Space-timie annel Simnulator using, An gular

Power DistributionsTerence Betlehemt and Thushara Abhayapalatt

Research School of Information Science and Engineering, Australian National University, Australiat Wireless Signal Processing Program, National ICT Australia Ltd.

Canberra, Australia[terence.betlehem,thushara.abhayc

Abstract- In this paper, we develop a channel simulator togenerate the channel gains to an arbitrary array of receiverantennas, for a general class of non-line-of-sight channels. Thechannel scattering environment is defined by the angular powerdistribution as seen by the receiver. We derive the second orderstatistics of the channel gains in terms of the parameters of theangular power distribution. As an illustration of the channelsimulator, we compare the performance of different direction-of-arrival techniques.

I. INTRODUCTION

With recent development of practical multiple-input-multiple-output (MIMO) systems, there is need to quantifyMIMO system performance over realistic channels. Manyoptions for channel models are now available. However manymodels either require ray-tracing, complicated parametriza-tions, or restrict simulation to a single array geometry. Wepresent here a simulator for arbitrary array geometries thatgenerates channel realizations from readily available angularpower distribution data.A number of schemes have been proposed for simulating

MIMO channels. Several authors propose ray tracing models[11]. However for non-line-of-sight channels, the channel gainsare dominated by their second order statistics [2]. To model thesecond order channel statistics, many use oversimplified mod-els such as Rayleigh fading and Kronecker models [3] whichpoorly estimate capacity [4]. Others use higher complexitydata-dependent models (e.g. [5], [6]) which learn statisticalparameters from a MIMO data set or geometric models [7]based on parametrizations of the directional power distribution.

In this paper, we describe a simulator to realize non-line-of-sight channels directly from directional power distributions.The advantage of our model is that (i) arbitrary distributions ofreceivers can be simulated from the same set of data, (ii) lowsimulator complexity and fast simulator time stemming froman efficient parametrization of the channel. Here we present a2-D model, but it extends naturally to 3-D. The simulator isused to predict performance of competing direction-of-arrival(DOA) algorithms to distributed sources.

In Section II, we describe a general channel model foruncorrelated scatterers. In Section III, we derive a low orderparametrization for such a channel and derive parameterstatistics. Section IV reviews common power distributions.Section V summarizes the structure of the proposed channel

ala]Canu.edu.au

Scattering ArrayEnvironment Geometry

s(t) Channel Zyt)

Simulator

Fig. 1. Black box representation of the charnnel simulator.

simulator. In Section VI, we simulate channels for severalpower distributions.

II. CHANNEL MODELConsider transmission of data from a transmitter over a flat

fading channel to nR receiver antennas. Let Xn be the nth re-

ceiver antenna position with respect to an arbitrary origin 0 (asshown in Fig. 2). Receiver antennas lie within a finite circularaperture B of radius RR. The transmitter transmits a signal s(t)over the time-varying channel with the transfer function hn (t)to receive a signal zr(t) at each antenna n. Collect transferfunctions into vector h(t) [h (t), h2 (t) hn (t)]T whereT is the vector transpose operator, and received signals zn(t)into vector z(t) - I1(t), Z2(t) *n..(t)ITn Letting w(t)uiW (t> W2(t)... Wn (t)]T be additive white Gaussian noise

at the receivers, we write.

z(t) = h(t)s(t) + w(t). (1)

Assume that all scatterers lie in the far-field. The scatteringenvironment causes the transmitter signal to propagate in asplane waves with a different amplitude for each direction.Define the scattering gain A(b, t) as the amplitude of the planewave propagating in from direction X at time t at the origin.We can then write:

J27tThn?(t)= A(O, t)e-t-ikx ,;b.do, (2)

where q5 is the unit vector of polar coordinates (1, §) and k iswave number. Assume slow fading so that the channel remainsstatic over the symbol time. hr and A are then not dependenton t over each symbol. For the remainder of the paper, thet-dependence is suppressed.

0-7803-9392-9/06/$20.00 (c) 2006 IEEE

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Similarly write A(X) as the Fourier expansion:

I C00,1~ 00A(X) = L Om 'imm=-oo

where J3. is the Fourier coefficient of A(Q),

2

,= A( e-iWdo.

Fig. 2. The charnel model. Scattering is modelled with the scattering gainA(, t). Each receiver is positioned at x, within a circular aperture B shapedof radius RR

To investigate the statistics of non-line-of-sight scattering,A(X) is assumed to be a zero mean Gaussian random variableat each angle . Further assume A(X) uncorrelated betweendifferent angles:

E A($)A ( ) } = Et{A(X) (3)

where 60,, is the Kronecker delta function and * is thecomplex conjugate. This assumption is referred to as the wide-sense stationary uncorrelated scatterer (WSSUS) assumption[8]. Without loss of generality, we assume A(X) is normalized:

E£ IA(X) }d( 1.

The SIMO channel is then characterized by the power densityof scatterers 'P(b) as seen by the receiver, defined by-

'P(0 -EfA(0)|} (4)

P(%) is interpreted as the average energy arriving from thechannel in directionL . Statistics of A(X) and hence h are

dependent entirely upon P(o).III. CHANNEL PARAMETRIZATION

We now derive a simple structure for the simulation of theabove channel model. By transforming angular functions intothe Fourier domain, we shall show how to represent the chan-nel mixing vector h with a minimal number of parameters.We then derive the statistics of the Fourier parameters.

A. Fourier ExpansionPerform the Fourier expansion of P(o),

1 00

lP ('f) = . E 7Yrn C,im5627r

wr=-ro

where am, is the Fourier series coefficient of P'(o),i27

'm = I W((P) -im(do.

(7)

Due to the limited size of B, (6) can be truncated and from(2) each hr can be written as a sum of a finite number of 3%coefficients. Drawing from [9], we state the following theorem:Theorem (General SIMO Parametrization)For a set of TIR receiver antennas, each positioned at xr withinthe circular aperture of radius RR, the vector of tra nsferfunctions h defined in SIMO model (1) can be decomposed:

h= JR/3, (8)

where 3 - [ -N, *Ni, ..*NT* is a vector of the

scatteringfuinction coeffcients defined in (6), JR is a functionof the antenna positions,

K-N (XI)

i-NR (X2)JR A

v-fNR (XznR )

JN, (XI)5NM (X )

5NIF (fXnR))

(9)

where for vector x defined in polar coordinates as (X, x),

Jr (x)-i JJ (h)eirklx (-10)

and J, (.) is the Bessel functioi of the first kind of order rnand NR = [ckRR12] is the dimensionality of the receiveraperture.

This theorem is proved in the Appendix.

B. Statistical Relationships

We are interested in simulating the channel mixing vectorh. From (8), this can be done by realizing a random variablewith the statistics of /.

Since the scattering gain is zero mean Gaussian, the statis-tics of are governed by its correlation matrix ftRE{3/3H From (7) each second order statistic [ftR3] =E{@l,i,n } can be calculated as:

z2.7 b2

E{fmf din}£{=/ A(O)A* (p) -i(m2-mi

Applying the WSSUS property (3).2{7

E{f3mo;3/I = j (m)-i(,n-m )Odo,(5)Since 'P() C R, we know that >y*n = V-m. Also note thatP(b) > 0 which corresponds to 2 m= Iy < -yo = 1-

dodo.

(11)

from which we see by comparison with (5) that E{803m03/,,from which is written:

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(6)

TABLE I

POPULAR ANGULAR POWER DISTRIBUTIONS. Im( ) IS THE MODIFIED BESSEL l( = e /(V2'm 1 FOR m EVEN, Fm = Mru FOR'

Theorem (WSSUS Parameter Correlation)In a WSSUS Gaussian channel possessing angalar power

distribution FP(y) the correlation rmatrix of defined in (7)possesses the Toeplitz structure:

Yo T-I ...

Ro =

rf2NF 2NR- I

Yf- 2NM-(2NR.- 1)

0

(12)

where /yrn is defined in (5). El

By virtue of the -y,r = -Y;rr property, to completely describethe channel statistics for any geometry of antennas withinEB, only 2NR + 1 complex parameters 02o,...,N+1 are

required.

IV. ANGULAR POWER DISTRIBUTIONS

In this section, we present some popular choices for theangular power distribution 'P(0) in the simulator. These dis-tributions are parametrised by the mean direction o0 and an

angular spreading parameter:. Jakes model [10] where scatterers are isotropically ar-

ranged to yield rich scattering.Uniform limited angular distribution [11] where energyarrives uniformly from a restricted range of angles (oo-A, ,oo + A).Von-Mises distribution [12] having degree of nonisotropyK >O.Laplaciati distribution [13] with measure a of angularspread.

The functional forms of angular power and corresponding 7mcoefficients are summarized in Table I. Derivation of thesecoefficients was made in [14].

V. SIMULATOR

We now describe a simple simulator algorithm that outputsrealizations of the channel mixing vector over a WSSUSchannel. This simulator generates a set of received sensor

data { (n)}N I statistically representative of the channelsimulation parameters, namely the angular power distributionP(q) , noise variance o,, and array geometry X1, X2 7 *XnR -

As summarized in Fig. 3, the procedure for producing thesensor data is:

'UNCTION OF THE FIRST KIND. FOR THE LAPLACIAN DISTRIBUTION,

m ODD AND Q IS A NORMALIZATION CONSTANT.

ChannelParametersjS P(j3)

r- -----|----------------

(1) Fourier analysis

f/=(27T ) d

ArrayGeometry{x n} 1

(2) Constructcorrelation matrix

(3) Generate realizations

(a) h(n) JRRq /2g3 (n)(b) iwv(n) =T,g, (n)

h(n) w(n)s(n) Z(n)

Fig. 3. Internal structure of the WSSUS channel simulator.

1) Calculate coefficients N>,}N N from (5).2) Construct the correlation matrix R, using (12).3) Generate N realizations of h and w, {h(n), i (n) }I

from the channel correlation parameters.4) Calculate signal realizations {z(n) }IN from (l)

To perform step 3), we define the oIR x I independent andidentically distributed Gaussian random variables g,3 and gwfrom which we generate realizations of and w as: 13(n) =RH1/2g-(n), and w(n) = cgTf,(n), where by calculatingE 3(71)3 (T) , we see that Rf HR, 3, so thatR1/2 is the Cholesky decomposition of Rf3. Then from (8),h(n) = JRO(n).

Comments.1) If we choose a scattering environment with a power

distribution in Table I, and associated mean angle andangular spread.

2) The simulator yields h vectors with the WSSUS statisti-cal properties preserved. The spatial correlation betweenpositions in the WSSUS field are presented in [14].

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Argular Power l 0 YJakes 1/2T6rrnoUniform linited O, 1h rwise. e-m9° sincmTnA)

Von-Mises 2eS- 1 po < enO, otherwise. IO(K

Laplacian ci < e-ii ( )F2n0_otlerwise 2

DesignSynthesizedSBF DOAMV DOA -

MUSIC DOA

50 100 150 200 250 300 350Angle of Arrival (°)

(a)

-~Dsgn-Synthesizd

0.4 - K= 100

0.35

0.3

0.25

2' 0.2 - K= 10 11

020 40 60 80 100 120 140 160 180Angle of Arrival

(b)

Fig. 4. Simulation of Synthesized versus Design angular power distribu-tions. (a) Laplacian distribution for a = 2° in apertures of radii RR=.5A, A, 2A, 5A]. (b) Von-Mises distribution for spreading parameter =[1,10, 100] in an aperture of radius RR = 2A.

3) Due to the finite size of the receiver aperture, themean synthesized power distribution 'Pyntl (X)E{lA(0; TI) 2} where A(0; T) is a realized scatteringgain, is a low bandwidth approximation to the designpower distribution P(O). Inserting the NR truncation of(6) for A(O; rn), we can show.

P)synth() =(2w)2 E{ 3rm }e (M m)mmP

2NRnX t(m)-m iemo27T Z-em=-2N9p

where in the first equation m and m/ are summedfrom -NR to NR and t(m) = 1- mj/(2NR + 1)is a triangular window function. Measuring with a finiteaperture, the higher order ym coefficients are attenuated.However in the limit of large NR, 'Psynth() = P(O)

VI. EXAMPLESIn this section, we use the WSSUS simulator to perform

Monte-Carlo simulations with N = 10000 trials for different

Fig. 5. Simulation of direction-of-arrival estination techniques from datagenerated from a Von-Mises power distribution with , = 10. Techniquesshown are steered beamforming (SBF), minimum variance (MV) and multiplesignal classification (MUSIC), using a 5 element uniform circular array ofradius 0.8A.

power distributions.In Fig. 4 we simulate the Laplacian and Von-Mises power

distributions described in Table I, plotting the synthesizedpower s ( ) in each case. Fig. 4(a) plots the Laplaciandistribution synthesized by a finite aperture. The cusp of theLaplacian distribution here possesses a large angular band-width that requires a large aperture to observe. Fig. 4(b) illus-trates its ability to reproduce different Von-Mises distributions.

In Fig. 5, we use of the simulator data to compare the perfor-mance of basic DOA estimation schemes. Due to underlyingassumptions (e.g. limited numbers of multipath directions),different schemes will perform better or worse in different en-vironments. In the case shown, MUSIC erroneously estimatesthe single peak as two distinct multipaths.

VII. CONCLUSION

In this paper, we present a single-input-multiple output(SIMO) channel simulator that generate channel data givenparticular parameters of an angular power distribution. Thissimulator implements a wide-sense stationary uncorrelatedscatterer flat fading channel. A Matlab implementation ofthis simulator is available at http: //rsise.anu.edu.au/terenceb/code/sirno sirn.zip. We are currently extending thissimulator to multiple-input-multiple output (MIMO) channels,using double directional angulLar power distributions, simulat-ing Doppler fading, and removing the uncorrelated scattererassumption.

ACKNOWLEDGMENT

This work was partially funded by Australian ResearchCouncil Discovery Grant number DP0343804.

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0.45 r-

APPENDIX

Proof: We start by writing for the vectors x` (a, ox) and(1, b) the Jacobi-Anger expression [15],

OC)

cikxao = r irJm(kx)eim(ox-o)rn=-oo

(13)

Substituting (6) and (13) into (2) followed by rearranging andintroducing J,, (x) as defined in (10)

cc00 00 nhn =rE-rX nv_tlzx) xrn=-oo m'=-oo

i(mt+,rn' )O¢do.27

Evaluating the integral:00

fin = : f3mSm (Xn).M=-ro)

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[16] H. M. Jones, R. A. Kennedy, and T. D. Abhayapala, "On dimensionalityof multipath fields: spatial extent and richness," in Proc. IEEE Inteina-tional Conference on Acoustics, Speech and Signal Processing, vol. III,pp. 2837 - 2840, 2002.

(14)

It has been shown in [:16] that (1 3) can accurately be truncatedto ri4 < NR for NR = rekRRf2] since successive termsof the series decay exponentially. For a bounded scatteringfunction, the 3n coefficients are bounded and (14) can betruncated to unt < NR. Writing truncated (14) in vectorproduct form hn J rn where

Jr [J-N, (Xn) J.* Ny (Xn)]

Result follows from noting that n is a row of (9).

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