Ristaniemi, Joutsensalo, Advanced ICA-based Receivers for Block Fading DS-CDMA Channels, 2002

Signal Processing 82 (2002) 417 431www.elsevier.com/locate/sigpro

Advanced ICA-based receivers for block fading DS-CDMAchannels

Tapani Ristaniemi , Jyrki JoutsensaloDepartment of Mathematical Information Technology, University of Jyvaskyla, P.O. Box 35 (Agora),

FIN-40351 Jyvaskyla, Finland

Received 4 September 2000; received in revised form 19 April 2001

Abstract

In this paper we propose two types of receivers, which perform blind interference suppression. The receivers are suitablefor the third generation mobile communications systems, which are based on direct-sequence code division multiple access(DS-CDMA). We especially consider the statistical technique called independent component analysis (ICA) as a tuningelement attached to an existing single-user or blind multiuser detector. By doing this, the statistical independence of theusers signals can be exploited. In addition, it makes it possible to alleviate the performance drop due to the erroneousparameters estimation in the receiver. From the ICA point of view, the receiver observes a mixture of users original signals,determined by the user-speci7c spreading codes, the data symbols, and the state of the propagation channel. Since all of themmight be complex valued by nature, complex ICA is needed. In the paper, a proof of global convergence is given for thekurtosis-based FastICA algorithm in the case of complex valued source signals with circular distribution, and considered asthe ICA component of the advanced receivers. Two types of receivers, RAKE-ICA and MMSE-ICA, are proposed and studiedin a Rayleigh block fading channel. Extensive numerical simulations indicate that the performance of RAKE and subspaceMMSE detector, respectively, can be greatly improved. Quite signi7cantly, their performance is close to the theoretical boundof an equal length MMSE detector. ? 2002 Elsevier Science B.V. All rights reserved.

Keywords: Code division multiple access; Independent component analysis; Blind multiuser detection; Block fading channel

1. Introduction

Code division multiple access (CDMA) technologyis a strong candidate for the evolving wireless commu-nications systems. Wideband CDMA (WCDMA) hasalready been selected for an air interface solution e.g.in UMTS, which will provide a multitude of services,especially multimedia, and high bit rate packet data.

Corresponding author. Tel.: +358-14-260-2727; fax:+358-14-260-4981.

E-mail addresses: [email protected] (T. Ristaniemi),[email protected] (J. Joutsensalo).

The existing commercial spread spectrum systems arebased on long spreading codes, where the period ofthe code sequence is much longer than the duration ofa symbol. However, along with the demand for higherdata rates, also short code modes, e.g. time-divisionduplex (TDD) mode in WCDMA [7], are becoming areality.In CDMA systems the users share the same fre-

quency band, and thus a good care must be takento limit mutual interference. In principle, orthogo-nal codes would be the simplest solution. However,orthogonality between the users is destroyed e.g. bymultipath propagation. This is why the traditional

0165-1684/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S 0165 -1684(01)00194 -3

418 T. Ristaniemi, J. Joutsensalo / Signal Processing 82 (2002) 417 431

RAKE receiver [20] might be inadequate, even inthe absence of background noise. Multiuser detec-tion (MUD) is a technique which tries to exploitthe structure of interference to be able to suppressit. Optimal MUD [25], however, is computationallyexhausting, and requires several system parametersto be known. As a consequence, many suboptimalmultiuser receivers and adaptive multiple access in-terference (MAI) suppression techniques have beenstudied extensively during the past 10 years [8,15,2629]. These techniques only utilize second orderstatistics. However, the assumption of independenceis also realistic, which can be utilized e.g. in constantmodulus (CM) algorithms [4,17].A more general approach than CM is a statisti-

cal technique called independent component analysis(ICA) [5,10,13]. Recently, ICA and the closely relatedblind source separation (BSS) problem have attracteda lot interest both in statistical signal processing andneural network communities. Several good algorithmsutilizing higher-order statistics either directly or indi-rectly via suitable nonlinearities are now available forsolving the basic linear ICA=BSS problem [1,19]. ICAis mainly considered for real valued signals, whereascomplex ICA is addressed less frequently [2,3,18]. Inpractice, however, the signal to be processed is of-ten complex valued. This problem arises by nature inthe reception of a CDMA system. This is because thespreading code might be complex valued, as well asthe symbols depending on the data modulation. Inaddition, the received signal strengths, which char-acterize channel variations in time, are modelled ascomplex valued fading processes.There exists many motivating reasons to use the

means of ICA in the reception of a CDMA system.First of all, ICA provides a near-far resistant receiver,being able to resist strong interferences. Resistanceis achieved by ICA quite naturally, since ICA onlyrequires the source signals to be statistically inde-pendent, but their strengths are allowed to diIer. InCDMA the sources are, roughly speaking, users sym-bol streams, and it is hence reasonable to assume thatthey are independent. Near-far resistance is one of thekey requirements of a receiver, and it becomes evenmore important as there is a demand for higher datarates. This is because many solutions for higher datarates, e.g. smaller spreading factors, and higher sym-bol constellations, tend to worsen the near-far situation

either directly or indirectly via e.g. power control im-perfections. Secondly, the propagation delay and thestate of the channel should be estimated prior to actualsymbol estimation. For subspace-type receivers alsothe estimation for the model order should be available.The estimation of these parameters will always includesome measurement errors, which degrade the accu-racy of symbol estimation. ICA, on the other hand,does not need that precise knowledge of the systemsparameters, since the estimation is based purely onthe (higher order) statistical properties of the signal.Therefore, with ICA we should expect some robust-ness against erroneous parameter estimation. Thirdly,an ICA block can be used as an add-on feature, tobe attached to any existing receiver structure. Thismakes it possible to consider hybrid receiver struc-tures, in which the ICA block could be intelligentlyactivated only when it is expected to improve perfor-mance.In this paper, we consider ICA to assist the symbol

estimation and interference suppression in DS-CDMAsystems by attaching an ICA block to RAKE [20]and a subspace-type MMSE receiver [28]. The goalis hence to exploit the independence of the sourcesignals, which is not possible by RAKE nor MMSEdetector alone. In addition, it gives the possibility tocompensate for any performance loss caused by erro-neous propagation delay or channel estimation, whichare prerequisite tasks for conventional receivers. Fas-tICA [3,9], which is one of the most promising ICAmethods, is especially considered as the ICA blockof the receivers. The contribution of the paper istwo-fold. Firstly, a proof of global convergence isgiven for the complex FastICA algorithm [3], when acubic nonlinearity is used. Then two types of receiverstructures, namely RAKE-ICA, and MMSE-ICA, areproposed. They consist of a RAKE receiver [20] anda subspace MMSE detector [28], respectively, fol-lowed by receiver adjustment by FastICA. Numericalexperiments are included, when the CDMA downlinkchannel is Rayleigh block fading. Compared to RAKEand the subspace MMSE detector, respectively, sig-ni7cant improvements in bit- and block-error-ratesare achieved. Quite signi7cantly, their performanceis close to the theoretical bound of an equal lengthMMSE detector.The rest of the paper is organized as follows. In

the next section we discuss ICA and its application to

T. Ristaniemi, J. Joutsensalo / Signal Processing 82 (2002) 417 431 419

CDMA. The data model is presented in Section 3. TheICA-based receiver structures are proposed in Sec-tion 4, followed by numerical experiments in Section5. Section 6 summarizes the paper.

2. Independent component analysis and CDMA

Independent component analysis (ICA) is a statis-tical technique where the goal is to represent a setof random variables as a linear transformation of sta-tistically independent component variables. The mainapplication of ICA is blind source separation (BSS)problem, which has become an attractive 7eld of re-search in the statistical signal processing and neuralnetwork communities. The growing interest in ICAis mainly due to emerging new practical applicationareas, where the assumption of independence is bothrealistic and powerful.In the conventional ICA approach, the data are as-

sumed to have a linear form

rm =Gbm + nm; (1)

where rm is the mth observed data vector, G is anunknown full rank mixing matrix (even its paramet-ric form is unknown), bm is a realization of an un-known non-Gaussian source, and nm is a realizationof a noise process. A goal is to estimate the originalsource process bm given only the observation rm, andthe assumption of the independence of the sources.This means that a set of 7lters w1;w2 : : : should be es-timated such that the 7ltered (i.e. separated) sourceprocesses wH1 rm;w

H2 rm : : : are independent, or in prac-

tice as independent as possible.The basic feature of any ICA method is that it is

able to estimate a set of independent source signals,but the order of those is somewhat unpredictable. If theICA method is hierarchic, the sources tend to becomeestimated in the order of decreasing non-Gaussianity(cf. [3]), but this is not in general enough to identifyeach source. With a CDMA communication applica-tion in mind, it is thus not meaningful to apply ICAon its own, since the source signals are well identi7edby the spreading codes. Rather, it would be meaning-ful to consider ICA as an additional element, attachedto some existing receiver structure. The existing re-ceiver would perform the task of user identi7cation,

after which ICA would oIer additional interference suppressioncapability, since also the independence of the sourcesignals is utilized.

ICA would mitigate the performance drop due toerroneous timing and channel estimation.

ICA block could be activated only when it is ex-pected to improve performance.One of the most promising solution for the linear

ICA=BSS problem is FastICA [9] due to its simplic-ity and fast convergence. The algorithm is actually aneural network learning rule transformed into a veryeMcient 7xed point iteration, which does not dependon any user-de7ned parameters. FastICA has been ap-plied to the blind interference suppression in CDMAfor the 7rst time in [11], and later on in [12,2124].Prior information can be utilized directly by addingadditional constraints for the ICA iterations, or indi-rectly by a proper initialization of the ICA iteration.The method in [24] is an example of the former case,where the ICA-solution of FastICA was forced to be-long to the same space as subspace MMSE 7lter. In[22,23], proper initializations by dedicated pilot sym-bols or matched 7lter output, respectively, were con-sidered. In those papers ICA was performed for realvalued signals, i.e. only for either I- or Q-branch,which naturally leads only to a suboptimal solution.In this paper, some of the former ideas are furtherre7ned towards more practical ICA-based receivers,which can handle complex valued signals and activatethe ICA block intelligently.

3. Data model

The channel model studied in this paper is aDS-CDMA downlink (e.g. base to mobile) multi-path channel model, which is depicted in Fig. 1. Indirect sequence (DS) CDMA each user spreads itsnarrowband information signal in frequency by directsequence modulation before transmission via com-mon channel. The spreading code is user-speci7c,and thus identi7es each user in the system. The datain the observation interval have thus the form

r(t)=Mm=1

Kk=1

bkmL

l=1

amlsk

(tmT dl TC

)+n(t);

(2)


Xa (t)

X X

+ +

Z

1 Z

2 Z

3

AWGN

r(t)

x(t)

a (t)1 2

a (t)3

Fig. 1. Channel model (L = 3 paths). x(t) is the data to be sent,zd stands for delaying by d chips, ai(t) denotes path i attenuationin time.

in which M symbols are sent to K users, and receivedvia L paths. The mth data symbol of the kth user isdenoted by bkm. sk() is the kth users binary chipsequence, supported by [0; T ), where T is the sym-bol duration. For notational simplicity, the path delaysare assumed to be discretized, and hence dl {0; : : : ;(C 1)=2}. The delays are assumed to remain con-stant during the block ofM data symbols. n(t) denotesadditive white gaussian noise (AWGN). The complexcoeMcient of the path l during the symbol m is de-noted by aml.Throughout the paper we assume Rayleigh block

fading channel. This is to say, the envelope of a fadingprocess of each path is Rayleigh distributed and variesslowly compared to the data rate. The latter meansthat the coherence time of the channel is much longerthan duration of a symbol, so that the channel is es-sentially constant during a block of symbols. This sce-nario arises when high data rates are used and=or usershave low mobility (e.g. indoor environment). As anexample of a slowly fading channel, assume a mobilespeed v = 3 km=h, a carrier frequency fc = 2:0 GHzand a symbol rate 1=T=16 kbits=s. Then the maximumDoppler spread equals fd = v=clightfc = 5:56 Hz, andconsequently, the channel coherence time is Tcoh 1=fd = 0:18 s, which is much longer than duration ofa symbol T = 62:5 s. Fig. 2 shows the autocorrela-tion function of the channel in this case, from whichit is seen that the states of the channel are highly cor-related within a period of, say, 200T .

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000_ 0.5

0

0.5

1

Delay [symbol intervals]

Nor

mal

ized

cor

rela

tion

Fig. 2. An example of autocorrelation function of a slowly fadingchannel.

As the continuous-time data is received, it is 7rstsampled by chip-matched 7ltering. Here we use chiprate sampling, soC equispaced samples are taken fromconsecutive time intervals of T seconds, whereC is theprocessing gain. Sampled data can then be processedusing suitable size of a window. For synchronous data,propagated through a single-path channel, a windowsize of T is suMcient for near-far resistant demodula-tion. This is because symbols of desired and interfer-ing transmissions fall entirely to the window. In caseof multipaths the window can be enlarged to collect allthe symbols energy of the desired transmission fromdiIerent paths.In this paper, the delay spread (T=2) is small enough

compared to the bit interval, and thus a window sizeof two symbols is used [16]. Therefore, the samplesare collected into 2C-vectors rm,

rm = [r[mC] r[mC + 1] r[(m+ 2)C 1]]T :(3)

Since sampling is usually symbol-asynchronous, thevector sample rm usually contains information aboutthree successive symbols, and the data thus have thewell known form [16]

rm =Kk=1

[bk;m1

Ll=1

algkl + bkmL

l=1

algkl

+ bk;m+1L

l=1

al Ngkl

]+ nm: (4)


Here nm denotes noise vector, and the code vectors oflength 2C are de7ned as

gkl= g

k(dl)

def= [sk [C dl + 1] : : : sk [C] 0T2Cdl]T;gkl = gk(dl)

def= [0Tdl sk [1] : : : sk [C] 0TCdl]

T;

Ngkl = Ngk(dl)def= [0TC+dl sk [1] : : : sk [C dl]]T:

(5)

The vector data Eq. (4) can be represented more com-pactly as

rm =Gbm + nm; (6)

where the 2C 3K dimensional code matrix G isassumed to be full rank, and contains the code vectorsand path strengths,

G=

[L

l=1

alg1l;L

l=1

alg1l;L

l=1

al Ng1l; : : : ;

Ll=1

algKl;L

l=1

algKl;L

l=1

al NgKl

](7)

and 3K-vector bm contains the symbols,

bm = [b1;m1; b1m; b1;m+1; : : : ; bK;m1; bKm; bK;m+1]T:

(8)

The similarity of the model Eq. (6) to ICA modelEq. (1) is now immediately seen. In Eq. (6) we alsohave an unknown mixing matrix (G) and an indepen-dent source process (bm).Notice that the ICA model arises regardless of the

selection of the window size. However, the size ofthe underlying mixing matrix G in Eq. (6) changes.Namely, the dimension of G is nC (n+1)K , wheren is the size of the window in symbols. From ICApoint of view, the enlargement of the window givespossibilities to support more users. This is because acommon requirement for ICA is to have more sensorsthat sources, which is ful7lled if K6 n=(n + 1)C.A disadvantage of the enlargement, however, is theincrease in computational load.

4. ICA-based receivers

4.1. Complex FastICA

The observed data are 7rst whitened, which is acommon preprocessing task in blind source separation.

It helps to reduce the number of unknowns in the mix-ing matrix, so that the remaining mixture can be mod-elled by a simpler orthonormal matrix. More precisely,whitening is a linear transform which de-correlatesthe observed mixtures, and normalizes the componentvariances to unity. This can be always performed e.g.by principal component analysis (PCA) as follows:

ymdef=1=2s U

Hs rm: (9)

Here Us corresponds to the principal eigenvec-tors of the data correlation matrix estimate R =(1=M)

m rmr

Hm ; R = E{rmrHm}, and the diagonal

matrix s contains the related eigenvalues on its diag-onal. If kurtosis is chosen as a contrast function, theFastICA [3] algorithm for complex signals performsas follows:1. Take a random unit-norm initial vector w(0), and

let t = 1.2. Update

w(t) = E{ym(w(t 1)Hym)|w(t 1)Hym|2} w(t 1): (10)

3. Divide w(t) by its norm.4. If |w(t)Hw(t 1)| is not close enough to 1, let

t = t + 1, and go back to step 2. Otherwise, outputthe vector w(t).

Notice that the existence of a simple factor in theupdate rule 2 is due to the choice of kurtic contrast.In fact, from [3] it is easy to see that = 2 if boththe mixing matrix and sources are complex valued. Inreal case, when both the mixing matrix and sourcesare real, = 3 [9]. In step 2, the expectation canbe estimated using a large set of sample vectors,y1; y2; : : : ; yM . This algorithm estimates one ICA ba-sis vector w, and hence an independent componentwHym. In a CDMA application, each whitened samplevector ym corresponds to the mth symbol. Hence, allthe M soft symbols of the desired user are estimatedaccording to wH [y1; y2; : : : ; yM ].The convergence of FastICA is shown to be cubic

[9] in the real case, where both the mixing matrix andthe sources are real valued. In [3], the complex casewas treated, but no proof of convergence was given.The global cubic convergence does also hold in thecomplex case with circular source distribution, and theproof is given in Appendix A. However, FastICA is


not considered at all in the case where the sources arereal valued but the mixing matrix is complex valued.In CDMA application this scenario easily arises. Forexample, BPSK modulation yields real valued sourcesignals (see Eq. (8)). In addition, if the channel variesslowly, as is the case in block fading channel, diIer-ent paths are strongly correlated, since they are justdelayed and scaled versions of each other. As a con-sequence, the mixing matrix (see Eq. (7)) is complexvalued due to the complex valued path gains. In thisscenario, the update rule Eq. (10) with = 2 does notresult in a desired separation. Interestingly, choosingthe same parameter value as in the pure real case, i.e.=3, turns out to be crucial. The global convergencein this speci7c case, however, is an open issue, see thecomments in Appendix A. Therefore, only numericalsimulations are given to endorse the choice of in thisspeci7c case.

4.2. Proposed ICA-based receivers

As one ICA basis vector w is estimated, it maycorrespond to any user since there is no control onthe component to be separated. This is because thestatistical properties of each transmission are the same:the symbols obey the same distribution, and so doesthe path gains up to variance, which reRects the powerused in the transmission. A CDMA application in mindit is thus not meaningful to use ICA on its own, sincethe desired signal is well identi7ed by the user-speci7cspreading code. What is proposed in the following isthat the FastICA iteration is properly initialized by themeans of some existing receivers, which thus wouldperform the task of user identi7cation.In the following, we consider in more detail two

ways of initializing FastICA. The starting point is tolook at the noiseless whitened data

ym =Wbm: (11)

HereW def= 1=2s UHs G, wheres andUs correspond to

the 3K principal eigenvalues,1 and -vectors of the dataautocorrelation matrix E{rmrHm}, respectively. First wenotice that the matrixW is orthonormal. This is due tothe uncorrelatedness of the symbols, i.e. E{bmbHm}=I,

1 From Eq. (8) it is seen that the sample vector rm contains 3Kindependent source signals, which is therefore also the dimensionof the signal subspace.

because then we have 2 E{ymyHm}=E{WbmbHmWH}=WWH = I. The goal hence is to estimate one columnofW, say the second column w2. This is because thenby w2 we can estimate the symbols of the desired user(user k=1): wH2 ym=w

H2 Wbm=[0100 0]bm=b1m.

Interestingly, from the de7nition of W and G we seethat

w2 = $1=2s U

Hs

Ll=1

a1lg1l; (12)

which is exactly the subspace MMSE detector [28] fordispersive channels. Using the terms of source sepa-ration, Eq. (12) can be used to separate the desiredsource. It should be noticed, however, that only un-correlatedness of the data symbols is assumed in thederivation of Eq. (12). In addition, the subspace pa-rameters, as well as the path gains and delays willalways be subject to estimation errors, which degradethe separation performance of Eq. (12). What is sug-gested now is that ICA would take a role of tuningelement, and thus to re7ne Eq. (12) to have betterseparation performance. This is possible, since inde-pendence is a stronger property than uncorrelatedness.In addition, it is meaningful to apply ICA having Eq.(12) as a starting point, since it already identi7es thedesired user. This identi7cation is not possible by ICAon its own.Alternatively, known or estimated symbols could

be used for the initialization of FastICA. This is dueto the uncorrelatedness of the symbols, since then wehave E{ymb1m}=E{Wbmb1m}=w2, i.e. the subspaceMMSE detector again. Since training symbols are notnecessarily implemented in all the systems, we preferthe use of RAKE receiver, or MMSE detector symboloutputs to be used for the initialization of ICA.The proposed receiver structures, MMSE-ICA,

RAKE-ICA and MMSEbit-ICA, are highlightedin Table 1. Steps 15 constitute the backbone ofthe receivers, including conventional detection andICA post-processing. In Step 6, a branch switchingbetween the conventional and ICA-assisted detec-tor is considered in order to take advantage of ICApost-processing only when it is expected to improveperformance. While there exists many ways to do theswitching (see e.g. [14]) we adopted the most suitable

2 Notice that E{ymyHm } = I, since whitening normalizes com-ponent variances.


Table 1ICA-based blind interference suppression schemes

Let k be the desired user, = 2; 3 for complex and real valued symbols, respectively, and rm; m= 1; : : : ; M the received blockof data according to Eq. (6). x denotes the estimate of x.1. Let either RAKE receiver or subspace MMSE detector to operate 7rst.

If RAKE [20] is chosen, denote bRAKEkm the symbol estimate of kth users mth symbol. Recall that this choice naturallyincludes channel and delay estimation.

If subspace MMSE detector [28] is chosen, denote mk the detector for user k, and bMMSEkm the related mth symbol. Recallthat in addition to channel and delay estimation, also signal subspace parameters, denoted $s and Us are to be estimatedfrom the data.

2. Start ICA post-processing as follows:Estimate signal subspace parameters ($s and Us) from the data block, if not yet done. Perform whitening of the data accor-ding to Eq. (9) to yield whitened data ym. By choosing one of the following options (ac), initialize w(0) = wk =||wk ||, where(a) MMSE-ICA: wk =mk .(b) RAKE-ICA: wk = E{ymbRAKEkm }.(c) MMSEbit-ICA: wk = E{ymbMMSEkm }.Expected values are estimated from the data block. Let t = 1.

3. Update

w(t)=E{ym(w(t1)Hym)|w(t1)Hym|2}w(t1): (14)4. Divide w(t) by its norm.5. If |w(t)Hw(t 1)| is not close enough to 1, let t = t + 1, and go back to step 3.6. Branch switching: Calculate the correlation = Re

(w(0)H

||w(0)||w(t)

||w(t)||). If || th, where th is a predetermined threshold value,

output ICA post-processed detector w= sign()w(t). Otherwise, choose the symbols estimate from step 1.

scheme to our problem. In this scheme a correla-tion between two detectors, for example RAKE andRAKE-ICA, is 7rst measured, and the latter detectoris chosen if the correlation is high enough. By doingso, we can trust the detector corresponds to the de-sired user. At the same time a possible phase reversalafter ICA 3 can be noticed and taken into account.

4.3. Computational considerations

In this section we discuss the computational com-plexity of the proposed methods, expressed as thenumber of multiply and add-operations needed. Wesuppose here that the length of the data vector is C,the number of users is K .All the proposed methods are subspace methods

in nature. The classical approach is either eigenvaluedecomposition of the data autocorrelation matrix, orsingular value decomposition of the data matrix. Forcomputing the eigenvectors of a C C dimensionalmatrix, several well-established O(C3) algorithms ex-ists in the literature [6]. If only K eigenvectors are

3 The source si is as independent from the others as si .

computed, their complexity reduces to O(C2K). Asa desire to lower the computational load, several re-cursive algorithms have also emerged (see [30], andreferences therein), which can reach a complexity ofO(CK) per iteration.In ICA, whitening of the data (see Eq. (11)) have

computational complexity O(CK)+O(K2)=O(CK),since K C. This is also the complexity of subspaceMMSE detector, when adaptive subspace tracking isused [27]. An additional complexity of the proposedschemes are also due to the FastICA iterations, seeEq. (10). The expected value can be estimated by sev-eral vector samples. SupposeM samples are used, and

form a data matrix Y def= [y1 : : : yM ]. Then the updaterule is

w(t) =1MY!(t 1)H w(t 1); (13)

where !(t) def= |w(t)HY|2 (w(t)HY). Here | |2 istaken elementwise, and stands for elementwise mul-tiplication. Now, wHY needs KM multiplications andsummations, and is thus of order O(KM). Conse-quently, !(t) is of order O(KM) + O(M), where thelast term is due to the elementwise multiplications.


Thus, the 7nal complexity of one FastICA iteration isof order O(KM). Putting all together, the computa-tional complexity of any proposed scheme is of orderO(CK) + O(KM) per iteration if adaptive subspacetracking is used, and O(C2K)+O(KM) if batch meth-ods are used for subspace estimation.

5. Numerical experiments

Since there does not exist an useful analytical per-formance measure for the iterative detectors, numer-ical simulations are given instead. We compare themethods in the downlink environment with Rayleighblock fading multipath channel. The methods forsymbol estimation are: RAKE, RAKE-ICA, subspaceMMSE detector, MMSE-ICA, and MMSEbit-ICA.In some experiments an exact MMSE detector ofthe same length as other detectors is also used as areference. Exact MMSE assumes also the codes ofinterfering user to be known.In one simulation, a block of M = 200 symbols

(of randomly chosen user) is estimated with randomlygenerated complex path gains, time delays, and noiserealization. QPSK modulation is assumed if not oth-erwise stated. The channel includes two paths (L=2)of equal average strength. Path delays are randomlychosen from {0; 1; : : : ; (C 1)=2}, and complex pathstrengths obey a zero mean normal distribution. Walshcodes of length C=8 are used together with a randomcomplex scrambling code. All the measured quantitiesare averaged over 10 000 simulations. No coding isconsidered, and thus the results are expressed in rawbit- and block-error-rates (BER and BLER, respec-tively).

5.1. Setup A: varying SNR level

The system includes K = 3 users, and all thesystem parameters are assumed to be known. Allthe interfering users are assigned with equal power.Signal-to-noise ratio (SNR) is varied from 0 to 20 dB.In Table 1, = 2 was used due to QPSK modulation.Fig. 3 shows the achieved bit-error-rates for the

methods as a function of average SNR. In general, itis seen that ICA can, indeed, improve the performanceeven if perfect knowledge of the system parameters,e.g. the channel and the delays, would be known. This

0 2 4 6 8 10 12 14 16 18 2010

_4

10_3

10_2

10_1

100

Signal_to_Noise Ratio

Bit_

Erro

r_R

ate

RAKE RAKE ICA subspace MMSEMMSE ICA Exact MMSE

Fig. 3. Bit-error-rate as a function of signal-to-noise ratio(0; : : : ; 20) in an equal energy two-path Rayleigh block fadingchannel. The system includes K = 3 users with equal strength.

performance enhancement is possible since ICAmethods are able to utilize independence of thesources unlike RAKE and subspace MMSE detectors.The improvement becomes visible after SRN of 78 dB. The improvement is much bigger with RAKE,because RAKE alone suIers from the multiple-accessinterference. For example, ICA gives a 8 dB gain forRAKE when considering raw BER of 102 as a tar-get. In addition, ICA enabled RAKE to achieve alsothe raw BER of 103. Attaching ICA to the MMSEdetector gave improvement, too, being at most 2 dB.Fig. 4 shows the corresponding block-error-rates

for the methods. Recall, that the block is correctly es-timated, if all the symbols in a block are estimatedcorrectly. It is seen that RAKE-ICA improves theperformance of RAKE quite remarkably in BLER. Infact, both the raw BLER of 101 and 102 are reachedwith RAKE-ICA unlike RAKE alone in this setup.The improvement with MMSE detector is also muchmore visible, being nearly 3 dB at raw BLER level of101 and at least 4 dB at raw BLER level of 102.It is also noticed how close to the MMSE bound theICA-assisted methods can go.Fig. 5 shows the average and the variance of the

normalized correlation coeMcient (see Table 1) ofthe conventional and ICA-assisted receiver as a func-tion of signal-to-noise ratio. Recall that the 7nal deci-sion of whether ICA branch is taken into account ornot is based on this coeMcient. Namely, a high enough


0 2 4 6 8 10 12 14 16 18 2010

_3

10_2

10_1

100


Bloc

k_Er

ror_

Rat

e

RAKE RAKE ICA subspace MMSEMMSE ICA Exact MMSE

Fig. 4. Block-error-rate as a function of signal-to-noise ratio(0; : : : ; 20) in an equal energy two-path Rayleigh block fadingchannel. The system includes K = 3 users with equal strength.

0 2 4 6 8 10 12 14 16 18 2010_1

100


Mea

n of

cor

rela

tion

RAKE_ICAMMSE_ICA

0 2 4 6 8 10 12 14 16 18 2010_5

10_4

10_3

10_2

10_1


Varia

nce

of c

orre

latio

n

RAKE_ICAMMSE_ICA

Fig. 5. The average (top) and variance (down) of the correla-tion coeMcient (see Table 1) of conventional and ICA-assistedreceiver as a function on average signal-to-noise ratio.

correlation is required which ensure the identi7cationof a desired user. It is seen that user identi7cation ispossible by this type of simple correlation approachgiven a moderate SNR level. A threshold value of 0:8for the correlation was used in all the simulations.Fig. 6 shows the convergence speed of FastICA in

the average number of iterations needed with diIerentSNR levels. With moderate SNR only less than 10iterations in average are needed.With low SNRs more

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80


Num

ber o

f ite

ratio

ns

RAKE_ICAMMSE_ICA

Fig. 6. The number of iterations (in average) as a function ofaverage signal-to-noise ratio needed by the FastICA algorithm inSetup A.

iterations are needed. This is because of the noise-freemodel behind FastICA [9].

5.2. Setup B: e:ect of erroneous delay estimation

The setup is otherwise the same as the previousone, but now SNR is set to 20 dB. In addition, thedelay estimation was assumed to be inaccurate byletting the error of the estimated delay to obey a zeromean normal distribution, whose variance is variedfrom 0 to 0:02 chips. Figs. 7 and 8 show the achievedBERs and BLERs for the methods as a function ofthe delay error variance. As expected, RAKE is morerobust against delay mismatch than subspace MMSEdetector. It is also seen that ICA can enhance quitesigni7cantly the performance of both RAKE andMMSE detector in the whole dynamic range of thesetup. For example, RAKE-ICA drops the raw BERfrom 102 to nearly 104, and raw BLER well be-low 102. On the other hand, when ICA is attachedto a subspace MMSE detector it allows much biggerdelay mismatch than subspace MMSE detector aloneto keep the same performance level. The achievedgain at the BER level of 103, for example, is thatMMSE-ICA allows four or two times bigger delayvariance depending whether 7nite alphabet propertyis utilized (MMSEbit-ICA) or not (MMSE-ICA), re-spectively. Also, it is quite important to notice that


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.0210

_5

10_4

10_3

10_2

10_1

Delay error variance

Bit_

Erro

r_R

ate

RAKERAKE_ICA subspace MMSEMMSE_ICA MMSEbit_ICA

Fig. 7. Bit-error-rate as a function of the variance of the delayestimation error (0; : : : ; 0:02) in an equal energy two-path Rayleighblock fading channel. The system includes K =3 users with equalstrength, and the average SNR is 20 dB.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.0210

_3

10_2

10_1

100

Delay error variance

Bloc

k_Er

ror_

Rat

e

RAKE RAKE_ICA subspace MMSEMMSE_ICA MMSEbit_ICA

Fig. 8. Block-error-rate as a function of the variance of the delayestimation error (0; : : : ; 0:02) in an equal energy two-path Rayleighblock fading channel. The system includes K =3 users with equalstrength, and the average SNR is 20 dB.

only the ICA-assisted detectors achieved the rawBLER of 102.

5.3. Setup C: e:ect of channel estimation errors

The setup is otherwise the same as Setup A butnow SNR is set to 20 dB. In addition, the channel

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.0210

_5

10_4

10_3

10_2

10_1

Path gain error variance

Bit_

Erro

r_R

ate


Fig. 9. Bit-error-rate as a function of the variance of the pathgain estimation error (0; : : : ; 0:02) in an equal energy two-pathRayleigh block fading channel. The system includes K = 3 userswith equal strength, and the average SNR is 20 dB.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.0210

_3

10_2

10_1

100

Path gain error variance

Bloc

k_Er

ror_

Rat

e


Fig. 10. Block-error-rate as a function of the variance of the pathgain estimation error (0; : : : ; 0:02) in an equal energy two-pathRayleigh block fading channel. The system includes K = 3 userswith equal strength, and the average SNR is 20 dB.

estimation is assumed to be inaccurate by letting therelative error of the absolute value of the estimatedpath gain to obey a zero mean normal distribution,whose variance is varied from 0 to 0:02.The characteristics of the results are very similar

to the Setup B, as can be seen from Figs. 9 and10. Namely, ICA gives an universal BER and BLER


_10 _5 0 5 10 15 2010

_6

10_5

10_4

10_3

10_2

10_1

100

Multiple access interference per user

Bit_

Erro

r_R

ate

RAKE RAKE_ICA subspace MMSEMMSE_ICA exact MMSE

Fig. 11. Bit-error-rate as a function of the multiple access interfer-ence per interfering user (10; : : : ; 20) in an equal energy two-pathRayleigh block fading channel. The system includes K = 3 users,and the average SNR is 20 dB.

improvement for the conventional detectors. In moredetails, at least twice an error variance is tolerated byICA-assisted MMSE receiver when using raw BER103 as a target, and raw BLER of 102 can, onceagain, be supported only by the ICA-assisted receivers.

5.4. Setup E: the e:ect of multiple accessinterference

This setup is to demonstrate the near-far resistanceof the methods. The setup is otherwise the same asSetup A, but the SNR is 7xed to 20 dB. In addition,the strengths of interfering users are varied from 10to 20 dB relative to the monitored user. The resultsare seen in Figs. 11 and 12.RAKE is not near-far resistant, which is seen as

increasing BER with increasing MAI. This naturallyalso limits the performance of RAKE-ICA in the highMAI region. However, if the raw BER of 102 is takenas a target, RAKE-ICA gives, in this setup, an im-provement of approximately 5 dB compared to RAKE.More remarkably, the raw BLER of 101 and 102

are achieved with 12 and 14 dB higher MAI per in-terfering user.The case of MMSE-ICA is more clear. Subspace

MMSE itself is near-far resistant which is seen as astable BER=BLER behaviour when incresing the levelof MAI. It is thus expected that ICA would intro-

_10 _5 0 5 10 15 2010

_4

10_3

10_2

10_1

100

Multiple access interference per user

Bloc

k_Er

ror_

Rat

e


Fig. 12. Block-error-rate as a function of the multiple accessinterference per interfering user (10; : : : ; 20) in an equal energytwo-path Rayleigh block fading channel. The system includes K=3users, and the average SNR is 20 dB.

duce a nearly constant improvement with respect toMAI. This improvement is noticeable especially inBLER behaviour since a fall well below 102 takesplace. Quite remarkably, the BLER performance ofMMSE-ICA is really close to the optimal MMSE de-tector of equal length.

5.5. Setup F: the e:ect of model order mismatch

In this setup the eIect of model order mismatch isevaluated. Again, the system includes K = 3 users ofequal strength in a channel with L = 2 paths. Sincepaths are essentially just delayed and scaled replicasof each other in the received block of data due to theblock fading channel, the true model order is 3K = 9when a two-symbol processing window is used. In thesimulations model order is varied from 3; : : : ; 16 to seethe eIect of the mismatch in the receiver performance.The results are seen in Figs. 13 and 14.RAKE does not need the model order, and hence

its performance remains at the same level. On theother hand, subspace MMSE is sensitive to the se-lection of the model order. The ICA-parts of bothRAKE-ICA and MMSE-ICA needs the estimate forthe model order. What is seen is that both RAKE-ICAand MMSE-ICA are able to improve the performanceof RAKE, and the subspace MMSE detector, respec-tively, even with quite a mismatched model order


4 6 8 10 12 14 1610

_5

10_4

10_3

10_2

10_1

Estimated model order

Bit_

Erro

r_R

ate


Fig. 13. Bit-error-rate as a function of the model order estimate(3; : : : ; 16) in an equal energy two-path Rayleigh block fadingchannel. The system includes K=3 users with equal strength, andthe true model order is 9. The average SNR is 20 dB.

4 6 8 10 12 14 1610

_3

10_2

10_1

100

Estimated model order

Bloc

k_Er

ror_

Rat

e


Fig. 14. Block-error-rate as a function of the model order estimate(3; : : : ; 16) in an equal energy two-path Rayleigh block fadingchannel. The system includes K=3 users with equal strength, andthe true model order is 9. The average SNR is 20 dB.

estimate. Only a considerable underestimation makesRAKE-ICA to perform worse that RAKE alone. Onthe other hand ICA is able to give a clear positiveimpact for the receivers even with highly overes-timated model order. This observation suggests tochoose a pessimistic rather than optimistic valuefor the model order estimate. This conclusion is even

0 2 4 6 8 10 12 14 16 18 2010

_4

10_3

10_2

10_1

100


Bit_

Erro

r_R

ate

RAKE RAKE_ICA subspace MMSEMMSE_ICA

Fig. 15. Bit-error-rate as a function of signal-to-noise ratio(0; : : : ; 20) in an equal energy two-path Rayleigh block fadingchannel. The system includes K = 3 users with equal strength.BPSK modulation is used.

more clearly justi7ed by the BLER performance,which shows nearly unaltered performance for theICA-assisted receivers when the model order is over-estimated. Finally, only RAKE-ICA and MMSE-ICAare able to respect the raw BLER of 102.

5.6. Setup G: real sources but complex mixing

This 7nal setup is to endorse the choice of =3 (seeTable 1) in a speci7c case when the sources are real butthe mixing is complex valued. The setup is otherwisethe same as Setup A but BPSK modulation is used forthe data. The results are seen in Figs. 15 and 16, fromwhich the conclusions about the enhancement eIectson receiver performance can be drawn analogously tothe preceding discussion.

6. Conclusions

In this paper we considered blind multiple accessinterference suppression in the DS-CDMA commu-nication system by means of independent componentanalysis. Modi7cations of FastICA [3,9] with com-plex valued data were proposed, and the proof ofglobal convergence was given in the case of complexvalued signals with circular distribution. Two types ofreceiver structures, RAKE-ICA and MMSE-ICA were


0 2 4 6 8 10 12 14 16 18 2010

_3

10_2

10_1

100


Bloc

k_Er

ror_

Rat

e

RAKE RAKE_ICA subspace MMSEMMSE_ICA

Fig. 16. Block-error-rate as a function of signal-to-noise ratio(0; : : : ; 20) in an equal energy two-path Rayleigh block fadingchannel. The system includes K = 3 users with equal strength.BPSK modulation is used.

proposed. The reasons for taking ICA as an additionaltuning element, were the following: a conventional receiver would perform the task ofuser identi7cation which is not possible by ICA onits own,

RAKE is not near-far resistant, but its outputs canbe good enough for ICA initialization,

even if all the users have equal strength, additionalmultiple access interference can be mitigated byICA, thus improving the performance of RAKE,

in the presence of timing, channel, or model orderestimation errors, the performance of the subspaceMMSE detector gets worse. This drop in perfor-mance can be mitigated by ICA.

the independence of the source signals of diIerentusers can be exploited.

These items were assessed by extensive numericalstudies, when the CDMA downlink channel wasRayleigh block fading. They indicated quite clearperformance gains over conventional and standardsubspace-based MMSE receivers. A price to pay isan increase in computational load, which is negligiblein MMSE-ICA. In RAKE-ICA, on the other hand,estimation of eigenvector and- values became alongwith ICA as an additional burden, although the ben-e7t in performance enhancement was evident. Also,a simple correlation procedure for adaptive branchswitching was noticed to be a suMcient tool for user

identi7cation and to guarantee almost universal per-formance enhancement.

Acknowledgements

This work was supported by the Research Pro-gramme for Telecommunication Electronics (Telec-tronics) of the Academy of Finland.

Appendix A. Proof of the global convergence

The proof is analogous to that of [9].First, the linear mixture to be analyzed has the form

y=Ws. Assume that the data are whitened already sothatW is an orthonormal mixing matrix with complexentries and the complex sources have unit variances.Sources are assumed to have zero mean. In addition,the sources are assumed to obey a circular distribu-tion, so the real and imaginary parts are uncorrelatedand have equal variance. These assumptions can berepresented in short as E{ssH} = I and E{ssT} = 0.Here I and 0 are identity and null matrix, respectively.First, a new variable is de7ned as z = WHw,

where W is the true mixing matrix and w is the ICAbasis vector to be estimated. The goal is to show thatz(t) t [0 0 v 0 0]T; |v|=1, i.e. w converges to onecolumn of W with a unit norm complex scalar ambi-guity. This indeterminacy is an inherent property ofcomplex ICA.For notational convenience, denote z+ the new

value for z. Applying Eq. (10), we thus have

z+ =WHw+ (A.1)

=WHE{y(wHy)|wHy|2} 2WHw (A.2)

= E{WHy(zHWHy)|zHWHy|2} 2z (A.3)

= E{s(zH s)|zH s|2} 2z: (A.4)In the third, and fourth equality we have used thefacts w = Wz, and s = WHy, respectively, since Wis orthonormal. To express the last equality in moredetail, the following relations are 7rst obtained:

zH s =i

zi si; (A.5)


|zH s|2 =i

|zi|2|si|2 +i =j

zi zjsisj ; (A.6)

(zH s)|zH s|2

=i; j

zi|zj|2si |sj|2 +h; i =j

zhzi zjsh sis

j : (A.7)

Therefore, the lth element of z+ is equal to

z+l = E{sl(zH s)|zH s|2} 2zl (A.8)

= zl|zl|2E{|sl|4}+ 2i =l

zl|zi|2 2zl (A.9)

= zl|zl|2E{|sl|4}+ 2zl(1 |zl|2) 2zl (A.10)

= zl|zl|2Jc(sl); (A.11)

where Jc(sl)def= E{|sl|4} 2, i.e. the kurtosis of

sl. Here the second equality is due to the as-sumptions of the sources s, since then for dif-ferent indices i; j; l; m we have: E{|si|2|sj|2} = 1,and E{sisj |sj|2} = E{s2i (sj )2} = E{sisj(sl )2} =E{sisj |sl|2}=E{si sj s2l}=E{sisjsl sm}=0. In the thirdequality, we have used the fact that ||z||=i |zi|2=1.This follows from the fact that w is normalized tounity after each iteration, and thus ||w|| = ||z|| = 1,since W is orthonormal.The rest of the proof is entirely similar to [9],

and therefore only the key steps are now shortlyhighlighted. First, j is chosen so that Jc(sj) =0, andzj(t 1) =0, so that|z+i ||z+j |

=|zi(t)||zj(t)| =

|Jc(si)||Jc(sj)|

( |zi(t 1)||zj(t 1)|

)3(A.12)

is well-de7ned. This leads to an explicit recursive for-mula for |zi(t)|=|zj(t)|:

|zi(t)||zj(t)| =

|Jc(si)||Jc(sj)|(|Jc(si)||zi(0)||Jc(sj)||zj(0)|

)3t: (A.13)

For j= argmaxp|Jc(sp)||zp(0)| this implies |zj|

1, and the other |zi| 0, since ||z||= 1.

The proof for the special case where the mixingmatrix is complex but the sources real is an openissue. Notice that due to the real sources we also have

E{s2i (sj )2}=E{s2i s2j}=1 for i = j, and thus the directaccess to the fourth order cumulants does not follow,as was the case in Eq. (A.11). However, motivatedby extensive numerical experiments, = 3 turns outto be a good choice. An argument for that choice,though heuristic, is the observation that both in purereal and complex case the choice of was entirely dueto the kurtosis of the sources. Recall that the choice = 2 in Eqs. (A.8)(A.11) was, in fact, to gain acompact representation Eq. (A.11), characterized bythe kurtosis, Jc(sl) = E{|sl|4} 2, of the source sl.This was the reason in [9], too, for the choice =3 in pure real case, when the kurtosis is de7ned as

Jr(sl)def= E{s4l} 3. In this respect, the choice = 3

for real sources seems justi7able despite of complexmixing.

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Advanced ICA-based receivers for block fading DS-CDMA channelsIntroductionIndependent component analysis and CDMAData modelICA-based receiversComplex FastICAProposed ICA-based receiversComputational considerations

Numerical experimentsSetup A: varying SNR levelSetup B: effect of erroneous delay estimationSetup C: effect of channel estimation errorsSetup E: the effect of multiple access interferenceSetup F: the effect of model order mismatchSetup G: real sources but complex mixing

ConclusionsAcknowledgementsAppendix A.References

Documents

Ristaniemi, Joutsensalo, Advanced ICA-based Receivers for Block Fading DS-CDMA Channels, 2002