1650 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Autoregressive Modeling for FadingChannel Simulation

Kareem E. Baddour, Student Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE

Abstract—Autoregressive stochastic models for the computersimulation of correlated Rayleigh fading processes are investi-gated. The unavoidable numerical difficulties inherent in thismethod are elucidated and a simple heuristic approach is adoptedto enable the synthesis of accurately correlated, bandlimitedRayleigh variates. Startup procedures are presented, which allowautoregressive simulators to produce stationary channel gain sam-ples from the first output sample. Performance comparisons arethen made with popular fading generation techniques to demon-strate the merits of the approach. The general applicability ofthe method is demonstrated by examples involving the accuratesynthesis of nonisotropic fading channel models.

Index Terms—Autoregressive processes, bandlimited stochasticprocesses, multipath channels, nonisotropic scattering, Rayleighchannels, simulation.

I. INTRODUCTION

THE bandlimited Rayleigh fading process, whose powerspectral density (PSD) is zero past the maximum Doppler

frequency, appears in many physical models of mobile radiochannels. Its emulation has been of theoretical and practicalinterest to the wireless community for many years, as the designand optimization of modern communication systems cannotbe carried out without computer simulations. The classicalfading simulator application is to generate a single sequenceof correlated Rayleigh variates in accordance with Clarke’swide-sense stationary (WSS) isotropic scattering model [1].However, the scattering encountered in many environments isnonisotropic, which strongly affects the second-order statisticsof the channel. A desirable simulator feature is the ability toemulate such directional fading scenarios, for which the realand imaginary Gaussian sequences underlying the sampledRayleigh channel can exhibit cross-correlations. This paperaddresses the development of a simulation methodology, which

Manuscript received April 24, 2003; revised October 24, 2003; accepted May4, 2004. The editor coordinating the review of this paper and approving it forpublication is C. Xiao. This work was supported by a postgraduate scholarshipfrom the Natural Sciences and Engineering Research Council of Canada(NSERC), by an Industry Canada Fessenden Postgraduate Scholarship, by anOntario Graduate Scholarship in Science and Technology, and by the AlbertaInformatics Circle of Research Excellence (iCORE). This paper was presentedin part at the 2001 Global Telecommunications Conference (GLOBECOM ’01),San Antonio, TX, November 2001.

K. E. Baddour is with the Department of Electrical and Computer En-gineering, Queen’s University, Kingston, ON K7L 3N6, Canada (e-mail:baddourk@ee.queensu.ca).

N. C. Beaulieu is with the Department of Electrical and Computer En-gineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail:beaulieu@ee.ualberta.ca).

Digital Object Identifier 10.1109/TWC.2005.850327

can be easily used to accurately synthesize such generalized flatRayleigh fading channels.

In the communications literature, a number of differentalgorithms have been proposed for the generation of corre-lated Rayleigh random variates (e.g., [2]–[5]). Among these,simulators based on either a sum-of-sinusoids approach, on awhite noise filtering method, or on the inverse discrete Fouriertransform (IDFT) algorithm have become popular. Recently,significant problems were found to exist with the stochasticbehavior of commonly used sum-of-sinusoids designs. In par-ticular, it was shown in [6] that the classical Jakes’ simulatorproduces fading signals that are not WSS. Careful redesignsof the sum-of-sinusoids models, such as those proposed in[7] and [8], are required to remove the stationarity problemwhile maintaining the accuracy of the correlation statistics. TheIDFT technique, on the other hand, is well known to be ahigh-quality and efficient fading generator [2]. Unfortunately,a disadvantage of the IDFT method is that all samples aregenerated with a single fast Fourier transform (FFT) operation.The storage requirements of this approach can make it unattrac-tive for the generation of a very large number of variates.The search for a fading simulator that can produce statisti-cally accurate variates “as they are needed” also motivates ourwork herein.

In this paper, we consider the use of a general autore-gressive (AR) modeling approach for the accurate generationof correlated Rayleigh processes. Essentially, this techniqueemploys all-pole infinite-impulse response (IIR) filtering toshape the spectrum of uncorrelated Gaussian variates. Unlikeprevious white noise filtering methods, precise matching of thetheoretical statistics is possible over the order of the model forpractical finite-length implementations. Furthermore, the modelparameters are easily computed. Previously, AR models havebeen used with success to predict fading channel dynamics forthe purposes of Kalman-filter-based channel estimation (e.g.,[9]–[12]) and for long-range channel prediction [13]. Theyhave also been used by several authors to simulate correlatedRayleigh fading [10], [14], though in [10] and [14], low-order AR processes were adopted, which do not provide agood match to the desired bandlimited correlation statistics.In [15], the authors also attempted to use an AR fadinggenerator, but ran into stability problems and abandoned thisapproach.

In this paper, we examine AR fading simulators in detail.First, we demonstrate that it is the highly deterministic natureof narrowband Doppler fading processes that leads to thenumerical problems faced by the AR method in this applica-tion. A simple heuristic approach is proposed to enable the

1536-1276/$20.00 © 2005 IEEE

BADDOUR AND BEAULIEU: AUTOREGRESSIVE MODELING FOR FADING CHANNEL SIMULATION 1651

synthesis of accurately correlated bandlimited Rayleighvariates. Startup procedures are described, which allow ARgenerators to produce stationary and statistically correctRayleigh variates from the first output sample. Performancecomparisons are then made with popular Rayleigh fadinggenerators to demonstrate the merits of the AR method. Theaccurate synthesis of nonisotropic Rayleigh fading models isalso demonstrated. To the best of our knowledge, no completestudy of using an AR approach to accurately generate band-limited Rayleigh random processes has been reported in theliterature.

The remainder of this paper is organized as follows.Section II briefly reviews correlated Rayleigh fading modelsand popular variate generation techniques. Section III describesthe AR method of generating stationary bandlimited Rayleighprocesses. Section IV compares the method to both the IDFTand a WSS sum-of-sinusoids simulator. The ability of theAR generator to accurately simulate nonisotropic models isalso demonstrated. Section V concludes the paper.

II. FADING SIMULATION

A. Correlated Fading Models

A Rayleigh characterization of the land mobile radio chan-nel follows from the Gaussian WSS uncorrelated scatteringfading model [16], where the fading process is modeled asa complex Gaussian process. In this model, the variabilityof the wireless channel over time is reflected in its auto-correlation function (ACF). This second-order statistic gen-erally depends on the propagation geometry, the velocityof the mobile, and the antenna characteristics. A commonassumption is that the propagation path consists of a two-dimensional isotropic scattering with a vertical monopoleantenna at the receiver [17]. In this case, the theoretical PSDassociated with either the in-phase or quadrature portion ofthe received fading signal has the well-known U-shaped band-limited form [17]

S( f) =

1

πfd

√1−

(f

fd

)2, |f | ≤ fd

0, elsewhere(1)

where fd is the maximum Doppler frequency in Hertz,given by fd = υ/λ, υ is the mobile speed and λ is the wave-length of the received carrier wave. The corresponding nor-malized (unit variance) continuous-time autocorrelation of thereceived signal under these conditions is R(τ) = J0(2πfdτ),where J0(·) is the zeroth-order Bessel function of the firstkind [17]. For the purposes of discrete-time simulation of thismodel, ideally generated in-phase and quadrature Gaussianprocesses should each have the autocorrelation sequence

R[n] = J0(2πfm|n|) (2)

where fm = fdT is the maximum Doppler frequency normal-ized by the sampling rate 1/T . Furthermore, in this model the

in-phase and quadrature processes must be independent andeach must have zero mean for Rayleigh fading.

Variations on the PSD in (1) have been proposed basedon more complicated propagation models. A PSD derivedwithout assuming that the incoming waves are only horizontalin three-dimensional space is given in [18]. This band-limited PSD is very similar to (1), but does not have as-ymptotes approaching infinity at the band edges of the fadingspectrum. Improvements to this spectrum have been proposedin [19]. A three-dimensional model with isotropic scatteringin all three directions is examined in [20]. In this model,the PSD has flat bandlimited characteristics with a normal-ized ACF

R[n] = sinc(2fm|n|) (3)

where sinc(x) = sin(πx)/πx. Complex Gaussian fadingmodels have also been proposed for nonisotropic scatteringscenarios (e.g., [21], [22]). In such environments, the DopplerPSD is not necessarily even symmetric, which results in achannel ACF that has an imaginary component. As a result, theunderlying real in-phase and quadrature fading processes ex-perience cross-correlations. In general, the theoretical Dopplerpower spectrum is bandlimited, since the maximum Dopplershift is a finite quantity that is proportional to the mobilevelocity. Thus, the focus in this paper is on the accurate syn-thesis of complex Gaussian processes with a specified band-limited spectrum.

B. Correlated Variate Generation

Exact generation of N Gaussian variates with an arbitrarycorrelation can be achieved in principle by decomposing the de-sired N × N covariance matrix R = GGH, where GH denotesthe Hermitian transpose of G, then multiplying N independentGaussian variates by G [23, pp. 254–256]. The Cholesky fac-torization, which can be performed with O(N2) operations forToeplitz matrices, is an efficient choice for this decomposition.For bandlimited processes, however, an approximation usinga singular value decomposition is typically required due tonumerical problems arising from an ill-conditioned covariancematrix. In this case factorization requires O(N3) operations,which restricts applicability of the method to the generation ofvery small sample sizes.

A popular method for modeling the Rayleigh flat fadingchannel is to sum the outputs from Ns complex sinusoidalgenerators [17]. In practice, the generated sequence closelyapproximates a complex Gaussian process provided a suffi-cient number of sinusoids are used. With proper choice ofthe distribution of the sinusoid frequencies (see, e.g., [6]),the process autocorrelation approaches that represented by (2)as Ns → ∞.

Correlated Rayleigh variates can also be generated by fil-tering two zero-mean independent white Gaussian processesand then adding the outputs in quadrature. Here, rational trans-fer function approximations of the nonrational PSD in (1)are typically used to shape the spectrum. The autocorrelation

1652 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

properties of the generated sequences are determined bythe choice of filter. Many possibilities for this choice exist(e.g., [3], [5], [15]). In general, these approaches do not provideprecise matching of the theoretical statistics.

Another popular technique for generating correlatedRayleigh variates is Smith’s IDFT algorithm [24]. Here, theIDFT operation is applied to sequences of uncorrelated com-plex Gaussian variates, each sequence weighted by appropriatefilter coefficients to shape the PSD. Young and Beaulieu[2] modified Smith’s algorithm for greater computationalefficiency and provided a statistical analysis of the method.By considering the quality of the generated variates and thecomputational effort, a comparison of the IDFT generator withthe sum-of-sinusoids technique and a finite-impulse response(FIR) filter method concluded that the IDFT generator issuperior [2].

III. AR MODELING OF BANDLIMITED RAYLEIGH

RANDOM PROCESSES

A. Model Computation

Autoregressive models are commonly used to approximatediscrete-time random processes [25]. This is due to the simplic-ity with which their parameters can be computed and due totheir correlation matching property. A complex AR process oforder p [AR(p)] can be generated via the time domain recursion

x[n] = −p∑

k=1

akx[n − k] + w[n] (4)

where w[n] is a complex white Gaussian noise process withuncorrelated real and imaginary components. For Rayleighvariate generation, w[n] has zero mean and the simulator outputis |x[n]|. The AR model parameters consist of the filter coeffi-cients {a1, a2, . . . , ap} and the variance σ2

p of the driving noiseprocess w[n]. The corresponding PSD of the AR(p) processhas the rational form [25]

Sxx(f) =σ2

p∣∣∣∣1 +p∑

k=1

ak exp(−j2πfk)∣∣∣∣2

. (5)

Although the Doppler spectrum models proposed for mobileradio are not rational, an arbitrary spectrum can be closelyapproximated by an AR model of sufficiently large order. Thebasic relationship between the desired model ACF Rxx[k] andthe AR(p) parameters is given by [25]

Rxx[k] ={−

∑pm=1 amRxx[k − m], k ≥ 1

−∑p

m=1 amRxx[−m] + σ2p, k = 0.

(6)

In matrix form this becomes for k = 1, 2, . . . , p

Rxxa = −v (7a)

where

Rxx =

Rxx[0] Rxx[−1] · · · Rxx[−p + 1]Rxx[1] Rxx[0] · · · Rxx[−p + 2]

......

. . ....

Rxx[p − 1] Rxx[p − 2] · · · Rxx[0]

a = [ a1 a2 · · · ap ]T

v = [Rxx[1] Rxx[2] · · · Rxx[p] ]T (7b)

and

σ2p = Rxx[0] +

p∑k=1

akRxx[−k]. (8)

Given the desired ACF sequence, the AR filter coefficientscan thus be determined by solving the set of p Yule–Walkerequations in (7a). These equations can in principle be solvedefficiently by the Levinson–Durbin recursion in O(p2). SinceRxx is an autocorrelation matrix, it is positive semidefinite andcan be shown to be singular only if the process is purely har-monic and consists of p − 1 or fewer sinusoids [27]. In all othercases, the inverse R−1

xx exists and the Yule–Walker equationsare guaranteed to have the unique solution a = −R−1

xxv. Thegenerated AR(p) process has the ACF [25]

R̂xx[k] ={

Rxx[k], 0 ≤ k ≤ p

−∑p

m=1 amR̂xx[k − m], k > p.(9)

That is, the simulated process has the attractive property thatits sampled ACF perfectly matches the desired sampled ACFup to lag p. The remaining ACF extension is characterized bythe property that the generated time series is the most randomone (maximum entropy) which has the assigned first p + 1ACF lags [25].

B. Ill Conditioning of the Yule–Walker Equations

In solving the Yule–Walker equations, the condition of theautocorrelation matrix is an important consideration in deter-mining the accuracy of the solution. A measure of the illconditioning is provided by the white noise variance parameterσ2

p. Moreover, it can be shown that [26, p. 354]

|Rxx| =p−1∏m=0

σ2m (10)

where |Rxx| denotes the determinant of Rxx and σ2m represents

the driving white noise variance corresponding to an AR(m)model of the process. Consequently, if the values of the σ2

m

are very small, Rxx is nearly singular so that significant errorsin the computed parameters are expected and unavoidable,regardless of the method used to solve (7a). In these cases,

BADDOUR AND BEAULIEU: AUTOREGRESSIVE MODELING FOR FADING CHANNEL SIMULATION 1653

Fig. 1. The one-step prediction MMSE for the Bessel ACF and sinc ACF versus the model order.

numerical problems in the solution typically yield unstablemodel filters.

To investigate the stability and accuracy of AR models ofbandlimited processes, in the sequel we consider the behaviorof σ2

p with increasing AR order. To begin, it is useful tonote that σ2

p in the AR formulation is equivalent to theminimum mean square error (MMSE) of a one-step-aheadlinear predictor of order p for the bandlimited process beingmodeled [27]. A lower bound on the MMSE is known fromprediction theory to be given by the infinite prediction memorycase, in which case the asymptotic σ2

p can be expressed via theKolmogoroff–Szego formula [28, p. 491]

limp→∞

σ2p = exp

1

2π

π∫−π

ln Sxx(e jω)dω

(11)

where ω = 2πf , and Sxx(e jω) =∑∞

k=−∞ Rxx[k]e−jkω isthe desired power spectrum of the process. From (11), wefind that a stochastic process whose spectrum is zero oversome finite frequency domain has a zero asymptotic predictionerror. Such a process is said to be deterministic since itsfuture can in principle be predicted exactly in a mean-square sense from knowledge of all its past samples takengreater than the Nyquist rate [29]. This implies that the rapidtime variation of a complex Gaussian fading channel process,which is due to bandlimited Doppler spreading, is in theorydeterministic.1

A more important consideration for our stability inves-tigation is how fast σ2

p converges to its zero asymptotic

1This paper focuses on discrete-time bandlimited processes. Related recentwork, which considers the predictability of continuous-time bandlimited fadingprocesses, can be found in [30] and [31].

value. In [32], a study of the behavior of linear predictionerrors as the predictor order increases was performed. TheMMSE of one-step-ahead linear prediction was observed to de-crease exponentially to zero with increasing predictor order formany processes with bandlimited spectra. The authors in [32]conjectured that this rapid decay occurs for all bandlimitedprocesses. In the signal processing literature, the exponentialdecay of the prediction error has been theoretically provenonly for flat bandlimited spectra (sinc autocorrelation) in [33].Although it does not appear to be well known, a more generalproof of this result for any real bandlimited process can befound in the mathematical literature in [34]. More precisely,it can be shown in this case that for large p and bandlimitfd, the MMSE of the one-step-ahead linear prediction van-ishes as [34]

σ2p ∼ k [sin(πfdT )]2p (12)

when the sampling rate 1/T is greater than the Nyquist rate2fd and where k is a constant. With the exception of adifferent constant term k, we have verified that this asymptoticrate of decay of σ2

p holds for the bandlimited processes ofinterest in mobile radio. For example, in Fig. 1, σ2

p is plottedversus the model order for various values of fdT assumingthat the random process has the ACFs in (2) and (3). Theexponential asymptotic rate of decay of σ2

p is only a functionof the normalized Doppler bandlimit and not of the actualprocess spectrum. Furthermore, since fd is typically severalorders of magnitude smaller than the sampling rate, it can beseen that σ2

p achieves its asymptotic rate of decay beginningat small AR orders and that this rate is very rapid. Thus,for narrowband Doppler processes, severely ill-conditionedYule–Walker equations are unavoidable for all but very smallAR model orders p. In light of this fact, and since the lower

1654 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Fig. 2. The one-step prediction MMSE for the Bessel ACF versus the model order for various ε when fm = 0.05.

order models result in a poor match to the desired band-limited ACF, it is not surprising that previous investigations(e.g., [15]) concluded that an AR parameterization cannotaccurately model the J0(·) autocorrelation sequence. In thesequel we overcome the numerical problems, making accu-rate AR simulation viable, including the difficult J0(·) auto-correlation sequence.

From the preceding discussion, it does not seem possible togenerate stable AR filters of large orders using finite word-length computations to accurately model bandlimited spec-tra. However, a simple heuristic approach that can be usedto resolve the numerical problems is to improve the condi-tioning of the autocorrelation matrix Rxx by increasing thevalues along its principal diagonal by a very small positiveamount ε. This is equivalent, of course, to adding white noise ofvariance ε to the original process. The addition of this spectralbias removes the bandlimitation of the original spectrum andcreates a nondeterministic or regular process that in somesense closely approximates the original process. As a result,σ2

p no longer decays exponentially to zero with increasingmodel order p. In Fig. 2, σ2

p is plotted versus the model orderfor various values of ε assuming that the random process hasthe Bessel autocorrelation in (2). Similar curves result forother bandlimited spectra [35]. We observe from Fig. 2 thatwhile σ2

p monotonically decreases with p, it now approachesa lower bound

limp→∞

σ2p = exp

1

2π

π∫−π

ln[Sxx(e jω) + ε

]dω

> 0 (13)

for asymptotically large p. The inequality in (13) follows sinceSxx(e jω) ≥ 0. The choice of ε represents a tradeoff between

the improved condition number of Rxx and the bias introducedin the model. By choosing ε appropriately, the lower bound onσ2

p can be made large enough to enable the stable computationof a large-order AR model. The first p + 1 autocorrelation lagsof the resulting AR(p) process will then be

R̂xx[m] ={

Rxx[0] + ε, m = 0Rxx[m], m = 1, 2, . . . , p.

(14)

That is, while the zeroth autocorrelation lag will contain a smalladditive error, the next p lags will match those of the desiredtheoretical ACF.

We note that the heuristic addition of a white noise compo-nent to overcome ill conditioning was recently used in [14].However, a proper justification for the inclusion of the biasε was not provided. The singularity at the band edge of thePSD in (1) is not the sole cause of the numerical problems,as suggested in [14]. Moreover, the “stiffness” of the autocor-relation matrix is manifested for any bandlimited PSD shapeand it is the deterministic nature of bandlimited processes thatis the main cause of the numerical problems experienced inthese cases.

C. Stationary Generation

The AR generator, like other rational transfer function ap-proximation methods, produces correlated variates by filter-ing white Gaussian noise sources. The common practice ofpassing white noise through a fixed filter produces transientsdue to nonstationary initial conditions. Theory dictates thatfor a stationary output to be strictly achieved, the filter musthave had an input of white noise for all of n ≥ −∞ [36]. Inpractice, it is necessary to run the filter for a while beforethe transient effects become negligible and for the output to

BADDOUR AND BEAULIEU: AUTOREGRESSIVE MODELING FOR FADING CHANNEL SIMULATION 1655

Fig. 3. The all-pole lattice representation of the AR filter.

achieve asymptotic stationarity [27]. In the sequel, we brieflydescribe two startup procedures that allow the AR generator toproduce stationary variates from the first output sample. Bothmethods use a time-varying IIR filter to generate the first pstationary outputs with the correct correlation values, where pis the desired model order. This provides the appropriate initialconditions for the fixed pth order AR filter to subsequentlygenerate a sequence which is truly stationary. Such a startupprocedure is particularly useful when a large AR order is chosento provide accurate correlation matching, as in these cases, alarge number of variates must typically be discarded before thetransient effects decay.

The first start-up technique appears implicitly in a methodfor synthesizing fractional Gaussian noise [37]. The procedureuses the result that for a zero-mean stationary Gaussian process,the conditional mean and variance of x[k], given the past valuesx[k − 1], x[k − 2], . . . , x[0] may be written as [38]

mk =E {x[k]|x[k − 1], x[k − 2], . . . , x[0]}

= −k∑

j=1

φkjx[k − j] (15)

vk = Var {x[k]|x[k − 1], x[k − 2], . . . , x[0]}

=Rxx[0]k∏

j=1

(1 − |φjj |2

). (16)

Here, φjj denotes the jth reflection coefficient [25] of {x[k]}and the φkj are the AR coefficients for a kth order model.For a particular k, the φkj coefficients for j = 1, 2, . . . , k canbe computed by an O(k2) Levinson–Durbin execution. Notethat the variance vk in (16) is equal to the variance σ2

k of thedriving noise in the context of a kth order AR process. Thestart-up procedure for generating the first p stationary correlatedcomplex Gaussian variates is then as follows.

1) Generate a starting value x[0], whose in-phase andquadrature components are drawn independently from aGaussian distribution N(0, Rxx[0]). Set v0 = Rxx[0].

2) For k = 1, . . . , p − 1, calculate the φkj coefficients forj = 1, 2, . . . , k using the Levinson recursion. Computemk = −

∑kj=1 φkjx[k − j] and vk = (1 − |φkk|2)vk−1.

Generate the next variate x[k], whose in-phase andquadrature components are drawn independently fromN(mk, vk).

TABLE IORDER OF ε OBSERVED TO YIELD THE MOST ACCURATELY CORRELATED

AR SIMULATOR OUTPUTS FOR THE J0(·) ACF MODEL

The remainder of the stationary sequence can then be generatedusing the fixed pth order AR filter as per (4). While thisprocedure requires solving for the AR coefficient parameter setof all orders less than and including the desired AR order p,these parameters can be obtained at the intermediate steps of asingle Levinson–Durbin execution.

An alternative start-up method can be achieved by usinga time-varying lattice filter [39] implementation of the pthorder AR filter. Here the optimum lattice filter coefficients aregiven by the reflection coefficients φjj , j = 1, . . . , p. Since thereflection coefficients of the lower order do not change as theAR model order is increased [28], only one set of parameters isrequired. The all-pole lattice representation of the AR filter isshown in Fig. 3. The start-up procedure consists of switchingon one stage of the lattice form at a time, with the real andimaginary components of the complex Gaussian white noiseinput at iteration k = 0, 1, . . . , p − 1 chosen independentlyfrom a Gaussian distribution N(0, vk), where v0 = Rxx[0]and vk = (1 − |φkk|2)vk−1. The signals in all modules greaterthan the time index k should have zero value until that stage isswitched on.

IV. PERFORMANCE EVALUATION

In this section, we evaluate the suitability of the AR gener-ator for producing high-quality bandlimited Rayleigh variates.Comparisons are made to a WSS sum-of-sinusoids simulatorand to the IDFT technique, which was demonstrated in [2]to be the most efficient and highest quality method amongseveral popular correlated Rayleigh variate generators. Thequantitative measures that are used for this comparison aredescribed first.

A. Quantitative Measures

Quantitative quality measures for generated random variatesthat are in a form familiar to communication engineers havebeen proposed in [40]. The proposed measures represent ap-proximately the difference in signal-to-noise ratio in decibelspredicted to meet a specified performance (error rate, outage,etc.) when an imperfect sequence of random variates is usedduring simulation rather than a statistically “ideal” sequence.See [40] for more details. Two quality measures have beendefined in [40] as follows. The first measure, called the meanbasis power margin, is given by

Gmean =1

σ2xL

trace{CxC−1

x̂ Cx

}(17)

1656 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

TABLE IIA COMPARISON OF THE AR, IDFT, AND SUM-OF-SINUSOIDS METHODS

OF GENERATING BANDLIMITED RAYLEIGH VARIATES FOR

COVARIANCE SEQUENCE LENGTH 200

and the second, the maximum basis power margin, is de-fined as

Gmax =1σ2x

max{diag

{CxC−1

x̂ Cx

}}. (18)

In (17) and (18), σ2x is the variance of the reference (ideal)

distribution, Cx̂ is the L × L covariance matrix of any length-Lsubset of adjacent variates produced by the stationary randomvariate generator, and Cx represents the desired covariancematrix of L ideally distributed variates. For some variate gen-eration schemes, Cx̂ can be determined directly. For the ARvariate generation method, this is easily accomplished using(9) and (14). Alternatively, empirical techniques can be usedto estimate this matrix from the generator output.

B. Tested Simulation Methods

1) AR Method: The method used was that of Section III.Based on our numerical work, the value of the bias ε that yieldsthe most accurate AR model computation was observed todepend mainly on the Doppler rate. Table I lists the order of theconstant ε that was empirically found to minimize (17) and (18)for the J0(·) ACF using MATLAB on a Pentium IV machine.The numerical conditioning of the Yule–Walker equations isworse for smaller Doppler bandwidths, which necessitates alarger ε. The dependence of the choice of ε on the AR modelorder is not significant for typical filter lengths. The values inTable I are recommended for model orders up to 1000. If theAR order is much greater, then a small increase in the ε valuesin Table II provides an improvement to the simulator accuracy.

Our AR implementation used the lattice start-up procedureof Section III to generate the first p stationary variates. Sincedirect IIR filtering requires fewer computations than latticefiltering, the remainder of the variates were generated by adirect structure using the MATLAB function filter. Theappropriate initial conditions for this filter, corresponding tothe first p generated stationary variates, were set using theMATLAB function filtic.

2) IDFT Method: The simulator used was implemented asdescribed in [2]. The MATLAB function inverse FFT (IFFT)was used for IDFT computation.3) Sum-of-Sinusoids: The method used was the WSS-

improved Jakes’ model of [7]. Following this model, the nor-malized low-pass discrete fading process is generated by

x[n] =xc[n] + jxs[n] (19a)

xc[n] =1√Ns

Ns∑k=1

cos(2πfmn cos αk + φk) (19b)

xs[n] =1√Ns

Ns∑k=1

cos(2πfmn sin αk + ϕk) (19c)

with

αk =2πk − π + θ

4Ns, k = 1, 2, . . . , Ns (20)

where φk, ϕk and θ are statistically independent and uni-formly distributed on [−π, π) for all k. The statistical propertiesof x[n] asymptotically approach those of Clarke’s isotropicmodel as the number of sinusoids approaches infinity, whilevery good approximation to the ensemble statistics has beenreported when Ns is not less than 8 [7]. For finite Ns, it isimportant to point out that this WSS simulator is not auto-correlation ergodic. That is, the infinite time-average autocor-relation is a random variable and unlike the IDFT and ARsimulators, the statistical information contained in each outputwaveform or sample function is not identical and may deviatefrom the desired statistics of the theoretical model [35].

C. Performance Comparisons

The quality measures described in [40] were used tocompare the AR variate generation method to the modifiedIDFT technique of [2] and to the WSS sinusoidal generatorof [7]. The results, which are presented in Table I, comparethe quality of the real part of the simulator outputs. Similarresults were achieved for the imaginary sequences and theseare omitted for brevity. Perfect variate generation correspondsto 0 dB for both measures. In all cases, the reference ACF is(2) with a normalized maximum Doppler of fm = 0.05. A biasof ε = 10−8 was used to condition the Yule–Walker equationsfor all AR model orders. An autocorrelation sequence lengthof 200 was considered for evaluation of (17) and (18). Thetheoretical (T) results for the IDFT routine were computedas proposed in [2] for the IDFT routine, and using (9) and(14) for the AR generator. For the empirical (E) results, time-average correlations were calculated based on 220 generatedsamples. The computed quality measures were then averagedover 50 independent simulation trials. Plots of the empiricalcorrelations of the IDFT and AR generator outputs are shownin Fig. 4. The empirical correlations for a typical simulation

BADDOUR AND BEAULIEU: AUTOREGRESSIVE MODELING FOR FADING CHANNEL SIMULATION 1657

Fig. 4. Empirical autocorrelations for the AR and IDFT methods.

Fig. 5. Empirical time-average autocorrelations for the sum-of-sinusoids method.

run of the nonergodic WSS sinusoidal generator are providedin Fig. 5.

The results demonstrate that the IDFT technique, whosetheoretical accuracy is only limited by small aliasing errors[41], generally provides closer ACF matching over a widerrange of lags. However, the correlation matching property ofthe AR method allows it to provide a more precise match to thedesired ACF over the order of the model used. Similar accuracycan be achieved by the sinusoidal generator when a largenumber of sinusoids is used. The mild benefits of this tradeoffon the variate quality can be observed by comparing thegenerator outputs with regards to other important statistics. For

example, in Fig. 6, the normalized level-crossing rate, definedas the rate at which the envelope crosses a specified level inthe positive direction, is computed for the IDFT, AR(50), andNs = 64 sinusoidal generator outputs and compared with thetheoretical value given by [17, eq. (1.3–37)]. Since the LCRstatistic depends strongly on the first two moments of theDoppler spectrum, or equivalently, the curvature of the ACFat the zeroth lag [17], the AR simulator provides an excellentmatch by virtue of its correlation matching property, betterthan the match achieved with the IDFT method. A comparablematch is achieved using the WSS sinusoidal generator, butonly when a large number of sinusoids is used. We report that

1658 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Fig. 6. Empirical level-crossing rates for the AR(50), IDFT, and Ns = 64 sum-of-sinusoids simulators.

Fig. 7. The time to generate 221 complex samples using the various generation methods.

to generate the level-crossing rate curves, 220 variates with anormalized Doppler of fm = 0.001 were used.

To assess the relative computational effort required togenerate variates using the AR, IDFT, and WSS sum-of-sinusoids methods, sequences of length 221 were generatedon a Pentium IV machine using routines coded in MATLAB.The sequence length was chosen as a power of two as thisis favorable for the IDFT method [2]. The reference ACF was(2) with fm = 0.05. Results for the time comparisons are

presented in Fig. 7. The IDFT method, owing to the inherentefficiency of the FFT operation, is superior in this regard.With the sinusoidal generator of [7], a large effort is requiredto ensure that the quadrature components of each generatedsample function have accurate correlation statistics. Almost30 times more time than the IDFT method was needed togenerate the variates using the accurate Ns = 64 simulator.As for the AR approach, it can only be performed fasterthan the IDFT method for short IIR filter lengths, with a

BADDOUR AND BEAULIEU: AUTOREGRESSIVE MODELING FOR FADING CHANNEL SIMULATION 1659

Fig. 8. The power spectral density of the nonisotropic model with fm = 0.05, κ = 5, µ = 0, and the corresponding complex AR(50) model fit with ε = 10−5.

corresponding loss in modeling accuracy. Almost three timesmore time was needed to computer the model coefficientsand generate the variates using a highly accurate AR(200)model. However, the AR method can be considered efficientwhen compared to the required effort of the sinusoidalgenerator. In practical applications, it is likely that theoreticallypredicted differences less than 0.2 dB will be vitiated byimplementation tolerances or limits to the theoretical models.From this point of view, the AR(50) model may be comparableto the IDFT method. Fig. 7 shows that the AR(50) modelrequires only 1.2 times more time than the IDFT method.The main advantage of the AR filtering and WSS sinusoidalmethods is that fading variates can be generated as they arerequired. In contrast, the computational efficiency of theIDFT approach comes at a cost in storage requirements asall variates are generated using a single IFFT. Thus, for verylong sequence lengths, the IDFT implementation may requireunavailable memory. The AR and sinusoidal generators,however, do not have such a limitation. One disadvantage ofthe AR approach is that the model coefficients need to berecomputed when fm is changed. For the WSS sinusoidalmethod, one does not have to recalculate any coefficients.

To reduce the computational load of the AR simulator,a multirate filtering implementation is recommended [5], [42].With this approach, the spectrum shaping filter is cascadedwith an interpolator. This allows for the generation of thecomplex channel gain at a low sampling rate, typically a fewmultiples of the Doppler frequency, with correspondinglyshorter AR shaping filters. A comparatively simple FIRinterpolator (e.g., MMSE, raised cosine, etc.) can thenaccurately provide the necessary rate conversion to the desiredsampling rate. When combined with the addition of a smallconstant as described in Section III, interpolation proves usefulin mitigating numerical problems that occur when fitting an

AR model to a very-small-bandwidth Doppler spectrum.Based on our simulations, interpolation is suggested for thesimulation of channels with normalized Doppler rates on theorder of fm = 0.001 or smaller.

D. Nonisotropic Fading Simulation

In this subsection, we examine the use of the ARmethod for accurately synthesizing nonisotropic Rayleighfading channels. For directional scenarios, the nonuniformprobability density function for the angle of arrival (AOA) atthe receiver can result in a baseband Doppler PSD which isnot symmetric around the f = 0 frequency or, correspondingly,a channel ACF, which is complex valued. The underlyingreal in-phase and quadrature Gaussian processes are cross-correlated in such cases. Such Rayleigh processes can bewell approximated using a complex AR process and followingthe methodology in Section III. The complex AR coefficientsare determined using the complex form of the Levinson–Durbin algorithm in this case. As an example, a plausiblemodel for the directional AOA, which conveniently resultsin closed-form correlation and Doppler PSD functions, is theparametric Von Mises/Tikhonov distribution function [22]

p(θ) =exp[κ cos(θ − µ)]

2πI0(κ), θ ∈ [−π, π) (21)

where I0(·) is the zeroth order modified Bessel function [43],µ represents the mean direction of the AOA, and κ controlsthe beamwidth. This model, which includes the uniform AOAdistribution as a special case (κ = 0), was corroborated withsome empirical measurements of narrowband fading channelsin [22]. For the AOA distribution in (21), the corresponding

1660 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Fig. 9. The empirical autocorrelation and I/Q cross-correlations corresponding to the nonisotropic simulation examples.

Fig. 10. The empirical level-crossing rates corresponding to the nonisotropic simulation examples.

sampled autocorrelation of the Rayleigh fading channel can beshown to be [22]

R[n] = RII [n] + jRIQ[n]

=I0

(√κ2 − (2πfm|n|)2 + 4jκ cos(µ)πfm|n|

)I0(κ)

(22)

where RII [n] = RQQ[n], RII [n], and RQQ[n] denote thesampled autocorrelation of the real in-phase and quadratureGaussian processes, respectively and RIQ[n] denotes the cross-correlation function. The corresponding PSD is given by[22, eq. (3)]. Accurate simulation of such nonisotropic Rayleighmodels can be easily accomplished using a complex AR model.For example, a complex AR(50) model was used to synthesizefading processes with fm = 0.05 and µ = 0 for scenarios withκ = 0 (isotropic), κ = 1 (slightly nonisotropic), and κ = 5(highly nonisotropic). In each case, a bias of ε = 10−5 was

BADDOUR AND BEAULIEU: AUTOREGRESSIVE MODELING FOR FADING CHANNEL SIMULATION 1661

used to condition the inputs to the complex Levinson recursion.The PSD of the nonisotropic model and the AR(50) approx-imation are shown in Fig. 8 for the directional κ = 5 case.The empirical autocorrelations and I/Q cross-correlations areplotted in Fig. 9 and these are found to provide a very accurateapproximation of the desired statistics. The empirical normal-ized level crossing rates are shown in Fig. 10 and comparedto the values in [44, eq. (19)]. The excellent agreement seenin these results serves to verify the theoretical nonisotropicRayleigh LCR expression derived in [44].

V. CONCLUSION

Autoregressive (AR) stochastic models were consideredfor the computer simulation of correlated Rayleigh fadingchannels. The numerical difficulties faced by this approachwere resolved by approximating the deterministic bandlimitedDoppler processes with regular processes. Two proceduresfor eliminating the need to discard, possibly many, initialgenerated samples due to transient distortion were presented.The proposed methods enable the synthesis of stationary andstatistically accurate Rayleigh channel gain samples as theyare needed. Furthermore, the fading channel ACF is easilyspecified, which makes the simulator especially suited for theemulation of generalized flat Rayleigh fading channels. Todemonstrate the general applicability of the method, an ARsimulator for generating nonisotropic fading channel variateswas derived. This nonisotropic simulator will be useful foremulating directional fading scenarios encountered in practicalmobile communication systems.

ACKNOWLEDGMENT

The authors wish to thank D. J. Young of the iCORE WirelessCommunications Laboratory at the University of Alberta forproviding the MATLAB implementation of the IDFT algorithmof [2].

REFERENCES

[1] R. H. Clarke, “A statistical theory of mobile-radio reception,” Bell Syst.Tech. J., vol. 47, no. 6, pp. 957–1000, Jul.–Aug. 1968.

[2] D. J. Young and N. C. Beaulieu, “The generation of correlated Rayleighrandom variates by inverse Fourier transform,” IEEE Trans. Commun.,vol. 48, no. 7, pp. 1114–1127, Jul. 2000.

[3] C. Loo and N. Secord, “Computer models for fading channels with appli-cations to digital transmission,” IEEE Trans. Veh. Technol., vol. 40, no. 4,pp. 700–707, Nov. 1991.

[4] P. Höher, “A statistical discrete-time model for the WSSUS multipathchannel,” IEEE Trans. Veh. Technol., vol. 41, no. 4, pp. 461–468, Jul.1992.

[5] D. Verdin and T. Tozer, “Generating a fading process for the simulationof land-mobile radio communications,” Electron. Lett., vol. 29, no. 23,pp. 2011–2012, Nov. 11, 1993.

[6] M. F. Pop and N. C. Beaulieu, “Limitations of sum-of-sinusoids fadingchannel simulators,” IEEE Trans. Commun., vol. 49, no. 4, pp. 699–708,Apr. 2001.

[7] Y. R. Zheng and C. Xiao, “Improved models for the generation of multipleuncorrelated Rayleigh fading waveforms,” IEEE Commun. Lett., vol. 6,no. 6, pp. 256–258, Jun. 2002.

[8] ——, “Simulation models with correct statistical properties for Rayleighfading channels,” IEEE Trans. Commun., vol. 51, no. 6, pp. 920–928,Jun. 2003.

[9] M. Tsatsanis and Z. Xu, “Pilot symbol assisted modulation in frequencyselective fading wireless channels,” IEEE Trans. Signal Process., vol. 48,no. 8, pp. 2353–2365, Aug. 2000.

[10] H. Wu and A. Duel-Hallen, “Multiuser detectors with disjoint Kalmanchannel estimators for synchronous CDMA mobile radio channels,” IEEETrans. Commun., vol. 48, no. 5, pp. 752–756, May 2000.

[11] Y. Ara and H. Ogiwara, “Adaptive equalization of selective fading channelbased on AR-model of channel impulse response fluctuation,” Electron.Commun. Jpn., pt. I, vol. 74, no. 7, pp. 76–86, 1991.

[12] V. Kafedziski and D. Morrell, “Optimal adaptive equalization of multi-path fading channels,” in Proc. IEEE Asilomar Conf. Signals, Systems,and Computers, Pacific Grove, CA, 1994, pp. 1443–1447.

[13] T. Eyceoz, A. Duel-Hallen, and H. Hallen, “Deterministic channel model-ing and long range prediction of fast fading mobile radio channels,” IEEECommun. Lett., vol. 2, no. 9, pp. 254–256, Sep. 1998.

[14] Q. T. Zhang, “A decomposition technique for efficient generation of cor-related Nakagami fading channels,” IEEE J. Sel. Areas Commun., vol. 18,no. 11, pp. 2385–2392, Nov. 2000.

[15] G. Colman, S. Blostein, and N. C. Beaulieu, “An ARMA multipath fadingsimulator,” in Wireless Personal Communications: Improving Capacity,Services and Reliability. Boston, MA: Kluwer, 1997.

[16] P. A. Bello, “Characterization of randomly time-variant linear channels,”IEEE Trans. Commun. Syst., vol. CS-11, no. 4, pp. 360–393, Dec. 1963.

[17] W. C. Jakes, Microwave Mobile Communications. New York: Wiley,1974.

[18] T. Aulin, “A modified model for the fading signal at a mobile radiochannel,” IEEE Trans. Veh. Technol., vol. 28, no. 3, pp. 182–203, Feb.1979.

[19] J. D. Parsons and A. D. Turkmani, “Characterisation of mobile radiosignals: Model description,” Proc. Inst. Elect. Eng., pt. I, vol. 138, no. 6,pp. 549–555, Apr. 1991.

[20] R. H. Clarke and W. L. Khoo, “3-D mobile radio channel statistics,” IEEETrans. Veh. Technol., vol. 46, no. 3, pp. 798–799, May 1997.

[21] J. Sadowsky and V. Kafedziski, “On the correlation and scattering func-tions of the WSSUS channel for mobile communications,” IEEE Trans.Veh. Technol., vol. 47, no. 1, pp. 270–282, Feb. 1998.

[22] A. Abdi, J. Barger, and M. Kaveh, “A parametric model for the distri-bution of the angle of arrival and the associated correlation function andpower spectrum at the mobile station,” IEEE Trans. Veh. Technol., vol. 51,no. 1, pp. 425–434, May 2002.

[23] A. Leon-Garcia, Probability and Random Processes for Electrical Engi-neering. Reading, MA: Addison-Wesley, 1990.

[24] J. I. Smith, “A computer generated multipath fading simulation formobile radio,” IEEE Trans. Veh. Technol., vol. VT-24, no. 3, pp. 39–40,Aug. 1975.

[25] S. M. Kay, Modern Spectral Estimation. Englewood Cliffs, NJ:Prentice-Hall, 1988.

[26] A. Giordano and F. Hsu, Least Square Estimation with Applications toDigital Signal Processing. New York: Wiley, 1985.

[27] S. Haykin, Adaptive Filter Theory, 2nd ed. Englewood Cliffs, NJ:Prentice-Hall, 1991.

[28] A. Papoulis, Probability, Random Variables, and Stochastic Processes.New York: McGraw-Hill, 1991.

[29] ——, “Predictable processes and Wold’s decomposition: A review,” IEEETrans. Acoust. Speech Signal Process., vol. ASSP-33, no. 4, pp. 933–938,Aug. 1985.

[30] R. J. Lyman, W. W. Edmonson, S. McCullough, and M. Rao, “Thepredictability of continuous-time, bandlimited processes,” IEEE Trans.Signal Process., vol. 48, no. 2, pp. 311–316, Feb. 2000.

[31] R. J. Lyman, “Optimal mean-square prediction of the mobile-radio fadingenvelope,” IEEE Trans. Signal Process., vol. 51, no. 3, pp. 819–824,Mar. 2003.

[32] B. Picinbono and J. Kerilis, “Some properties of prediction and interpola-tion errors,” IEEE Trans. Acoust. Speech Signal Process., vol. 36, no. 4,pp. 525–531, Apr. 1988.

[33] D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, anduncertainty—V: The discrete case,” Bell Syst. Tech. J., vol. 57, no. 5,pp. 1371–1430, May 1978.

[34] M. Rosenblatt, “Some purely deterministic processes,” J. Math. Mech.,vol. 6, no. 6, pp. 801–810, Nov. 1957.

[35] K. E. Baddour, “Simulation, estimation and prediction of flat fading wire-less channels,” Ph.D. dissertation, Dept. Elect. Comput. Eng., QueensUniv., Kingston, ON, Canada.

[36] D. G. Lampard, “The response of linear networks to suddenly ap-plied random noise,” IRE Trans. Circuit Theory, vol. CT-2, pp. 49–57,Mar. 1955.

[37] J. R. M. Hosking, “Some models of persistence in time series,” in Time

1662 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Series Analysis: Theory and Practice I, O. D. Anderson, Ed. Amster-dam, The Netherlands: North Holland, 1982, pp. 641–653.

[38] F. L. Ramsey, “Characterization of the partial autocorrelation function,”Ann. Stat., vol. 6, no. 6, pp. 1296–1301, 1974.

[39] B. Friedlander, “Lattice filters for adaptive processing,” Proc. IEEE,vol. 70, no. 8, pp. 829–867, Aug. 1982.

[40] D. J. Young and N. C. Beaulieu, “Power margin quality measures forcorrelated random variates derived from the normal distribution,” IEEETrans. Inf. Theory, vol. 49, no. 1, pp. 241–252, Jan. 2003.

[41] N. C. Beaulieu and C. C. Tan, “FFT based generation of band-limited Gaussian noise variates,” Eur. Trans. Telecommun., vol. 10, no. 1,pp. 540–550, Sep./Oct. 1999.

[42] H. Brehm, W. Stammler, and M. Woerner, “Design of a highly flexibledigital simulator for narrowband fading channels,” in Signal ProcessingIII: Theories and Applications, EURASIP. North Holland, The Nether-lands: Elsevier, 1986, pp. 1113–1116.

[43] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions.New York: Dover, 1965.

[44] A. Abdi, K. Wills, H. Barger, S. Alouini, and M. Kaveh, “Comparisonof level crossing rate and average fade duration of Rayleigh, Rice, andNakagami fading models with mobile channel data,” in Proc. IEEEVehicular Technology Conf., Boston, MA, 2000, pp. 1850–1857.

Kareem E. Baddour (S’93) received the B.Eng. de-gree in electrical engineering in 1996 from MemorialUniversity of Newfoundland, St. John’s, NF, Canada,and the M.Sc. (Eng.) degree in 1998 from Queen’sUniversity, Kingston, ON, Canada, where he is cur-rently working towards the Ph.D. degree in the De-partment of Electrical and Computer Engineering.

Previously, he was with Northern Telecom, New-bridge Networks, Microtel Pacific Research, as wellas with the Canadian Coast Guard’s Telecommuni-cations and Electronics Branch. He is also affiliated

with the iCORE Wireless Communications Laboratory at the University ofAlberta, Edmonton, AB, Canada. His current research interests lie in the generalarea of wireless communications, with a focus on fading channel simulation,estimation, prediction, and equalization.

Mr. Baddour has received numerous scholarships, including two NaturalSciences and Engineering Research Council of Canada (NSERC) PostgraduateScholarships, two Industry Canada Fessenden Postgraduate Scholarships, andan Ontario Graduate Scholarship in Science and Technology. He received theMemorial University Medal for Electrical Engineering in 1996.

Norman C. Beaulieu (S’82–M’86–SM’89–F’99)received the B.A.Sc. (honors), M.A.Sc., and Ph.D.degrees in electrical engineering from the Universityof British Columbia, Vancouver, Canada, in 1980,1983, and 1986, respectively.

He was a Queen’s National Scholar Assistant Pro-fessor from September 1986 to June 1988, an Asso-ciate Professor from July 1998 to June 1993, and aProfessor from July 1993 to August 2000 with theDepartment of Electrical Engineering, Queen’s Uni-versity, Kingston, ON, Canada. In September 2000,

he became the iCORE Research Chair in Broadband Wireless Communicationsat the University of Alberta, Edmonton, Canada, and, in January 2001, hebecame the Canada Research Chair in Broadband Wireless Communications.His current research interests include broadband digital communications sys-tems, fading channel modeling and simulation, interference prediction andcancellation, decision-feedback equalization, and space-time coding.

Dr. Beaulieu is a Member of the IEEE Communication Theory Committeeand served as its Representative to the Technical Program Committee for the1991 International Conference on Communication and as Co-Representative tothe Technical Program Committee for the 1993 International Conference onCommunications and the 1996 International Conference on Communications.He was General Chair of the Sixth Communication Theory Mini-Conference inassociation with GLOBECOM’97 and Co-Chair of the Canadian Workshop onInformation Theory 1999. He has been an Editor of the IEEE TRANSACTIONS

ON COMMUNICATIONS for wireless communication theory since January 1992and has served on the Editorial Board of the PROCEEDINGS OF THE IEEE sinceNovember 2000. He served as Editor-in-Chief of the IEEE TRANSACTIONS ON

COMMUNICATIONS from January 2000 to December 2003 and as an AssociateEditor of the IEEE COMMUNICATIONS LETTERS for wireless communicationtheory from November 1996 to August 2003. He received the Natural Scienceand Engineering Research Council of Canada (NSERC) E. W. R. SteacieMemorial Fellowship in 1999. He was awarded the University of BritishColumbia Special University Prize in Applied Science in 1980 as the higheststanding graduate in the Faculty of Applied Science. He is a Fellow of TheRoyal Society of Canada.