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Modelling and Simulation in Engineering

Guest Editors: Matthias Ptzold, Neji Youssef,

and Carlos A. Gutierrez

Modeling and Simulation of

Mobile Radio Channels

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Modeling and Simulation ofMobile

Radio Channels

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Copyright 2012 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in Modelling and Simulation in Engineering. All articles are open access articles distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

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Editorial Board

Driss Aboutajdine, MoroccoK. Al-Begain, UK

Adel M. Alimi, Tunisia

Zeki Ayag, Turkey

Sergio Baragetti, Italy

Andrzej Bargiela, UK

Joaquim Barros, Portugal

Salim Belouettar, LuxembourgJean-Michel Bergheau, France

Tianshu Bi, China

Philippe Boisse, France

Agostino Bruzzone, Italy

Hing Kai Chan, UK

Jinn Kuen Chen, USA

Min-Chie Chiu, Taiwan

Weizhong Dai, USAMingcong Deng, Japan

Lei Ding, China

Tadashi Dohi, Japan

Dimitris Drikakis, UK

Andrzej Dzielinski, PolandAbdelali El Aroudi, SpainEnmin Feng, China

Huijun Gao, China

F. Gao, UK

Xiaosheng Gao, USA

Huijun Gao, China

Dariusz J. Gawin, PolandRatan K. Guha, USA

Chung-Souk Han, USA

Shinsuke Hara, Japan

Joanna Hartley, UK

Tasawar Hayat, Pakistan

Jing-song Hong, ChinaQinglei Hu, China

Bassam A. Izzuddin, UK

MuDer Jeng, Taiwan

Chia-Feng Juang, Taiwan

Iisakki Kosonen, Finland

Stavros Kotsopoulos, Greece

Nikos D. Lagaros, Greece

Hak-Keung Lam, UKHongyi Li, China

Dimitrios E. Manolakos, Greece

Shengwei Mei, China

Laurent Mevel, France

Azah Mohamed, MalaysiaCarlos A. Mota Soares, Portugal

Antonio Munjiza, UKJavier Murillo, Spain

Petr Musilek, Canada

Tomoharu Nakashima, Japan

Gaby Neumann, Germany

Alessandra Orsoni, UKJavier Otamendi, Spain

Anna Pandolfi, Italy

Bei Peng, China

Ricardo Perera, Spain

Evtim Peytchev, UK

Zhiping Qiu, ChinaLuis Carlos Rabelo, USAAhmed Rachid, France

Franco Ramrez, Spain

Rubens Sampaio, Brazil

Aiguo Song, China

Rajan Srinivasan, India

S. Taib, Malaysia

Mohamed B. Trabia, USAJoseph Virgone, France

Bauke Vries, The Netherlands

Guowei Wei, USA

Ligang Wu, China

Feng Xiao, ChinaFarouk Yalaoui, France

Shigeru Yamada, JapanGuang-Hong Yang, China

ShengKai Yu, Singapore

Qingling Zhang, China

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Contents

Modelingand Simulation ofMobile RadioChannels, Matthias Patzold, Neji Youssef,

and Carlos A. Gutierrez

Volume 2012, Article ID 160297, 2 pages

DesignofWidebandMIMOCar-to-Car ChannelModels Based on theGeometrical Street Scattering

Model, Nurilla Avazov and Matthias Patzold


A Three-DimensionalGeometry-Based StatisticalModelof 2 2 Dual-PolarizedMIMO

Mobile-to-MobileWideband Channels, Jun Chen and Thomas G. Pratt


AnAccurate Hardware Sum-of-Cisoids FadingChannel Simulator for Isotropic andNon-Isotropic

Mobile Radio Environments, L. Vela-Garcia, J. Vazquez Castillo, R. Parra-Michel, and Matthias PatzoldVolume 2012, Article ID 542198, 12 pages

Filter-Based FadingChannelModeling, Amirhossein Alimohammad, Saeed Fouladi Fard,

and Bruce F. Cockburn


Wideband andUltrawidebandChannelModels inWorkingMachineEnvironment,Attaphongse Taparugssanagorn, Matti Hamalainen, and Jari Iinatti


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Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2012, Article ID 160297,2pagesdoi:10.1155/2012/160297

EditorialModeling and Simulation of Mobile Radio Channels

Matthias Patzold,1 Neji Youssef,2 and Carlos A. Gutierrez3

1 Department of Information and Communication Technology, University of Agder, 4604 Kristiansand, Norway2 Departement EPP (Electronique, Physique et Propagation), Ecole Superieure des Communications de Tunis, 2083 Ghazala, Tunisia3 Faculty of Science, Universidad Autonoma de San Luis Potosi, 78290 San Luis Potosi, SLP, Mexico

Correspondence should be addressed to Matthias Patzold,[email protected]

Received 2 December 2012; Accepted 2 December 2012

Copyright 2012 Matthias Patzold et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The aggregate demand for multimedia services, high mobil-ity, and global connectivity has resulted in recent years in anexplosion of new technologies for wireless communicationsystems. All components of a wireless communication systemranging from digital modulation schemes over channelcoding techniques up to higher layer protocols are more

or less influenced by the characteristics of the mobileradio channel. A thorough understanding of the mobileradio channel is therefore crucial for the development,performance optimization, and testing of present as well asnext-generation mobile radio systems. This is one of thereasons why exploring the mobile radio channel has alwaysbeen a key research topic from the very beginning of mobilecommunications to the present. Currently, the research onmobile radio channels involves a variety of challenging topicssuch as the modeling of car-to-car channels, multiple-inputmultiple-output (MIMO) channels, cooperative channels,satellite channels, and ultrawideband channels, only to namea few. The objective of this special issue is to give an overviewof the state-of-the-art research in the fascinating area ofmodeling and simulation of mobile radio channels. Thisspecial issue is composed of five papers.

The paper entitled A Three-Dimensional Geometry-Based Statistical Model of 2 2 Dual-Polarized MIMOMobile-to-Mobile Wideband Channels uses a geometricscattering approach to derive a three-dimensional (3D)parametric reference model for wideband dual-polarizedMIMO mobile-to-mobile (M2M) channels. The authorsassume nonisotropic scattering, where the distribution of thescatterers is characterized in the azimuth direction throughthe von Mises distribution and in the elevation directionby the cosine distribution. The statistical properties of the

reference channel model are controlled by a large numberof physical model parameters, including the velocities ofthe transmitter and the receiver, the distance between thetransmitter and the receiver, the 3D antenna pattern gains,the azimuth and elevation angles of arrival (departure),the geometrical distribution of the scatterers, the Rician

K-factor, the maximum Doppler frequency, the scatteringloss factor, the cross-polar power discrimination ratio, andthe copolarization power ratio. The spotlight here is onthe derivation and analysis of the time-frequency corre-lation function of the proposed reference model for 3Dnonisotropic scattering environments. With the knowledgeof the time-frequency correlation function, an essentialprerequisite is provided for the performance analysis andoptimization of mobile-to-mobile communication systemsemploying 2 2 dual-polarized architectures.

The paper An Accurate Hardware Sum-of-CisoidsFading Channel Simulator for Isotropic and NonIsotropicMobile Radio Environments presents a hardware simulatorfor mobile fading channels characterized by Doppler powerspectral densities with arbitrary symmetrical and asymmet-rical shapes. The hardware simulators architecture describedin this paper is based on an efficient implementation ofthe sum-of-cisoids principle. According to this principle, afinite number of complex sinusoids (cisoids) are combinedto generate complex-valued waveforms with statistical prop-erties resembling those of a given reference channel model.Each of the cisoids that constitute the sum-of-cisoids modelis synthesized in hardware using a piecewise polynomialapproximation technique, which offers a high-performancealternative to traditional solutions using look-up tables.The architecture of the hardware simulator is capable

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2 Modelling and Simulation in Engineering

of handling any configuration of the cisoids amplitudes,frequencies, and phases. The flexibility and accuracy of theauthors implementation scheme are demonstrated througha variety of numerical results of the first- and second-orderstatistics of emulated Rayleigh fading channels assumingdifferent isotropic and nonisotropic scattering conditions.

The authors analyze the computational complexity of thehardware simulator in an FPGA implementation with 32cisoids and different data path bit-widths. To better assessthe complexity of their proposal, they provide a comparisonwith two alternative hardware fading channel simulators,which are also built on the sum-of-sinusoids principle. Asidefrom its efficiency and accuracy, the main feature of thehardware fading channel simulator presented in this paperis the capability of simulating mobile fading channels undera wide range of scattering conditions.

The paper Filter-Based Fading Channel Modeling out-lines a filter-based approach for the design of fading channelsimulators. As distinct from the sum-of-sinusoids concept,which has been studied extensively in the last two decades forthe modeling and simulation of multipath fading channels,the filter concept has attracted only little attention in thepast. This is mainly due to the design complexity of digitalfilters and its restriction to channel models described byrational Doppler power spectral densities. This third paperaddresses the design of Rayleigh fading channel simulatorusing infinite impulse response (IIR) filters for shaping theDoppler spectrum. Specifically, the authors proposed aniterative technique for the IIR filter design with fixed-pointarithmetic by considering both isotropic and nonisotropicscattering scenarios. The proposed iterative algorithm forthe determination of the filter coefficients is described in asystematic manner. The authors also provide an overview ofalternative methods for the design of digital filters and theydiscuss their respective advantages and disadvantages.

The authors of the paper, Wideband and UltrawidebandChannel Models in Working Machine Environment, focuson a rather new topic in the area of mobile radio channelmodeling. This paper presents statistical models for wide-band and ultrawideband (UWB) radio channels in specificworking environments, such as working machine cabins.From the analysis of measurement data collected in suchkinds of rather small and confined propagation environ-ments, it turned out that the channel impulse response ismainly composed of a sum of diffuse multipath componentsrather than of specular components. In wideband channels,

the path gains associated with the various delay bins aregenerally Rayleigh distributed, whereas only a few path gainsexhibit a Rician distribution. This is in contrast to UWBchannels, in which the path gains associated with the delaybins show a tendency to follow the log-normal distribution.The results of the measurements were used to estimate thekey channel parameters for the development of stochasticchannel simulators. The developed mobile radio channelmodels provide important information to the designers ofwireless communication systems dedicated to operate inworking machine cabins.

In the paper Design of Wideband MIMO Car-to-CarChannel Models Based on the Geometrical Street Scattering

Model, the authors deal with the modeling and simulationof channel models for car-to-car (C2C) communicationsystems. The starting point is an environment-specificgeometrical street scattering model, which assumes that thelocal scatterers are uniformly distributed over two separatedrectangular scattering areas located on opposite sides of the

street. From the geometrical scattering model, the authorsderive a reference model for a wideband MIMO C2C channelmodel by assuming that the number of scatterers is infinite.The statistical properties of the reference model are analyzed,where the emphasis lies on the space-time-frequency cross-correlation function from which many other importantsystem functions and characteristic quantities can be easilyderived. To validate the proposed channel model, the meanDoppler shift and the Doppler spread of the reference modelhave been matched to measurement data. In a furtherstep, an efficient sum-of-cisoids channel simulator has beenderived from the reference model. It is demonstrated that thestatistical properties of the sum-of-cisoids channel simulatormatch very well those of the reference model. The proposedgeometry-based channel simulator is useful for studying theeffect of local scatterers along the street on the performanceof C2C communication systems.

Matthias PatzoldNeji Youssef

Carlos A. Gutierrez

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Research ArticleDesign of Wideband MIMO Car-to-Car Channel Models Based onthe Geometrical Street Scattering Model

Nurilla Avazov and Matthias Patzold

Faculty of Engineering and Science, University of Agder, P.O. Box 509, 4898 Grimstad, Norway

Correspondence should be addressed to Nurilla Avazov,[email protected]

Received 2 May 2012; Revised 28 August 2012; Accepted 5 September 2012

Academic Editor: Neji Youssef

Copyright 2012 N. Avazov and M. Patzold. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We propose a wideband multiple-input multiple-output (MIMO) car-to-car (C2C) channel model based on the geometrical streetscattering model. Starting from the geometrical model, a MIMO reference channel model is derived under the assumption ofsingle-bounce scattering in line-of-sight (LOS) and non-LOS (NLOS) propagation environments. The proposed channel modelassumes an infinite number of scatterers, which are uniformly distributed in two rectangular areas located on both sides of thestreet. Analytical solutions are presented for the space-time-frequency cross-correlation function (STF-CCF), the two-dimensional(2D) space CCF, the time-frequency CCF (TF-CCF), the temporal autocorrelation function (ACF), and the frequency correlationfunction (FCF). An efficient sum-of-cisoids (SOCs) channel simulator is derived from the reference model. It is shown that thetemporal ACF and the FCF of the SOC channel simulator fit very well to the corresponding correlation functions of the referencemodel. To validate the proposed channel model, the mean Doppler shift and the Doppler spread of the reference model havebeen matched to real-world measurement data. The comparison results demonstrate an excellent agreement between theory andmeasurements, which confirms the validity of the derived reference model. The proposed geometry-based channel simulatorallowsus to study the effect of nearby street scatterers on the performance of C2C communication systems.

1. Introduction

C2C communications is an emerging technology, whichreceives considerable attention due to new traffic telematicapplications that improve the efficiency of traffic flow andreduce the number of road accidents [1]. The development ofC2C communication technologies is supported in Europe by

respected organizations, such as the European Road Trans-port Telematics Implementation Coordinating Organization(ERTICO) [2] and the C2C Communication Consortium(C2C-CC) [3]. In this context, a large number of researchprojects focussing on C2C communications are currentlybeing carried out throughout the world.

In C2C communication systems, the underlying radiochannel differs from traditional fixed-to-mobile and mobile-to-fixed channels in the way that both the transmitter and thereceiver are in motion. In this connection, robust and reliabletraffic telematic systems have to be developed and tested,which calls for new channel models for C2C communicationsystems. Furthermore, MIMO communication systems can

also be of great interest for C2C communications due to theirhigher throughput [4]. In this regard, several MIMO mobile-to-mobile (M2M) channel models have been developed andanalyzed under different scattering conditions induced by,for example, the two-ring model [5], the elliptical model[6], the T-junction model [7], and the geometrical streetmodel [8,9]. A 2D reference model for narrowband single-

input single-output (SISO) M2M Rayleigh fading channelshas been proposed by Akki and Haber in [10,11]. Simulationmodels for SISO M2M channels have been reported in [12,13]. In [5,14,15], the 2D reference and simulation modelshave been presented for narrowband MIMO M2M channels.The proposed model in [15] combines the two-ring modeland the elliptical model, where a combination of single- anddouble-bounce scattering in LOS propagation environmentsis assumed.

All aforementioned channel models are narrowbandM2M channel models. In contrast with narrowband chan-nels, a channel is called a wideband channel or frequency-selective channel if the signal bandwidth significantly exceeds

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the coherence bandwidth of the channel. Owing to increasingdemands for high data rate wideband communicationsystems employing MIMO technologies, such as MIMOorthogonal frequency division multiplexing (OFDM) sys-tems, it is of crucial importance to have accurate andrealistic wideband MIMO M2M channel models. According

to IEEE 802.11p [16], the dedicated frequency bands forshort-range communications [17] will be between 5770 MHzand 5925 MHz depending on the region. The range 57955815 MHz will be devoted to Europe, while 58505925MHzand 57705850 MHz will be assigned to North America andJapan, respectively. Consequently, a large number of C2Cchannel measurements have been carried out at differentfrequency bands, for example, at 2.4 GHz [18], 3.5GHz[19], 5 GHz [20,21], 5.2 GHz [22], and 5.9 GHz [23]. Real-world measurement campaigns for wideband C2C channelscan be found in [2427]. In the literature, there existseveral papers [2830] with the focus on the modelingof wideband MIMO M2M channels. A reference modelderived from the geometrical T-junction scattering modelhas been proposed in [7] for wideband MIMO vehicle-to-vehicle (V2V) fading channels. In [29], a three-dimensional(3D) model for a wideband MIMO M2M channel has beenstudied. Its corresponding first- and second-order statisticshave been investigated and validated on the basis of real-world measurement data. In the same paper, it has beenshown that 3D scattering scenarios are more realistic than 2Dscattering scenarios. However, 2D scattering models are morecomplexity efficient, and they provide a good approximationto 3D scattering models [31]. For those reasons, we proposein our paper a 2D street scattering model.

In the literature, numerous fundamental channel modelswith different scatterer distributions, such as the uniform,Gaussian, Laplacian, and von Mises distribution, have beenproposed to characterize the angle-of-departure (AOD) andthe angle-of-arrival (AOA) statistics. In [32], the authorstudied the effect of Gaussian distributed scatterers on thechannel characteristics in a circular scattering region arounda mobile station. The spatial and temporal properties of thefirst arrival path in multipath environments have also beenanalyzed in [32]. The authors of [9] assume rectangularscattering areas on both sides of the street, in which aninfinite number of scatterers are uniformly distributed. It hasbeen observed that the shape of the Doppler power spectraldensity (PSD) resembles a Gaussian function if the width ofthe scattering area is very large.

In contrast to our previous work in [9], where the focuswas on the derivation of a reference channel model fornarrowband SISO C2C channels, we design in this paper awideband MIMO C2C channel model by starting from thesame geometrical street scattering model. We focus on thestatistical characterization of a wideband reference channelmodel assuming that an infinite number of scatterers areuniformly distributed within two rectangular areas. Theradio propagation phenomena in street environments aremodelled by a wide-sense stationary uncorrelated scatteringprocess, where in addition a LOS component is taken intoaccount. The reference model has been derived from thegeometrical street scattering model assuming that the AOD

and the AOA are dependent due to single-bounce scattering.To account for the nature of C2C channels, we take themobility of both the transmitter and the receiver for granted.

In our model, we consider a 2D street scattering environ-ment to reduce the computational cost by still guaranteeinga good match between the reference model and measured

channels. A typical propagation scenario for the proposedmodel is illustrated inFigure 1, where the buildings and thetrees are considered as scattering objects. Such a typical denseurban environment scenario allows us to assume that thelocal scatterers are uniformly distributed in a specific area.An analytical expression will be derived for the STF-CCFfrom which the 2D space CCF, the TF-CCF, the temporalACF, and the FCF can be obtained directly. To validate theproposed reference model, the mean Doppler shift and theDoppler spread of the reference model have been matchedto the corresponding quantities of the measured channeldescribed in [25] for different propagation environments,such as urban, rural, and highway areas. Furthermore, wehave derived an SOC channel simulator from the referencemodel. It is shown that the designed channel simulatormatches the underlying reference model with respect to thetemporal ACF and the FCF.

The rest of this paper is organized as follows.Section 2describes the geometrical street scattering model. InSection 3, the reference channel model is derived from thegeometrical street model.Section 4analyzes the correlationproperties of the reference model, such as the STF-CCF,the 2D space CCF, the TF-CCF, the temporal ACF, and theFCF. The computation of the measurement-based modelparameters and the characteristic quantities describing theDoppler effect are discussed inSection 5.Section 6describesbriefly the simulation model derived from the referencemodel. The illustration of some numerical results found forthe correlation functions of the reference model and thecorresponding simulation model is the topic ofSection 7.Finally,Section 8draws the conclusion of the paper.

2. The Geometrical Street Scattering Model

This section briefly describes the geometrical street scatteringmodel for wideband MIMO C2C channels. The proposedgeometrical model describes the scattering environment inan urban area, where the scatterers are located in tworectangular areas on both sides of the street as illustrated inFigure 2. We consider rectangular grids formed by rows and

columns, where the length and the width of the rectangulargrids are denoted by LA = A1 + A2 and Bi (i = 1,2),respectively. The scatterer located in the mth column ofthe nth row is denoted by S(mn) (m = 1,2, . . . ,M, n =1,2, . . . , N). It is assumed that the local scatterers S(mn) areuniformly distributed in the rectangles. The symbolsMSTand M SR inFigure 2stand for the mobile transmitter andthe mobile receiver, respectively. The symbol D representsthe scalar projection of the distance between the transmitterand the receiver onto thex-axis. The transmitter (receiver) islocated at a distance yT1 (yR1 ) from the left-hand side of thestreet and at a distance yT2 (yR2 ) from the right-hand side ofthe street. Both the transmitter and the receiver are in motion

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Modelling and Simulation in Engineering 3

MSR

MST

Figure1: A typical propagation scenario along a straight street in urban areas.

and equipped withMTtransmitter antenna elements andMRreceiver antenna elements, respectively. The antenna elementspacings at the transmitter and the receiver are denoted byTandR, respectively. The symbols

(mn)T and

(mn)R denote the

AOD and the AOA, respectively. The angle T(R) describesthe tilt angle of the transmitter (receiver) antenna array.Moreover, it is assumed that the transmitter (receiver) moveswith speedvT (vR) in the direction determined by the angleof motionTv (

Rv).

3. The Reference Model

3.1. Derivation of the Reference Model. In this section, we

derive the reference model for the MIMO C2C channel underthe assumption of LOS and NLOS propagation conditions.From Figure 2, we realize that the (mn)th homogeneous

plane wave emitted from the lth antenna element A(l)T (l=

1,2, . . . ,MT) of the transmitter travels over the local scatterer

S(mn) before impinging on thekth antenna elementA(k)R (k =

1,2, . . . ,MR) of the receiver. The reference model is based onthe assumption that the number of local scatterers withinboth rectangular areas is infinite, that is,M, N . Thetemporal, spatial, and frequency characteristics of the refer-ence model are determined by theMR MTchannel matrixH(f, t) = [Hkl(f, t)]MRMT, where Hkl(f, t) denotes thetime-variant transfer function (TVTF) of the channel for the

link between thelth transmitter antenna elementA(l)Tand the

kth receiver antenna elementA(k)R . The TVTFHkl(f

, t) canbe expressed as a superposition of the diffuse component andthe LOS component as follows:

Hklf, t

= HDIFkl f, t +HLOSkl f, t, (1)whereHDIFkl (f

, t) and HLOSkl (f, t) represent the diffuse and

the LOS components of the channel, respectively.Note that the single-bounce scattering components bear

more energy than the double-bounce scattering components.Hence, in our analysis, we model the diffuse componentHDIFkl (f

, t) by only taking into account the single-bounce

scattering effects, which is in accordance with the assump-tions made in [28,33]. From the geometrical street scatteringmodel shown in Figure 2, we can derive the TVTF of thediffuse component, which results in the following expression:

HDIFklf, t

= limM,N

1(cR+ 1)MN

M,Nm,n=1

al,mnbk,mncmn

ej[mn+2(f(mn)T +f(mn)R )t2 f(mn)kl ],

(2)

where

al,mn=

ej(T/)(MT2l+1)cos((mn)T T), (3)

bk,mn= ej(R/)(MR2k+1)cos((mn)R R), (4)

cmn= ej(2/)(yT1/sin((mn)T )+yR1/sin(

(mn)R )), (5)

f(mn)T = fTmaxcos

(mn)T Tv

, (6)

f(mn)R = fRmaxcos

(mn)R Rv

, (7)

kl(mn) = 1

c0

D

(l,mn)T +D

(mn,k)R

. (8)

In (6) and (7), the symbols fTmax = vT/ and fRmax =vR/ denote the maximum Doppler frequencies associ-ated with the movement of the transmitter and thereceiver, respectively, and is the wavelength. The symbolcR in (2) represents the Rice factor, which is definedas the ratio of the power of the LOS component tothe power of the diffuse component, that is, cR =E{|HLOSkl (f, t)|2}/E{|HDIFkl (f, t)|2}. The phases mn in (2)denote the phase shift introduced by the scatterer S(mn). Itis assumed that the phases mn are independent, identicallydistributed (i.i.d.) random variables, which are uniformly

distributed over the interval [0,2). The symbolskl(mn)

andc0represent the propagation delays of the diffuse component

and the speed of light, respectively. In (8), the quantityD(l,mn)T

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A1 A2

A1 A2

B1

y

S(mn)

yT1

yR1

yR2

D(mn,1)R

(mn)T

D(mn)T

A(1)R

R

T

A(1)T

A(2)T

R

D(1,mn)T

Rv

VR

VT

(mn)R

D(mn)R

Tv

T

0

A(MT)

TD

B2

yT2

A(MR)R

...

..

.

.

.

.

.

.

.. . .

.

.

....

.

.

....

.

.

.

..

....

..

....

..

....

..

.

MSR

MST

x

Figure2: The geometrical street scattering model with local scatterers uniformly distributed in two rectangular areas on both sides of the

street.

stands for the distance from the lth transmitter antennaelement A

(l)T to the scatterer S

(mn), whereas D(mn,k)R is the

distance between the scatterer S(mn) and the kth receiverantenna element A

(k)R . It is assumed that (MT 1)T

min{yT1,yT2} and (MR 1)R min{yR1,yR2}. Theseassumptions, together with the approximation

1 +x

1 + x/2 (x 1), allow us to approximate the two distancesD

(l,mn)T andD

(mn,k)R as follows:

D(l,mn)T D(mn)T

(MT 2l+ 1)T2 cos(mn)T T,(9)

D(mn,k)R D(mn)R (MR 2k+ 1)

R2

cos

(mn)R R

,

(10)

whereD(mn)T andD

(mn)R are given byD

(mn)T = yT1/sin((mn)T )

andD(mn)R = yR1/sin((mn)R ), respectively.

It is noteworthy that one can also find articles [11,34],in which only double-bounce scattering is assumed for M2Mcommunications. However, by following a similar approachas in [15], one can easily extend our analysis on the basisof single-bounce scattering to the case of double-bounce

scattering, and thus also to a combination of single- anddouble-bounce scattering.

The TVTF of the LOS component is given by

HLOSklf, t

= cR(cR+ 1)

ej[2(f(0)T +f

(0)R )t(2/)Dkl2 fkl

(0)],

(11)

where

f(0)T = fTmaxcos

(0)T Tv

, (12)

f(0)R = fRmaxcos(0)R Rv, (13)

Dkl= D0 (MT 2l+ 1) R2

cos

T

+(MR 2k+ 1) R2

cos

R

,

(14)

D0=

D2 +yT1 yR1

2. (15)

In (11), f(0)T and f

(0)R denote the Doppler shifts of the

LOS component caused by the movement of the transmitter

and the receiver, respectively. The symbols (0)T and

(0)R

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in (12) a nd (13) represent the AOD and the AOA of

the LOS component, respectively. Finally, kl(0)

denotes thepropagation delay of the LOS component. The delay of the

LOS component is defined bykl(0) = Dkl/c0 withDklbeing

the length of the direct path from thelth transmitter antenna

elementA(l)T to the kth receiver antenna element A

(k)R . The

symbolD0 in (14) denotes the Euclidean distance betweenthe transmitter and the receiver. According to [35], the LOScomponentHLOSkl (f

, t) is assumed to be a deterministic pro-cess, while the diffuse componentHDIFkl (f

, t) is a stochasticprocess.

3.2. Derivation of the AOD and the AOA. The position of alllocal scatterersS(mn) is described by the Cartesian coordinates(xm,yn). In the reference model, the coordinatesxm and ynare independent random variables, which are determinedby the distribution of the local scatterers. With referencetoFigure 2, we take into account that due to single-bounce

scattering, the AOD(mn)

T and the AOA(mn)

R are dependent.By using the trigonometric identities, we can express the

AOD(mn)T and the AOA

(mn)R in terms of the coordinates

(xm,yn) of the local scatterersS(mn) as follows:

(mn)T

xm,yn

=

g

xm,yn

, ifynJi, xm [0,A2](1)i+1+gxm,yn, ifynJi, xm [A1, 0]

(16)

(mn)R

xm,yn=

f

xm,yn

, ifynJi, xm [D,A2](1)i+1+ fxm,yn, ifynJi, xm [A1, D]

(17)

for i = 1,2, whereJ1= [yT1 ,yT1 +B1], J2= [yT2 B2, yT2 ],and

g

xm,yn = arctan yn

xm

fxm,yn = arctanyn yT1+ yR1

xm D .

(18)

4. Correlation Properties ofthe Reference Model

In this section, we derive a general analytical solution for theSTF-CCF, from which other correlation functions, such asthe 2D space CCF, the TF-CCF, the temporal ACF, and theFCF can easily be derived.

4.1. Derivation of the STF-CCF. According to [10], the STF-

CCF of the linksA(l)TA(k)R andA(l

)T A(k

)R is defined as the

correlation between the channel transfer functionsHkl(f, t)andHk l (f, t), that is,

kl,k l (T, R, , ) = E

Hkl

f, t

Hk l

f+ , t+

=DIFkl,k l (T, R,

, )

+LOSkl,k l (T, R, , ),

(19)

where () denotes the complex conjugate operator andE{}stands for the expectation operator that applies to all randomvariables: the phases{mn} and the coordinates (xm,yn)of the scatterers S(mn). The first term DIFkl,k l (T, R, ,

)represents the STF-CCF of the diffuse component. Thiscorrelation function can be expressed, after substituting (2)in (19), by

DIFkl,kl (T, R, , )

= limM,N

1

(cR+ 1)MN

M,N

m,n=1E

c(mn)ll d

(mn)kk e

j2[(f(mn)T +f

(mn)R )kl

(mn)]

,

(20)

where

c(mn)ll = ej2(T/)(ll

)cos((mn)T T),

d(mn)kk = ej2(R/)(kk

)cos((mn)R R).(21)

The quantities f(mn)T , f

(mn)R , and

kl

(mn)are given by (6), (7),

and (8), respectively. We recall that the AOD(mn)T and the

AOA(mn)R can be expressed in terms of the random variables

xmand ynaccording to (16) and (17), respectively.In Section 2, it has been mentioned that all scatterers

are uniformly distributed in the two rectangular areas onboth sides of the street, as illustrated in Figure 2. Hence, therandom variablesxm and yn are also uniformly distributedover the rectangular areas. If the number of scatterers tendsto infinity, that is, M, N , then the discrete randomvariables xm and yn become continuous random variablesdenoted by x and y, respectively. Thus, the probability

density functions (PDFs) px(x) and py(y) of x and y,respectively, are given by

px(x) = 1LA

, ifx [A1,A2],

pyy =

1

2B1, ify yT1 , B1+ yT1

1

2B2, ify B2 yT2 , yT2,

(22)

whereLA= A1 + A2. Assuming that the random variablesx and y are independent, the joint PDF pxy(x,y) of the

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random variablesxand ycan be expressed as a product ofthe marginal PDFspx(x) and py(y), that is,

pxy

x,y

= px(x) pyy

=

12LAB1, ifx [A1,A2], y yT1 , B1+ yT1

1

2LAB2, ifx [A1,A2], y

B2 yT2 , yT2.(23)

The infinitesimal power of the diffuse component cor-responding to the differential axes dx and d y is propor-tional to pxy(x,y)dxdy. As M, N , this infinitesimalcontribution must be equal to 1/MN = pxy(x,y)dxdy.Consequently, it follows from (20) that the STF-CCF of thediffuse component can be expressed as

DIFkl,kl (T, R, , )

= 12LAB1(cR+ 1)

yT1 +B1yT1

A2A1

cDIFll

T, x,y

dDIFkk

R, x,y

ej2[(fT(x,y)+fR(x,y))kl(x,y)]dxdy

+ 1

2LAB2(cR+ 1)

yT2

B2yT2

A2A1

cDIFll

T, x,y

dDIFkk

R, x,y

ej2[(fT(x,y)+fR(x,y))kl(x,y)]dxdy,(24)

where

cDIFll

T, x,y = ej2(T/)(ll)cos(T(x,y)T),

dDIFkk

R, x,y = ej2(R/)(kk)cos(R(x,y)R),

fT

x,y = fTmaxcosTx,y Tv,

fR

x,y = fRmaxcosRx,y Rv,

kl

x,y = 1

c0

D

(l)T

x,y

+D

(k)R

x,y

.

(25)

Using the functions in (9) and (10), the distancesD(l)

T

(x,y)

andD(k)

R (x,y) can be expressed as

D(l)T

x,y

yT1sin

T

x,y

(MT 2l+ 1)

T2

cos

T

x,y T,

D(k)R

x,y

yR1sin

R

x,y

(MR 2k+ 1)

R2

cos

R

x,y R.

(26)

In (19), the quantityLOSkl,kl (T, R, , ), which represents the

STF-CCF of the LOS component, can be written as

LOSkl,kl (T, R, , ) = cR

(cR+ 1)c

(0)ll (T)

d(0)kk (R)ej2[(f(0)T +f

(0)R )kl(0)],

(27)

where

c(0)ll (T) = ej2(T/)(ll

)cos(T), (28)

d(0)kk (R) = ej2(R/)(kk

) cos(R). (29)

The Doppler shifts f(0)T and f

(0)R are given by (12) and (13),

respectively.

4.2. Derivation of the 2D Space CCF. The 2D spaceCCF kl,kl (T, R) is defined as kl,k l (T, R) =E{Hkl(f, t)Hkl (f, t)}, which is equal to the STF-CCF

kl,k l (T, R, , ) in (19) by setting and to zero, that is,

kl,k l (T, R) =kl,kl (T, R,0 ,0)

= 12LAB1(cR+ 1)

yT1 +B1yT1

A2

A1cDIFll

T, x,y

dDIFkk

R, x,y

dxdy

+ 1

2LAB2(cR+ 1)

yT2B2yT2

A2

A1cDIFll

T, x,y

dDIFkk

R, x,y

dxdy

+ cR

(cR+ 1)c

(0)ll (T)d

(0)kk (R).

(30)

4.3. Derivation of the TF-CCF. The TF-CCF of the trans-

mission link from A(l)T (l = 1,2, . . . ,MT) to A(k)R (k =

1,2, . . . ,MR) is defined byrkl(, ) := E{Hkl(f, t)Hkl(f+

, t+)} [36]. The TF-CCF can be obtained directly from theSTF-CCF [see (19)] by setting the antenna element spacingsTandRto zero, that is,

rkl(, ) =DIFkl,kl (0,0, , )+LOSkl,kl (0,0, , )

= 12LAB1(cR+ 1)

yT1 +B1yT1

A2

A1ej2[(fT(x,y)+fR(x,y))

kl(x,y)]dxdy

+ 1

2LAB2(cR+ 1)

yT2B2yT2

A2

A1ej2[(fT(x,y)+fR(x,y))

kl(x,y)]dxdy

+ cR

(cR+ 1)ej2(f

(0)T +f

(0)R )ej2

kl(0)

.

(31)

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4.4. Derivation of the Temporal ACF and the Doppler PSD.

The temporal ACF of the transmission link fromA(l)T (l=

1,2, . . . ,MT) toA(k)R (k = 1,2, . . . ,MR) is defined byrkl() :=

E{Hkl(f, t)Hkl(f, t+ )} [36, Page 376]. The temporal ACFcan be obtained directly from the TF-CCF (see (31)) bysetting at to zero, that is,rkl()

=rkl(, 0), which gives

rkl() = 12LAB1(cR+ 1)

yT1 +B1yT1

A2

A1ej2[fT(x,y)+fR(x,y)]dxdy

+ 1

2LAB2(cR+ 1)

yT2B2yT2

A2

A1ej2[fT(x,y)+fR(x,y)]dxdy

+ cR(cR+ 1)

ej2(f(0)T +f(0)R ).

(32)

Notice that the expression in (32) reveals that the ACFrkl()is independent ofkand l.

Computing the Fourier transform of the temporal ACFrkl() results in the Doppler PSDSkl(f), that is,

Sklf =

rkl()e

j2 f d. (33)

The two most important statistical quantities character-izing the Doppler PSD Skl(f) are the average Doppler shift

B(1)kl and the Doppler spreadB

(2)kl [35]. The average Doppler

shiftB(1)kl is defined as the first moment ofSkl(f), which can

be expressed as follows:

B(1)kl =

f Skl

f

df Skl

f

df . (34)

The Doppler spreadB(2)kl is defined as the square root of

the second central moment ofSkl(f), which can be writtenas

B(2)kl =

f B(1)kl 2SklfdfSkl

f

df . (35)

4.5. Derivation of the FCF. The frequency characteristics ofthe reference model are described by the FCFrkl().The FCFrkl() of the transmission link fromA

(l)T toA

(k)R is defined by

rkl() := E{Hkl(f, t)Hkl(f+ , t)} for alll= 1,2, . . . ,MTandk= 1,2, . . . ,MR. This function can be obtained directly

from the TF-CCF [see (31)] by setting to zero, that is,rkl() = rkl(0, ), which results in

rkl() = 1

2LAB1(cR+ 1)

yT1 +B1yT1

A2A1

ej2kl(x,y)dxdy

+ 1

2LAB2(cR+ 1) yT2

B2yT2 A2

A1 ej2

kl(x,y)

dxdy

+ cR

(cR+ 1)ej2

kl(0)

.

(36)

In contrast to the temporal ACF rkl(), the FCF rkl()depends onkandl.

5. Measurement-Based Computation ofthe Model Parameters

The objective of this section is to determine the set of model

parameters P = {A1,A2, B1, B2,yT1 ,yT2 ,yR1 ,yR2 , D,fTmax,fRmax, cR}describing the reference model in such a way thatthe average Doppler shiftB

(1)kl and the Doppler spreadB

(2)kl

of the reference model match the corresponding quantities

(B(1)kl andB

(2)kl ) of the measured channel reported in [25].

To determine the set of model parameters P, we minimizethe following error:

Emin= W1EB(1)kl +W2EB(2)kl , (37)

whereW1andW2denote the weighting factors. The symbolsEB(1)kl

and EB(2)klin (37) stand for the absolute errors of

the average Doppler shift and Doppler spread, respectively,

which are defined as

EB(1)kl= argmin

P

B(1)kl B(1)kl , (38)EB(2)kl

= arg minP

B(2)kl B(2)kl . (39)In (38) and (39), the notation arg minxf(x) stands for

the argument of the minimum, which is the set of pointsof the given argument for which f(x) reaches its minimumvalue. At the beginning of the optimization procedure, theweighting factors W1 and W2 are selected arbitrarily, butsuch that they satisfy the equality W1 + W2 = 1. If theerror EB(i)kl

(i

= 1 ,2) in (37) is large, then we reduce

the corresponding weighting factor Wi and vice versa. Wecontinue the optimization procedure until the result in (37)reaches an error floor, meaning that the average Doppler shiftand the Doppler spread of the reference model best match themeasured average Doppler shift and the measured Dopplerspread, respectively.

For the measured channels in [25], the resulting opti-mized model parameters and the corresponding averageDoppler shift and Doppler spread are listed in Table 1.The results found for the reference model demonstrate anexcellent fitting to real-world measured channels for rural,urban, and highway propagation areas, which validates theusefulness of the proposed reference model. It is worth

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Table1: Measurement-based parameters of the geometrical street scattering model and the resulting average Doppler shift and the Dopplerspread.

Model parameters Propagation environment

Urban LOS Urban NLOS Rural LOS Highway LOS Highway NLOS

A1(A2) (m) 546.28 (1249) 537.03 (908.3) 546.52 (1236) 547.69 (1207) 546.88 (1193)

B1(B2) (m) 198.96 (198.77) 76.46 (1.1113) 20.89 (18.25) 199.8 (200) 0.01 (0.01)fTmax(fRmax) (Hz) 223.55 (219.77) 262.1 (209.97) 463.72 (491.65) 511.68 (442.62) 491.67 (481.97)

yT1(yT2) (m) 10.42 (7) 2.12 (1.18) 15.28 (4.63) 17.62 (19.78) 1.3 (1.3)

yR1(yR2) (m) 19.82 (6.6) 20 (7.06) 14.57 (9.4) 19.63 (25) 20 (9.4)

D(m) 238.6 236.7 186.77 896.7 749.6

cR 0.485 0 0.27 0.4 0

Measured

average Doppler 20 103 201 209 176shiftB

(1)kl (Hz) [25]

Theoretical

average Doppler 20 102.67 200.55 208.8 110shiftB

(1)

kl

(Hz)

Measured

Doppler 341 298 782 761 978

spreadB(2)kl (Hz) [25]

Theoretical

Doppler 341 298 782.03 760.88 941

spreadB(2)kl (Hz)

mentioning that the computed average Doppler shiftB(1)kl =

110 Hz and the Doppler spread B(2)kl = 941Hz do notclosely agree with the measured channel (B

(1)kl = 176Hz

andB(2)kl = 978 Hz) in case of the highway NLOS scenario.

For this scenario, a close agreement can be found forsufficiently small values ofcR /= 0.

6. The Simulation Model

The reference model described above is a theoretical model,which is based on the assumption that the number ofscatterers (M, N) is infinite. Owing to an infinite realizationcomplexity, the reference model is non-realizable. However,the reference model can serve as a ground for the derivationof stochastic and deterministic simulation models. Accordingto the generalized principle of deterministic channel mod-

eling [35, Sec. 8.1], a stochastic simulation model can bederived from the reference model introduced in (1) by usingonly a finite number of scatterers. In the literature, severaldifferent models exist that allow for a proper simulationof mobile channels. The SOC model is an appropriatesimulation model for mobile radio channels under non-isotropic scattering conditions. A detailed description andthe design of SOC models can be found in [37, 38],respectively. In [38], several parametrization techniques forSOC models have been discussed and analyzed. Here, we usetheLp-norm method (LPNM), which is a high-performanceparameter computation method for the design of SOCchannel simulators.

7. Numerical Results

This section illustrates the analytical results given by (30),(31), (32), and (36). The correctness of the analytical

results will be verified by simulations. The performance ofthe channel simulator has been assessed by comparing itstemporal ACF and the FCF to the corresponding systemfunctions of the reference model (see (32) and (36)).

As an example for our geometrical street scatteringmodel, we consider rectangular scattering areas on both sidesof the street with a length ofLA = A1 +A2, where A1 =50m and A2= 450m, and a width ofB1 = B2 = 100m.With reference toFigure 2, the position of the transmitterand the receiver are defined by the distances D = 400m,yT1= yR2= 20m, and yT2= yR1= 10 m. For the referencemodel, all theoretical results have been obtained by choosingthe following parameters: T= 90, R = 90, Tv = 0,Rv= 180, and fTmax= fRmax= 91 Hz. The Rice factorcRwaschosen from the set{0, 0.5, 1}. The scatterers are uniformlydistributed over the considered rectangular areas. The Lp-norm method has been applied to optimize the simulationmodel parameters by using a finite number of scatterers(cisoids). For the simulation model, we useMN= 50 25scatterers (cisoids) within the rectangle on the left-hand sideas well as on the right-hand side.

In Figure 3, the absolute value of the 2D space CCF|11,22(T, R)| of the reference model is presented for theNLOS propagation scenario (cR= 0). The results have beenobtained by using (30). FromFigure 3, we can observe thatthe 2D space CCF decreases as the antenna element spacings

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0.8

0.6

0.4

0.2

00

0

1

1

122

3

3

A1 = 50mA2 = 450mB1 = B2 = 100mD = 400m

yT1 =

yR2 =

20myT2 =yR1 = 10m

cR = 0

R/T/

2DspaceCCF,

|11,2

2(T,R

)|

Figure3: Absolute value of the 2D space CCF |11,22(T, R)| of thereference model for a NLOS propagation scenario (cR= 0).

0.8

0.6

0.4

0.20

0

1

1

122

3

3

A1 = 50mA2 = 450mB1 = B2 = 100mD = 400myT1 =yR2 = 20myT2 =yR1 = 10m

cR = 1

R/T/

2DspaceCCF,

|11,2

2(T,R

)|

Figure4: Absolute value of the 2D space CCF |11,22(T, R)| of thereference model for a LOS propagation scenario (cR= 1).

increase. For comparison reasons, the absolute value of the2D space CCF|11,22(T, R)| is depicted in Figure 4 for aLOS propagation scenario (cR = 1). From Figure 4, onecan see that the channel transfer functions Hkl(f, t) andHkl (f, t) are highly correlated over a large range of antennaelement spacings T and R. This can be concluded fromthe fact that even for large antenna element spacings, forexample, T = R = 3, the absolute value of the 2Dspace CCF

|11,22(T, R)

|equals approximately one half of

its maximum value. Comparing Figures3 and 4 shows thatby increasing the Rice factor cR, the 2D space CCF alsoincreases.

Figures 5 and 6 illustrate the TF-CCFs of the refer-ence model under NLOS and LOS propagation conditions,respectively. FromFigure 5, we can observe that the TF-CCFdecreases as the time and frequency lags increase in NLOSpropagation environments. A comparison of Figures5 and6shows that the absolute value of the TF-CCF under LOSconditions is in general higher than under NLOS.

Figure 7depicts the absolute value of the temporal ACF|rkl()| according to (32) if both the transmitter and thereceiver are moving towards each other. A good match


cR = 0

Timelag, (s)

TF-CCF,

|r1

1(A

,)|

1

0.8

0.6

0.4

0.2

00

0.010.02

0.030.04 10

5

0

Frequen

cylag, A

(MHz)

Figure5: Absolute value of the TF-CCF |r11(, )| of the referencemodel for a NLOS propagation scenario (cR= 0).


cR = 1

Timelag, (s)

TF-CCF,

|r11

(A

,)|

1

0.8

0.6

0.4

0.2

00

0.010.02

0.030.04 10

5

0

Frequen

cylag,

A (MHz)

Figure6: Absolute value of the TF-CCF |r11(, )| of the referencemodel for a LOS propagation scenario (cR= 1).

between the temporal ACF of the reference model and that ofthe simulation model can be observed in Figure 7. This figuredemonstrates also that the experimental simulation resultsof the temporal ACF match very well with the theoreticalresults.

Finally,Figure 8illustrates the absolute value of the FCF|rkl()| for different Rice factorscR= {0, 0.5, 1} if both thetransmitter and the receiver are moving towards each other.A close agreement between the reference model and thesimulation model can be seen inFigure 8for all chosen Ricefactors. One can realize that the experimental simulationresults of the FCF match very well with the theoretical results.

8. Conclusion

In this paper, a reference model for a wideband MIMOC2C channel has been derived by starting from the geomet-rical street scattering model. Taking both LOS and NLOS

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Research ArticleA Three-Dimensional Geometry-Based Statistical Model of2 2Dual-Polarized MIMO Mobile-to-Mobile Wideband Channels

Jun Chenand Thomas G. Pratt

Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

Correspondence should be addressed to Thomas G. Pratt, [email protected]

Received 27 April 2012; Accepted 27 July 2012

Academic Editor: Carlos A. Gutierrez

Copyright 2012 J. Chen and T. G. Pratt. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A three-dimensional (3D) model for wide-band dual-polarized (DP) multiple-input-multiple-output (MIMO) mobile-to-mobile(M2M) channels is proposed. Using geometrical scattering based on concentric spheres at the transmitter (Tx) and at the receiver(Rx), a 3D parametric reference model for 2 2 M2M DP multipath fading channels is developed. The channel model assumes theuse of colocated half-wavelength dipole antennas for vertical and horizontal polarizations at both transmit and receive stations.Model parameters include the velocities of the Tx and Rx nodes, the distance between the nodes, the 3D antenna pattern gains,the azimuth and elevation angles of arrival and departure, the geometrical distribution of the scatterers, the Rician K-factorsdefining the fading envelope distributions, the maximum Doppler frequency, the scattering loss factors, the cross-polar powerdiscrimination ratio (XPD), and the copolarization power ratio (CPR). Using the proposed model, expressions for joint time-frequency correlation functions (TFCFs) are derived which are used to investigate system behavior over different wide sensestationary uncorrelated scattering (WSSUS) channel realizations. The numerical results illustrate the sensitivities of the TFCFto simultaneous time and frequency offsets for the 2 2 DP-MIMO architectures.

1. Introduction

For nearly two decades, multiple-input-multiple-output(MIMO) communications systems [13] have been rig-orously studied, leading to the adoption of MIMO ina number of wireless communications standards such asLong Term Evolution (LTE) and 802.11n. The LTE provides

high-speed wireless communications for mobile phonesand data terminals, while 802.11n supports MIMO, frameaggregation, and security improvements for wireless localarea networks (WLAN). In these systems, multiple antennascan be used at both the transmitter (Tx) and receiver (Rx)to exploit the diversity and channel capacity afforded byMIMO architectures. In array-based MIMO systems, CPantennas at mobile stations and access points are ideallyseparated by at least one-half wavelength and at cellular basestations by at least ten wavelengths to achieve significantdiversity or multiplexing gains [4]. In space-constrainedimplementations such as with handheld devices, use ofantenna spacings less than one-half wavelength can lead to

antenna correlations that lend themselves to beamforminggains, but ohmic coupling effects can deplete any gains thatwould otherwise be achieved with these strategies [5]. Analternative architecture for space-constrained deploymentsinvolves the use of dual-polarized (DP) antennas. This

strategy has the advantage of offering colocated antennas

that provide largely uncorrelated fading responses, althoughusually with asymmetrical average powers among the MIMOchannel matrix entries. Hence, the channel behaviors forDP systems are substantively different from conventional

copolarized (CP) channels, the latter which exhibit similaraverage powers. The relative performance of DP-MIMO incomparison to traditional MIMO with CP arrays will bedictated by the disparities in these responses. While DParchitectures are receiving increasing attention, they havethus far been studied with much less rigor than CP systems.

To characterize DP systems, especially in widebandsignaling applications, modeling of dispersive input-to-output polarimetric channel behavior is important. Many

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theoretical models [611] provide statistical representationsof polarimetric channels based on cross-polarizationcoupling (XPC), but do not characterize polarization-frequency behavior that is needed to represent wider-band, frequency-selective channels for the analysis of DPcommunications systems. For example, XPC is most often

analyzed in the narrowband sense and yet it has been shownthat the XPC can vary with frequency. It also exhibitscorrelation in the frequency dimension in a manner thatdepends on the channels temporal dispersion propertiesand polarization coupling [12]. Statistical channel models inliterature that address time-selective and frequency-selectivebehavior often assume that the time dispersion (due totime delays) and the frequency dispersion (due to Dopplerspread) are statistically independent, where the time delaysdepend on the relative locations of the random scatterers andthe Doppler spreads depend on the motion of the Tx andRx antennas [13, 14]. However, this assumption is brokenif both the time delays and the Doppler spreads dependon the relative location of the random scatterers (e.g., theangles of departure and the angles of arrivals). Becausethese phenomena would not be expected to be statisticallyindependent, it is of interest to examine joint correlationfunctions in time and frequency, which is an application ofthe model that we consider in our paper.

Analytical approaches such as Kronecker or eigenbeammodels have also been used for model simulations, andthese have dealt with narrowband channels. An analyticalframework based on a Kronecker model is presented in[15] to model narrowband DP Rayleigh and Rician fadingchannels for arbitrary array sizes. The framework uses arelatively small number of physical parameters to analyzethe benefits of multiple polarization architectures. In otherwork, a 3D polarized narrowband spatial channel modelis presented in [7], and the impact of elevation angle oncapacity is reported in [16]. These models assume idealdipole antennas and unity cross-polar discrimination ratios.

Another approach to characterize channels for DPsystems involves the use of geometric scattering models.Geometrical scattering models [1719] have the advantagethat they are able to represent important channel behav-iors, including mutual effects that arise from polarizationcoupling, time-dispersion, angle-dispersion, and frequency-dispersion. However, they also are more complex. Early two-dimensional (2D) geometric scattering models were devel-oped for narrowband CP single-input single-output (SISO)

mobile-to-mobile (M2M) Rayleigh fading channels [2022].In [23], the polarisation-sensitive geometric modelling isdeveloped with a direction-of-arrival (DOA) distributionthat depends on the polarisation states of the transmittingantennas, the receiving antennas and the polarisation prop-erties of the scatterers. These models were 2D in the sensethat the models treated electromagnetic propagation only ina fixed-elevation plane and did not model elevation antennapattern dependencies or distributions of scatterers in theelevation dimension.

Some three-dimensional (3D) models have been pro-posed to overcome shortcomings of 2D M2M channel mod-els, particularly in environments where deployed antenna

heights are lower than surrounding buildings and obstacles.In [18], a 3D wideband M2M mathematical reference modelwas proposed based on a concentric-cylinders geometryusing a superposition of line-of-sight (LOS), single bounceat the transmit side (SBT), single bounce at the receiveside (SBR), and double-bounce (DB) rays in a variety of

urban environments. The analysis approach proved usefulto characterize M2 M correlations, but to date has only beenapplied to the analysis of CP MIMO channels with unity-gain idealized dipole antennas. Recently, a 3D polarizedchannel model has been proposed to treat spatially-separatedorthogonally polarized elements [19]. The model dealsexclusively with the double-bounce ray and narrowbandchannels.

In this paper, we develop a 3D geometric scatteringmodel for 2 2 M2M DP wideband channels based onconcentric spheres to evaluate joint time frequency cor-relation functions associated with the subchannel fadingenvelopes. The use of concentric spheres is motivated bythe ease with which scatterer locations may be identifiedfor a given radius, azimuth AOD, and elevation AOD,and by its capability to support the modeling of overheadreflectors. The DP-MIMO channel is constructed assuming aDP antenna at both the Tx and the Rx, where the polarizationbasis at the Rx is matched to the polarization basis usedat the Tx. The received signals can be translated to anarbitrary orthogonally-polarized basis by applying a unitarytransformation to the received signal vectors [24] withoutimpacting theoretical performance measures such as capacityand diversity.

We assume a channel with wide sense stationary uncor-related scattering (WSSUS) where the channel correlationfunction is invariant over time, and the scatterers withdifferent path delays are uncorrelated [25, 26]. Channeltransfer functions for each transmit antenna/receive antennapair are derived as a superposition of LOS, SBT, SBR, andDB component rays. The transfer functions are used toform time-frequency correlation functions (TFCF) for theassumed 3D nonisotropic scattering environment, wherescattering distributions are characterized in the azimuthdimension through the von Mises distribution and inthe elevation dimension by a cosine distribution. The3D scattering propagation model used to evaluated theTFCFs simulates the effects of the antenna pattern gains,the geometrical distribution of scatterers and the asso-ciated azimuth/elevation angles of arrival and departure,

the K-factor of the fading distributions, the maximumDoppler frequency, the scattering loss factors, the cross-polar discrimination (XPD), and the copolarization ratio(CPR).

The remainder of the paper is organized as follows.The 3D DP 2 2 MIMO M2M model is presented inSection 2. In Section 3, we derive transfer functions andTFCFs for each subchannel of the DP-MIMO channeland for each ray path type for 3D nonisotropic scatteringenvironments. Numerical results of the joint correlationfunctions associated with the 2 2 DP channel models arepresented in Section4. We conclude with a summary of ourfindings in Section5.

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(m)r

Z

S(m)t

RtY

(n)t(m)tvtvt A

(p)t

Ot

X

A( p)t

D

Z

S(n)r

Y

vr (m)r

(n)rRr

vr A(q)r

OrX

LOSt LOSr A(q)r

Tx

Rx

p,m

vt

LOSt

(m)t

(n)t

m,n

vr

LOSr

n,q

(n)r

Figure 1: Concentric-spheres for 3D colocated DP channel model. The signals from the Tx antennas arrive in parallel at the Rx antennaarray.

2. 2 2DP-MIMO Channel Model Based onConcentric Spheres

The concentric sphere geometry-based scattering modelassumes a mobile Tx and a mobile Rx, both equipped

with DP antennas using a matched polarization basis. Radiopropagation between the Tx and the Rx is characterized by3D WSSUS under channels that can include line-of-sight(LOS) and non-line-of-sight (NLOS) components, wherescattering centers in the latter case reside on concentricspheres about either the Tx, the Rx, or both. The linearlytime-variant MIMO channel can be represented by a 2 2impulse response matrix in terms of timetand delay

H(t, ) =

hqp(t, ) hq p(t, )hqp(t, ) hq p(t, )

. (1)

Figure 1 illustrates the concentric sphere model for aMIMO M2M channel with DP antennas. One DP antenna

with vertically-polarized and horizontally-polarized compo-

nents denoted byA(p)t and A

(p)t , respectively, is located at

the center of the Tx sphere (Ot). A second DP antenna

with corresponding components denoted byA(q)r and A

(q)r

is located at the center of the Rx sphere (Or). At the Tx,Mfixed scatterers reside within the volume of a sphere defined

by a radiusRt. Themth transmit scatterer is denoted byS(m)t ,

where 1 m Mand resides on the surface of a sphere withradiusRm, whereRm < Rt. Similarly at the Rx,Nscatterersoccupy the spherical volume with a radiusRr. The nth receive

scatterer is denoted byS(n)r , where 1 n N and resides

on the surface of a sphere with radius Rn, where Rn < Rr.

The set of scatterers is comprised ofHscatterers (reflectinghorizontally polarized waves) and V scatterers (reflectingvertical polarized waves). We assume that the distribution ofthe H scatterers and theVscatterers are identical and thatthe number of scatterers for each are the same, although this

may not be true in general.The center of the Tx sphere serves as the global origin of

a rectangular coordinate system. At time t= 0, the Rx is adistanceD from the Tx withXYZcoordinates denoted by(x, y, z). The height difference between the Tx and theRx antennas is included in the offset z. The symbols p,m,

m,q, p,n, n,q , m,n and p,q denote distances d(A(p)t , S

(m)t ),

d(S(m)t ,A

(q)r ), d(A

(p)t , S

(n)r ), d(S

(n)r ,A

(q)r ), d(S

(m)t , S

(n)r ), and

d(A(p)t ,A

(q)r ), respectively, where d() denotes the distance

between the two coordinates. The symbols(m)t ,

(n)t are the

elevation angles of departure (EAoD, relative to the XY

plane) to the scatterers S(m)t and S

(n)r , respectively, whereas

(m)t , (n)t are the azimuth angles of departure (AAoD, in

the XY plane relative to the Z-axis) to the scatterers S(m)t

andS(n)r respectively. Similarly, the symbols

(m)r ,

(n)r denote

the elevation angles of arrival (EAoA, relative to the x-y

plane) reflected from the scatterers S(m)t and S

(n)r , respectively,

whereas (m)r , (n)r denote the azimuth angles of arrival

(AAoA, in the XY plane relative to the z-axis) reflected

from the scatterersS(m)t and S

(n)r respectively. For the LOS

component between the Tx and the Rx, the symbols LOSt andLOSt are the elevation angle of departure (relative to theXYplane) and the azimuth angle of departure in theXYplanerelative to theZ-axis, respectively. Similarly, the symbolsLOSr

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andLOSr are the elevation angle of arrival and the azimuthangle of arrival, respectively. The Tx and Rx are moving withspeeds vt and vr in directions described by the elevationangles vt and vr relative to the XY plane, respectively,and by the azimuth angles vt and vr in the XY plane,respectively.

2.1. Scatterer Distributions. The directions of scatterers aredescribed by azimuth and elevation angle distributions.Several different distributions, such as uniform, Gaussian,Laplacian, and von Mises, have been used in the literature tocharacterize the azimuth angles of departure and arrival. vonMises Fisher (VMF) distributions are effective to model bothAAoD and AAoA for spatial fading correlation models [27]and have the advantage that the probability density functionapproximates many of distributions and admits closed-formsolutions [18]. To simplify the modeling, it is assumed thatthe azimuth and elevation angles are independent, enablingthe use of a product of distribution functions instead of a

joint distribution. The scatterer distributions are synthesizedusing different angle distributions for the azimuth and theelevation dimensions. The Von Mises Fisher probabilitydensity function (pdf) distribution is used for azimuth

dimensions, including for angles of departure ((m)t and(n)r )

and angles of arrival ((n)t and

(m)r ). The von Mises pdf is

defined as [28]

f

= exp cos

2I0() , [,), (2)

where [,) is the mean value of the scattererdirections in the azimuth plane, controls the spread ofscatterers around the mean , and I0(

) is the zero-order

modified Bessel function of the first kind. When =0, f()=1/2 is a uniform distribution, corresponding toisotropic scattering in azimuth. Asincreases, the scatterersbecome more clustered about the mean angle , andscattering is nonisotropic.

The random elevation angles of departure and arrivalcan be characterized by a uniform, cosine or Gaussiandistribution. In [18,29], a cosine pdf is used as it may fit withthe typical propagation in M2M communications, where theTx and Rx are in motion and equipped with low elevationantennas. A cosine distribution employed for the elevationangles of departure and arrival is given by [29]:

f() =

4|max|

cos2

max

, || |max| 2

,

0, otherwise.

(3)

where maxis the maximum elevation angle which we assumeto have a value near 20. This maximum elevation angle istypical of M2M wireless communications where both the Txand Rx are equipped with low elevation antennas.

The above-mentioned distributions are used to representthe response from a cluster, which describes a group of scat-terers located within an isolated solid angle, and the responsefrom multiple clusters compose the aggregate response from

the channel. In [27], a mixture of Von Mises distributionsis proposed for modeling the 3D direction of scatterersin the presence of multiple clusters of scatterers over thepropagation channel. Mathematically, each distribution inthe mixture of M VMFs at the Tx or N VMFs at the Rxcan be described as f(

(m)t |

(m)

t , m)(1 m M) or

f((n)r |

(n)

r , n)(1 n N). Note that (m)

t or (n)

r is themean azimuth angle of the mth or nth cluster; m or nis the concentration of the mth or nth cluster. Hence, theoverall density functions of the mixture model consisting

of M VMF distributions for the cluster azimuth angles(m)tat the transmitter and N VMF distributions for the clusterazimuth angles(n)r at the Rx can be described as [27]

f

t = M

m=1m

f (m)t |

(m)

t , m

,

fr =N

n=1 nf (n)r

|

(n)

r , n,(4)

whereM and N the number of clusters at the Tx and Rx,respectively; m or n is defined as the a priori probabilitythatmth ornth cluster was generated. Similarly, the overalldensity functions of the mixture model consisting of the

cosine distributions for the elevation angles (m)t at the Tx

and(n)r at the Rx can be expressed as

f(t) =Mm=1

m

(m)t | f

(m)

t

,

f(r) =Nn=1

n(n)r | (n)r ,(5)

where(m)

t and(n)

r are the mean elevation angle of themthornth cluster at the Tx and Rx, respectively.

Signals from the Tx antenna elements that propagatedirectly to the Rx antenna elements form the LOS compo-nent. Signals reflected exclusively from the scatterers locatedaround the Tx before arriving at the Rx antenna elementsare collectively called the SBT component. Similarly, transmitsignals reflected only by scatterers located around the Rxbefore arriving at the Rx antenna elements form the SBR

component. The DB component is formed from the signalsthat are reflected from scatterers about both the Tx andthe Rx before arriving at the Rx antenna elements. Foreach realization of the WSSUS channel, the channel impulseresponses and TFCFs can be written as superpositions of theLOS, SBT, SBR, and DB signal components.

The channel impulse responses for the subchannels

A(p)t A(q)r ,A(p)t A(q)r ,A(p)t A(q)r andA(p)t A(q)r

can be written as a superposition of the LOS, SBT, SBR andDB signals,

hab(t, ) = hSBRab (t, )+hSBRab (t, )+hDBab(t, )+hLOSab (t, ),(6)

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0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

2

01

2

01

21

21

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4X

Z

Y

Figure 3: Gain patterns of half-wavelength dipole antenna withinclination angle of 0: G(v)(, ). The horizontal componentG(h)(, ) is zero.

2

1

0

1

22

10

12

0.02

0.04

0.06

0.08

0.1

0.122

10

12

Z

Y X

(a)

2

1

0

1

22

10

12

0.02

0.04

0.06

0.08

0.1XY

Z

21

01

2

0

(b)

Figure 4: Gain patterns of half-wavelength dipole antenna with

inclination angle of/2: (a)G(v)

(, ), and (b)G(h)

(, ).

The random variable (m)ab represents the uniformly-distributed random phase offset associated with the path to

each scatterer S(m)t between the V or H component of the

Tx antenna elements and the V or H component of the Rx

antenna elements. Similarly, the random variable(n)

ab is theuniformly-distributed phase offset associated with the path

to each scattererS(n)r between the V or H component of the

Tx antenna elements and the V or H component of the Rxantenna elements.

r(m)pq , r(n)pq, r(n)pq,and r(n)pq represent the inverse of the channel

cross-polar discrimination ratios along the single bouncepath to scattererm or scatterern, and have statistical meansthat follow

Er(m)pq = Er(n)pq = 1XPDp ,Er

(m)pq

= Er

(n)pq

= 1XPDp .

(12)

Here, the channel cross-polar discrimination for eachtransmit polarization is defined as

XPDp=Ehqp2

E

h

qp

2

,

XPDp=Ehq p2

Ehq p2 ,

(13)

whereE{} denotes the expectation operator.The XPD depends on channel parameters and the

environment, such as the distance between the Tx and the Rx,the angles of arrival and departure (both azimuth andelevation), the delay spread of the multipath components,and the transmit and receive antenna polarization basis.

We define a parameter CPRqp q p as the copolar powerratio between the average powers transmitted through thevertical-vertical subchannel and the average powers trans-mitted through the horizontal-horizontal subchannel:

CPRqp ,q p=Ehq p2

Ehq p2 . (14)

The CPR depends on the Brewster angle phenomenon [31].

The XPD and CPR, when expressed in decibel (dB),

are often observed as having the normal distribution withN(, ). In [7], the mean of XPD varies from 0 to 18 dB,with the standard deviation in order of 3 8 dB. Normally,the received power in the vertical-to-vertical transmissionis reported to be greater than that in the horizontal-to-horizontal transmission (CPR > 0 dB). For example, themean of the CPR is reported to vary between 0 and6dB [15]. Depending on the propagation environment andtransmission configuration, the CPR may be less than 0 dBwhen the amplitude of vertically polarized waves is degradedmore than that of horizontally polarized waves or thetransmission power of horizontally polarized waves is greaterthan that of vertically polarized waves. When the XPD or

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The random variable (m,n)ab is the phase offset of the

double-bounce path scattered by both the Tx scattererS(m)t

and the Rx scattererS(n)r between theVorHcomponent of

the Tx antenna elements and the VorHcomponent of theRx antenna elements.

2.4. LOS Channel Impulse Response. The LOS components ofchannel impulse responses are

hLOSab (t, ) =

K

K+ 1LOSa,b g

LOSa,b (t)

LOSa,b

Ga

LOSt , LOSt

Gb

LOSr , LOSr

,

(36)

whereLOSa,b (a {p,p},b {q, q}) denotes the amplitudeand LOSa,b denotes the time delay of the LOS components

between the antenna elementA(a)

t at the Tx and the antenna

elementA(b)r at the Rx. The time-varying functiongLOSa,b (t) is

defined as

gLOSp,q (t) =gLOSp,q (t) =gLOSp,q (t) =gLOSp,q (t)= exp

j 2

cp,q

exp

j2tftmaxcos

LOSt vt

cosLOSt cosvt

exp

j2tftmaxsin

LOSt sinvt

expj2tfrmaxcosLOSr vr cosLOSr cosvrexp

j2tfrmaxsin

LOSr sinvr

.

(37)

The amplitude of the LOS componentsLOSa,b is given by

LOSp,q = LOSp,q D

2= Am,

LOS

p,

q = LOS

p,q

1

CPRqp ,q pD

2= Am

1

CPRqp,q p.

(38)

The time delayLOSa,b is the travel time of the signal from

the Tx antenna element A(a)t and the Rx antenna element

A(b)r . Consider the following:

LOSp,q = LOSp,q = LOSp,q = LOSp,q = p,qc . (39)

It is assumed that the elevation angles (LOSt ,LOSr ,

(m)t ,

(n)t ,

(m)r , and

(n)r ) and the azimuth angles (LOSt ,

LOSr ,

(m)t ,

(n)t ,(m)r and

(n)r ) are independent random variables. The

radii R t of the Tx sphere and Rrof the Rx sphere are also

independent. The phase offsets (m)

ab , (n)ab , and

(m,n)ab are

assumed to be uniformly random variables on the interval[,) that are independent from the elevation angles,azimuth angles, and radii of the scattering spheres. Usingthe Central Limit Theorem [33, 34], we posit that the

delay-spread functions hSBTab (t, ), h

SBRab (t, ), h

LOSab (t, ), and

hDBab(t, ) are zero-mean complex Gaussian random pro-cesses.

2.5. Time-Variant Transfer Functions. The time-varianttransfer function is the Fourier transform of the channelimpulse response with respect to the delay. Consider thefollowing:

Tabt,f

=F{hab(t, )} = TSBTab t,f + TSBRab t,f+ TDBab

t,f

+ TLOSab

t,f

,

(40)

where TSBTab (t,f), TSBRab (t,f), T

DBab(t,f), and T

LOSab (t,f) (a

{p,p}, b {q, q}) are time-variant transfer functions for theSBT, SBR, DB and LOS components, respectively.

3. Polarization Matched Time-FrequencyCorrelation Functions

Wide-sense stationarity and uncorrelated scatterers are oftenassumed to be valid for mobile radio channels [18]. In thispaper, time and frequency dispersion are modeled depen-dently over a wide sense stationary uncorrelated scatteringchannel for a 3D nonisotropic scattering environment. For

such channels, the time-frequency correlation function is aneffective way of characterizing the statistical dependenciesin the temporal and frequency domains associated with themobile-to-mobile channels. Since the polarization states ofantennas at both the Tx and Rx are matched, we derivethe matched polarization-basis time-frequency correlationfunction to show the relationship between the frequency andthe time of the 3D statistical model for 2 2 DP antennas.

For a WSSUS channel, the TFCF of the time-varyingtransfer function Tab(a {p,p}, b {q, q}) is defined interms of the time difference tand the frequency separationf

abt,f = ETabt,fTabt+ t,f + f, (41)where a {p,p}, b {q, q} and p,p, q,q correspondto vertical and horizontal antenna polarizations.E[] is theexpectation operator, and denotes the complex conjugateoperation. The TFCFs in (41) can be rewritten as thesuperposition of the TFCFs of the SBT, SBR, DB and LOScomponents. Consider

abt,f

=SBTab

t,f

+SBRab

t,f

+DBab

t,f+LOSab

t,f.

(42)

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At the Rx, the azimuth angle(s)r and elevation angle(s)r

induced by the scattererS(m)t are approximated as

(s)r 3

2 +

RtD

sin (s)t ,

(s)r RtD

(s)t +

z

D.

(49)

The angle of incidence(i)t with the surface of Tx scatterer

can be determined by using trigonometric identities in the

triangleOt S(m)t Oras follows:

(i)t

1

2arccos

R2t+d

2m,rD2

2Rtdm,r

, (50)

wheredm,ris given by

d2m,r=x Rtcos (s)t cos (s)t

2+y Rtcos (s)t sin (s)t

2+z Rtsin (s)t

2.

(51)

The TFCFs along the SBR path are given by

SBRabt,f

= r,abA

2m

K+ 1 ISBRab

t,f

, (52)

whereISBRab (t,f) is defined as

ISBRabt,f

= 1

8|rm|I0(s)r Rr2

Rr1 rm

rm

1

2

Rr

D22

expj 2

c f

p,n+ n,q

3R2rR3r2 R3r1

cos

2

(s)r

rm

exp

(s)r cos

(s)r (s)r

exp

j2tftmaxcos

(s)t vt

cos

(s)t cosvt

exp

j2tftmaxsin

(s)t sin vt

exp

j2tfrmaxcos

(s)r vrcos(s)r cosvr

exp

j2tfrmaxsin (s)r sin vr

exp8hsin (i)r

c

2I08hsin (i)r

c

2

E 1CPRqp ,q pG(v)a (s)t , (s)t G(h)a (s)t , (s)t

1 E

r(n)pq

E

r(n)pq

1

G(v)

b

(

s)r ,

(s)r

G(h)b

(s)r , (s)r

d(s)r d(s)r dRr,(53)

whereE[r

(n)p

q] and E[

r

(n)

pq] represent the expectations of

the random variables r(n)pq and r(n)pq, respectively. They arecalculated over the interval [20 dB, 20 dB] using the pdf ofthe normal XPD model in (16).

At the Tx, the azimuth angle(s)t and elevation angle(s)t

induced by the scattererS(n)r are approximated as

(s)t RrD

sin (s)r ,

(s)t

RrD(s)r

z

D.

(54)

The angle of incidence(i)r with the surface of Rx scatterer

can be determined by using trigonometric identities in the

triangleOt S(n)r Oras follows:

(i)r 1

2arccos

R2r+d

2t,n D2

2Rrdt,n

, (55)

wheredt,nis given by

d2t,n= x+ Rrcos (s)r cos (s)r 2+y+ Rrcos

(s)r sin

(s)r

2+z+ Rrsin

(s)r

2.

(56)

The TFCFs along the DB path are given by

DBab

t,f

= tr,abA

2m

K+ 1 IDBab

t,f

, (57)

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10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

40

Radius (m)

Coherencebandw

idth(MHz)

Figure12: The minimum coherence bandwidth of 2 2 DP systemvaries with upper radii of spheres of scatterers in both Tx and Rx.

0 2 4 6 8 10 12 14 16 18 2030

25

20

15

10

5

0

TFCF|

|

(dB)

of XPD distribution (dB)

SBT-V-V

DB2-V-V

SBT-V-H

DB2-V-H

Figure 13: Normalized time-correlation functions of SBT and DBrays vary with the mean XPD and the configuration is set up inTable1.

matched polarization-basis time-frequency correlation func-tions were formulated and numerically computed forWSSUS 3D non-isotropic scattering environments. Thenumerical results show that the joint TFCFs are not sep-arable into independent time-correlation and frequency-correlation functions. The normalized TFCFs can be parsedinto bounce path components for matched polarization andcopolarization links, leading to power-normalized marginalTFCFs for the channel realization. The flexibility of themodel enables control of channel parameters to achieve avariety of multipath fading environments to investigate 2 2DP architectures.

Our intention in future research is to validate theproposed channel model using measurements. A series ofexperiments are planned to verify and justify our model.

Disclaimer

The views and conclusions contained in this paper are thoseof the authors and should not be interpreted as representingthe official policies, either expressed or implied,of the Officeof Naval Research

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