmetic mean of eigenvalues ofD(e, c)†D(e, c) with equality achieved when all eigenvaluesare equal.

Ionescu recognized that it might be difficult to enforcethis condition for all pairs of code words c, e thus proposed


a suboptimal solution to enforce the condition of the codewords corresponding to the shortest error event paths in thecode trellis. In the next section, we demonstrate a simplemethod for constructing near optimal space-time trellis code(STTC) using the equal eigenvalue criterion.

4. SIMPLE STTC CONSTRUCTION FOR EQUALEIGENVALUE DESIGNS

Here, we postulate that concatenation of an orthogonalspace-time block code (STBC) with bandwidth-efficientcodes designed for additive white Gaussian noise (AWGN)channels yield near optimum codes in terms of coding gain.It is well known that the STBC [9, 10] are spectrally less ef-ficient than STTC. To compensate for the poor spectral effi-ciency of STBC, the input symbols s1, s2, . . . , sk of the STBCcan be generated from a spectrally efficient modulation suchas trellis-coded modulation (TCM) or block-coded modu-lation (BCM). Note that the use of an orthogonal STBCguarantees satisfaction of the equal eigenvalue criterion byIonescu [8]. Furthermore, the symbols s1, s2, . . . , sk shouldbe chosen to maximize the Euclidean distance for all pairof code words c, e. Since STTC assumes flat fading chan-nels in its development, consider space-time block codingof an OFDM system using the Radon-Hurwitz (R-H) uni-tary transform, popularized by Alamouti [11], defined overOFDM symbols as

R-H

Xo

Xe

= Xo Xe

−X∗e X∗

o

, (10)

where the rows in (10) index transmit antennas and thecolumns index symbol epochs. If two consecutive OFDMsymbols are referred to as Xo and Xe, then at the first an-tenna, Xe is transmitted in the first time epoch followed byXo in the second time epoch while, at the second transmitter,X∗o is transmitted in the first time epoch followed by −X∗

e

in the second time epoch. The appeal of the R-H transformand other orthogonal transmit diversity schemes is becausethey allow the individual symbols at the receiver to be sepa-rated. Denote the diagonal matrices containing the channelfrequency response vectors, H1 and H2, for a two transmitterand one receiver configuration by Λ1 and Λ2, respectively.Assuming the channel is constant over the two consecutivesymbol epochs, the baseband received signals (Y1,Y2) in therespective symbol periods are given by

Y1 = Λ1Xo + Λ2Xe + V1,

Y2 = −Λ1X∗e + Λ2X∗

o + V2.(11)

Using simple substitution methods, the noise corrupted esti-mates of the consecutive OFDM symbols are

Xo = Λ∗1 Y1 + Λ2Y∗2 ,

Xe = −Λ1Y∗2 + Λ∗2 Y1.(12)

Substituting (11) into (12) yields

Xo =(∣∣Λ1

∣∣2+∣∣Λ2

∣∣2)

Xo,

Xe =(∣∣Λ1

∣∣2+∣∣Λ2

∣∣2)Xe.

(13)

The question that remains is what values of Xo and Xe max-imize the coding advantage for the system. Recall that, it wasnoted that the performances of STC are dominated by min-imum distance for the code. Also, recall that for any cosetcode, the minimum distance for the code [12, page 113] isdetermined by

dmin = min(dfree, dcoset

), (14)

where dfree is the free distance of the code used to select acoset and dcoset is the minimum distance within a coset. Nowconsider the code difference matrix D(e, c) of our concate-nated code for the mth subcarrier of an OFDM system withtwo transmit antennas,

D(e, c) =

−[c1(0)− e1(0)]∗ [

c2(0)− e2(0)]∗

c2(0)− e2(0) c1(0)− e1(0)

......

−[c1(l)− e1(l)]∗ [

c2(l)− e2(l)]∗

c2(l)− e2(l) c1(l)− e1(l)

(15)

which leads to

D(e, c)†D(e, c) =d2

min 0

0 d2min

. (16)

Clearly, (16) satisfies the equal eigenvalue criterion and thusits coding gain is guaranteed to achieve its upper bound.Hence near-optimum performance is achievable providedthat the eigenvalues are made as large as possible. Otherresearchers [13, 14] have arrived to this same conclusionbased on different arguments. The main restriction on thecode words c, e is that the channel must be static overthe two symbol epochs defining the Radon-Hurwitz trans-form. Hence, we can construct a space-time block cosetcode (STBCC) by simply selecting a well-known coset codewith the desired Euclidean distance, dfree, and complex-ity. One limitation to these codes is that there is no fullrate complex orthogonal STBC for more than two antennas[9, 10].

We construct a few simple STBCCs using Ungerboeckcodes and compare them to some previously publishedSTTCs [4] of similar complexity.

Example 1. Consider the fully connected 2-STC 4-PSK, 4states, 2 bit/s/Hz in Figure 1 from [4]. Since the code is fullyconnected, we can reach any state from any other state.Therefore, its outer product of the code word difference


δ23 = 4Es

δ22 = (2 +

√2)Es ∼ 3.414Es

δ21 = 2Es

δ20 = (2 − √

2)Es ∼ 0.586Es

(110)S6

S7(111)

√Es

S0(000)

S1(001)

(010)

(011)S3

(100)S4

(101)S5

δ3

δ2δ1 δ0δ0

δ1√Es

δ20 = 2Es

δ21 = 4Es

S3

S0

S2

S2

S1

(00)

(01)

(10)

(11)

Figure 1: Distance between constellation points for 4-PSK and 8-PSK.

00 01 02 03

10 11 12 13

20 21 22 23

30 31 32 33

Figure 2: Tarokh et al. [4]: a fully connected 4-state STC usingquadrature phase shift keying (QPSK).

matrix is given by

D(e, c)†D(e, c) =δ2

0 0

0 δ20

, (17)

where δ20 = 2 for 4-PSK as shown in Figure 2. Now consider a

2 bit/s/Hz Ungerboeck code with 4 state, defined by the octalgenerator polynomials g1 = 2 and g2 = 5, in [12, page 120],with δ2

3 = 4 for 8-PSK. Hence, the coding advantage usingthis simple construction is 3 dB better than the 4-state STCof the same rate.

Example 2. Now consider the 2-space-time code, 4-PSK, 8states, 2 bit/s/Hz shown in Figure 3. The code word differentmatrix is equal to

D(e, c) =δ2

1 0

0 δ21

, (18)

where δ21 = 4 for 4-PSK. Now consider a 2 bit/s/Hz with 8

states, with octal generator polynomials g1 = 04, g2 = 04, andg3 = 11. The minimum free distance for this code is d2

free =4.586, which is 1.15 dB better than the equivalent STC codeby Tarokh [4].

10, 11, 12, 1302, 03, 00, 0130, 31, 32, 33

22, 23, 20, 2130, 31, 32, 33

22, 23, 20, 21

10, 11, 12, 1302, 03, 00, 01

12, 13, 10, 11

00, 01, 02, 03

32, 33, 30, 3120, 21, 22, 23

32, 33, 30, 3120, 21, 22, 23

12, 13, 10, 1100, 01, 02, 0300, 01, 02, 03

10, 11, 12, 1320, 21, 22, 23

30, 31, 32, 3322, 23, 20, 2132, 33, 30, 3102, 03, 00, 01

12, 13, 10, 11

Figure 3: Tarokh et al. [4]: trellises for 8 states and 16 states withdiversity order 2.

Example 3. The final example of this section considers a 2-space-time code, with 16 states, 2 bit/s/Hz. The trellis for thiscode is given in Figure 3. The code word difference matrix forthis code is

D(e, c) = 6 2

√2

2√

2 6

(19)

with eigenvalues of D(e, c)†D(e, c) equal to 3.17 and 8.83.The geometric mean of the eigenvalues equals 5.29. AnUngerboeck code—g1 = 16, g2 = 04, g3 = 23 octal gener-ator polynomials—with 16 states and rate of 2 bit/s/Hz hasa minimum distance of 5.17, which is slightly less than theSTTC. However, the performance of this code should be bet-ter than the STTC since half of the data is transmitted over achannel whose eigenvalue is 3.17 and it should dominant theperformance for the overall code while, for our construction,each path to the receiver has eigenvalues of 5.17 associatedwith it.

Perhaps, the most attractive property of these STBCCsis their simpler decoder complexity compared to their STTCcounterparts. When multiple receive antennas are used, max-imal ratio combining of each estimate of the transmittedsymbols in (12) can be performed for all the receive anten-nas.

4.1. Improving diversity gain

In Section 3, the design criterion is optimum in terms of Eu-clidean distance. However, for fast fading channels, the Ham-ming distance rather than the Euclidean distance is the ap-propriate design metric. Recall (see [15]) that the design of aTCM scheme that achieves diversity of order p requires allpairs of possible encoded sequences to differ in at least psymbols. A simple method of improving the Hamming dis-tance of a coset scheme would be to use a signal constellationwhose symbols differ in every coordinate yet maintain the


desired minimum Euclidean distance dfree amongst all signalpoints. The authors in [16] show that rotation of well-knownconstellations, rather than designing such a constellation, canachieve the same purpose. Once each symbol has distinct co-ordinates over the whole signal constellation or within a sub-constellation, then the bound on the pairwise error probabil-ity is computed over coordinates, that is,

P(

s −→ s) ≤ 1(

SNRc /4)pc Πpc

k=11∣∣sk − sk

∣∣2 , (20)

where pc is the number of distinct coordinates between pairsof code sequences, and SNRc = SNR /2 for 2D constellations(SNR stands for signal-to-noise ratio). In other words, if eachcoordinate experiences independent fading, then we antici-pate a two-fold improvement in diversity level at moderate-to-high SNR. To ensure independent fading, we interleaveover coordinates rather than symbols.

We revisit Example 1 with respect to Hamming distance.For Example 1, the symbol Hamming distance for the STCand STBCC are both two. However, if this coordinate inter-leaving technique was employed, then the coordinate Ham-ming distance for the STBCC would increase to 4. Similararguments can be made for Examples 2 and 3. The remain-ing sections in this paper consider cases when CSI is availableat the transmitter.

5. POWER OPTIMIZATION AND BIT ALLOCATIONALGORITHM

In this section, power optimization and bit allocation algo-rithm for MIMO systems are considered. Information theorydetermines the theoretical maximum informational data ratethrough parallel channels via a water-filling power allocationscheme. This theoretic solution relies on a channel-codingscheme that achieves Shannon’s channel capacity for eachsubcarrier. For the past fifty years, designing, coding, andmodulation schemes with practical implementation com-plexity that achieve this theoretic upper bound are still anopen research topic. In Section 6, we outline the problem.

Let the singular value decomposition of H be H = UΣVH ,where U and V are unitary matrices and Σ is a diagonal ma-trix with positive real values on diagonal elements standingfor the singular values of the channel. If the transmitted vec-tor is premultiplied by V in transmitter, and received vectorbe postmultiplied by UH in receiver, then (3) could be rewrit-ten as

y = Σs + η, (21)

where y and η denote the received vector and noise vectorafter postmultiplication by U. Note that because U is a uni-tary matrix, there is no noise amplification and the noisevectors remain spatially white. Thus, the error rate of max-imum likelihood decoder remains the same as it was fordecoding (3), but its complexity is completely reduced be-cause the entries of s are decoded separately. If r denotes therank of Σ, then this MIMO channel is a set of parallel SISO

channels,

yk = σksk + ηk, k = 1, . . . , r, (22)

where σk is the gain of kth channel and ηk is white Gaussiannoise. For an OFDM framework, (22) becomes

Yk(m) = σkSk(m) + Vk(m), k = 1, . . . , r; m = 1, . . . , L.(23)

The total capacity of the MIMO channel is equal to theaggregate capacity of all these SISO channels. If spatial water-filling is applied for this MIMO channel, then the capacity ofthis MIMO channel is the sum of the individual capacitiesper subcarrier.

For practical implementation, designers must choose abaseband modulation scheme such as MPSK or QAM for bitassignments. Thus, the symbol error rate (SER) is boundedstrictly away from zero for finite transmission power in thepresence of random noise. This still leaves us the question onhow to maximize the total data rate R,

max[R] =L∑

m=1

bm, (24)

subject to a total power constraint,

L∑m=1

r∑k=1

E(∣∣Skm∣∣2

)≤ Ptotal, (25)

as well as to an upper bound on the symbol error rate persubcarrier ρm

ρm ≤ ς, m = 1, . . . , L, (26)

where bm is the number of bits per subcarrier, ς is the up-per bound on the SER across subcarrier, and Ptotal is the totalpower allocated for the system. Our proposed power opti-mization and bit-loading algorithms for MIMO configura-tions are derived in Section 6.

Throughout this section, we have only considered M-aryQAM constellations for subcarrier bit mappings with two ormore bits, bm ≥ 2, and BPSK signaling for single bit sub-carrier allocation. In addition, the analysis presented hereinis for the kth path and the dependence on k is suppressed.Thus, we are able to upper bound the SER per subcarrier [17]as

ρm ≤ 4Q

√√√√ 3

∣∣Hm

∣∣Pm(2bm − 1

)N0

, m = 1, . . . , L, (27)

where N0 is the noise power density and Q is complemen-tary error function. This bound is tight for high SNRs. Us-ing this upper bound as an equality on SER and letting thebit values be continuous, we make the aforementioned prob-lems mathematically tractable in order to suggest a solutionfor each. Due to the “looseness” of this upper bound for low


SNRs, any solution derived from the equality approximationwill be suboptimal, but from existing literature, the loss inperformance is small (∼ 0.1 dB) and still one of the bettertechniques.

To maximize the total rate with a constraint on to-tal power, we rewrite (24) using the Lagrange multipliermethod, that is,

JR =L∑

m=1

bm + λL∑

m=1

Pm (28)

which can be solved relatively straightforward for a powerallocation and bit allocation as

Pm = Ptotal

L− Q−1

(ς/4

)2

3CNRm+

1L

L∑j j=1

Q−1(ς/4

)2

3CNR j j,

bm = log2

(1 +

3CNRmPm

Q−1(ς/4

)2

), for m = 1, . . . , L,

(29)

where the channel-to-noise-ratio (CNR) is

CNRm =∣∣Hm

∣∣2

N0. (30)

During implementation, we must take the necessary care thatwe also satisfy positive power constraints and finite granular-ity (i.e., integer number of bits per subcarrier). Obviously,for unused subcarriers determined by the bit allocation ex-pression above, the transmitter cannot apply power to thesubcarriers with no bits assigned, thus in practice we set

Pm = 0 ∀m, where bm = 0. (31)

Furthermore, we do not use the subcarriers with poor gaincharacteristics which are identified by negative solutions dueto the equality constraint in our Lagrange multiplier formu-lation. Thus, we remove these subcarriers by not allocatingpower to those subcarriers,

Pm = max[0, Pm

], (32)

and due to logarithm of one (in any base) in our bit expres-sion, we do not allocate any rate to the poorly operating sub-carrier. Although through creative means fractional rates areachievable for QAM mapping, we consider here only an inte-ger number of bits per subcarrier, so we floor the result fromthe above bit allocation equation,

bm = max[0,⌊bm

⌋]. (33)

By flooring the bit assignment value, we ensure that the SERfor each subcarrier meets the upper bound requirement inthe problem formulation. Similar expressions can be derivedfor minimization of the average SER and transmission powerunder rate constraints. In Section 6, bit allocation concate-nated with coding and modulation, referred to as adaptivemodulation, is considered.

AdaptivemodulatorM(γ), S(γ)

Signalpoints

One of M(γ)constillation

points

Signal pointselector

Uncodeddata bits

n(γ) − k bits

Buffer

One of 2k+r

cosets

Uncodeddata bits

Modulation

Channel coding

Cosetselector

k + r bits

CodedbitsBinary

encoderk bits

Uncodeddata bits

Figure 4: Goldsmith [6] proposed adaptive TCM scheme.

6. REVIEW OF ADAPTIVE MODULATION

Recall that, as mentioned in previous sections, when channelstate information (CSI) is not available in transmitter, space-time codes should be used to get diversity and coding gain.Likewise for the SISO systems, when CSI is available in trans-mitter, we can increase the efficiency of the system optimalpower and rate adaptation in transmitter. Provided that thefading channel gains are available in transmitter, codes canbe constructed such that it adapts its output power and datarate to channel variation. Such codes can potentially achievemaximum channel capacity and are termed adaptive modu-lation. With adaptive modulation, rather than transmittingthe same information rate for good channel and degradedchannel, power optimization and bit allocation [6, 18, 19]are adapted to transmit more information when channel isgood, and less information when channel is degraded. Thus,without sacrificing bit error rate (BER), these schemes pro-vide high average spectral efficiency by transmitting at highspeeds under favorable channel conditions, and reducing thethroughput as the channel degrades. Adaptive coded mod-ulation does not require interleaving since error bursts areeliminated by adjusting the power, size, and duration of thetransmitted signal constellation relative to the channel fad-ing. However, adaptive modulation does require accuratechannel estimates at the receiver which are fed back to thetransmitter with minimal latency.

The general structure of an adaptive coded modulation,which was introduced by Goldsmith and Chua [6] is shownin Figure 4. Specifically, a binary encoder operates on k un-coded data bits to produce (k + r) coded bits, and the cosetselector uses these coded bits to select one of the 2k+r cosetsfrom a partition of the signal constellation. For the nonadap-tive modulation, (n−k) additional bits are used to select oneof the 2n−k signal points in the selected coset, while in theadaptive modulator (n(γ)− k) additional bits are used to se-lect one of the 2n(γ)−k preselected signal points out of the 2n−k

available signal points in the selected coset. The preselectionis done in such a way that maximizes the minimum distancebetween signals.

In a fading channel the instantaneous SNR varies withtime, which will cause the distance between signals varies.The basic premise for using adaptive modulation with cosetcodes is to keep these distances constant by varying the size


of the signal constellation relative to γ, subject to an aver-age transmit power constraint P. Therefore, maintaining aconstant minimum distance dmin, the adaptive coded mod-ulation exhibits the same coding gain as coded modulationdesigned for an AWGN channel with minimum coded dis-tance dmin.

6.1. Improved adaptive modulation

We outline methods for improving the adaptive TCM modu-lation proposed by Goldsmith [6, 18], and others [14] whenused in an OFDM framework. Goldsmith [6, 18] argues thatburst error events are eliminated when power optimization isused. In constrast, we found that adaptive interleaving basedon CSI can considerably improve the channel statistics suchthat greater throughputs are achievable. Goldsmith [6, 18]implemented adaptive modulation in a time-varying channelenvironment. OFDM systems, on the other hand, are typi-cally deployed in very slowly varying fading environments.

In these scenarios, it is possible to benefit from inter-leaving. Typically, interleavers are designed to convert slowlyvarying channels into rapidly changing channel to exploittime diversity. For OFDM systems, the interleavers shouldexploit the frequency selective of the channel to benefit fromfrequency diversity. Normal block or convolutional inter-leaver can be used for this purpose. However, the fullest ben-efit from interleaving is achieved when the CSI is used to de-sign it. Such interleaving is referred to as adaptive interleav-ing.

6.1.1 Adaptive interleaving

We outline a simple interleaver based on CSI at the trans-mitter, which improves the performance of an adaptive TCMscheme such that more uncoded bits per subcarrier can besupported than without interleaving. Our adaptive coordi-nate interleaver algorithm is outlined as follows:

(i) given an estimate of H(m);(ii) define Z(m) = |H(m)|;

(iii) [ f f , kk] = SortZ(m), where kk are indexes associ-ated with the weakest to strongest amplitudes;

(iv) let j j = [1, . . . , L];(v) map real[S( j j)] −→ H(kk);

(vi) map imag[S( j j)] −→ H(reverse(kk)), wherereverse(

[1 2 3 4

]) = [

4 3 2 1];

(vii) combine coordinators as order transmit over channel.

The resulting channel statistics after de-interleaving atthe receiver has fewer deep null. A comparison of the chan-nel statistic with and without our new interleaver is shownin Figure 5. Based on this figure, the number of supportableuncoded bits per subcarrier improves considerably. An alter-native interleaving scheme based on CSI at the transmittertakes advantage of the pad bits used to generate integer num-ber of OFDM symbol within a packet. The frame format foran OFDM packet for the IEEE 802.11a standard is shownin Figure 6. On average, the number of pad bits required ishalf an OFDM symbol, which is quite significant. Consid-ering that the pad bits are discarded at the receiver, a better

50454035302520151050

Subcarrier number in OFDM symbol

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Ch

ann

elam

plit

ude

s

After interleavingBefore interleaving

Figure 5: Virtual channel statistics at the decoder after de-interleaving.

Data #of OFDM symbols

Signalone OFDM symbol

Preamble12 symbols

Rate(4 bits)

Reserved(1 bit)

Length(12 bits)

Parity(1 bit)

Tail(6 bits)

Service(16 bits) PSDU

Tail(6 bits)

PadBits

Header

Figure 6: Physical layer protocol data unit (PPDU) in 802.11a, in-cluding the preamble, header, and PSDU.

utilization of the power and channel allocation for those padbits would be to interchange data symbols on weak subcar-riers for pad bits on strong subcarriers. To achieve the bestperformance, the adaptive interleaving should be performedin conjunction with a power optimization and bit allocationalgorithm. Our proposed algorithm is outlined as follows:

(i) sort the channel amplitudes for all the OFDM datasubcarriers in the packet in order of weakest tostrongest;

(ii) reverse the order of the bits in data portion of thepacket, for example, pad bits, tail bits, and physicallayer service data unit (PSDU);

(iii) map the modulation symbols onto the subcarriers byfilling the set of weakest subcarriers for each OFDMsymbol first;

(iv) continue mapping the modulation symbols onto thegroup of the next weakest subcarriers for each OFDMsymbol;


(v) continue in this fashion until the packet is filled;(vi) using a power optimization algorithm, water-fill the

packet.

This algorithm guarantees that the data symbols aremapped onto the most reliable subcarriers; the pad bits, theleast reliable subcarriers. The subcarriers, which are allo-cated no power, are assigned to the pad bits first. Depend-ing on channel conditions, some of the data subcarriers maystill not be allocated any power; however, there are fewer ofthose data subcarriers than if the adaptive interleaver was notused. A better choice for adaptive modulation for this pro-posed adaptive interleaving is to use a block-coded modula-tion scheme. In particular, a Reed-Solomon (RS) constituentencoder provides great flexibility in error protection, codeword size, and code rate, which is well suited for this ap-plication. The adaptive TCM scheme discussed thus far inthis paper has been designed to minimize error probabilityand maximize average data rate. It is desirable to maintaina fixed number of bits per symbol to facilitate the need formultiple demodulators at the receiver. In this scenario, thedata rate is guaranteed for each packet, and the error rateis driven as small as possible for any instance of the chan-nel. That is, these codes minimize the outage probability fora given rate.

6.1.2 Adaptive error correction coding

A code that minimizes the outage capacity for a given rateshould be able to easily adjust its coding gain to the chan-nel statistics. The adaptive TCM scheme outlined in this pa-per assumes a certain number of coded bits per subcarrier,and the number of uncoded bits per subcarrier is adapted.We allow both numbers of uncoded bits per subcarrier andcoded bits per subcarriers to varying, under the restrictionthat the number of bits per symbol per subcarrier is fixed.This is a very interesting problem that stems from practi-cal considerations. For an OFDM system perspective, it isdesirable to set the rate for the system prior to transmis-sion. Furthermore, the packet error rate, not the bit errorrate, must be minimized. Hence, each packet must meetthe rate requirement first and the error requirement sec-ond. Since, there are a fixed number of parity symbols avail-able determined by the rate, it is wise to use the adap-tive interleaver scheme described in this section to spec-ify erasures for the code. Consider a shortened RS linearblock code. Adding erasures, up to the length of the orig-inal code, does not reduce the parity symbols available forrandom symbol errors. The caveat to this statement is thatthere need to be available sufficient pad bits for the specifiederasures.

Now, we focus on applying bit and power allocation ideasto improve error correction codes. If hard erasure decodingis implemented, RS codes can correct up to (NN −KK) era-sures or (NN − KK)/2 random errors, where KK is thenumber of information symbols and NN is the length of theRS code. Thus for an AWGN channel (spectrally flat chan-nel), the word error rate (WER) is upper bounded by the

expression

WER ≤NN∑

k=(NN−KK+1)

NN

k

ρk(1− ρ)NN−k, (34)

which assumes that the erasure locations are known or can bepredicted. Note that for a spectrally shaped system such as anOFDM system, the SER ρm varies from subcarrier to subcar-rier. Hence, the average SER over the NN−χ best subcarriersρχ is defined by

ρχ = 1NN − χ

NN∑m=χ+1

ρm, (35)

and (34) becomes

minχε[0,...,(NN−KK+1)]

(WER), (36)

where

WER ≤NN−χ∑

k=(NN−χ−KK+1)

NN − χ

k

ρkχ(1− ρχ

)NN−χ−k. (37)

Although the erasures are not actually transmitted across thechannel, there need to be enough “empty” or “unmodulated”subcarriers to account for those symbols. Power saved by nottransmitting the erasures should be reallocated amongst thedata bearing subcarriers in a manner that minimizes (37). Anoutline for the algorithm is given as follows:

(1) given an estimate of H(m), sort the channel ampli-tudes for all the OFDM data subcarriers in the packetin order of weakest to strongest;

(2) determine the zero power locations from power andbit allocation algorithms and specify them as erasuresfor the shortened RS code;

(3) determine WERi from (37);(4) specify the next weakest subcarrier as an erasure and

reallocate power;(5) recalculate WERi+1;(6) if WERi > WERi+1, repeat steps 4 and 5 until this is no

longer true;(7) using targeted SER bound ς, compute the SER per

subcarrier ρm using (27), determine the location andnumber of symbols, which can be unprotected;

(8) finally, determine the number of bits needed to meetthe total data rate R and allocate bits using the adaptivecoded modulation structure introduced by Goldsmithand Chua [6], under the restriction of a fixed constel-lation size.

Note that the coset selector for this problem is the short-ened RS code and not a convolutional code. An exampleof allocation of variable coded bits and uncoded bits for afixed constellation size is depicted in Figure 7. In Figure 7, thenumber of uncoded bits supportable per subcarriers varies


250200150100500

Ordered subcarriers

0

0.5

1

1.5

2

2.5

3

3.5

4

Bit

sin

erro

rp

ersy

mbo

l

Erasuresregion

Random errorregion – 4 coded2 bits uncoded

Uncoded errorregion

Figure 7: An adaptive bit allocation per subchannel for BCMscheme with 6 bits per symbol.

over the packet. The first 48 subcarriers of the 240 subcarri-ers in the packet are specified as erasures. The overall rate forthe system is 2/3 not including the erasures specified by thestuff bits. When the power is re-distributed, the single errorin the uncoded region is corrected. A summary of steps forperforming adaptive error.

In the next section, we validate some of the algorithmspresented in this document within the IEEE 802.11a frame-work. Note, that the IEEE 802.11a framework assumes quasi-static channels. That is, the channel is constant over a packetbut changes from packet to packet. Hence, this is akin or sim-ilar to the environment used for outage probability.


We present a performance evaluation of several MIMO con-figurations within the IEEE 802.11a framework. The con-stituent TCM scheme uses a 128-state, rate 2/3 convolutionalencoder for all the data rates considered. The IEEE 802.11abaseline system parameters are

(i) 64 pt. FFT: 48 data carrier and 4 pilot carriers,(ii) 20 MHz sampling frequency,

(iii) 3.2 microsecond FFT period and 0.8 microsecondcyclic prefix,

(iv) symbol rate of 12 Msymbols/s.

First, we examined the packet error rate performance(PER) of STBCC schemes, 128 states, 4 bit/s/Hz over a five-tap Rayleigh channel. The channel taps were spaced 50 nsecapart with an exponential channel delay profile, which is lessthan the cyclic prefix. The first observation concerning Fig-ures 8 and 9 is that the diversity order of the STBCC with nointerleaving is two not one. Considering that this code hastwo parallel transitions per state; this simulation validatesthat STBCCs are guaranteed to have a diversity order of atleast two, which was determined from their code word dif-ference matrices. A second observation is that coordinating

2524232221201918171615

SNR

10−6

10−5

10−4

10−3

10−2

10−1

100

BE

R

No interleaverCoordinate interleavingSTBC+CC

Figure 8: Diversity improvement for STBCC using coordinate in-terleaving in a Rayleigh fading channel.

2524232221201918171615

SNR

10−4

10−3

10−2

10−1

100

PE

R

No interleaverCoordinate interleavingSTBC+CC

Figure 9: PER performance improved with coordinate interleavingfor TCM 48 Mbit/s.

interleaving with constellation rotation definitely improvesthe diversity order of the code. In fact, the diversity or-der is double at moderate-to-high SNR as was predicted inSection 4.1. Furthermore, the bit error rate shown in Figure 8is greatly improved as well as the error event probability illus-trated by the PER in Figure 9. For comparison, a STBC con-catenated with the IEEE 802.11a 48 Mbit/s mode is shownalso in the figures. The purpose was to illustrate how wellthe coordinate interleaving improves the Hamming distance.The IEEE 802.11a uses a bit interleaver prior to the STBC.


30282624222018

SNR

10−3

10−2

10−1

100

PE

R

Adaptive PER R.S.128 Stage TCM PER rate

Figure 10: Adaptive modulation PER comparison: BCM versusTCM for five-tap Rayleigh fading and 48 Mbit/s mode.

To illustrate our proposed adaptive BCM, we considerthe same TCM scheme with two parallel transitions perstate. Both schemes are adaptively interleaved according tothe algorithms specified in Section 6.1.1. The adaptive BCMscheme uses shortened versions of RS(63, 31) and uncodedsymbols across the packet in a similar fashion, as depictedin Figure 7. The power optimization algorithm specified 48erasures, of which 43 were pad symbols to fill the packet.Also, ninety-four of the subcarriers had four coded bits andtwo coded bits. The remaining subcarriers were uncoded.The power from the 48 erasures were distributed in a water-filling fashion across the data bearing subcarriers, for exam-ple, the uncoded symbol were placed on the strongest subcar-riers and distributed a greater percentage of the power. Theadaptive BCM performs significantly better in terms of PERas shown in Figure 10, although the adaptive TCM schemeperforms better in terms of BER performance as shown inFigure 11. This validate our assertion that an adaptive BCMis better suited for environments where the outage probabil-ity is the appropriate capacity measure.

Our final example illustrates STBCC with coordinate in-terleaving using the TCM scheme described earlier and 6uncoded bits per symbol. The overall rate is 96 Mbit/s in a20 MHz band, which has a spectral efficiency of 4.8 bit/s/Hz.For comparison, the 54 Mbit/s mode of the IEEE 802.11ais shown in Figure 12, which has a spectral efficiency of2.7 bit/s/Hz. At high SNR, the STBCC would eventually out-performs the 54 Mbit/s although it carries, on average, anadditional 2.1 bit/s/Hz capacity. If receive diversity is avail-able, power saving of the order of 10 dB is possible with onlytwo-receive antennas. Further, because of the structure of theSTBCC, the additional signal processing required for multi-ple receiver is greatly diminished when compared to the max-imum likelihood decoders need for STTC.

30282624222018

SNR

10−4

10−3

10−2

10−1

BE

R

Adaptive BER R.S.128 state TCM

Figure 11: Adaptive modulation BER comparison: BCM versusTCM for five-tap Rayleigh fading and 48 Mbit/s mode.

35302520151050

Eb/N0

10−3

10−2

10−1

100

PE

R

TCM 96Mb, baseTCM 96Mb, symbol interleaverTCM 96Mb, rotation, coord interleaverIEEE 54Mbits

Figure 12: Comparison of STBCC with symbol interleaving to aSTBCC with coordinate interleaving, and constellation rotationsand 54 Mbit/s mode of the IEEE 802.11a in five-tap Rayleigh fad-ing.

8. CONCLUSIONS

In this paper, we demonstrated the benefits of the concate-nating coset codes with orthogonal STBC, which we termedSTBCC. It was found that STBCC with outer codes designedfor AWGN channels produces suitable codes for fading chan-nels. In fact, the coding advantage of these codes, based onthe code word difference matrix, is as good or better than


the STTC found from computer search. Although STBCCsare guaranteed a minimum diversity level of 2, we showedthat 2D constellation rotation and coordinate interleavingcan double that diversity order. Performance evaluation overa frequency selective fading channel was demonstrated us-ing Monte Carlo simulations, which showed substantial im-provements in PER performances. Our final example showeda STBCC configuration for 96 Mbit/s in 20 MHz bandwidth,which performed comparably to the 54 Mbit/s mode of theIEEE 802.11a in terms of Eb/N0.

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[18] A. J. Goldsmith and S. Chua, “Variable-rate variable-powerMQAM for fading channels,” IEEE Trans. Communications,vol. 45, no. 10, pp. 1218–1230, 1997.

[19] K.-K. Wong, S.-K. Lai, R. S.-K. Cheng, K. B. Letaief, and R. D.Murch, “Adaptive spatial-subcarrier trellis coded MQAM andpower optimization for OFDM transmission,” in Proc. IEEEInternational Conference on Vehicular Technology, pp. 2049–2053, Tokyo, Japan, May 2000.

John D. Terry was born on September 29,1966 in Norfolk, Virginia. He received hisB.S. in electrical engineering from Old Do-minion University and M.S. in the samefield from Cleveland State University, in1988 and 1993, respectively. In the spring of1999, he received his Ph.D. degree in elec-trical and computer engineering from theGeorgia Institute of Technology. Dr. Terryjoined Nokia Research Center (NRC) inDallas in January of 1999 as a senior research engineer. Currently,he manages the OFDM modulation and coding project in theWireless Data Group at NRC Dallas. His IEEE activities includetechnical reviews for several conferences and journal publicationsrelated to wireless communications, and serving as vice-chair ofIEEE 802.11g task group. He is the coauthor of the book entitled,“OFDM WLANs: A Theoretical and Practical Guide.” In 2002, Dr.Terry was honored with a national award as the Black Engineerof the Year for Outstanding Technical Contributions in Industry.His research interests include array processing, space-time coding,WLAN technology, diversity techniques, and error correction cod-ing.

Juha Heiskala received his M.S. in electri-cal engineering in 1996 from Helsinki Uni-versity of Technology. He is in the electri-cal engineering Ph.D. program at SouthernMethodist University. He joined Nokia Re-search Center 1995, where he has worked onseveral different areas of digital communi-cations, for example, digital audio broad-casting, satellite radios, and wireless LANtechnology. His current research interestsinclude multiple transmitter and receiver antenna technologies,and error correcting code systems. He is the coauthor of the bookentitled, “OFDM WLANs: A Theoretical and Practical Guide.”

Victor Stolpman received his B.S. degreein electrical engineering with honors fromTexas A&M University, College Station, Tex,in 1995, and his M.S. degree from SouthernMethodist University, Dallas, Tex in 1999.Currently at Southern Methodist Univer-sity, he is working on a Ph.D. degree in elec-trical engineering. From 1992 to 1998, heworked as a design engineer for Dresser In-dustries, and since 1998 he has held researchpositions at both Texas Instruments, Richardson, Tex and NokiaResearch Center, Irving, Tex, investigating information theoretic


and signal processing applications for wireless data communica-tions systems. His current research includes incorporating adap-tive modulation techniques for multicarrier systems with error cor-rection coding for improved performance for high-speed wirelessLAN applications.

Majid Fozunbal was born in Tehran, Iran,in 1974. He received his B.S. and M.S.both in electrical engineering in 1996 and1998, respectively. He is currently pursinghis Ph.D. studies in electrical engineering atGeorgia Institute of Technology. From 1998to 2000 he was with Mobile Telecommuni-cation Technology Inc., Tehran, Iran, work-ing in the area of GSM technology. Duringthe summer of 2001, he was with Nokia Inc.,Dallas, Tex, USA, working in the area of wireless communication.His main research areas lie within digital communication, statisti-cal signal processing, and information theory.


Dual Switched Predictive DIR MLSD Receiverfor Dynamic Channels

Michael BoyleDepartment of Electronic and Electrical Engineering, University College Dublin, Belfield, Dublin 4, IrelandEmail: [email protected]

Anthony D. FaganDepartment of Electronic and Electrical Engineering, University College Dublin, Belfield, Dublin 4, IrelandEmail: [email protected]


A linear prefilter can be used to reduce the required complexity of a maximum likelihood sequence detector Viterbi algorithm(MLSD-VA) by shortening the overall channel and prefilter impulse response in dynamic communication systems. The combi-nation of channel and prefilter should have the effect of producing a desired impulse response (DIR) at the detector. Falconerand Magee (1973) showed that for a finite length DIR there are a limited number of possible DIRs that are optimal. For a DIRof length two symbols, there are only two optimal DIRs, for a length three DIR there exists a range of possibly optimal DIRs. Inthis paper, we present a novel receiver architecture in which we use two equalisers and two Viterbi detectors. Each equaliser has adifferent target DIR. A selection device chooses between the output of the two VAs. It is demonstrated that, using the two optimallength two DIRs can be preferable to both switched triple DIR system and adaptive DIR strategies. It is also demonstrated in thispaper that there exists a range of environments where adaptive DIR MLSD-VA receivers fail, however the proposed dual switchedDIR MLSD-VA is successful in these environments. The efficacy of the switched dual DIR MLSD-VA is also shown using doublyselective fading channels.

Keywords and phrases: maximum-likelihood sequence detection, fading channels, equalisation.

1. INTRODUCTION

The detection of a signal transmitted through a communica-tion channel that contains intersymbol interference (ISI) andadditive Gaussian noise has been widely studied for a broadrange of channel models. Maximum likelihood sequence de-tection (MLSD) implemented using the Viterbi algorithm,proposed by Forney [1], is an optimal equalisation methodto combat ISI. The computational complexity of the Viterbialgorithm (VA) grows exponentially with the length of thechannel impulse response (CIR). One technique [2] that hasbeen used to reduce the complexity of the VA is to use a pre-filter to truncate the CIR. The cascade of the prefilter and thechannel produces an equivalent channel impulse response(ECIR) at the input to the VA, which is close to the adap-tive desired impulse response (DIR) being used by the VA.The DIR is shorter than the CIR and hence the complexityof the VA is reduced. In [3], different length DIRs were in-vestigated using the mean square error (MSE) criterion. In[4], the optimal DIR using the effective signal-to-noise ratio(SNR) criterion was determined. In both cases, it was shown

that for the case of a length two DIR there exist just two opti-mal DIRs. If the power of the DIRs is constrained to one, thenthe two optimal DIRs are (1/

√2, 1/

√2) and (1/

√2,−1/

√2).

In [4], optimal length three DIRs were shown to have a par-tial response of type (α, 2α, α).

In a dynamic multipath fading channel environment,adaptive channel estimation has been used with MLSD-VA totrack the channel variation. Adaptive MLSD-VA refers to us-ing MLSD-VA along with adaptive channel estimation. Theadaptive MLSD using a single channel estimator has been in-vestigated for fading channels in [5, 6, 7]. Per-survivor pro-cessing (PSP), proposed in [8, 9], when applied to adaptiveMLSD, uses the same number of channel estimators as statesin the trellis of the VA.

In a dynamic channel environment, the adaptive length-two DIR tracks the channel between the two optimal DIRs.As channel activity dictates the optimal DIR at any giveninstant, the adaptive length-two DIR MLSD-VA can be re-quired to change target DIR. This can result in a periodof gross misadjustment as the adaptive DIR changes fromone optimal DIR to the other optimal DIR. It has been

Dual Switched Predictive DIR MLSD Receiver for Dynamic Channels 845

shown using simulation that the adaptive DIR MLSD-VAsystem may perform badly during this transition. This leadsto the idea of using both of the optimal DIRs in a receiverstructure.

It has been shown [10, 11] that rather than using a singleadaptive prefilter together with an adaptive DIR, it is prefer-able to simultaneously operate two adaptive prefilters, eachattempting to equalise the channel to one of the possible op-timum DIRs. Each equaliser is followed by its own VA andhence at every symbol interval two decisions are produced.The final decision is made by a device that selects the symbolcorresponding to the smaller of the two VA metrics. We referto this type of equaliser as switched dual DIR Viterbi equal-isation. The main advantage of such a system is that, as thechannel conditions vary there is no period of gross equalisermisadjustment.

In the case of length-three and higher DIRs, it was shownin [4] that the optimal DIRs were of a partial response type.This implies that a large number of optimal DIRs exist forthese higher order DIRs. Therefore, it seems plausible thata satisfactory switched DIR equaliser, where the DIR is oflength three or greater, can be constructed using a finitenumber of adaptive prefilters each attempting to equalise toa fixed partial response type DIR. The final symbol decisionagain is based on selecting the symbol produced by the VAhaving the smallest metric.

The adaptive DIR MLSD-VA and the switched DIRMLSD-VA will be examined using two different types ofchannels. A channel consisting of a time varying determin-istic channel and AWGN (additive white Gaussian noise) willbe used to examine the tracking ability of both systems indifficult operating conditions. It will be shown that the adap-tive system fails in a range of environments where the pro-posed dual switched system is successful. The adaptive andswitched systems will be investigated using frequency selec-tive Rayleigh fading channels to obtain bit error rates (BER).The BER will show that the dual switched DIR systems out-perform the triple switched systems and comparable adap-tive systems.

The paper is organised as follows. In Section 2, a briefdescription of the mathematical background that results inthe optimal DIR is outlined and the basis for a switched DIRMLSD-VA is explained. In Section 3, the predictive switchedDIR MLSD-VA is presented as a possible solution to the de-lay associated with the VA. Simulation results are presentedin Section 4, which demonstrate the efficacy of the proposedswitched dual DIR MLSD-VA. Finally, in Section 5, conclu-sions are drawn.

2. OPTIMAL DIRs

It is well known that the transmitter filter, the band-limitedchannel containing additive white noise, the matched filter,the symbol rate sampler, and the whitening filter can be rep-resented as an equivalent discrete time white noise filter withwhite Gaussian noise added at the output of the filter. Thereceived signal at the output of the equivalent white noisefilter is

r(T) =K∑

l=−Kcl(T)a(T − l) + n(T), (1)

where the equivalent white noise filter, sequence of informa-tion symbols, and sequence of uncorrelated noise samples arerepresented, respectively, by ci(T)Ki=−K , a(lT)∞l=−∞, andn(lT)∞l=−∞. The error signal used to update the prefilter canbe expressed as

e(T −D) =N∑

l=−Npl(T −D)r(T −D − L)

−L∑l=0

ql(T)a(T −D − l),

(2)

where the vectors PT = (p−N (T), . . . , p0(T), . . . , pN (T)) andQT = (q0(T), . . . , qL(T)) represent the tap coefficients of theprefilter and DIR, respectively, and D is the delay in the up-date error associated with the VA. In a manner similar to thatof [3], the error can be minimised to reduce the noise vari-ance seen at the input to the VA. The complete derivation canbe found in [12]. Minimising the error in this manner allowsfor the error to be expressed in a quadratic form

E0 = QTBQ, (3)

where B is a square matrix of dimension (L + 1); B can beshown to be a positive definite symmetric Toeplitz matrix.Minimisation of E0, with an appropriate energy constrainton Q, is accomplished by making Q that normalised eigen-vector of B corresponding to its minimum eigenvalue. In thecase of L = 1, B is a 2 × 2 matrix with values b0 (the diag-onal entries) and b1 (the off diagonal entries). The eigenval-ues associated with B are b0 + b1 and b0 − b1 correspond-ing to just two optimal eigenvectors, either (1/

√2, 1/

√2) or

(1/√

2,−1/√

2), when |Q|2 = 1. Therefore, the eigenvaluesb0 + b1 and b0 − b1 have associated eigenspaces containingthe optimal eigenvectors. The value of b1 determines whichof the eigenvalues is the minimum eigenvalue at any giventime instant. A case of interest arises when b1 = 0. Whenthis occurs, B becomes b0I (I is the identity matrix) result-ing in all 1 × 2 vectors with |Q|2 = 1 being optimal. Thisis a highly undesirable state of operation. In practical im-plementations the receiver is unable to determine the exactchannel state, due to factors including delay in the receiver,additive noise, and round off error, producing inaccuracy indetermining B. Therefore, there exists a region between thetwo eigenspaces, each containing a single optimal eigenvec-tor, where all 1 × 2 vectors (with |Q|2 = 1) are optimal. Toremove the possibility of the receiver operating in this un-desirable region, prompted the development of the switcheddual DIR receiver.

In the case of L = 2, the optimal eigenvectors can be ei-ther (a, b, a) or (a, 0,−a). In [13] a length-three DIR with|Q|2 = 1.5 was used as an example. Eigenvectors that com-ply with these constraints are (0.5, 1, 0.5), (0.5, −1, 0.5), and(√

1.5/2, 0,−√1.5/2). These DIRs are used in the switchedtriple DIR MLSD-VA receiver. In [4], the optimal DIR was


obtained using the effective SNR criterion to minimise (2).This criterion requires that the prefilter decorrelates the noiseseen at the input to VA. The form of the optimal DIR wasfound to be either (a,−a) or (a, a) for L = 1, or (a, 2a, a) forL = 2.

The choice of DIR in a given situation depends on thechannel amplitude characteristics. For a length-two DIR, thepossible choices are (1/

√2,−1/

√2) or (1/

√2, 1/

√2). These

two DIRs reflect ECIRs that have either a high-pass or a low-pass frequency response.

It was shown in [14] that there exists a catastrophic er-ror mode in the switched dual DIR MLSD-VA when theDIRs are either (1/

√2, 1/

√2) or (1/

√2,−1/

√2). It was also

shown that this error mode can be prevented using theDIRs (1/

√2, α/

√2) and (1/

√2,−α/√2). Using α = 0.99 has

been found to work well in dynamic environments. BER areshown in the appendix that demonstrate the performanceimprovement as a result of the suggested modification to theDIRs.

3. NEW RECEIVER ARCHITECTURE

In this section the predictive switched DIR MLSD-VA re-ceiver is introduced. This receiver is proposed to reduce theimpact of the delay associated with the VA. This section startsby highlighting the relationship between delay and excessMSE in a normalised adaptive algorithm.

It was shown in [13] that the delay inherent in the con-ventional adaptive DIR MLSD-VA resulted in reduced track-ing of dynamic channels. In [13], the prefilter and DIR wereupdated using an adaptive algorithm. The effect of the delayin the update prefilter will now be determined in the case of anormalised adaptive algorithm. The update equation for theprefilter is

P(T) = P(T − 1) +βe(T −D)R(T −D)R′(T −D)R(T −D)

, (4)

where β is the normalised step size [15], R(T) is the vector ofreceived signals at time T , and e(T − D) is the error in (2).The MSE can be expressed as

ε(T) = ⟨e2(T)

⟩ = εmin + εex(T − 1), (5)

where εmin is the minimum MSE and εex is the excess MSE.Using [16, 17], it can be shown that assuming that the algo-rithm converges, εex can be related to the delay D by

εex= βεmin

(2−β)−2Ds+s2D(D+1)−(s3/3)D(D+1)(2D+1)+· · · ,(6)

where s can be assumed constant. It can be seen that, for con-stant step size β, if D is increased from zero, the excess meansquare error also increases from zero monotonically until thedenominator in (6) becomes zero. In that case, εex rapidly in-creases and the algorithm diverges. The complete derivationcan be found in [12].

The concept of using prediction in adaptive MLSD is wellknown [5, 6, 18]. Employing a prefilter to shorten the chan-

Prefilter

Error

+ + −FixedDIR

Del

ay copy Pb

PrefilterViterbi

detector

Dec

isio

nde

vice

[r(T −D)] [a(T)]

Prefilterv(T −D)

Del

ay copy Pa

Prefilter + + −

Error

FixedDIR

Viterbidetector

Figure 1: Predictive switched dual DIR MLSD-VA.

nel impulse response duration, to compensate for channeldistortion, and to supply the Viterbi detector with predictedvalues of the input signal, was proposed in [18]. This reducesthe effect of the delay associated with the VA. Figure 1 il-lustrates the structure of the proposed dual DIR predictiveequaliser. Since the embedded VA operates on predicted sig-nals, the detected symbols at the output of the VA have ashorter delay.

The principle of the switched DIR MLSD-VA will be in-vestigated using Monte Carlo methods to compare the per-formance of switched fixed DIRs with adaptive DIRs in thepredictive receiver strategy. The use of a switched dual DIRsystem implies that there will be two branches within thereceiver structure that is to be tested. Similarly, there arethree branches in the switched triple DIR system. The re-ceiver contains two VAs, each using one of the optimal DIRs.The use of a prefilter constrains the ECIR preceding theMLSD-VA. Each branch uses two prefilters, both prefiltershave identical tap coefficients, one of the prefilters has adelay D at its input, and is used to determine the updateerror for the prefilter adaptation algorithm. Each branchcontains a two-state Viterbi detector using one of the op-timum length-two DIRs. At each symbol interval, each ofthe Viterbi detectors supplies the symbol it decides uponand its associated metric to a selection device. The selec-tion device compares the metrics of the symbols from eachof the two Viterbi detectors. As Euclidean distance is used toobtain the incremental metrics, the symbol with the small-est metric is chosen as the receiver output for that symbolinterval. As this paper is concerned with investigating thedual DIR principle in a dynamic environment, an exponen-tial weighting factor has been incorporated into the metriccalculation as

Mj(n) = λMk(n− 1) + I j,k(n), (7)


where Mj(n) is the metric of state j at time step n given thetransition from state k to j, I j,k(n) is the incremental met-ric from state k to state j at time step n, and λ is the forget-ting factor. This improves the responsiveness of the receiverto channel variations.


The predictive DIR MLSD-VA was tested using two types ofchannels, the swept notch channel and the doubly selectivefading channel. To examine the tracking and switching ca-pabilities of the proposed system in the predictive architec-ture, a channel, consisting of the deterministic swept notchchannel (SNC) and AWGN, was used. BER were obtainedfor several different systems using the predictive DIR MLSD-VA architecture with time varying frequency selective fadingchannels.

4.1. Tracking properties of predictive DIR MLSD-VAsystems

The SNC is a three-tap dynamic channel consisting of twozeros with a trajectory within the unit disc in the z-plane.This channel requires that the prefilter performs, in the caseof length-two DIRs, its three tasks

(i) channel shortening,(ii) prediction,

(iii) reduction of distortion.

One of the zeros of the SNC moves in a clockwise direc-tion while the other moves in a counterclockwise direction.This results in the two zeros coinciding as they cross the realaxis. Therefore, the SNC changes from being a high-pass toa low-pass channel (or vice versa) as the zeros pass throughthe imaginary axis. This property of the SNC makes it partic-ularly suitable in examining the tracking and switching prop-erties of the predictive dual switched system as the two opti-mal DIRs represent high and low-pass channels.

The following examples show some scenarios where theadaptive predictive DIR MLSD-VA failed and the switcheddual DIR MLSD-VA was successful.

Figure 2 shows the update-error (2) for the dual switchedand adaptive predictive DIR MLSD-VA systems for a chan-nel consisting of the SNC and AWGN. In this case, the SNCwas implemented with the zeros having a circular trajectoryof radius 0.88. The zeros were rotated around the unit discat a constant rate π/(5 × 104) radians per symbol interval.The SNR was 10 dB. Figure 2 shows the update-error pow-ers for the adaptive and dual switched systems for 105 sym-bols. The systems were trained for the first 5 × 103 sym-bols. The SNC started at π radians and therefore the chan-nel had a low-pass characteristic. The zeros passed throughthe imaginary axis after 2.5 × 103 symbol instants and thechannel starts to change to a high-pass channel. As thechannel is low-pass initially, the (1/

√2, α/

√2) DIR is opti-

mal and as shown in Figure 2, the update-error for the pre-filter, using (1/

√2, α/

√2) as a target ECIR, has the small-

est error of the dual switched system. The adaptive system

0 1 2 3 4 5 6 7 8 9 10×104Symbols

0 1 2 3 4 5 6 7 8 9 10×104Symbols

−30

−20

−10

0

Upd

ate-

erro

r(d

B)

−30

−20

−10

0

Upd

ate-

erro

r(d

B)

Adaptive DIR

DIR (1/√

2, α/√

2) DIR (1/√

2,−α/√2)

Figure 2: Update-error powers of dual switched and adaptive re-ceivers in SNC and high AWGN power environment. Parameters:radius = 0.88, dθ/dt = π/(5× 104), SNR = 10 dB.

converges to this DIR as shown by the adaptive update-errorin Figure 2.

The update-error power for the prefilter using (1/√

2,−α/√2) is initially much larger than that of the other prefilterbut as the channel approaches the transition from low-passto high-pass, the update-error powers. As the zeros approachthe imaginary axis, the update-error signals become quitenoisy. The high additive noise power reduces the ability ofthe prefilters to have an accurate ECIR as the channel changesfrom low to high pass. After the transition at the 2.5× 104thsymbol instant, Figure 2 shows that the update-error powerfor the (1/

√2,−α/√2) DIR branch of the switched dual sys-

tem is considerably less noisy and has a smaller magnitudeindicating that (1/

√2,−α/√2) is the optimal DIR in this en-

vironment. The update-error power of the (1/√

2,−α/√2)DIR branch of the system continues to decrease as the chan-nel becomes more high pass until the 5 × 104th symbol in-stant, when the two zeros coincide as they cross the real axis.Then the update-error power starts to increase as the channelmoves to the transition to a low-pass channel.

Figure 2 shows that the adaptive system performs poorlyin this high AWGN power environment. The update-error ofthe adaptive system indicates that the adaptive system trainedto the (1/

√2, α/

√2) DIR successfully. The adaptive system

tracks the channel as it approaches the transition from low-pass to high-pass, however it fails to track the channel af-ter transition as shown in Figure 2 by the large oscillationsin the update-error power. These oscillations in the update-error power are the result of the ECIR having collapsed tobecome quite small in magnitude. Figure 2 is an example ofthe robustness of the dual switched DIR MLSD-VA operat-ing successfully in a robust dynamic environment, where theadaptive DIR MLSD-VA failed.

Figure 3 shows the update-error powers for the dualswitched and adaptive DIR MLSD-VA for a dynamic chan-


0 2 4 6 8 10 12×104Symbols

0 2 4 6 8 10 12×104Symbols

0 2 4 6 8 10 12×104Symbols

−20−15−10−505

Upd

ate-

erro

r(d

B)

−20−15−10−505

Upd

ate-

erro

r(d

B)

−20−15−10−505

Upd

ate-

erro

r(d

B)

Switching instants

Adaptive DIR

DIR (1/√

2, α/√

2)

DIR (1/√

2,−α/√2)

Figure 3: Update-error powers of dual switched and adaptive re-ceivers for a SNC with increasing trajectory. Parameters: initialradius = 0.4, final radius = 0.912, dθ/dt = π/(4 × 104), SNR =9.5 dB.

nel. The deterministic channel that was used to obtainFigure 3 consists of an SNC whose zeros have a trajectorysuch that the channel has increasing ISI. Increased ISI isachieved by increasing the radius of the trajectory of the ze-ros. Also, the trajectory of the zeros was such that the channelremained in the vicinity of the boundary between a high-passand a low-pass channel. This allowed for the examination ofthe behaviour of the switched dual system and the adaptivesystem in an environment where the channel was near theboundary of the two optimal DIRs.

The dual switched and adaptive systems were trained forthe first 5 × 103 symbols. As can be seen from Figure 3,the channel passed through the boundary between the op-timal DIRs nine times. The two update-error powers of thedual switched system indicate the repeated transition of thechannel through the imaginary axis in the z-plane. However,Figure 3 shows that the update-error power of the adaptivesystem is very similar to that of the update-error power of theprefilter with (1/

√2,−α/√2) as DIR. This indicates that the

adaptive system was unable to detect the channel transitiondue to the high noise power (SNR = 9.5 dB). The adaptivesystem was initially able to track the channel as the chan-nel ISI was low (initial radius = 0.4), however as the ISI in-creased, Figure 3 shows that the adaptive system failed afterthe 1.2 × 105th symbol instant (indicated by the large os-cillations in the adaptive update-error power), yet the dualswitched system continued to track successfully. Figure 3 il-lustrates an example of an environment where ISI resulted inthe failure of the adaptive system yet the dual switched sys-tem was successful.

Figure 4 shows the update-error power for the dualswitched DIR MLSD-VA and the adaptive DIR MLSD-VA

0 1 2 3 4 5 6 7 8×104Symbols

0 1 2 3 4 5 6 7 8×104Symbols

−40

−30

−20

−10

0

Upd

ate-

erro

r(d

B)

−40

−30

−20

−10

0

Upd

ate-

erro

r(d

B)

Adaptive DIR

DIR (1/√

2,−α/√2) DIR(1/√

2,α/√

2)

Figure 4: Update-error powers of dual switched and adaptive re-ceivers for a SNC with constant trajectory. Parameters: radius =0.85, dθ/dt = π/(5× 104), SNR = 20 dB.

systems operating in a channel consisting of the determin-istic SNC and AWGN. The zeros of the SNC were moved ina circular trajectory. In this case, the channel was slowly timevarying and had a low noise power. Figure 4 shows that thedual switched DIR MLSD-VA successfully tracked the chan-nel, however, the adaptive system fails to track the channelresulting in failure of the adaptive system. This was termed acatastrophic error mode in adaptive predictive DIR MLSD-VA in [19].

The ability of the proposed predictive switched dual DIRMLSD-VA to track a dynamic channel was investigated us-ing the SNC. Figures 2, 3, and 4 show examples of envi-ronments where the proposed dual system was successfulin tracking the dynamic channel and the adaptive systemfailed. A more rigorous examination of the performance ofthe fixed and adaptive DIR MLSD-VA can be obtained fromBER.

4.2. BER for predictive DIR MLSD-VA systems

BER were obtained for the receivers using a BPSK transmis-sion system. These were obtained for four different configu-rations of the predictive receiver:

(i) system 1 is the dual switched DIR MLSD-VA;(ii) system 2 is the length two adaptive DIR MLSD-VA;

(iii) system 3 is the triple switched DIR MLSD-VA;(iv) system 4 is the length three adaptive DIR MLSD-VA.

The four systems were used so that comparison betweenchannel shortening to length-two DIRs (as in the case of sys-tems 1 and 2) and channel shortening to length three DIRs(in the cases of systems 3 and 4) could be made. The fixedDIRs used in system 3 are listed in Section 2.

The channel model used consisted of time and fre-quency selective channels with continuous power delay pro-files (PDPs) as proposed by Hoeher [20] that explicitly


account for outdoor mobile channel characteristics at 1 GHz.Hoeher suggested that the equivalent baseband model can bewritten as

c(t; τ) = 1√P

P∑ν=1

exp[j(2π fD,νt + θν

)] · δ(τ − τν), (8)

where P is the number of elementary echo paths and δ(·) isDirac’s delta function. Impulse responses can easily be ob-tained from (8) by independently obtaining the following:

(a) P Doppler frequencies fD,ν from a random variablewith Jakes probability density function in (− fD,max,fD,max);

(b) P initial phases θν from a uniformly distributed ran-dom variable in [0, 2π);

(c) P echo delay times τν .

The echo delay times were exponentially distributed thusallowing the length of the power delay profile (PDP) to be de-termined by altering the decay value of the exponential dis-tribution. Only Rayleigh fading is considered throughout thispaper.

To test the relative abilities of the four systems to suc-cessfully shorten a CIR, the four systems were compared us-ing samples of c(t, τ) that had two different mean echo delaytimes (impulse response durations). Motivated by the factthat systems 1 and 2 have DIRs of length 2T and systems 3and 4 have DIRs of length 3T , we used channels with meanecho delay times of 5T and 7T . The four systems were alsotested using fD,max = 1 Hz, 3 Hz, 12 Hz, and 25 Hz with eachof the two different echo delay times. In order to ensure (1)meaningful measurements of BER and (2) a satisfactory ap-proximation of the channel by a sample impulse responseof c(t, τ), we used a large observation period. The channelwas faded for each bit. Let Bi denote the ith burst containing1.5× 105 bits with values from −1, 1. The first 3× 104 bitsof each burst were used for training. For each BER diagram,a total of 1.3× 105 bursts were transmitted. A signalling rateof 7.5 MBd was used. It is important to note that the resultswould be significantly improved with the use of coding andinterleaving.

Figures 5 and 6 show BER for the four systems being in-vestigated. The mean echo delay time (MEDT) of the chan-nels used to obtain the results in Figures 5 and 6 was 5T . Fig-ures 5a and 5b show that system 3 is the least suitable of thesystems in a slowly time varying channel. Figures 6a and 6bshow that as the maximum Doppler spread increases, system2 becomes the least successful of the four systems. Figure 6ashows that at high SNR and fD,max = 12 Hz, the performanceof systems 2 and 4 become almost identical. It can be seenfrom Figures 5 and 6 that as fD,max increases the most suitableadaptive DIR changes from length 2T to length 3T . System 1performs considerably better than any of the other systems asthe Doppler spread increases.

Figures 7 and 8 show the BER for the four systems beinginvestigated for channels with mean echo delay times of 7Tfor a range of maximum Doppler spreads. As expected, theperformance of each of the systems has decreased in com-

5 10 15 20 25 30

Eb/N0 (dB)

10−5

10−4

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erro

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Dual switchedAdaptive (L2)

Triple switchedAdaptive (L3)

(a) Maximum Doppler frequency = 1 Hz.

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10−4

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(b) Maximum Doppler frequency = 3 Hz.

Figure 5: BER of DIR MLSD-VA, MEDT = 5T .

parison to the BER for channels with mean echo delay timesof 5T .

Figures 7a and 7b show that for slowly time varyingchannels, the adaptive DIR system of length 2T (adaptive2T DIR) is again more suitable than the adaptive 3T DIR.This indicates that in slowly time varying environments re-quiring channel shortening, the adaptive 2T DIR is able tosuccessfully converge to either of the two optimal DIRs. Allthe BER indicate that system 3 performs poorly in low SNR


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channels. The only observable merit in using system 3 isshown in Figures 7 and 8 in high SNR environments. Sys-tem 1 is again shown to offer the best performance of thefour systems under consideration in these difficult operatingconditions.

The length-two adaptive DIR MLSD-VA has a VA imple-mentation cost C, the dual switched system has a VA cost 2C,the length-three adaptive system also has a VA cost of 2C,and the triple switched system has a VA cost of 6C. The BER

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show that the dual switched DIR system offers the best per-formance for the VA cost.

A surprising result is that the dual switched system out-performs the triple switched system that has three times theVA cost. However, the result can be explained by noting thatwith the length two system all optimum DIRs are imple-mented, whereas with the length-three DIR triple switchedsystem, only three DIRs out of an infinite number of possi-bilities are covered.


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5. CONCLUSIONS

The performance of switched DIR systems to reduce thecomplexity of an MLSD-VA required for a range of dynamicchannels has been investigated. It is readily seen from theresults that the switched dual DIR MLSD-VA exhibits supe-rior performance in each of the dynamic environments whencompared with similar adaptive 2T DIR systems. For dou-bly selective multipath radio channels where the adaptive 3T

5 10 15 20 25 30

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Dual switched (DS)Modified DS

Triple switched (TS)Modified TS

Figure A.1: BER for modified and original DIR systems withMEDT = 7T . Maximum Doppler frequency = 1 Hz.

DIR system is more suitable than the adaptive 2T DIR sys-tem, BER curves show that the dual switched DIR system forthe same VA implementation cost offers a considerable im-provement in performance.

The behaviour of switched dual DIR MLSD-VA andadaptive 2T DIR systems have been studied using severaldifferent channel models. It has been shown that there ex-ists a range of dynamic environments where the adaptiveDIR system fails because of either ISI or high noise power.It was shown that in these dynamic environments, the dualswitched DIR MLSD-VA was successful in tracking the chan-nel behaviour.

APPENDIX

BER are presented in Figures A.1 and A.2, which indi-cate the efficacy of using the modified DIRs [14] for thedual switched DIR MLSD-VA system in comparison tothe original DIRs suggested by Falconer and Magee [3]and Fredricsson [4]. The BERs shown in Figures A.1 andA.2 were obtained using time varying frequency selectiveRayleigh fading channels as described previously. The chan-nels had a mean PDP of length 7T and a range of maxi-mum Doppler spreads. The results show that in slowly timevarying channels the effect of the catastrophic error that re-sults from the use of the original DIRs reduces the effec-tiveness of the switched DIR MLSD-VA systems consider-ably. As the maximum Doppler spread increases, the effectof the catastrophic error mode decreases as can be seen inFigure A.2b.


5 10 15 20 25 30

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Figure A.2: BER for modified and original DIR systems withMEDT = 7T .

REFERENCES

[1] G. D. Forney, “Maximum likelihood sequence estimation ofdigital sequences in the presence of intersymbol interference,”IEEE Transactions on Information Theory, vol. 18, no. 3, pp.363–378, 1972.

[2] S. U. H. Qureshi and E. E. Newhall, “Adaptive receiver for datatransmission over time-dispersive channels,” IEEE Transac-tions on Information Theory, vol. 19, no. 4, pp. 448–457, 1973.

[3] D. D. Falconer and F. R. Magee Jr, “Adaptive channel memorytruncation for maximum likelihood sequence estimation,”Bell Syst. Tech. J., vol. 52, no. 9, pp. 1541–1562, 1973.

[4] S. A. Fredricsson, “Optimum transmitting filter in digitalPAM systems with a Viterbi detector,” IEEE Transactions onInformation Theory, vol. 20, no. 4, pp. 479–489, 1974.

[5] E. Dahlman, “New adaptive Viterbi detector for fast-fadingmobile-radio channels,” Electronics Letters, vol. 26, no. 19, pp.1572–1573, 1990.

[6] M.-C. Chiu and C.-C. Chao, “Analysis of LMS-adaptive MLSEequalization on multipath fading channels,” IEEE Trans. Com-munications, vol. 44, no. 12, pp. 1684–1692, 1996.

[7] J. G. Proakis, “Adaptive equalization for TDMA digital mobileradio,” IEEE Transactions on Vehicular Technology, vol. 40, no.2, pp. 333–341, 1991.

[8] N. Seshadri, “Joint data and channel estimation using blindtrellis search techniques,” IEEE Trans. Communications, vol.42, no. 2–4, pp. 1000–1011, 1994.

[9] R. Raheli, A. Polydoros, and C.-K. Tzou, “Per-survivor pro-cessing: a general approach to MLSE in uncertain environ-ments,” IEEE Trans. Communications, vol. 43, no. 2–4, pp.354–364, 1995.

[10] M. R. Boyle and A. D. Fagan, “Dual DIR equalisation of dy-namic channels,” in IST 2000 Mobile Summit, pp. 719–724,Galway, Ireland, 2000.

[11] M. R. Boyle and A. D. Fagan, “Switched DIR Viterbi equalisa-tion for frequency selective fading channels,” in COMCON 8,Crete, Greece, 2001.

[12] M. R. Boyle, Switched desired impulse response maximum like-lihood sequence detectors for dynamic channels, Ph.D. thesis,University College Dublin, Ireland, 2001.

[13] D. D. Falconer and F. R. Magee Jr, “Evaluation of deci-sion feedback equalisation and Viterbi algorithm detectionfor voiceband data transmission-Part I,” IEEE Trans. Com-munications, vol. 4, pp. 1130–1139, October 1979.

[14] M. R. Boyle and A. D. Fagan, “Switched DIR equalisation ofdynamic channels and a degenerative error mode,” in EURO-CON 2001, pp. 18–21, Bratislava, Slovak Republic, July 2001.

[15] D. R. Morgan and S. G. Kratzer, “On a class of computa-tionally-efficient, rapidly-converging, generalized NLMS al-gorithms,” IEEE Signal Processing Letters, vol. 3, no. 8, pp.245–247, 1996.

[16] R. D. Gitlin and S. B. Weinstein, “On the required tap-weightprecision for digitally implemented, adaptive, mean squaredequalisers,” Bell Syst. Tech. J., vol. 58, no. 2, pp. 301–321, 1979.

[17] G. Long, F. Ling, and J. G. Proakis, “The LMS algorithm withdelayed coefficient adaptation,” IEEE Trans. Acoustics, Speech,and Signal Processing, vol. 37, no. 9, pp. 1397–1405, 1989.

[18] Y. Gu and T. Le-Ngoc, “Adaptive combined DFE/MLSE tech-niques for ISI channels,” IEEE Trans. Communications, vol.44, no. 7, pp. 847–857, 1996.

[19] M. R. Boyle and A. D. Fagan, “A catastrophic error mode inadaptive predictive DIR equalisation of dynamic channels,” inSIPS 2001, Antwerp, Belgium, 2001.

[20] P. Hoeher, “A statistical discrete time model for the WSSUSmultipath channel,” IEEE Trans. Vehicular Technology, vol. 41,no. 4, pp. 461–468, 1992.


Michael Boyle received his B.Eng. degree inelectronic engineering from University Col-lege Dublin in 1993. He joined the Digi-tal Signal Processing Group, University Col-lege Dublin. He received his Ph.D. in 2001.His current research interests are in the fieldof signal processing, equalisation, reducedcomplexity maximum likelihood sequencedetection, and channel modelling.

Anthony D. Fagan received the Ph.D. de-gree from University College Dublin (UCD)in 1978. He held a position as a research en-gineer at Marconi Research Laboratories, inEssex from 1977 to 1980 where he workedon digital signal processing (DSP) for ad-vanced communication systems. In 1980, hetook up a position of Lecturer in the De-partment of Electronic and Electrical En-gineering at UCD where he established theDSP Research Group. The group carries out a balanced mix of the-oretical and applied research in the areas of digital communica-tions (wireline and wireless), speech and audio processing, imageprocessing, pattern recognition, and biomedical signal processing.Professor Fagan’s main research interest is in the application of DSPtechniques to advanced digital communications. He is widely con-sulted by many international communication companies. ProfessorFagan is a reviewer for many international journals and is currentlyon the editorial board of the Academic Press journal DIGITAL SIG-NAL PROCESSING. On behalf of the European Commission, heacted as a technical auditor of mobile communications projects inthe RACE and ACTS programmes. He is currently technical evalu-ator of projects in the ESPRIT programme.


Spatial Block Codes Based on Unitary TransformationsDerived from Orthonormal Polynomial Sets

Giridhar D. MandyamNokia Research Center, 6000 Connection Drive, Irving, TX 75039, USAEmail: [email protected]

Received 1 September 2001 and in revised form 15 March 2002

Recent work in the development of diversity transformations for wireless systems has produced a theoretical framework for space-time block codes. Such codes are beneficial in that they may be easily concatenated with interleaved d trellis codes and yet stillmay be decoded separately. In this paper, a theoretical framework is provided for the generation of spatial block codes of arbi-trary dimensionality through the use of orthonormal polynomial sets. While these codes cannot maximize theoretical diversityperformance for given dimensionality, they still provide performance improvements over the single-antenna case. In particular,their application to closed-loop transmit diversity systems is proposed, as the bandwidth necessary for feedback using these typesof codes is fixed regardless of the number of antennas used. Simulation data is provided demonstrating these types of codes’ per-formance under this implementation as compared not only to the single-antenna case but also to the two-antenna code derivedfrom the Radon-Hurwitz construction.

Keywords and phrases: spatial block codes, closed-loop transmit diversity, space-time codes.

1. INTRODUCTION

In wireless communications systems, fading transmissionchannels are problematic due to the fact that fading chan-nels are nonstationary, and therefore the design of effectivechannel codes based on assumed channel statistics becomesdifficult. As a result, diversity is essential for addressing theproblem of fading in wireless channels. Diversity essentiallyentails receiving several replicas of the same signal over in-dependently fading channels [1]. Diversity may take manyapproaches. For instance, frequency diversity methods em-ploy transmission of multiple symbol replicas over multi-ple carriers, each of the carriers separated in frequency bya sufficiently large amount to ensure independent fading.This approach is accompanied with the additional cost of in-creased complexity at the transmitter and receiver, along withthe fact that it may be difficult to implement in bandwidth-limited systems (such as common public wireless systemsthat must conform to electromagnetic compatibility require-ments). Temporal diversity entails transmission of signalreplicas in different time slots, each slot sufficiently spacedin time to ensure independent fading. This approach suffersfrom reduced throughput due to multiple transmissions ofthe same symbol over time. Another instance of temporal di-versity may be achieved in multipath channels where the sig-nal bandwidth is larger than the coherence time of the chan-nel; in this case the multipaths are resolvable and may be re-covered by a rake receiver.

However, flat fading channels are troublesome forbandwidth-limited systems where neither frequency nortemporal diversity is possible. In such conditions, antennadiversity is a concept that has gained much interest. Trans-mission of signal replicas over multiple antennas using sep-arable waveforms essentially results in a received signal thatmay be demodulated with a rake receiver. Usually, to achievesuch diversity, a spatial separation of at least ten wavelengthsbetween antennas is required to ensure independent-fadingconditions for signals associated with each antenna.

While antenna diversity is a desirable alternative for pub-lic wireless systems, the actual requirements for achievingoptimal diversity over such systems have recently been thesubject of several studies. The two questions at hand are

(1) can coding over multiple antennas have benefits oversimple diversity schemes, where multiple copies of thesame signal are transmitted over multiple antennas atdiscrete instances in time? In other words, can a space-time code be designed?

(2) If so, how do we optimally code to achieve the full ben-efits of diversity in these systems?

There have been several approaches to answer both ofthese questions. For instance, in [2] (which is an extensionof the authors’ earlier work in [3]), the authors provide acriterion for block code design for transmitter diversity sys-tems and demonstrate their benefits when proper channel

Spatial Block Codes Based on Unitary Transformations Derived from Orthonormal Polynomial Sets 855

estimation is possible through the use of pilot-symbol as-sisted modulation. This work has been addressed from aslightly different viewpoint in [4], wherein the authors con-struct generalized orthogonal space-time block codes basedon the Radon-Hurwitz construction for unitary matrices ofindeterminates. The performance data for these codes aregiven in [5].

Other approaches have been taken with respect to trans-mitter code design where trellis coding is incorporated. Forinstance, in [6] the authors derive several space-time trel-lis codes, which were found with respect to the product cri-terion, wherein the minimum of the product of distancesbetween all distinct code word pairs is maximized assum-ing that the rank of the code word difference matrices aremaximized. In [7], the authors provide a criterion for thedesign of space-time trellis codes by forming a search cri-terion different from [6] based on the assumption that op-timal codes will satisfy the same criterion for their distancespectra as traditional trellis codes used in the single antennacase.

In [8], the author derives a criterion for space-time codedesign based on the Euclidean distance between all possiblecode word pairs. This criterion is different from the productdistance used in [6, 7], but is shown to be a true metric.

In this paper, a general design methodology is pre-sented for spatial block codes based on orthogonal designs.The reason why these codes are referred to as spatial blockcodes rather than space-time block codes is that, as will beshown, these codes primarily involve spatial processing butnot temporal processing. Although the new design method-ology does not satisfy design criteria for diversity maxi-mization, they can be shown to be useful in closed-looptransmit diversity application. Simulation results are pro-vided to verify the benefits of these codes in closed-loopscenarios.

This paper is organized as follows. Section 2 provides anoverview of the design criteria for space-time block codes.Section 3 presents a general framework for construction ofspatial block codes from unitary transform matrices and in-troduces an application of these codes to closed-loop trans-mit diversity systems. Section 4 provides simulation resultsusing the proposed codes. Section 5 includes a discussion onthe significance of the results and directions for future work.

2. SPACE-TIME BLOCK CODES: DESIGN CRITERIA

In this section, the criterion for optimal space-time blockcodes are derived and presented. This criterion has been de-rived and presented in previous work (e.g., [2, 6]). Given aspace-time block code designed for L antennas for durationof K epochs, the transmitted code words may be defined bya K × L matrix D

D(t) =

d1t d2

t · · · dLtd1t+1 d2

t+1 · · · dLt+1...

.... . .

...d1t+K−1 d2

t+K−1 · · · dLt+K−1

, (1)

where the matrix entries dit represent the modulation symboltransmitted over the ith antenna at time t (t being in multi-ples of the symbol duration). Given a single-antenna receiver,the received signal may be represented as

x(t) =L∑i=1

ditci(t) + n(t), (2)

where ci(t) is the complex channel gain at time t of the signaltransmitted from the ith antenna and ni(t) is the associatedGaussian noise. If it is assumed that the channel estimate,corresponding to the channel as seen from each antenna, isseparable at the receiver (by means of orthogonal waveformcoding, for instance) and that the complex channel gain andnoise for each antenna remain constant over K epochs, thenthe signal corresponding to the entire code matrix receivedover the K-epoch duration of the space-time code may berepresented as

x(t) = D(t)c(t) + n(t), (3)

where the vector x(t) is a K×1 observation vector, D(t) is theK × L code word matrix, and c(t) is the L × 1 channel gainvector defined as

c(t) = [c1(t), c2(t), . . . , cL(t)

], (4)

and n(t) is the K × 1 noise vector defined as

n(t) = [n(t), n(t + 1), . . . , n(t + K − 1)

]. (5)

Given the received signal vector x(t) and assuming perfectchannel estimation at the receiver, the maximum a posterioridetector is given as

D(t) = maxDγ∈S

p(

Dγ | c(t), x(t)), (6)

where S is the set of all possible codematrices, D(t) is the de-tected codematrix, and p(arg) is the probability density func-tion of arg. If it is assumed that each Gaussian noise sample isindependent, zero-mean with variance σ2, then the pdf usedin (6) may be found as

p(

Dγ | c(t), x(t))

= (2πσ2)−K/2e−(1/2σ2)(x(t)−Dγc(t))H (x(t)−Dγc(t)).

(7)

Using (7), the probability of decoding error may be foundas

Perror = P(

D(t) = Dα | D(t) = Dβ)

= Pp(

Dα | c(t), x(t))> p

(Dβ | c(t), x(t)

)= P

ln(p(

Dα | c(t), x(t)))< ln

(p(

Dβ | c(t), x(t)))

.(8)


The relationship of (8) may be simplified to

Perror = P(

x(t)−Dαc(t))H(

x(t)−Dαc(t))

<(

x(t)−Dβc(t))H(

x(t)−Dβc(t)).

(9)

Noting that in the original expression in (7), it was assumedthat if D(t) = Dβ, then x(t) = Dβc(t) + n(t). Therefore, theerror probability may be simplified to

Perror = P

2 Re[

nH(t)(

Dβ −Dα)

c(t)]

> cH(t)(

Dβ −Dα)H(

Dβ −Dα)

c(t).

(10)

Observing (10), it is clear that the probability of errordecreases as the term on the right-hand side of the inequalityincreases. Noting that (Dβ − Dα) is a K × L matrix, it maybe decomposed using singular value decomposition. As a re-sult, (Dβ − Dα) is equivalent to VHΣW, where V is a K × Kunitary matrix, W is an L × L unitary matrix, and Σ is aK × L matrix whose diagonal entries are the singular val-ues in order of value of (Dβ − Dα) (i.e., the eigenvalues of(Dβ − Dα)H(Dβ − Dα)). Therefore, the following equationmay be derived:

cH(t)(

Dβ −Dα)H(

Dβ −Dα)

c(t)

= cH(t)VHΣWWHΣHVc(t)

= cH(t)VHΣΣHVc(t).

(11)

If we assume that Σ has the structure diag[λ1, λ2, . . . ,λr , 0, . . .], where λk denotes the kth nonzero eigenvalue of(Dβ−Dα)H(Dβ−Dα), then taking into account the unitarityof V, then (11) may be further simplified as

cH(t)(

Dβ −Dα)H(

Dβ −Dα)

c(t)

= cH(t)VHΣΣHVc(t)

= cH(t)ΣΣHc(t)

=r∑

i=1

λ2i c

2i (t).

(12)

Quite clearly, the larger the rank of the L × L matrix(Dβ − Dα)H(Dβ − Dα), the lower the decision error prob-ability. If this matrix is full rank, then the maximum gainsfrom diversity are achieved. However, this criterion is gen-eral, and it would be of interest for code design to find a nar-rower criterion. This may be accomplished by examining ofcH(t)(Dβ − Dα)H(Dβ − Dα)c(t) in the mean sense. Firstly,it is assumed that the transmitted symbol energy from eachantenna is Es. Since the channel itself does not create or de-stroy energy, the mean energy from the complex channel gaincoefficients as seen at the receiver should be Ec2

i (t) = Es.

Therefore, the following equations may be derived:

E

cH(t)(

Dβ −Dα)H(

Dβ −Dα)

c(t) | Dβ,Dα

= E

r∑i=1

λ2i c

2i (t)

=r∑

i=1

Eλ2i c

2i (t)

=r∑

i=1

Eλ2i

Ec2i (t)

=r∑

i=1

Eλ2i

Es.

(13)

In [8], the author proposes that EcH(t)(Dβ−Dα)H(Dβ−Dα)c(t) | Dβ,Dα may be bounded using the Cauchy-Schwartz inequality assuming that the singular values of (Dβ−Dα) are deterministic. As a result, the relationship in (13) maybe bounded as

E

cH(t)(

Dβ −Dα)H(

Dβ −Dα)

c(t) | Dβ,Dα

=

r∑i=1

Eλ2i

Es

=r∑

i=1

λ2i Es

≤√√√√( r∑

i=1

λ4i

)√rE2

s .

(14)

Therefore, if (Dβ − Dα)H(Dβ − Dα) is a diagonal ma-trix with all entries of the diagonal being equal, the boundof (14) becomes tight. However, even the singular values of(Dβ−Dα) are in fact not deterministic in the mean-sense, dueto the fact that for all given code words these values are func-tions of the mean code word differences. Therefore, given aset of code word symbols which may be transmitted, one mayuse the distribution of all possible code word symbol differ-ences to form an expression for EcH(t)(Dβ − Dα)H(Dβ −Dα)c(t). It is clear that if (Dβ −Dα)H(Dβ −Dα) is diagonaland all entries along the diagonal are nonzero, then the maxi-mum gain from diversity is achieved. In this case, the singularvalues of (Dβ − Dα) are functions of the symbol differencesbetween the code words.

3. SPATIAL BLOCK CODE DESIGN FROM UNITARYTRANSFORM MATRICES: A GENERAL DESIGNFRAMEWORK

As established in Section 2, the desired criterion for space-time block code design is to find codes whose differencematrices satisfy the condition that there exists the maxi-mum number of singular values associated with these matri-ces. One such construct, as discussed in [4], is the Radon-Hurwitz unitary matrix construction. A set of k unitary


matrices BI of size L×L is part of the Radon-Hurwitz fam-ily if the following three rules hold:

BTi Bi = I,

BTi = −Bi, 1 ≤ i ≤ k,

BiBj = −BjBi, 1 ≤ i, j ≤ k.

(15)

In [9], the author presented a code that corresponded toa special case of the Radon-Hurwitz family

D(t) =[

s1 s2

−(s2)∗ (s1)∗

], (16)

where si are the set of symbols to be transmitted overthe K time epochs of the code (in this case, K = 2). Thecode in (16) is an example of a rate 1 code, where the num-ber of symbols transmitted is equal to the number of timeepochs required for the code. Such codes are desirable forbandwidth-limited systems. In [4], the authors show that forthe Radon-Hurwitz family of code constructs, rate 1 designsexist for real constellations only for L = K = 2, 4, and 8.The authors conclude that real orthogonal designs thereforeexist only for these dimensions. The singular values for thecode word difference matrix (Dβ − Dα) can be shown to be(see [4])

λ2i =

L∑l=1

∣∣sαl − sβl∣∣2 ∀i. (17)

In (17),[sα1 sα2 · · · sαL

]is the first row of Dα and[

sβ1 sβ2 · · · sβL]

is the first row of Dβ. Since it is assumedthat Dα differs from Dβ in at least one position, all the sin-gular values of the code word difference matrix are nonzero,that is, r = L in (13). As a result, the relationship of (13) maybe found as

E

cH(t)(

Dβ −Dα)H(

Dβ −Dα)

c(t) | Dβ,Dα

=

r∑i=1

Eλ2i

Es

= EsLL∑l=1

∣∣sαl − sβl∣∣2.

(18)

However, given an L × L unitary matrix U whose ele-ments are denoted by Ui j , then by forming a diagonal matrixG = diag[s1, s2, . . . , sL], a code word matrix may be formedas D(t) = GU. This matrix will be unitary assuming thatthe symbol constellation points have equal magnitude, and un-der this assumption the singular values of the code worddifference matrix (Dβ − Dα) are simply [|sα1 − sβ1|2, |sα2 −sβ2|2, . . . , |sαK − sβK |2]. This type of design was demonstratedin [2] for a specific code, but taking into account the fact thatunitary matrices may be constructed from sets of orthonor-mal polynomials, a general method for designing block codesbased on unitary matrices may be specified. As a result, therelationship in (13) for this type of code, herein denoted as

a simple orthogonal code, becomes

E

cH(t)(

Dβ −Dα)H(

Dβ −Dα)

c(t) | Dβ,Dα

=L∑i=1

Eλ2i

Es

= Es

r∑i=1

∣∣sαi − sβi∣∣2.

(19)

Comparing (18) to (19), it can be seen that at best, theperformance of the simple orthogonal code can match thatof the Radon-Hurwitz code for any given code word pair.This is due to the fact that although the first column ofDα and that of Dβ are distinct, they may differ in at leastone position. Therefore, to analyze the diversity gain of asimple orthogonal code, we must analyze it in the mean-sense. This would mean that we should look at the aver-age rank of (Dβ − Dα)H(Dβ − Dα) rather than the rank of(Dβ − Dα)H(Dβ − Dα) for a particular code word pair. Thiswould be determined by the average number of positions inwhich the first column of Dα and that of Dβ differ for all pos-sible distinct code word pairs (Dα,Dβ).

Given a symbol alphabet of dimensionality M, the set ofall possible code words that make up the first column of thecode word matrix derived from a simple orthogonal code of

dimension L × L is ML. Given that there are(ML

2

)distinct

code word pairs, the average rank (i.e., Er, where r is thenumber of singular values of (Dβ − Dα)H(Dβ − Dα) for anycode word pair (Dα,Dβ)) is

Er =L∑i=1

ipi =∑L

i=1 iAi(ML

2

) , (20)

where Ai is the number of code word pairs differing in i posi-tions and pi is the probability that any two code words differin i positions,

pi = Ai(ML

2

) . (21)

Finding the general form for Ai is cumbersome; however,we may derive an upper bound for Er based on the valueof AL. This value may be shown to be

AL =M−2∑i=0

ML−1[(M − 1)L − i(M − 1)L−1]. (22)

Given this value, we can find the value of pL. In addition,since pi ≥ 0,

L∑i=1

pi=1=⇒ pL+L−1∑i=1

pi=1=⇒ pi ≤(1− pL

) ∀1 ≤ i < L.

(23)Therefore, the maximum value of Er based on the value ofAL may be derived as


Er =L∑i=1

ipi

= LpL +L−1∑i=1

ipi

≤ LpL +L−1∑i=1

i(1− pL)

≤ LpL + (L− 1)(1− pL).

(24)

Substituting in for pL, (24) may be expressed as

Er ≤ LAL(ML

2

) + (L− 1)

1− AL(ML

2

) , (25)

where the first term on the right-hand side of the inequalityrepresents the likelihood that a code word pair differs in allL positions, while the second term represents the likelihoodthat a code word pair differs in at most (L− 1) positions.

Similarly, a lower bound may be derived for Er. Wemay first derive the number of code word pairs that differin only one position

A1 = LML−1M−2∑i=0

(M − 1)− i. (26)

Therefore, the lower bound is

Er ≥ p1 + 2(1− p1

)=⇒ Er ≥ A1(

ML

2

) + 2

1− A1(ML

2

) .

(27)

In (27), the first term on the right-hand side of the inequal-ity represents the probability of a code word pair differingin only one position, and the second term represented theprobability of a code word pair differing in at least two posi-tions. The derivations of A1 and AL are explained further inthe appendix.

As an example, consider a 3-antenna code using QPSKsymbols. In this case, M = 4 and L = 3. Using (25) and (27),the following bounds are derived:

1.86 ≤ Er ≤ 2.43. (28)

To see how meaningful these bounds are, a simulation wasrun under one-path Rayleigh fading conditions at a velocityof 1 km/h using a 19.2 kbit/s transmission rate (2 bits/QPSK,symbol) and a carrier frequency of 1960 MHz. The demodu-lated bit error rate (BER) as a function of QPSK symbol, SNRwas compared between three-transmission methods: no di-versity, use of the 2 × 2 Radon-Hurwitz code as in (16), anda 3-antenna orthogonal code based on the discrete Fouriertransform (DFT) matrix (see (45)). The results are depictedin Figure 1. The simple orthogonal code provided nearly thesame performance as the 2-antenna Radon-Hurwitz code,which has diversity order 2. The bounds given in (28) pre-dict a mean diversity order near 2 as well.

0 2 4 6 8 10 12 14SNR (dB)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Bit

erro

rra

te

Single path3-antenna DFT2-antenna Radon-Hurwitz

Figure 1: Performance of simple orthogonal code.

Now that a general description of simple orthogonal spa-tial block codes has been presented, a general method for de-riving these codes may be formulated starting with a generalmethod for deriving U, the unitary transformation matrix.

3.1. The Gauss-Jacobi procedure for unitary transformmatrix derivation

A set of real polynomials Pk(x), each of degree k, is saidto be orthonormal with respect to the weighting functionp(x) over the support space Ω, if∫

ΩPi(x)Pj(x)p(x)dx = δi j , (29)

where δ is the Dirac delta operator. In order to ensure or-thogonality, the polynomials Pk(x) of degree greater thanzero must satisfy [10]∫

ΩPk(x)xmp(x)dx = 0, 0 ≤ m < k. (30)

The Lagrange interpolating polynomial for a set of n dis-crete sample points f (xk) of a function f (x) is defined as

Fn(x) =n∑

k=1

ω(x)(x − xk

)ω′(xk) f (xk), (31)

where ω(x) = (x − x1)(x − x2) · · · (x − xn) and ω′(xk)is the polynomial given by (xk − x1) · · · (xk − xk−1)(xk −xk+1) · · · (xk−xn). If f (x) is a polynomial of degree less thann, then Fn(x) = f (x). Otherwise, we can form the interpola-tory polynomial Fn(x) and represent f (x) as

f (x) = Fn(x) + r(x), (32)

where r(x) is a remainder polynomial. Integrating f (x) overΩ with respect to p(x), and assuming that the remainder


polynomial r(x) is negligible, we obtain

∫Ωf (x)p(x)dx ≈

n∑k=1

Ak f(xk), (33)

where

Ak =∫Ω

ω(x)(x − xk

)ω′(xk) p(x)dx. (34)

The right-hand side of (33) is commonly referred to asa quadrature formula, and leads to two known theorems (theproofs may be found in [11]). The first theorem is that thequadrature formula in (33) is interpolatory if and only if it isexact for all possible polynomials f (x) of degree less thanor equal to n− 1. The second theorem is that the quadratureformula in (33) is exact for all polynomials of degree less thanor equal to 2n − 1, if and only if (i) the quadrature formulain (33) is interpolatory and (ii) for all polynomials Q(x) ofdegree less than n,∫

Ωω(x)Q(x)p(x)dx = 0. (35)

Assuming a set of orthonormal polynomials Pk(x) oversupport space Ω, a discrete unitary transform matrix cannow be constructed. Assume that the polynomials Pk(x) arearranged in order of increasing degree, that is, deg(P1(x)) ≤deg(P2(x)) ≤ · · · ≤ deg(Pk(x)) ≤ · · · . If an N ×N unitarymatrix is desired, it can be generated by first taking the dis-cretization points xk as the roots of PN+1(x). If we form ω(x)from these points, we know that PN+1(x) is directly propor-tional to ω(x) and therefore, any polynomials orthogonal toPN+1(x) will also be orthogonal to ω(x). We also know thatif we define f (x) in (33) as the product of PN+1(x) and thearbitrary polynomial d(x) of degree less than N , then f (x) isa polynomial of degree less than or equal to (2N − 1) and

∫ΩPN+1(x)d(x)p(x)dx = 0 =

N∑k=1

AkPN+1(xk)d(xk), (36)

since PN+1(xk) = 0 for all k and since we can always find Ak

such that (33) is exact. Therefore, all polynomials of degreeless than n are orthogonal to ω(x) and thus by the secondtheorem, previously mentioned, (33) is exact for all polyno-mials of degree less than 2N−1. Thus, by the orthonormalitycondition of the polynomials Pk(x), we conclude that

∫ΩPi(x)Pj(x)p(x)dx = δi j =

N∑k=1

AkPk(xk)Pj(xk)

(37)

for i, j less than (N + 1). Therefore, if the (i, k) entry of anN × N matrix is formed by the value Pi(xk), we can find Ak

such that this matrix is unitary, that is, all the row vectors aremutually orthonormal.

To that end, we first consider the Christoffel-Darbouxidentity [11], which is defined as follows: given a set of or-thonormal polynomials Pn(x), each of order n, with the

nth-order term Pn(x) in each being of the form anxn, itcan be shown that

(x − t)n∑s=0

Ps(x)Ps(t)=− anan+1

[Pn+1(x)Pn(t)− Pn(x)Pn+1(t)

].

(38)In (38), if we set t to be the roots of Pn(x), that is, xk, then itcan be shown that

n−1∑s=0

Ps(x)Ps(xk) = − an

an+1

Pn(x)Pn+1(xk)

x − xk. (39)

Multiplying both sides by p(x) and integrating over Ω, we get

1 = − anan+1

Pn+1(xk) ∫

Ωp(x)

Pn(x)x − xk

dx. (40)

The result in (40) follows from the fact that the quantityPs(xk)

∫Ω Ps(x)p(x)dx equals 0 for s > 0, as a result of the

orthogonality condition in (30), and equals 1 for s = 0, as aresult of the orthonormality of P0(x). We note that the inte-gral on the right-hand side of (40) is similar to the definitionof Ak in (34), from which it follows that

Ak = − anan+1

1P′n(xk)Pn+1

(xk) . (41)

If we refer to the desired N × N unitary matrix as U, wecan define the elements of U as

Ui j =√AjPi

(xj). (42)

It should be noted that the theory presented here is not di-rectly applicable to complex orthonormal polynomials. Thiswill be discussed in more detail in Section 3.2.

3.2. Sample orthogonal designs

Returning to the code construct D(t) = GU, where G =diag[s1, s2, . . . , sL] and U is a unitary matrix, several codesmay be derived, which satisfy the rank criterion for the codeword difference matrix. We primarily concentrate on the 3×3case, as this is the lowest order where Radon-Hurwitz codesdo not exist. For instance, the discrete Fourier transform ma-trix of dimension L× L is derived from the rule

Flm = e− j2π(l−1)(m−1)/L√L

, (43)

where l is the row index ranging from 0 to (L − 1) and m isthe column index also ranging from 0 to (L − 1). Thus, re-turning to the terminology presented in Section 3.1, the set oforthonormal polynomials for the DFT matrix are simply de-scribed by Pn(x = e− j2πi/L) = xn = e− j2πin/L, where nis the order of the polynomial, and the normalization factorsare simply Aj = 1/L. Clearly, since the polynomial set is fullydescribed by a complex exponential, roots of zero do not ex-ist for any of these polynomials in the conventional case. Thismeans that much of the analysis presented in Section 3.1 isnot directly applicable to the DFT. However, we may find the


so-called roots of unity for these polynomials; it can be shownthat for the Lth degree complex polynomial xL that there areexactly L roots of unity for these complex exponentials [12].These roots of unity may be found at i = 0, 1, . . . , L − 1.These values will satisfy, for any k, l < L,

L−1∑r=0

Pk(xk)(Pl(xk))∗ = L−1∑

r=0

e− j2π(l−m)r/L = δ(l −m). (44)

In the 3 × 3 case, this relationship generates the transformmatrix

F3×3 =

1√3

1√3

1√3

1√3

e− j2π/3√

3e− j4π/3√

31√3

e− j4π/3√

3e− j8π/3√

3

. (45)

This matrix is a transpose of the one presented in [2, SectionIV.A]. This type of simple orthogonal code will be denotedas distance preserving. This means that at any given instantin time, for two distinct code words Dα = sα1, . . . , sαL andDβ = sβ1, . . . , sβL, the expected value of |sαi− sβi|2 does notchange for 1 ≤ i ≤ L. Since the DFT-derived simple orthog-onal code involves only phase shifts and any symbol trans-mitted at any instant in time over any antenna has a constantmagnitude, this code is in fact distance preserving.

Although the DFT matrix is well known, the fact that acomplex phase shift needs to be performed may not be desir-able. As a result, real-number transformations may be used.For instance, the discrete cosine transform, which is derivedfrom the roots of the Tchebychev polynomials Pn(x) =cos(n cos−1(x)), yields only real matrix entries. The generalform for the discrete cosine transform (DCT) matrix is

Clm =

1√L, l = 0,√

2L

cosπ(2m + 1)l

2L, l > 0.

(46)

The 3× 3 matrix associated with the DCT is

C3×3 =

1√3

1√3

1√3

1√2

0 − 1√2

1√6−√

23

1√6

. (47)

This matrix is not distance preserving, and as a result, theinstantaneous code word symbol differences will be differentfrom codes derived from the DFT. Assuming a QPSK con-stellation, an average code symbol difference may be derived,assuming perfect synchronization, however. Due to the factthat the matrix is unitary, the mean code word difference willbe equivalent to the block code derived from the DFT. How-ever, this does not imply how diversity will affect the perfor-mance of this code when other elements of a typical digital

communications system are considered (e.g., trellis coding,interleaving). This particular matrix may also not be desir-able, due to the fact that one of the entries is zero; this resultsin large peak-to-average ratios for the transmitted data. An-other possible transform matrix is based on the discrete La-guerre transform [10], which is based on the Laguerre poly-nomials. Due to the fact that this transform is not a sinu-soidal transform, no general closed form solution exists forthis transform. The 3×3 discrete Laguerre transform is givenbelow (rounded to 4 digits):

L3×3 = 0.8433 0.5277 0.1019−0.4927 0.6831 0.53920.2149 −0.5049 0.8360

. (48)

This matrix, which avoids the complex phase rotation ofthe DFT matrix, yet does not suffer from the same power-balancing problems from which DCT-derived matrix does.However, both of these codes are not distance preserving.

Many other codes based on unitary transform matricesexist. Considering that all these codes have identical perfor-mance in terms of diversity, the code chosen would be basedon not just raw symbol error rate but other criteria as well.

3.3. Enhancing diversity of simple orthogonal codes

Although (24) places an upper bound on the maximum di-versity achievable by simple orthogonal codes as defined inthis section, we may enhance the diversity performance ofthe code by implementing the code in a closed-loop trans-mit diversity method. Closed-loop transmit diversity meth-ods are methods that rely on feedback so that using a com-plex weighting of each of the symbols to be transmitted fromeach antenna, a coherent combination is possible at the re-ceiver. Essentially, this approach is used to pre-equalize thechannel prior to transmission. In contrast, transmit diversitymethods such as the Radon-Hurwitz space-time block codeof (16), which do not require receiver feedback, are also clas-sified as open loop.

One of the first approaches to this problem was providedin [13], where the authors proposed transmitting trainingsequences to several users in the network. These sequencesare transmitted over L antennas. If we assume that the chan-nel, as seen by a single user k with respect to L antenna el-ements at time t, may be represented by the channel vectorak(t) = [ak1(t)ak2(t) · · · akL(t)], where aki(t) is the complexchannel response for antenna i with respect to user k at timet, then the transmitter can make use of this information toscale each antenna input accordingly so that a coherent com-bination of the signals from each antenna is possible at thereceiver. Thus, if the receiver estimates the channel from eachantenna as ak(t) = [ak1(t)ak2(t) · · · akL(t)], then these esti-mates may be relayed to the transmitter. Thus, if we assumethat the signal d(t) is transmitted from each antenna at timet, the received signal after scaling would be

r(t)= a∗k1(t)ak1(t) + a∗k2(t)ak2(t) + · · · + a∗kL(t)akL(t) + n(t),(49)

where n(t) is an additive Gaussian noise term. Clearly, if


aki(t) ≈ aki(t) then the received SNR is maximized. Thisapproach has been narrowed to include quantized relativephase feedback in [14]. This approach has also been ad-dressed for two antennas in CDMA systems in [15].

However, the amount of coding given to the feedback in-formation and the latency of the feedback information be-come critical to performance of these systems. As a result,these systems tend to actually degrade performance with re-spect to space-time block coded systems such as the 2 × 2Radon-Hurwitz transformation at high mobile speeds. Forinstance, in [16] the authors present a theoretical frameworkfor the performance of closed-loop transmit diversity anddemonstrate how the performance degrades at high Dopplerwith respect to the Radon-Hurwitz code as a result of feed-back latency. More specifically, in [17] the authors show asevere degradation in performance of a 2-antenna closed-loop method versus a 2-antenna Radon-Hurwitz transfor-mation at speeds of 30 km/h or greater at 2 GHz carrier fre-quency in a CDMA system. Moreover, with respect to one-path Rayleigh fading conditions in a CDMA system, resultspresented in [15] actually demonstrated worse performancefor closed-loop transmit diversity methods with respect tonot using any diversity methods at all at speeds of 100 km/hunder certain high SNR conditions due to the additionaldegradation provided by fast power control.

In addition, closed-loop systems require increased band-width for feedback information as the number of antennasincrease. Balancing this need with the need for reliability onthe feedback information could result in suboptimal perfor-mance for a large number of antennas.

However, the use of simple orthogonal block codes couldbe used to address these problems in a closed-loop imple-mentation. Assume that we have a block of K transforma-tions to choose from for modulating the input data matrixinto the transmit antenna array. If each of these L × L blocktransformations can be grouped as T = [

T1 T2 · · · TK],

then knowing the channel estimates from each antenna,the receiver may make a prediction of the best availabletransform and feed this information back to the transmit-ter. Since the transforms may be generated for arbitrarydimensionality (as shown in Section 3), the feedback re-quires log2 K bits for an arbitrary number of antennas. As-suming that the estimated channel vector is still ak(t) =[ak1(t) ak2(t) · · · akL(t)

]and that this channel estimate

remains relatively constant over the L time epochs of theblock code, then the receiver transform selection T(t) forfeedback that maximizes SNR would be

T(t) = maxTi∈T

∥∥TiaTk (t)∥∥. (50)

This transform selection may be sent to the transmitter forapplication in the ensuing data sequence.

Each data sequence to which a transform is appliedshould include a means of error detection, for example, acyclic redundancy check (CRC). This is necessary due to thefact that the feedback of the transform selection may not beimplemented due to feedback error. However, using an errordetection mechanism such as a CRC, the receiver may decode

the received data sequence using multiple hypotheses testing,with up to K hypotheses. A simple decoding algorithm at thereceiver may be attained:

(1) determine appropriate transform for the next data se-quence and relay selection to transmitter;

(2) for the next received data sequence, apply selectedtransform and decode. If CRC passes, return to step(1) for next data sequence;

(3) if CRC fails, sequentially apply each of the other K − 1possible transforms to the received data and decode.If a CRC passes for a transform, return to step (1) fornext data sequence;

(4) classify the received data sequence as an erasure. Re-turn to step (1) for next data sequence.

The drawback of this type of method is that using CRCsfor short data sequences could severely impact throughput.As a result, this type of feedback mechanism would in prac-tice perform relatively slowly with respect to channel condi-tions. On the other hand, this method is not as sensitive tofeedback errors as the method described in [15] due to theuse of multiple hypotheses testing. More importantly, how-ever, this method will still provide diversity gains at fast fad-ing conditions, despite the fact that the feedback mechanismis highly inaccurate in these types of channel conditions. Thisis due to the fact that these simple orthogonal block codesprovide at least the diversity order given in (27). Therefore,for instance, a 3-antenna code for a QPSK constellation willalways provide mean diversity order of nearly 2, regardless offeedback error.

4. EXAMPLE: QPSK SYSTEM

The 3 × 3 block codes presented in Section 3.2 were simu-lated in a simple QPSK system under single-path Rayleighfading conditions [18]. It was not merely of interest to de-termine the benefit of the proposed codes versus no diver-sity, but also to measure the difference in performance be-tween the dual and triple antenna cases. For the closed-loopmethod, codes used were based on the DFT, DCT, and dis-crete Laguerre transform (DLT) as described in Section 3.2.In addition, the conjugate transposes of these matrices werealso used for the closed-loop method. For comparison, the2 × 2 Radon-Hurwitz code and 2 × 2 closed-loop methodresults were provided, in addition to ideal triple-diversity re-sults. It should be noted that only two transforms were usedfor the 2-antenna closed-loop method, the DLT and the DFT.This is due to the fact that the 2× 2 DCT transform is iden-tical to the 2× 2 DFT.

The system under consideration was a QPSK system thatmapped two bits to each constellation point. It is assumedthat pilot signals from different antennas arrive at the receiversimultaneously and are separated using orthogonal wave-form modulation; for simulation purposes, however, perfectchannel knowledge at the receiver was assumed. If the signalsfrom different antennas did not arrive simultaneously, thenself-interference would occur due to imperfect suppression


−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0QPSK symbol SNR (dB)

Velocity 1 km/h

102

101

100

Bit

erro

rra

te

One path2-antenna perfect CL3-antenna perfect CLRadon-Hurwitz2-antenna new3-antenna new

Figure 2: 1 km/h results.

of other antenna signals when demodulating the signal froma particular antenna.

An information source at 12.8 kbit/s was assumed. Thissource was passed into a rate 1/3, constraint length 9, con-volutional encoder, and block interleaved. The interleaveddata was then modulated using a QPSK constellation. Nopower control was assumed. The carrier frequency assumedwas 1960 MHz. Under such conditions, the diversity perfor-mance for different space-time block coding methods maybe isolated for evaluation. The metric for performance, how-ever, was BER after decoding. 192000-bit simulations wererun for each given code, velocity, and QPSK symbol SNR.The results for 1 km/h, 10 km/h, and 100 km/h are shown inFigures 2, 3, and 4. In these figures, single path results areprovided, and the proposed closed-loop method results fortwo and three antennas are designated as 2-antenna new and3-antenna new. In addition, perfect closed-loop transmit di-versity results are provided that emulate closed-loop trans-mit diversity with no feedback delay or error. The results for2 and 3 antennas are designated as 2-antenna perfect CL and3-antenna perfect CL, respectively.

The simulation results show benefits not only to space-time block coding but also to increasing from two anten-nas (Radon-Hurwitz) to three (using the proposed method)in certain situations, particularly at low SNR (as much as4 dB performance improvement at 1 km/h velocity). The newclosed-loop methods did start to degrade in the 3-antennacase versus the Radon-Hurwitz block code at high veloci-ties and high SNR, but this is most likely due to the limitedset of transform choices. At high speeds, the 3-antenna codenot only performed well with respect to the Radon-Hurwitzat low SNR, but also provided very little degradation (less

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

QPSK symbol SNR (dB)

Velocity 10 km/h

102

101

100

Bit

erro

rra

te



−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

QPSK symbol SNR (dB)

Velocity 100 km/h

103

102

101

100

Bit

erro

rra

te



than 0.5 dB) at high SNR. Therefore, the proposed methodshows promise in increasing the crossover Doppler frequency[16], that is, the Doppler frequency at which an open-loopmethod such as the Radon-Huwitz transform outperforms aclosed-loop method. This is a result of the proposed methodreverting back to the performance bounds described by themean diversity performance derived in (28). The 2-antenna


closed-loop method did not provide quite the gains of the3-antenna closed-loop method, but this was also most likelydue to an even more limited transform set size than the 3-antenna case (once again, due to the fact that many of thetransform kernels used provide the exact same transformmatrix in the 2× 2 case).

5. CONCLUSIONS

A general framework for deriving space-time block codes waspresented. This framework involves starting with sets of or-thonormal polynomials and deriving unitary transform ma-trices from these sets. These transform matrices may in turnbe used to generate orthogonal spatial block codes. Simu-lation results in a closed-loop deployment show benefit forthis approach to code generation as opposed to the approachpresented in [4] under certain scenarios, as these codes maybe defined for arbitrary dimensions and their usage in theproposed closed-loop framework did not result in a signifi-cant degradation in performance at high velocities. However,since these codes do not maximize diversity in the mean-sense for a given dimensionality, further analysis should beperformed on methods for increasing the diversity of thesecodes in typical wireless environments.

APPENDIX

CODE WORD PAIR DIFFERENCE PROBABILITY

Assume a diversity transformation of rate 1 using L anten-nas, and a symbol constellation set of cardinality M. Eachpossible code word may be represented as a base-M numberconsisting of L digits. All possible code words may be listedas follows:

0

L−2︷︸︸︷0 · · · 0 0

0

L−2︷︸︸︷0 · · · 0 1

......

...

0

L−2︷︸︸︷0 · · · 0 (M − 1)

0

L−2︷︸︸︷0 · · · 1 0

......

...

1

L−2︷︸︸︷0 · · · 0 0

......

...

1

L−2︷︸︸︷0 · · · 0 (M − 1)

1

L−2︷︸︸︷0 · · · 1 0

......

...

(M − 1)

L−2︷︸︸︷(M − 1) · · · (M − 1) (M − 1).

(A.1)

If we examine only the code words designated by the nu-merals

0

L−2︷︸︸︷0 · · · 0 0 through 0

L−2︷︸︸︷0 · · · 0 (M − 1),

it can be seen that there are∑M−2

i=0 (M−1)− i code word pairsthat differ in only one position. Similarly, if we examine onlythe code words designated by the numerals

0

L−2︷︸︸︷0 · · · 1 0 through 0

L−2︷︸︸︷0 · · · 1 (M − 1),

it can be seen that there are still∑M−2

i=0 (M − 1)− i code wordpairs that differ in only one position. In fact, if we examinethe code words designated by the numerals

s0

L−2︷︸︸︷s1 · · · sL−2 0 through s0

L−2︷︸︸︷s1 · · · sL−2 (M − 1)

for arbitrary symbols s0, s1, . . . , sL−2, then the same num-ber of code word pairs differing in one position remains as∑M−2

i=0 (M−1)− i. As a result, there are ML−1∑M−2

i=0 (M−1)− icode word pairs that only differ in the last position. This rela-tionship also holds true for code word pairs differing only inthe second-to-last position, and so on for all remaining L− 2positions. As a result, the total number of code word pairsdiffering in only one position isA1 = LML−1

∑M−2i=0 (M−1)−i.

The next case to be examined is the number of codeword pairs that differ in all L positions. For instance,take the code words defined by 0 s1 · · · sL−2 sL−1, thatis, code words which have 0 for the first digit. For anygiven value of 0 s1 · · · sL−2 sL−1, there exist (M − 1)L

code word pairs which differ from 0 s1 · · · sL−2 sL−1in L positions. Since there are ML−1 possible values for0 s1 · · · sL−2 sL−1, there exist ML−1(M − 1)L codeword pairs that differ in L positions for all possible val-ues of 0 s1 · · · sL−2 sL−1. Now examine the code wordsdefined by 1 s1 · · · sL−2 sL−1, that is, code wordswhich have 1 for the first digit. For any given value of1 s1 · · · sL−2 sL−1, there exist (M− 1)L code word pairsthat differ in all L positions. Among these code words, (M −1)L−1 have 0 as the first digit. If we assume that these codeword pairs were already accounted for when we examinedcode words of the structure 0 s1 · · · sL−2 sL−1, thenfor all possible values of 1 s1 · · · sL−2 sL−1 there existML−1(M − 1)L − (M − 1)L−1 additional code word pairsthat differ in all L positions. Using this reasoning, we cansay that, for a given i (0 ≤ i < M), for all possible values ofi s1 · · · sL−2 sL−1, there exist ML−1(M − 1)L − i(M −1)L−1. As a result, the total number of code word pairs dif-fering in L positions is

AL =M−2∑i=0

ML−1[(M − 1)L − i(M − 1)L−1]. (A.2)


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Giridhar D. Mandyam is the ResearchManager of the Wireless Data Access Groupat Nokia Research Center, Irving, Texas.He received his B.S. degree Magna CumLaude in electrical engineering from South-ern Methodist University (Dallas, Texas) in1989, the M.S. degree in the same field fromthe University of Southern California (LosAngeles, California) in 1993, and the Ph.D.degree in electrical engineering from theUniversity of New Mexico (Albuquerque, New Mexico) in 1996.He has worked for several companies on wireless communicationsequipment, including Qualcomm and Texas Instruments. In 1998,he joined Nokia, where he has worked on standardization and im-plementation concepts for cdma2000, 1X-EV, and WCDMA. Hehas authored or coauthored over 40 journal and conference pub-lications and four book chapters. He also holds four US patents inthe area of wireless communications technology.

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