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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006 3667 Novel Sum-of-Sinusoids Simulation Models for Rayleigh and Rician Fading Channels Chengshan Xiao, Senior Member, IEEE, Yahong Rosa Zheng, Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE Abstract— The statistical properties of Clarke’s fading model with a finite number of sinusoids are analyzed, and an improved reference model is proposed for the simulation of Rayleigh fading channels. A novel statistical simulation model for Rician fading channels is examined. The new Rician fading simulation model employs a zero-mean stochastic sinusoid as the specular (line-of- sight) component, in contrast to existing Rician fading simulators that utilize a non-zero deterministic specular component. The sta- tistical properties of the proposed Rician fading simulation model are analyzed in detail. It is shown that the probability density function of the Rician fading phase is not only independent of time but also uniformly distributed over [-π,π). This property is different from that of existing Rician fading simulators. The statistical properties of the new simulators are confirmed by extensive simulation results, showing good agreement with theoretical analysis in all cases. An explicit formula for the level-crossing rate is derived for general Rician fading when the specular component has non-zero Doppler frequency. Index Terms— Fading channel simulator, Rayleigh fading, Rician fading, statistics. I. I NTRODUCTION M OBILE radio channel simulators are commonly used in the laboratory because they make system tests and evaluations less expensive and more reproducible than field trials. Many different techniques have been proposed for the modeling and simulation of mobile radio channels [1]-[25]. Among them, the well known Jakes’ model [3], which is a simplified simulation model of Clarke’s model [1], has been widely used for frequency nonselective Rayleigh fading channels for about three decades. Various modifications [9], Manuscript received February 3, 2005; revised October 6, 2005; accepted November 27, 2005. The editor coordinating the review of this paper and approving it for publication is X. Shen. The work of C. Xiao was supported in part by the National Science Foundation under Grant CCF-0514770 and the University of Missouri-Columbia Research Council under Grant URC-05-064. The work of Y. R. Zheng was supported in part by the University of Missouri System Research Board. The work of N. C. Beaulieu was supported in part by the Alberta Informatics Circle of Research Excellence (iCORE). Parts of this paper were previously presented at the IEEE Wireless Communications and Networking Conference (WCNC) 2003, New Orleans, LA, and the IEEE International Conference on Communications (ICC) 2003, Anchorage, AK. C. Xiao is with the Department of Electrical and Computer Engi- neering, University of Missouri, Columbia, MO 65211 USA (e-mail: [email protected]; http://www.missouri.edu/xiaoc/). Y. R. Zheng is with the Department of Electrical and Computer Engineering, University of Missouri, Rolla, MO 65409 USA (e-mail: [email protected]; http://web.umr.edu/zhengyr/). N. C. Beaulieu is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada T6G 2G7 (e-mail: [email protected]; http://www.ee.ualberta.ca/beaulieu/). Digital Object Identifier 10.1109/TWC.2006.05068 [16]-[19] and improvements [22], [24], [25] of Jakes’ simu- lator for generating multiple uncorrelated fading waveforms needed for modeling frequency selective fading channels and multiple-input multiple-output (MIMO) channels have been reported. Since Jakes’ simulator needs only one fourth the number of low-frequency oscillators as needed in Clarke’s model, it is commonly perceived that Jakes’ simulator (and its modifications) is more computationally efficient than Clarke’s model. However, it was recently established by Pop and Beaulieu [19] that Jakes’ simulator and its variants (e.g., [3] and [16]) are not wide sense stationary (WSS) and that “reduction in the number of simulator oscillators based on azimuthal symmetries is meritless”. They proposed a Clarke’s model-based simulator design having the WSS property in [19], [21]. The Pop-Beaulieu simulator has been employed in a number of diverse applications [26]-[29]. In the first part of this paper, we give a statistical analysis of Clarke’s model with a finite number of sinusoids and show that the Pop- Beaulieu simulator has deficiencies in some of its higher-order statistics (as warned in [19, Section III.B]). We then propose an improved version of the Pop-Beaulieu simulator based on Clarke’s model for Rayleigh fading channels. All the existing Rician channel simulation models in the literature assume that the specular (line-of-sight) component is either constant and non-zero [13], or time-varying and deterministic [4], [16]. These assumptions may not reflect the physical nature of specular components, particularly when a specular component is random, changing from time to time and from mobile to mobile. Furthermore, according to [4], all these Rician fading models are nonstationary in the wide sense and the probability density function (PDF) of the fading phase is a function of time [4], [16]. In the second part of this paper, a novel statistical simulation model will be proposed for Rician fading channels. The specular component will employ a zero-mean stochastic sinusoid with a pre-chosen angle of arrival and a random initial phase. This assumption implies that different specular components in different channels may have different initial phases. The remainder of this paper is organized as follows. In Section II, we present the statistical properties of Clarke’s model with a finite number of sinusoids and show that the Pop-Beaulieu simulator has limitations in its higher-order statistics. An improved simulator for Rayleigh fading channels is proposed. In Section III, we present a novel statistical simulation model for Rician fading channels, and analyze the statistical properties of the new Rician fading model. 1536-1276/06$20.00 c 2006 IEEE
Transcript

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006 3667

Novel Sum-of-Sinusoids Simulation Models forRayleigh and Rician Fading Channels

Chengshan Xiao, Senior Member, IEEE, Yahong Rosa Zheng, Member, IEEE,and Norman C. Beaulieu, Fellow, IEEE

Abstract— The statistical properties of Clarke’s fading modelwith a finite number of sinusoids are analyzed, and an improvedreference model is proposed for the simulation of Rayleigh fadingchannels. A novel statistical simulation model for Rician fadingchannels is examined. The new Rician fading simulation modelemploys a zero-mean stochastic sinusoid as the specular (line-of-sight) component, in contrast to existing Rician fading simulatorsthat utilize a non-zero deterministic specular component. The sta-tistical properties of the proposed Rician fading simulation modelare analyzed in detail. It is shown that the probability densityfunction of the Rician fading phase is not only independent oftime but also uniformly distributed over [−π, π). This propertyis different from that of existing Rician fading simulators.The statistical properties of the new simulators are confirmedby extensive simulation results, showing good agreement withtheoretical analysis in all cases. An explicit formula for thelevel-crossing rate is derived for general Rician fading when thespecular component has non-zero Doppler frequency.

Index Terms— Fading channel simulator, Rayleigh fading,Rician fading, statistics.

I. INTRODUCTION

MOBILE radio channel simulators are commonly usedin the laboratory because they make system tests and

evaluations less expensive and more reproducible than fieldtrials. Many different techniques have been proposed for themodeling and simulation of mobile radio channels [1]-[25].Among them, the well known Jakes’ model [3], which isa simplified simulation model of Clarke’s model [1], hasbeen widely used for frequency nonselective Rayleigh fadingchannels for about three decades. Various modifications [9],

Manuscript received February 3, 2005; revised October 6, 2005; acceptedNovember 27, 2005. The editor coordinating the review of this paper andapproving it for publication is X. Shen. The work of C. Xiao was supportedin part by the National Science Foundation under Grant CCF-0514770 and theUniversity of Missouri-Columbia Research Council under Grant URC-05-064.The work of Y. R. Zheng was supported in part by the University of MissouriSystem Research Board. The work of N. C. Beaulieu was supported in partby the Alberta Informatics Circle of Research Excellence (iCORE). Parts ofthis paper were previously presented at the IEEE Wireless Communicationsand Networking Conference (WCNC) 2003, New Orleans, LA, and the IEEEInternational Conference on Communications (ICC) 2003, Anchorage, AK.

C. Xiao is with the Department of Electrical and Computer Engi-neering, University of Missouri, Columbia, MO 65211 USA (e-mail:[email protected]; http://www.missouri.edu/∼xiaoc/).

Y. R. Zheng is with the Department of Electrical and ComputerEngineering, University of Missouri, Rolla, MO 65409 USA (e-mail:[email protected]; http://web.umr.edu/∼zhengyr/).

N. C. Beaulieu is with the Department of Electrical and ComputerEngineering, University of Alberta, Edmonton, Canada T6G 2G7 (e-mail:[email protected]; http://www.ee.ualberta.ca/∼beaulieu/).

Digital Object Identifier 10.1109/TWC.2006.05068

[16]-[19] and improvements [22], [24], [25] of Jakes’ simu-lator for generating multiple uncorrelated fading waveformsneeded for modeling frequency selective fading channels andmultiple-input multiple-output (MIMO) channels have beenreported. Since Jakes’ simulator needs only one fourth thenumber of low-frequency oscillators as needed in Clarke’smodel, it is commonly perceived that Jakes’ simulator (and itsmodifications) is more computationally efficient than Clarke’smodel. However, it was recently established by Pop andBeaulieu [19] that Jakes’ simulator and its variants (e.g.,[3] and [16]) are not wide sense stationary (WSS) and that“reduction in the number of simulator oscillators based onazimuthal symmetries is meritless”. They proposed a Clarke’smodel-based simulator design having the WSS property in[19], [21]. The Pop-Beaulieu simulator has been employed ina number of diverse applications [26]-[29]. In the first part ofthis paper, we give a statistical analysis of Clarke’s modelwith a finite number of sinusoids and show that the Pop-Beaulieu simulator has deficiencies in some of its higher-orderstatistics (as warned in [19, Section III.B]). We then proposean improved version of the Pop-Beaulieu simulator based onClarke’s model for Rayleigh fading channels.

All the existing Rician channel simulation models in theliterature assume that the specular (line-of-sight) componentis either constant and non-zero [13], or time-varying anddeterministic [4], [16]. These assumptions may not reflect thephysical nature of specular components, particularly when aspecular component is random, changing from time to timeand from mobile to mobile. Furthermore, according to [4],all these Rician fading models are nonstationary in the widesense and the probability density function (PDF) of the fadingphase is a function of time [4], [16]. In the second part of thispaper, a novel statistical simulation model will be proposed forRician fading channels. The specular component will employa zero-mean stochastic sinusoid with a pre-chosen angle ofarrival and a random initial phase. This assumption impliesthat different specular components in different channels mayhave different initial phases.

The remainder of this paper is organized as follows. InSection II, we present the statistical properties of Clarke’smodel with a finite number of sinusoids and show that thePop-Beaulieu simulator has limitations in its higher-orderstatistics. An improved simulator for Rayleigh fading channelsis proposed. In Section III, we present a novel statisticalsimulation model for Rician fading channels, and analyzethe statistical properties of the new Rician fading model.

1536-1276/06$20.00 c© 2006 IEEE

3668 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

Section IV gives extensive performance evaluations of the newRayleigh and Rician fading simulators. Section V concludesthe paper.

II. AN IMPROVED RAYLEIGH FADING SIMULATOR

Clarke’s Rayleigh fading model is sometimes referred to asa mathematical reference model, and is commonly consideredas a computationally inefficient model compared to Jakes’Rayleigh fading simulator. In this section, we show thatClarke’s model with a finite number of sinusoids can bedirectly used for Rayleigh fading simulation, and that itscomputational efficiency and second-order statistics are asgood as those of improved Jakes’ simulators. We then brieflyshow that the Pop-Beaulieu simulator has some higher-orderstatistical deficiencies and improve the model by introducingrandomness to the angle of arrival, which leads to improvedhigher-order statistics.

A. Clarke’s Rayleigh Fading Model

The baseband signal of the normalized Clarke’s two-dimensional (2-D) isotropic scattering Rayleigh fading modelis given by [1], [30]

g(t) =1√N

N∑n=1

exp[j(wdt cosαn + φn)], (1)

where N is the number of propagation paths, wd is themaximum radian Doppler frequency and αn and φn are,respectively, the angle of arrival and initial phase of the nthpropagation path. Both αn and φn are uniformly distributedover [−π, π) for all n and they are mutually independent.

The central limit theorem justifies that the real part, gc(t) =Re[g(t)], and the imaginary part, gs(t) = Im[g(t)], of the fad-ing g(t) can be approximated as Gaussian random processesfor large N . Some desired second-order statistics for fadingsimulators are manifested in the autocorrelation and cross-correlation functions which are given in [30] for the case whenN approaches infinity. However, the statistical properties ofClarke’s model with a finite value of N (number of sinusoids)are not available in the literature. These properties are veryimportant for justifying the suitability of Clarke’s model asa valid Rayleigh fading simulator. Thus, we present some ofthese key statistics here.

Theorem 1: The autocorrelation and cross-correlation func-tions of the quadrature components, and the autocorrelationfunctions of the complex envelope and the squared envelopeof fading signal g(t) are given by

Rgcgc(τ) = Rgsgs(τ) =12J0(wdτ) (2a)

Rgcgs(τ) = Rgsgc(τ) = 0 (2b)

Rgg(τ) = Eα,φ[g∗(t)g(t+ τ)] = J0(wdτ) (2c)

R|g|2|g|2(τ) = 1 + J20 (wdτ) − J2

0 (wdτ)N

, (2d)

where Eα,φ[·] denotes expectation w.r.t. α and φ, and J0(·) isthe zero-order Bessel function of the first kind [31].

Proof: The autocorrelation function of the real part ofthe fading g(t) is proved as follows

Rgcgc(τ) = Eα,φ [gc(t)gc(t+ τ)]

=1N

N∑n=1

N∑i=1

Eα,φ {cos(wdt cosαn + φn)

· cos[wd(t+ τ) cosαi + φi]}

=1

2N

N∑n=1

Eα[cos(wdτ cosαn)]

=1

2N

N∑n=1

∫ π

−π

cos [wdτ cosαn]dαn

=1

2N

N∑n=1

J0(wdτ) =12J0(wdτ).

Similarly, one can prove the second part of (2a) and equations(2b)-(2c). The proof of equation (2d) is lengthy and can betreated as a special case of the proof of equation (8d) given inthe next subsection. The details are omitted here for brevity.

It is noted here that when N approaches infinity, all thederived statistical properties in equations (2) become identicalto the desired ones of Clarke’s reference model given in [30].

In simulation practice, time-averaging is often used in placeof ensemble averaging. For example, the autocorrelation ofthe real part of the fading signal for one trial (sample of theprocess) is given by

Rgcgc(τ) = limT→∞

1T

∫ T

0

gc(t)gc(t+ τ)dt

=1

2N

N∑n=1

cos(wdτ cosαn).

Clearly, this time averaged autocorrelation changes fromtrial to trial due to the random angle of arrival. Notethat the variance of the time average, Var{Rgcgc(τ)} =E[|Rgcgc(τ)−0.5J0(wdτ)|2

], carries important information

indicating the closeness between a single trial with finite Nand the ideal case with N = ∞. We now present the time-averaged variances of the aforementioned correlation statistics.

Theorem 2: The variances of the autocorrelation and cross-correlation of the quadrature components, and the varianceof the autocorrelation of the complex envelope of the fadingsignal g(t) are given by

Var{Rgcgc(τ)} = Var{Rgsgs(τ)}=

1 + J0(2wdτ) − 2J20 (wdτ)

8N(3a)

Var{Rgcgs(τ)} = Var{Rgsgc(τ)}=

1 − J0(2wdτ)8N

(3b)

Var{Rgg(τ)} =1 − J2

0 (wdτ)N

. (3c)

Proof: We start with the first equality of eqns. (3a) and(3b) and derive

XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3669

Var{Rgcgc(τ)}

= E

[∣∣∣∣Rgcgc(τ) −J0(wdτ)

2

∣∣∣∣2]

= E

[∣∣∣Rgcgc(τ)∣∣∣2]− J2

0 (wdτ)4

=1

4N2E

[N∑

n=1

N∑m=1

cos(wdτ cosαn) cos(wdτ cosαm)

]

−J20 (wdτ)

4

= −J20 (wdτ)

4+

14N2

{N∑

n=1

E[cos2(wdτ cosαn)

]

+N∑

n=1

N∑m=1m �=n

E [cos(wdτ cosαn)]E [cos(wdτ cosαm)]

⎫⎪⎪⎬⎪⎪⎭

=1

4N2

[N · 1 + J0(2wdτ)

2+ (N2 −N)J2

0 (wdτ)]

−J20 (wdτ)

4

=1 + J0(2wdτ) − 2J2

0 (wdτ)8N

.

Var{Rgcgs(τ)}

= E

[∣∣∣Rgcgs(τ) − 0∣∣∣2]

=1

4N2E

[N∑

n=1

N∑m=1

sin(wdτ cosαn) sin(wdτ cosαm)

]

=1

4N2

{N∑

n=1

E[sin2(wdτ cosαn)

]

+N∑

n=1

N∑m=1m �=n

E [sin(wdτ cosαn)]E [sin(wdτ cosαm)]

⎫⎪⎪⎬⎪⎪⎭

=1

4N2

[N · 1 − J0(2wdτ)

2+ 0

]

=1 − J0(2wdτ)

8N.

Similarly, we can validate the second equality of eqns. (3a)and (3b). Thus, we have

Var{Rgg(τ)} = E

[∣∣∣Rgg(τ) − J0(wdτ)∣∣∣2]

= E

[∣∣∣2Rgcgc(τ) + j2Rgcgs(τ) − J0(wdτ)∣∣∣2]

= 4E[∣∣∣Rgcgc(τ)

∣∣∣2]+ 4E[∣∣∣Rgcgs(τ)

∣∣∣2]−J2

0 (wdτ)

=1 − J2

0 (wdτ)N

.

This completes the proof of Theorem 2.

The results given in Theorems 1 and 2 show that thosestatistics considered that depend on N , depend on N exclu-sively as N−1. Therefore, the dependence on N is reducedby increasing N . We shall see later that Clarke’s model usinga number of sinusoids, N ≥ 8, can be usefully employed asa Rayleigh fading simulator, in some applications (typicallyshort simulation runs). In applications where the asymptoticvariance must be small (typically for long simulation runs),larger values of N (say, 40) can be used for greater simulationaccuracy. Its computational efficiency and statistics are similarto those of the recently improved Jakes’ models [22], [24],[25], which have removed some statistical deficiencies ofJakes’ original model [3] and various modified Jakes’ modelsproposed in [9], [16], [17] and [19].

Before proceeding to further discussion, we make a remarkto acknowledge and correct a mistake in [25], which wasoriginally discovered by Sun, Ye and Choi [32]. Specifically,the complex fading process defined by eqn. (14) of [25] maynot be a Gaussian random process when the duration of time isvery short, and the autocorrelation of the squared envelope ofthis fading process is nonstationary. However, the problemswith this fading process, which arise from a slight over-simplification of earlier results in an associated conferenceversion of the paper, can be easily solved by changing φ of(14) in [25] to φn with φn being statistically independent anduniformly distributed over [−π, π) for all n. Actually, in theconference version of [25], the complex fading process wasdefined correctly; details can be found in (14) of [22]. It isalso noted that inspired by Sun et al [32], we revisited andcorrected the expression for the squared envelope correlationfunction of the complex fading processes we defined. After thesubmission of this paper, we also noticed that Patel, Stuber andPratt [33] have independently discovered the aforementionedmistake in [25].

B. The Pop-Beaulieu Simulator

Based on Clarke’s model given by (1), Pop and Beaulieu[19], [21] recently developed a class of wide-sense stationaryRayleigh fading simulators by setting αn = 2πn

N in g(t). Thus,the lowpass fading process becomes

X(t) = Xc(t) + jXs(t) (4a)

Xc(t) =1√N

N∑n=1

cos(wdt cos

2πnN

+ φn

)(4b)

Xs(t) =1√N

N∑n=1

sin(wdt cos

2πnN

+ φn

). (4c)

They warned, however, that while their improved simulatoris wide sense stationary (contrary to previous sum-of-sinusoidssimulators such as, for example, [3], [16]), it may not modelsome higher-order statistical properties accurately. Reference[26] reported outstanding agreement between results obtainedfrom one implementation of the Pop-Beaulieu simulator andtheory in some turbo decoding applications. However, ingeneral, the quality required of a simulator will depend onthe application and some higher-order behaviors may not beaccurately modeled using this simulator. To further reveal the

3670 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

statistical properties of this model, we present the followingcorrelation statistics of this model.

RXcXc(τ) = RXsXs(τ) (5a)

=1

2N

N∑n=1

cos(wdτ cos

2πnN

)(5b)

RXcXs(τ) = −RXsXc(τ) (5c)

=1

2N

N∑n=1

sin(wdτ cos

2πnN

)(5d)

RXX(τ) = 2RXcXc(τ) + j2RXcXs(τ) (5e)

R|X|2|X|2(τ) = 1 + 4R2XcXc

(τ) + 4R2XcXs

(τ) − 1N.

(5f)

The proof of these statistics shown above is a special caseof the proof of Theorem 3 given in the next subsection. Thedetails are omitted here for brevity.

We make three remarks based on (5): 1) These second-order statistics of this modified model with N = ∞ arethe same as the desired ones of the original Clarke’s model.However, when N is finite, the statistics of this model aredifferent from the desired ones derived from Clarke’s model;2) the statistics of this model do not converge asymptoticallyto the desired ones when N increases as was discussed in[21] for the real part of RXX(τ); 3) when N is finite andodd, the imaginary part of RXX(τ), along with RXcXs(τ)and RXsXc(τ), can significantly deviate from zero (the desiredvalue), which implies that the quadrature components of thismodel are statistically correlated when N is odd.

C. An Improved Rayleigh Fading Channel Simulator

Based on the statistical analyses of Clarke’s model and thePop-Beaulieu simulator, we propose an improved simulationmodel as follows.

Definition 1: The normalized lowpass fading process ofan improved sum-of-sinusoids statistical simulation model isdefined by

Y (t) = Yc(t) + jYs(t) (6a)

Yc(t) =1√N

N∑n=1

cos(wdt cosαn + φn) (6b)

Ys(t) =1√N

N∑n=1

sin(wdt cosαn + φn) (6c)

with

αn =2πn+ θn

N, n = 1, 2, · · · , N (7)

where φn and θn are statistically independent and uniformlydistributed over [−π, π) for all n. It is noted that the differencebetween this improved model and the Pop-Beaulieu simulatoris the introduction of random variables θn to the angle ofarrival. Randomizing θn slightly decreases the efficiency ofthe simulator, but significantly improves the statistical qualityof the simulator. This model differs from Clarke’s model inthat it forces the angle of arrival, αn, to have a value restrictedto the interval

[2nπ−π

N , 2nπ+πN

). The angle of arrival is random

and uniformly distributed inside this sector, in contrast tobeing fixed as it is in Jakes’ model and in the Pop-Beaulieusimulator. Clarke’s model and a simulator proposed by Hoeher[7], assume independent αn, each uniformly distributed on[−π, π). Although our simulator design requires generatingthe same number of random αn, it ensures a more uniformempirical distribution of αn, particularly for small valuesof N , (but does not fix the values of αn). We shall seesubsequently that this modification reduces the variances ofthe empirical simulator statistics. It can be shown that thefirst-order statistics of this improved model are the same asthose of the Pop-Beaulieu simulator. However, some second-order statistics of this improved model are different, and theyare presented below.

Theorem 3: The autocorrelation and cross-correlation func-tions of the quadrature components, and the autocorrelationfunctions of the complex envelope and the squared envelopeof fading signal Y (t) are given by

RYcYc(τ) = RYsYs(τ) =12J0(wdτ) (8a)

RYcYs(τ) = RYsYc(τ) = 0 (8b)

RY Y (τ) = J0(wdτ) (8c)

R|Y |2|Y |2(τ) = 1 + J20 (wdτ) − fc(wdτ,N)

−fs(wdτ,N), (8d)

where

fc(wdτ,N) =N∑

k=1

[12π

∫ 2πk+πN

2πk−πN

cos(wdτ cos γ)dγ

]2

(9a)

fs(wdτ,N) =N∑

k=1

[12π

∫ 2πk+πN

2πk−πN

sin(wdτ cos γ)dγ

]2

. (9b)

The proof of this theorem is lengthy; a proof is outlined inAppendix I.

We now present the time-averaged variances of some keycorrelation statistics of Y (t) in Theorem 4.

Theorem 4: The variances of the autocorrelation and cross-correlation of the quadrature components, and the varianceof the autocorrelation of the complex envelope of the fadingsignal Y (t) are given by

Var{RYcYc(τ)} = Var{RYsYs(τ)}=

1 + J0(2wdτ)8N

− fc(wdτ,N)4

(10a)

Var{RYcYs(τ)} = Var{RYsYc(τ)}=

1 − J0(2wdτ)8N

− fs(wdτ,N)4

(10b)

Var{RY Y (τ)} =1N

− fc(wdτ,N) − fs(wdτ,N).

(10c)Proof: The proof of this theorem is similar to that of

Theorem 2; details are omitted for brevity.As can be seen from Theorems 1 and 3, the correlation

statistics, except the autocorrelation of the squared envelope,of the improved model are the same as those of Clarke’smodel when both models have the same number of sinusoids.Fig. 1 shows that the autocorrelations of the squared envelopefor Clarke’s model (2d) and for the new model (8d) aresimilar, and that this statistic for N = 8 is closer to the ideal

XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3671

0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

1

Squared envelope autocorrelation

Nor

mal

ized

R|g

|2 |g|2(τ

) an

d R

|Y|2 |Y

|2(τ)

Normalized time: fdτ

Clarke’s model with N = 8Clarke’s model with N = ∞Improved model with N = 8

Fig. 1. Theoretical autocorrelations of the squared envelopes of Clarke’smodel and our improved model.

0 2 4 6 8 10

0.05

0.1

0.15Variance of autocorrelation of the complex envelope, N = 8

Var

{Rgg

(τ)}

or

Var

{RY

Y(τ

)}

Normalized time: fdτ

SimulationTheory

Clarke’s model

Improved model

Fig. 2. Variances of autocorrelations of the complex envelope of Clarke’smodel and our improved model.

value (N = ∞) for the improved simulator than for Clarke’smodel. However, the variances of the empirical correlationsof the improved model are smaller than the empirical corre-lation variances of Clarke’s model. Using Theorems 2 and 4,Fig. 2 shows, as an example, some theoretical results and thecorresponding simulation results for the correlation variancesof Clarke’s model and the improved model. Obviously, thevariances of the autocorrelation of the complex envelope ofour improved model are smaller than those of Clarke’s model.This implies that the improved simulator converges faster thanClarke’s model (and Hoeher’s simulator) to an average valuefor a finite number of simulation trials.

III. A NOVEL RICIAN FADING SIMULATOR

In this section, we present a statistical Rician fading simu-lation model and its statistical properties.

Definition 2: The normalized lowpass fading process of anew statistical simulation model for Rician fading is defined

by

Z(t) = Zc(t) + jZs(t) (11a)

Zc(t) =[Yc(t) +

√K cos(wdt cos θ0 + φ0)

]/√

1 +K

(11b)

Zs(t) =[Ys(t) +

√K sin(wdt cos θ0 + φ0)

]/√

1 +K

(11c)

where K is the ratio of the specular power to scattered power,θ0 and φ0 are the angle of arrival and the initial phase,respectively, of the specular component, and φ0 is a randomvariable uniformly distributed over [−π, π).

A Rician fading simulator having a specular componentwith a non-zero Doppler frequency was studied in [16]. Oursimulator model (11) is different from the simulator in [16] be-cause in our model the initial phase of the specular componentis considered a random variable uniformly distributed over[−π, π), while the initial phase of the specular component in[16] is assumed to be constant. This is an important differencesince it results in a wide-sense stationary model for our case,whereas the model in [16] is nonstationary.

We present the ensemble correlation statistics of the fadingsignal, Z(t), in the following theorem.

Theorem 5: The autocorrelation and cross-correlation func-tions of the quadrature components, and the autocorrelationfunctions of the complex envelope and the squared envelopeof fading signal Z(t) are given by

RZcZc(τ) = RZsZs(τ)= [J0(wdτ) +K cos(wdτ cos θ0)] /(2 + 2K)

(12a)

RZcZs(τ) = −RZsZc(τ)= K sin(wdτ cos θ0)/(2 + 2K) (12b)

RZZ(τ) = [J0(wdτ) +K cos(wdτ cos θ0)+jK sin(wdτ cos θ0)] /(1 +K) (12c)

R|Z|2|Z|2(τ) ={1+J2

0 (wdτ)+K2−fc(wdτ,N)−fs(wdτ,N)+2K [1+J0(wdτ) cos(wdτ cos θ0)]} /(1+K)2.

(12d)The proof of Theorem 5 is given in Appendix II.Based on Definition 2 and Theorems 4 and 5, we present

the following corollary omitting the proof.Corollary: The variances of the autocorrelation and cross-

correlation of the quadrature components, and the varianceof the autocorrelation of the complex envelope of the fadingsignal Z(t) are given by

Var{RZcZc(τ)} = Var{RZsZs(τ)}=[1+J0(2wdτ)

8N− fc(wdτ,N)

4

]/(1+K)2

(13a)

Var{RZcZs(τ)} = Var{RZsZc(τ)}=[1−J0(2wdτ)

8N− fs(wdτ,N)

4

]/(1+K)2

(13b)

Var{RZZ(τ)} =[1/N−fc(wdτ,N)−fs(wdτ,N)]

(1+K)2(13c)

3672 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

where fc(wdτ,N) and fs(wdτ,N) are given by (9). Note thatwhen the number of sinusoids, N , is fixed, the variances ofthe aforementioned correlation statistics tends to be smaller asthe Rice factor, K , increases.

We now present the PDF’s of the fading envelope |Z(t)|and phase Ψ(t) = arctan [Zc(t), Zs(t)]1.

Theorem 6: When N approaches infinity, the envelope|Z(t)| is Rician distributed and the phase Ψ(t) is uniformlydistributed over [−π, π), and their PDF’s are given by

f|Z|(z) = 2(1 +K)z · exp[−K − (1 +K)z2

]·I0

[2z√K(1 +K)

], z ≥ 0 (14a)

fΨ(ψ) =12π, ψ ∈ [−π, π) (14b)

respectively, where I0(·) is the zero-order modified Besselfunction of the first kind [31].

Proof: Since the random sinusoids in the sums of Yc(t)and Ys(t) are statistically independent and identically distrib-uted, Yc(t) and Ys(t) tend to Gaussian random processes as thenumber of sinusoids, N , increases without limit, according to acentral limit theorem [34]. Moreover, since RYcYs(τ) = 0 andRYsYc(τ) = 0, Yc(t) and Ys(t) are uncorrelated and asymptot-

ically independent. Let mc(t) =√

K1+K cos(wdt cos θ0 + φ0)

and ms(t) =√

K1+K sin(wdt cos θ0 + φ0). Then, [Zc(t) −

mc(t)] and [Zs(t)−ms(t)] are uncorrelated and asymptoticallyindependent.

Given an initial phase φ0 of the specular component, theconditional joint PDF of Zc(t) and Zs(t) can be derived asfollowsf

Zc,Zs(zc, zs|φ0)

=1+Kπ

exp{−(1+K) [zc−mc]

2−(1+K) [zs−ms]2}

=1 +K

πexp

{−(1 +K)(z2c + z2

s) −K

+2(1 +K)[zcmc + zsms]} .Since the initial phase φ0 is uniformly distributed over

[−π, π), the joint PDF of Zc(t) and Zs(t) is given by

fZc,Zs

(zc, zs) =∫ π

−π

fZc,Zs

(zc, zs|φ0) · 12π

· dφ0

=1 +K

πexp

[−(1 +K)(z2c + z2

s) −K]

·∫ π

−π

exp {2(1+K)[zcmc + zsms]} dφ0

=1 +K

πexp

[−(1 +K)(z2c + z2

s) −K]

·I0[2√K(1 +K)(z2

c + z2s)]

where the last step uses the identity∫ π

−π exp [a cos(t+ x) + b sin(t+ x)] dx = 2πI0(√a2 + b2)

[31, p.336].Transforming the Cartesian coordinates (zc, zs) to polar

coordinates (z, ψ) with zc = z · cosψ and zs = z · sinψ, weobtain the transformation’s Jacobian J = z; therefore, the joint

1The function arctan(x, y) maps the arguments (x, y) into a phase in thecorrect quadrant in [−π, π).

PDF of the envelope |Z| and the phase Ψ = arctan(zc, zs) isgiven by

f|Z|,Ψ(z, ψ) =(1 +K)z

π· exp

[−K − (1 +K)z2]

·I0[2z√K(1 +K)

], z ≥ 0, ψ ∈ [−π, π).

Then, the marginal PDF’s of the envelope and the phasecan be obtained by the following two integrations

f|Z|(z) =∫ π

−π

f|Z|,Ψ(z, ψ)dψ

= 2(1 +K)z · exp[−K − (1 +K)z2

]·I0

[2z√K(1 +K)

], z ≥ 0

fΨ(ψ) =∫ ∞

0

f|Z|,Ψ(z, ψ)dz =12π, ψ ∈ [−π, π)

where the last equality utilizes the identity∫∞0x exp(−ax2)I0(bx)dx = 1

2a exp(

b2

4a

)[31, p.699].

This completes the proof.

We now highlight Theorem 6 with three remarks. First,both the fading envelope and the phase are stationary becausetheir PDF’s are independent of time t. This is very differentfrom the previous Rician models [4], [16], where the PDFof the fading phase is a very complicated function of timet, and therefore the fading phase is not stationary as pointedout in [4]. Here, the fading phase of our new model is notonly stationary but also uniformly distributed over [−π, π).Second, the fading envelope and phase of our new Ricianmodel are independent. As usual, the PDF’s of the envelopeand the phase of our Rician channel model include Rayleighfading (K = 0) as a special case. Third, the PDF of the fadingenvelope of our Rician model can be derived by using thetheory of two-dimensional random walks described in [35]and [36]. Details are omitted.

Two other important properties associated with the fadingenvelope are the level-crossing rate (LCR) and the averagefade duration (AFD). Both of these represent higher-orderbehaviors that a high quality simulator should emulate accu-rately. The LCR is defined as the rate at which the envelopecrosses a specified level with positive slope. The AFD is theaverage time duration that the fading envelope remains belowa specified level after crossing below that level. Both theLCR and AFD provide important information for the statisticsof burst errors [37], [38], which facilitates the design andselection of error correction techniques. Also, both representpractical behaviors of the simulator that depend on the higher-order statistics of the simulator. We now present explicitformulas for the LCR and AFD for a general Rician fadingchannel whose specular component has non-zero Dopplerfrequency. The following result (15a) is original while result(15b) represents a minor extension of a known result [30, p.66]for the case when the specular component is a constant.

Theorem 7: When N approaches infinity, the level-crossingrate L|Z| and the average fade duration T|Z| of the new

XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3673

simulator output are given by

L|Z| =

√2(1 +K)

πρfd · exp

[−K − (1 +K)ρ2]

·∫ π

0

[1 +

√K

1 +Kcos2 θ0 · cosα

]

· exp[2ρ√K(1 +K) cosα− 2K cos2 θ0 · sin2 α

]dα

(15a)

T|Z| =1 −Q

[√2K,

√2(1 +K)ρ2

]L|Z|

(15b)

where ρ is the normalized fading envelope level given by|Z|/|Z|rms with |Z|rms being the root-mean-square envelopelevel, and Q(·) is the first-order Marcum Q-function [39].

Proof: When N approaches infinity, the fading envelopeis Rician distributed as shown in Theorem 6. Therefore, wecan use the formula provided in [40] to obtain the LCR, L|Z|,viz

L|Z| =∫ ∞

0

rf(|Z|, r)dr,

where r is the envelope slope, f(r, r) is the joint PDF of theenvelope r and its slope r given by [40], [30]

f(r, r) =r√

(2π)3Bb0exp

(−r

2 + s2

2b0

)

·∫ π

−π

exp[rs cosαb0

− (b0r + b1s sinα)2

2Bb0

]dα

where, for our model defined in Definition 2, s, B = b0b2−b21,b0, b1 and b2 are given by

s =

√K

1 +K, b0 =

12(1 +K)

b1 = 2πb0∫ π

−π

(fd cosα− fd cos θ0)dα

2π= −2πb0fd cos θ0

b2 = (2π)2b0∫ π

−π

(fd cosα− fd cos θ0)2 dα

2π= 2π2b0f

2d

(1 + 2 cos2 θ0

)B = 2π2b20f

2d .

Using the procedure provided in [40] for deriving the LCR,we can validate (15a). Employing the procedure proposed in[30] for the AFD, we can obtain (15b). Details are omittedhere for brevity.

It is noted here that if θ0 = π2 or θ0 = −π

2 , which meansthat the specular component has zero Doppler frequency, thenthe LCR given by (15a) has a closed-form solution as follows

L|Z| =√

2π(1 +K)ρfd exp[−K − (1 +K)ρ2

]·I0

[2ρ√K(1 +K)

]. (16)

This is the same solution as that given in [40] and [30] for thecase of the specular component being deterministic. If K = 0,Z(t) = Y (t) becomes a Rayleigh fading process; then boththe LCR and the AFD have closed-form solutions given by

L|Y | =√

2πρfde−ρ2

(17a)

T|Y | =eρ2 − 1ρfd

√2π. (17b)

Before concluding this section, it is important to point outthat the new simulation model can be directly used to generatemultiple uncorrelated fading sample sequences for simulatingfrequency selective Rayleigh and/or Rician channels, MIMOchannels, and diversity combining techniques. Let Zk(t) be thekth Rician (or Rayleigh with Kk = 0) fading sample sequencegiven by

Zk(t) =√

11+Kk

√1N

N∑n=1

exp[jwd,kt cos

(2πn+ θn,k

N

)]

· exp (jφn,k)+√

Kk

1+Kkexp [j (wd,kt cos θ0,k+φ0,k)]

(18)

where wd,k, Kk and θ0,k are, respectively, the maximumradian Doppler frequency, the Rice factor and the specularcomponent’s angle of arrival of the kth Rician fading samplesequence, and where θn,k, φn,k and φ0,k are mutually inde-pendent and uniformly distributed over [−π, π) for all n and k.Then, Zk(t) retains all the statistical properties of Z(t) definedby eqn. (11). Furthermore, Zk(t) and Zl(t) are statisticallyindependent for all k �= l, due to the mutual independence ofθn,k, φn,k, φ0,k, θn,l, φn,l and φ0,l when k �= l.

IV. EMPIRICAL TESTING

Verification of the proposed fading simulator is carried outby comparing the corresponding simulation results for finite Nwith those of the theoretical limit when N approaches infinity.Throughout the following discussions, the newly proposedstatistical simulators have been implemented by choosingN = 8 unless otherwise specified. It is noted that if wechoose a larger value for N , then the statistical accuracy ofthe simulator will be increased.

A. Correlation Statistics

We have conducted extensive simulations of the autocorrela-tions and cross-correlations of the quadrature components, andthe autocorrelation of the complex envelope of both Rayleighand Rician (with various Rice factors) fading signals. Thesimulation results of these correlation statistics match thetheoretically calculated results with high accuracy even forsmall N . For example, Figs. 3 and 4 show the good agreementfor the real part and imaginary part of the autocorrelation ofthe complex envelope of the fading. The simulation results andthe theoretically calculated results for the autocorrelation ofthe squared envelope of the fading signals are slightly differentwhen N = 8 as can be seen from Fig. 5. The differencesdecrease if we increase the value of N , as expected.

B. Envelope and Phase PDF’s

Figs. 6 and 7 show that the PDF’s of the fading envelope andphase of the simulator with N = 8 are in very good agreementwith the theoretical ones. It is noted that when N > 8, thesePDF’s will have even better agreement with the theoreticallydesired ones. It is also noted that the more random samplesused for the ensemble average in the simulations, the smallerthe difference between the simulated curves and the desiredreference curves for the phase PDF.

3674 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

0 2 4 6 8 10−1

−0.5

0

0.5

1

Real part of RZZ(τ), N = 8

Re[

RZZ

(τ)]

Normalized time: fdτ

SimulationTheory

K = 1 K = 3

K = 0 (Rayleigh)

Fig. 3. The real part of the autocorrelation of the complex envelope Z(t);θ0 = π/4 for K = 1 and K = 3 Rician cases.

0 2 4 6 8 10−1

−0.5

0

0.5

1

Imaginary part of RZZ(τ), N = 8

Im[R

ZZ(τ

)]

Normalized time: fdτ

SimulationTheoryK = 1 K = 3

K = 0 (Rayleigh)

Fig. 4. The imaginary part of the autocorrelation of the complex envelopeZ(t); θ0 = π/4 for K = 1 and K = 3 Rician cases.

C. LCR and AFD

The simulation results for the normalized level-crossing rate(LCR),

L|Z|fd

, and the normalized average fade duration (AFD),fdT|Z|, of the new simulators are shown in Figs. 8 and 9,respectively, where the theoretically calculated LCR and AFDfor N = ∞ are also included in the figures for comparison,indicating generally good agreement in both cases. Again, ifwe increase the number of sinusoids, N , the simulation resultsfor the case of finite N approach the theoretical N = ∞results.

For the region of ρ < 0 dB, it is interesting to note thatthe average fade duration for θ0 = 0 (or θ0 < π/4) tendsto be smaller for larger values of the Rice factor K . Thisis different from the AFD for θ0 = π/2, which tends to belarger with larger Rice factors [30]. The main reason for thisphenomenon is that when θ0 = 0, the Doppler frequency ofthe specular component is equal to the maximum Dopplerfrequency, fd. For a given ρ < 0 dB and K > 0, the LCR isat its largest value and the AFD is at its smallest value. When

0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

1

Squared envelope autocorrelation, N = 8

Nor

mal

ized

R|Z

|2 |Z|2(τ

)

Normalized time: fdτ

SimulationTheory

K = 3

K = 1

K = 0 (Rayleigh)

Fig. 5. The autocorrelation of the squared envelope |Z(t)|2 with θ0 = π/4for K = 1 and K = 3 Rician cases.

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2f |Z

|(z)

z

PDF of the fading envelope

Simulation (N=8)Theory (N=∞)

K = 1

K = 3

K = 5

K = 10

K = 0 (Rayleigh)

Fig. 6. The PDF of the fading envelope |Z(t)|.

the value of K is increased, the specular component becomesmore dominant over the Rayleigh scatter components, and theAFD tends to be even smaller. However, when θ0 = π/2, theDoppler frequency of the specular component is zero, for eachsingle trial and the AFD becomes larger when the value of Kis increased.

V. CONCLUSION

In this paper, it was shown that Clarke’s model with afinite number of sinusoids can be directly used for simulatingRayleigh fading channels, and its computational efficiency andsecond-order statistics are better than those of Jakes’ originalmodel [3] and as good as those of the recently improvedJakes’ Rayleigh fading simulators [22], [24] and [25]. Animproved Clarke’s model was proposed to reduce the varianceof the time averaged correlations of a fading realization froma single trial. A novel simulation model employing a randomspecular component was proposed for Rician fading channels.The specular (line-of-sight) component of this Rician fadingmodel is a zero-mean stochastic sinusoid with a pre-chosen

XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3675

−1 −0.5 0 0.5 10.14

0.15

0.16

0.17

0.18

f Ψ(ψ

)

ψ, ( × π)

PDF of the fading phase

Theory (N=∞)Simulation: K=0Simulation: K=1Simulation: K=3Simulation: K=5Simulation: K=10N=8 for all simulations

Fig. 7. The PDF of the fading phase Ψ(t).

−25 −20 −15 −10 −5 0 5 1010−3

10−2

10−1

100

Nor

mal

ized

leve

l cro

ssin

g ra

te

Normalized fading envelope level ρ (dB)

LCR of the fading, θ0 = π/4

Simulation (N=8)Theory (N=∞)

K = 0

K = 1

K = 3

K = 5 K = 10

Fig. 8. The normalized LCR of the fading envelope |Z(t)|, where θ0 = π/4for all K > 0 Rician fading.

Doppler frequency and a random initial phase. Compared toall the existing Rician fading simulation models, which havea non-zero deterministic specular component, the new modelbetter reflects the fact that the specular component is randomfrom ensemble sample to ensemble sample and from mobileto mobile. Additionally and importantly, the fading phase PDFof the new Rician fading model is independent of time anduniformly distributed over [−π, π).

This paper has also analyzed the statistical properties of thenew simulation models. Mathematical formulas were derivedfor the autocorrelation and cross-correlation of the quadraturecomponents, the autocorrelation of the complex envelope andthe squared envelope, the PDF’s of the fading envelope andphase, the level-crossing rate and the average fade duration. Ithas been shown that all these statistics of the new simulatorseither exactly match or quickly converge to the desired ones.Good convergence can be reached even when the number ofsinusoids is as small as 8.

−20 −15 −10 −5 0 5

10−1

100

101AFD of the fading, θ0 = 0

Nor

mal

ized

ave

rage

fade

dur

atio

n

Normalized fading envelope level ρ (dB)

Simulation (N=8)Theory (N=∞)

K = 0

K = 1

K = 3

K = 5

K = 10

Fig. 9. The normalized AFD of the fading envelope |Z(t)|, where θ0 = 0for all K > 0 Rician fading.

ACKNOWLEDGMENT

The first author, C. Xiao, is grateful to Drs. Q. Sun, H.Ye and W. Choi of Atheros Communications Inc. for pointingout an error in an earlier version of the manuscript. He is alsoindebted to Dr. F. Santucci and Professor G. L. Stuber forhelpful discussions at the early stage of this work.

APPENDIX IPROOF OF THEOREM 3

Proof: The autocorrelation function of the real part ofthe fading is proved first. One has

RYcYc(τ) = Eα,φ [Yc(t)Yc(t+ τ)]

=1N

N∑n=1

N∑i=1

E {cos(wdt cosαn + φn)

· cos[wd(t+ τ) cosαi + φi]}

=1

2N

{N∑

n=1

E[cos(wdτ cosαn)]

}

=1

2N

{N∑

n=1

∫ π

−π

cos[wdτ cos

(2πn+ θn

N

)]dθn

}

=1

2N

{N∑

n=1

∫ 2πn+πN

2πn−πN

cos (wdτ cos γn)N

2πdγn

}

=14π

∫ 2π+ πN

πN

cos (wdτ cos γ) dγ

=14π

∫ 2π

0

cos(wdτ cos γ)dγ

=12J0(wdτ).

Similarly, we can obtain the autocorrelation of the imagi-

3676 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

nary part of the fading signal in (8a) as

RYsYs(τ) = E [Ys(t)Ys(t+ τ)]

=14π

∫ 2π

0

cos(wdτ cos γ)dγ

=12J0(wdτ).

We are now in a position to prove equation (8b) startingfrom

RYcYs(τ) = E [Yc(t)Ys(t+ τ)]

=1N

N∑n=1

N∑i=1

E {cos(wdt cosαn + φn)

· sin[wd(t+ τ) cosαi + φi]}

=1

2N

N∑n=1

E [sin(wdτ cosαn)]

=14π

∫ 2π

0

sin(wdτ cos γ)dγ = 0.

The second part of eqns. (8b) and (8c) are proved in a similarmanner. The proof of equation (8d) is different and lengthy.A brief outline with some salient details is given below. Onehas

R|Y |2|Y |2(τ) = E[Y 2

c (t)Y 2c (t+ τ)

]+ E

[Y 2

s (t)Y 2s (t+ τ)

]+E

[Y 2

c (t)Y 2s (t+ τ)

]+ E

[Y 2

s (t)Y 2c (t+ τ)

].

(19)

The derivation of the first term on the right side of (19) indetail starts as

E[Y 2

c (t)Y 2c (t+ τ)

]

=1N2

· E{

N∑n=1

cos(wdt cosαn + φn)

·N∑

i=1

cos(wdt cosαi + φi)

·N∑

p=1

cos[wd(t+ τ) cosαp + φp]

·N∑

q=1

cos[wd(t+ τ) cosαq + φq]

}. (20)

Since the random phases φk and φl are statistically indepen-dent for all k �= l, the right side of (20) is zero except forfour different cases: a) n = i = p = q; b) n = i, p = q, andn �= p; c) n = p, i = q, and n �= i; and d) n = q, i = p, andn �= i. Subsequently, E

[Y 2

c (t)Y 2c (t+ τ)

]is derived for each

of the four cases.

For the first case, n = i = p = q, we have

E[Y 2

c (t)Y 2c (t+ τ)

]1st

=1N2

N∑n=1

E{cos2(wdt cosαn + φn)

· cos2[wd(t+ τ) cosαn + φn]}

=1N2

{N∑

n=1

E

[1 + cos(2wdt cosαn + 2φn)

2

· 1 + cos[2wd(t+ τ) cosαn + 2φn]2

]}

=1N2

{N

4+

18

N∑n=1

E[cos(2wdτ cosαn)]

}

=1

4N+

18N

J0(2wdτ).

For the second case, n = i, p = q, and n �= p, we haveE[Y 2

c (t)Y 2c (t+ τ)

]2nd

=1N2

⎧⎪⎪⎨⎪⎪⎩

N∑n=1

N∑p=1

p�=n

E[cos2(wdt cosαn + φn)

]

· E (cos2[wd(t+ τ) cosαp + φp]

)}=

1N2

[N2 −N

4

]=

14− 1

4N.

For the third case, n = p, i = q, and n �= i, we haveE[Y 2

c (t)Y 2c (t+ τ)

]3rd

=1N2

N∑n=1

N∑i=1i�=n

E {cos(wdt cosαn + φn)

· cos[wd(t+ τ) cosαn + φn]}·E {cos(wdt cosαi + φi)· cos[wd(t+ τ) cosαi + φi]}

=1N2

{N∑

n=1

12E[cos(wdτ cosαn)]

}2

− 1N2

N∑n=1

{12E[cos(wdτ cosαn)]

}2

=14J2

0 (wdτ) − fc (wdτ,N)4

.

For the fourth case, n = q, i = p, and n �= i; in a mannersimilar to that used in the third case, one can prove

E[Y 2

c (t)Y 2c (t+ τ)

]4th

=14J2

0 (wdτ) − fc (wdτ,N)4

.

Since these four cases are the exclusive and exhaustivepossibilities for E

[Y 2

c (t)Y 2c (t+ τ)

]being non-zero, adding

them together we haveE[Y 2

c (t)Y 2c (t+ τ)

]= E

[Y 2

c (t)Y 2c (t+ τ)

]1st

+ E[Y 2

c (t)Y 2c (t+ τ)

]2nd

+E[Y 2

c (t)Y 2c (t+ τ)

]3rd

+ E[Y 2

c (t)Y 2c (t+ τ)

]4th

=14

+12J2

0 (wdτ) +1

8NJ0(2wdτ) − fc (wdτ,N)

2.

XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3677

This completes the derivation of E[Y 2

c (t)Y 2c (t+ τ)

].

Using the same procedure for the second, third and fourthterms on the right side of (19), one obtains

E[Y 2

s (t)Y 2s (t+ τ)

]=

14

+12J2

0 (wdτ) +1

8NJ0(2wdτ)

−fs (wdτ,N)2

E[Y 2

c (t)Y 2s (t+ τ)

]=

14− 1

8NJ0(2wdτ) − fc (wdτ,N)

2

E[Y 2

s (t)Y 2c (t+ τ)

]=

14− 1

8NJ0(2wdτ) − fs (wdτ,N)

2.

Therefore,

R|Y |2|Y |2(τ) = 1 + J20 (wdτ) − fc (wdτ,N) − fs (wdτ,N) .

This completes the proof of Theorem 3.

APPENDIX IIPROOF OF THEOREM 5

Proof: Based on the assumption that the initial phase ofthe specular component is uniformly distributed over [−π, π),and independent of the initial phases of the scattered compo-nents, one can prove eqns. (12a)-(12c) by using the results ofTheorem 3. The details are omitted for brevity. The proof ofequation (12d) is outlined as follows. One has

R|Z|2|Z|2(τ) = E[Z2

c (t)Z2c (t+ τ)

]+ E

[Z2

s (t)Z2s (t+ τ)

]+E

[Z2

c (t)Z2s (t+ τ)

]+ E

[Z2

s (t)Z2c (t+ τ)

].

Then,E[Z2

c (t)Z2c (t+ τ)

]=

1(1 +K)2

E

{[Yc(t) +

√K cos (wdt cos θ0 + φ0)

]2·(Yc(t+ τ) +

√K cos [wd(t+ τ) cos θ0 + φ0]

)2}

=E[Y 2

c (t)Y 2c (t+ τ)

](1 +K)2

+K ·E [

Y 2c (t)

] · E {cos2 [wd(t+ τ) cos θ0 + φ0]

}(1 +K)2

+K ·E [

Y 2c (t+ τ)

] ·E [cos2 (wdt cos θ0 + φ0)

](1 +K)2

+4K ·E [Yc(t)Yc(t+ τ)]

(1 +K)2·E {cos (wdt cos θ0+φ0)

· cos [wd(t+ τ) cos θ0 + φ0]}+

K2

(1 +K)2·E {

cos2 (wdt cos θ0 + φ0)

· cos2 [wd(t+ τ) cos θ0 + φ0]}

=E[Y 2

c (t)Y 2c (t+ τ)

](1 +K)2

+K

2(1 +K)2[1 + 2J0(wdτ) · cos(wdτ cos θ0)]

+K2

4(1 +K)2

[1 +

cos(2wdτ cos θ0)2

].

Similarly, we have

E[Z2

s (t)Z2s (t+ τ)

]=

1(1 +K)2

E

{[Ys(t) +

√K sin (wdt cos θ0 + φ0)

]2·(Ys(t+ τ) +

√K sin [wd(t+ τ) cos θ0 + φ0]

)2}

=E[Y 2

s (t)Y 2s (t+ τ)

](1 +K)2

+K

2(1 +K)2[1 + 2J0(wdτ) · cos(wdτ cos θ0)]

+K2

4(1 +K)2

[1 +

cos(2wdτ cos θ0)2

]

andE[Z2

c (t)Z2s (t+ τ)

]=

1(1 +K)2

E

{[Yc(t) +

√K cos (wdt cos θ0 + φ0)

]2·(Ys(t+ τ) +

√K sin [wd(t+ τ) cos θ0 + φ0]

)2}

=E[Y 2

c (t)Y 2s (t+ τ)

](1 +K)2

+K · E [

Y 2c (t)

] ·E {sin2 [wd(t+ τ) cos θ0 + φ0]

}(1 +K)2

+K · E [

Y 2s (t+ τ)

] ·E [cos2 (wdt cos θ0 + φ0)

](1 +K)2

+K2

(1 +K)2· E {

cos2 (wdt cos θ0 + φ0)

· sin2 [wd(t+ τ) cos θ0 + φ0]}

=E[Y 2

c (t)Y 2s (t+ τ)

](1 +K)2

+K

2(1 +K)2

+K2

4(1 +K)2

[1 − cos(2wdτ cos θ0)

2

]

andE[Z2

s (t)Z2c (t+ τ)

]=

1(1 +K)2

E

{[Ys(t) +

√K sin (wdt cos θ0 + φ0)

]2·(Yc(t+ τ) +

√K cos [wd(t+ τ) cos θ0 + φ0]

)2}

=E[Y 2

c (t)Y 2s (t+ τ)

](1 +K)2

+K

2(1 +K)2

+K2

4(1 +K)2

[1 − cos(2wdτ cos θ0)

2

].

Therefore,R|Z|2|Z|2(τ)

=R|Y |2|Y |2(τ)+K2+2K [1+J0(wdτ) cos(wdτ cos θ0)]

(1 +K)2

=1+J2

0 (wdτ)+K2+2K [1+J0(wdτ) cos(wdτ cos θ0)](1 +K)2

−fc(wdτ,N) + fs(wdτ,N)(1 +K)2

This completes the proof.

3678 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

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Chengshan Xiao (M’99-SM’02) received the B.S.degree from the University of Electronic Scienceand Technology of China, Chengdu, China, in 1987,the M.S. degree from Tsinghua University, Beijing,China, in 1989, and the Ph.D. degree from theUniversity of Sydney, Sydney, Australia, in 1997,all in electrical engineering.

From 1989 to 1993, he was on the Research Staffand then became a Lecturer with the Departmentof Electronic Engineering at Tsinghua University,Beijing, China. From 1997 to 1999, he was a Senior

Member of Scientific Staff at Nortel Networks, Ottawa, ON, Canada. From1999 to 2000, he was an faculty member with the Department of Electrical andComputer Engineering at the University of Alberta, Edmonton, AB, Canada.Since 2000, he has been with the Department of Electrical and ComputerEngineering at the University of Missouri-Columbia, where he is currentlyan Associate Professor. His research interests include wireless communicationnetworks, signal processing, and multidimensional and multirate systems. Hehas published extensively in these areas. He holds three U.S. patents inwireless communications area. His algorithms have been implemented intoNortel’s base station radios with successful technical field trials and networkintegration.

Dr. Xiao is a member of the IEEE Technical Committee on PersonalCommunications (TCPC) and the IEEE Technical Committee on Commu-nication Theory. He served as a Technical Program Committee member fora number of IEEE international conferences including WCNC, ICC andGlobecom in the last few years. He was a Vice Chair of the 2005 IEEEGlobecom Wireless Communications Symposium. He is currently the ViceChair of the TCPC of the IEEE Communications Society. He has been anEditor for the IEEE Transactions on Wireless Communications since July2002. Previously, he was an Associate Editor for the IEEE Transactions onVehicular Technology from July 2002 to June 2005, the IEEE Transactionson Circuits and Systems-I from January 2002 to December 2003, and theinternational journal of Multidimensional Systems and Signal Processing fromJanuary 1998 to December 2005.

XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3679

Yahong Rosa Zheng (S’99-M’03) received the B.S.degree from the University of Electronic Scienceand Technology of China, Chengdu, China, in 1987,the M.S. degree from Tsinghua University, Beijing,China, in 1989, both in electrical engineering. Shereceived the Ph.D. degree from the Department ofSystems and Computer Engineering, Carleton Uni-versity, Ottawa, ON, Canada, in 2002.

From 1989 to 1997, she held Engineer positionsin several companies. From 2003 to 2005, she was aNatural Science and Engineering Research Council

of Canada (NSERC) Postdoctoral Fellow at the University of Missouri,Columbia, MO. Currently, she is an Assistant Professor with the Departmentof Electrical and Computer Engineering at the University of Missouri,Rolla, MO. Her research interests include array signal processing, wirelesscommunications, and wireless sensor networks.

Dr. Zheng has served as a Technical Program Committee member for the2004 IEEE International Sensors Conference, the 2005 IEEE Global Telecom-munications Conference, and the 2006 IEEE International Conference onCommunications. Dr. Zheng is currently an Editor for the IEEE Transactionson Wireless Communications.

Norman C. Beaulieu (S’82-M’86-SM’89-F’99) re-ceived the B.A.Sc. (honors), M.A.Sc., and Ph.D de-grees in electrical engineering from the University ofBritish Columbia, Vancouver, BC, Canada in 1980,1983, and 1986, respectively. He was awarded theUniversity of British Columbia Special UniversityPrize in Applied Science in 1980 as the higheststanding graduate in the faculty of Applied Science.

He was a Queen’s National Scholar Assistant Pro-fessor with the Department of Electrical Engineer-ing, Queen’s University, Kingston, ON, Canada from

September 1986 to June 1988, an Associate Professor from July 1988 to June1993, and a Professor from July 1993 to August 2000. In September 2000, hebecame the iCORE Research Chair in Broadband Wireless Communicationsat the University of Alberta, Edmonton, AB, Canada and in January 2001, theCanada Research Chair in Broadband Wireless Communications. His currentresearch interests include broadband digital communications systems, ultra-wide bandwidth systems, fading channel modeling and simulation, diversitysystems, interference prediction and cancellation, importance sampling andsemi-analytical methods, and space-time coding.

Dr. Beaulieu is a Member of the IEEE Communication Theory Committeeand served as its Representative to the Technical Program Committee of the1991 International Conference on Communications and as Co-Representativeto the Technical Program Committee of the 1993 International Conference onCommunications and the 1996 International Conference on Communications.He was General Chair of the Sixth Communication Theory Mini-Conferencein association with GLOBECOM 97 and Co-Chair of the Canadian Workshopon Information Theory 1999. He has been an Editor for Wireless Communica-tion Theory of the IEEE Transactions on Communications since January 1992,and was Editor-in-Chief from January 2000 to December 2003. He servedas an Associate Editor for Wireless Communication Theory of the IEEECommunications Letters from November 1996 to August 2003. He has alsoserved on the Editorial Board of the Proceedings of the IEEE since November2000. He received the Natural Science and Engineering Research Council ofCanada (NSERC) E.W.R. Steacie Memorial Fellowship in 1999. ProfessorBeaulieu was elected a Fellow of the Engineering Institute of Canada in 2001and was awarded the Medaille K.Y. Lo Medal of the Institute in 2004. Hewas elected Fellow of the Royal Society of Canada in 2002 and was awardedthe Thomas W. Eadie Medal of the Society in 2005.


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