Novel Sum-of-Sinusoids Simulation Models for Rayleigh and Rician Fading Channels

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 Clarkes 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 [, ). 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 andevaluations 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 Clarkes 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 Clarkesmodel, it is commonly perceived that Jakes simulator (and itsmodifications) is more computationally efficient than Clarkesmodel. 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 thatreduction in the number of simulator oscillators based onazimuthal symmetries is meritless. They proposed a Clarkesmodel-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 Clarkes 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 onClarkes 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 Clarkesmodel 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

Clarkes Rayleigh fading model is sometimes referred to asa mathematical reference model, and is commonly consideredas a computationally inefficient model compared to JakesRayleigh fading simulator. In this section, we show thatClarkes 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. Clarkes Rayleigh Fading Model

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

g(t) =1N

Nn=1

exp[j(wdt cosn + n)], (1)

where N is the number of propagation paths, wd is themaximum radian Doppler frequency and n and n are,respectively, the angle of arrival and initial phase of the nthpropagation path. Both n and n are uniformly distributedover [, ) 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 ofClarkes 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 Clarkes 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() = Rgsgs() =12J0(wd) (2a)

Rgcgs() = Rgsgc() = 0 (2b)Rgg() = E,[g(t)g(t + )] = J0(wd) (2c)

R|g|2|g|2() = 1 + J20 (wd) J20 (wd)

N, (2d)

where E,[] denotes expectation w.r.t. and , 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() = E, [gc(t)gc(t + )]

=1N

Nn=1

Ni=1

E, {cos(wdt cosn + n)

cos[wd(t + ) cosi + i]}

=1

2N

Nn=1

E[cos(wd cosn)]

=1

2N

Nn=1

cos [wd cosn]dn2

=1

2N

Nn=1

J0(wd) =12J0(wd).

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 Clarkes 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() = limT

1T

T0

gc(t)gc(t + )dt

=1

2N

Nn=1

cos(wd cosn).

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()} =E[|Rgcgc()0.5J0(wd)|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()} = Var{Rgsgs()}=

1 + J0(2wd) 2J20 (wd)8N

(3a)Var{Rgcgs()} = Var{Rgsgc()}

=1 J0(2wd)

8N(3b)

Var{Rgg()} = 1 J20 (wd)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()}

= E

[Rgcgc() J0(wd)22]

= E[Rgcgc()2

] J

20 (wd)

4

=1

4N2E

[N

n=1

Nm=1

cos(wd cosn) cos(wd cosm)

]

J20 (wd)

4

= J20 (wd)

4+

14N2

{N

n=1

E[cos2(wd cosn)

]

+N

n=1

Nm=1m =n

E [cos(wd cosn)]E [cos(wd cosm)]

=1

4N2

[N 1 + J0(2wd)

2+ (N2 N)J20 (wd)

]

J20 (wd)

4

=1 + J0(2wd) 2J20 (wd)

8N.

Var{Rgcgs()}

= E[Rgcgs() 02

]

=1

4N2E

[N

n=1

Nm=1

sin(wd cosn) sin(wd cosm)

]

=1

4N2

{N

n=1

E[sin2(wd cosn)

]

+N

n=1

Nm=1m =n

E [sin(wd cosn)]E [sin(wd cosm)]

=1

4N2

[N 1 J0(2wd)

2+ 0

]

=1 J0(2wd)

8N.

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

Var{Rgg()} = E[Rgg() J0(wd)2

]

= E[2Rgcgc() + j2Rgcgs() J0(wd)2

]

= 4E[Rgcgc()2

]+ 4E

[Rgcgs()2]

J20 (wd)=

1 J20 (wd)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 N1. Therefore, the dependence on N is reducedby increasing N . We shall see later that Clarkes 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 of(14) in [25] to n with n being statistically independent anduniformly distributed over [, ) 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 SimulatorBased on Clarkes model given by (1), Pop and Beaulieu

[19], [21] recently developed a class of wide-sense stationaryRayleigh fading simulators by setting n = 2nN in g(t). Thus,the lowpass fading process becomes

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

Xc(t) =1N

Nn=1

cos(wdt cos

2nN

+ n

)(4b)

Xs(t) =1N

Nn=1

sin(wdt cos

2nN

+ n

). (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


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

RXcXc() = RXsXs() (5a)

=1

2N

Nn=1

cos(wd cos

2nN

)(5b)

RXcXs() = RXsXc() (5c)

=1

2N

Nn=1

sin(wd cos

2nN

)(5d)

RXX() = 2RXcXc() + j2RXcXs() (5e)R|X|2|X|2() = 1 + 4R2XcXc() + 4R

2XcXs()

1N

.

(5f)The proof of these statistics shown above is a special case

of 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 Clarkes model.However, when N is finite, the statistics of this model aredifferent from the desired ones derived from Clarkes 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(); 3) when N is finite andodd, the imaginary part of RXX(), along with RXcXs()and RXsXc(), 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 SimulatorBased on the statistical analyses of Clarkes model and the

Pop-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) =1N

Nn=1

cos(wdt cosn + n) (6b)

Ys(t) =1N

Nn=1

sin(wdt cosn + n) (6c)

with

n =2n + n

N, n = 1, 2, , N (7)

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

[2n

N ,2n+

N

). 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. Clarkes model and a simulator proposed by Hoeher[7], assume independent n, each uniformly distributed on[, ). Although our simulator design requires generatingthe same number of random n, it ensures a more uniformempirical distribution of n, particularly for small valuesof N , (but does not fix the values of n). 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() = RYsYs() =12J0(wd) (8a)

RYcYs() = RYsYc() = 0 (8b)RY Y () = J0(wd) (8c)

R|Y |2|Y |2() = 1 + J20 (wd) fc(wd,N)fs(wd,N), (8d)

where

fc(wd,N) =N

k=1

[12

2k+N

2kN

cos(wd cos )d

]2(9a)

fs(wd,N) =N

k=1

[12

2k+N

2kN

sin(wd cos )d

]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()} = Var{RYsYs()}=

1 + J0(2wd)8N

fc(wd,N)4

(10a)Var{RYcYs()} = Var{RYsYc()}

=1 J0(2wd)

8N fs(wd,N)

4(10b)

Var{RY Y ()} = 1N

fc(wd,N) fs(wd,N).(10c)

Proof: The proof of this theorem is similar to that ofTheorem 2; details are omitted for brevity.

As can be seen from Theorems 1 and 3, the correlationstatistics, except the autocorrelation of the squared envelope,of the improved model are the same as those of Clarkesmodel when both models have the same number of sinusoids.Fig. 1 shows that the autocorrelations of the squared envelopefor Clarkes model (2d) and for the new model (8d) aresimilar, and that this statistic for N = 8 is closer to the ideal


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(

) an

d R

|Y|2 |

Y|2()

Normalized time: fd

Clarkes model with N = 8Clarkes model with N = Improved model with N = 8

Fig. 1. Theoretical autocorrelations of the squared envelopes of Clarkesmodel 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

()}

or

Var

{RY

Y(

)}

Normalized time: fd

SimulationTheory

Clarkes model

Improved model

Fig. 2. Variances of autocorrelations of the complex envelope of Clarkesmodel and our improved model.

value (N = ) for the improved simulator than for Clarkesmodel. However, the variances of the empirical correlationsof the improved model are smaller than the empirical corre-lation variances of Clarkes model. Using Theorems 2 and 4,Fig. 2 shows, as an example, some theoretical results and thecorresponding simulation results for the correlation variancesof Clarkes model and the improved model. Obviously, thevariances of the autocorrelation of the complex envelope ofour improved model are smaller than those of Clarkes model.This implies that the improved simulator converges faster thanClarkes model (and Hoehers 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 0 + 0)

]/

1 + K

(11b)Zs(t) =

[Ys(t) +

K sin(wdt cos 0 + 0)

]/

1 + K

(11c)where K is the ratio of the specular power to scattered power,0 and 0 are the angle of arrival and the initial phase,respectively, of the specular component, and 0 is a randomvariable uniformly distributed over [, ).

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[, ), 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() = RZsZs()= [J0(wd) + K cos(wd cos 0)] /(2 + 2K)

(12a)RZcZs() = RZsZc()

= K sin(wd cos 0)/(2 + 2K) (12b)RZZ() = [J0(wd) + K cos(wd cos 0)

+jK sin(wd cos 0)] /(1 + K) (12c)R|Z|2|Z|2() =

{1+J20 (wd)+K

2fc(wd,N)fs(wd,N)+2K [1+J0(wd) cos(wd cos 0)]} /(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()} = Var{RZsZs()}=[1+J0(2wd)

8N fc(wd,N)

4

]/(1+K)2

(13a)Var{RZcZs()} = Var{RZsZc()}

=[1J0(2wd)

8N fs(wd,N)

4

]/(1+K)2

(13b)Var{RZZ()} = [1/Nfc(wd,N)fs(wd,N)](1+K)2 (13c)


where fc(wd,N) and fs(wd,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 PDFs of the fading envelope |Z(t)|and phase (t) = arctan [Zc(t), Zs(t)]1.

Theorem 6: When N approaches infinity, the envelope|Z(t)| is Rician distributed and the phase (t) is uniformlydistributed over [, ), and their PDFs are given by

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

I0[2z

K(1 + K)], z 0 (14a)

f() =12

, [, ) (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() = 0 andRYsYc() = 0, Yc(t) and Ys(t) are uncorrelated and asymptot-ically independent. Let mc(t) =

K

1+K cos(wdt cos 0 + 0)

and ms(t) =

K1+K sin(wdt cos 0 + 0). Then, [Zc(t)

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

Given an initial phase 0 of the specular component, theconditional joint PDF of Zc(t) and Zs(t) can be derived asfollows

fZc,Zs

(zc, zs|0)

=1+K

exp

{(1+K) [zcmc]2(1+K) [zsms]2

}=

1 + K

exp{(1 + K)(z2c + z2s)K

+2(1 + K)[zcmc + zsms]} .Since the initial phase 0 is uniformly distributed over

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

fZc,Zs

(zc, zs) =

fZc,Zs

(zc, zs|0) 12 d0

=1 + K

exp

[(1 + K)(z2c + z2s)K]

exp {2(1+K)[zcmc + zsms]} d02=

1 + K

exp[(1 + K)(z2c + z2s)K]

I0[2

K(1 + K)(z2c + z2s)]

where the last step uses the identity exp [a cos(t + x) + b sin(t + x)] dx = 2I0(

a2 + b2)

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

coordinates (z, ) with zc = z cos and zs = z sin, weobtain the transformations Jacobian J = z; therefore, the joint

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

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

f|Z|,(z, ) =(1 + K)z

exp [K (1 + K)z2]

I0[2z

K(1 + K)], z 0, [, ).

Then, the marginal PDFs of the envelope and the phasecan be obtained by the following two integrations

f|Z|(z) =

f|Z|,(z, )d

= 2(1 + K)z exp [K (1 + K)z2]I0[2z

K(1 + K)], z 0

f() = 0

f|Z|,(z, )dz =12

, [, )

where the last equality utilizes the identity0

x exp(ax2)I0(bx)dx = 12a 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 PDFs 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 [, ).Second, the fading envelope and phase of our new Ricianmodel are independent. As usual, the PDFs 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


simulator output are given by

L|Z| =

2(1 + K)

fd exp

[K (1 + K)2] 0

[1 +

2

K

1 + Kcos2 0 cos

]

exp[2

K(1 + K) cos 2K cos2 0 sin2 ]d

(15a)

T|Z| =1Q

[2K,

2(1 + K)2

]L|Z|

(15b)

where 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

(2)3Bb0exp

(r

2 + s2

2b0

)

exp[rs cos

b0 (b0r + b1s sin)

2

2Bb0

]d

where, for our model defined in Definition 2, s, B = b0b2b21,b0, b1 and b2 are given by

s =

K

1 + K, b0 =

12(1 + K)

b1 = 2b0

(fd cos fd cos 0) d2 = 2b0fd cos 0

b2 = (2)2b0

(fd cos fd cos 0)2 d2= 22b0f2d

(1 + 2 cos2 0

)B = 22b20f

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 0 = 2 or 0 = 2 , which meansthat the specular component has zero Doppler frequency, thenthe LCR given by (15a) has a closed-form solution as follows

L|Z| =

2(1 + K)fd exp[K (1 + K)2]

I0[2

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 | =

2fde2 (17a)

T|Y | =e

2 1fd

2

. (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

Nn=1

exp[jwd,kt cos

(2n + n,k

N

)]

exp (jn,k)+

Kk1+Kk

exp [j (wd,kt cos 0,k+0,k)]

(18)where wd,k, Kk and 0,k are, respectively, the maximumradian Doppler frequency, the Rice factor and the specularcomponents angle of arrival of the kth Rician fading samplesequence, and where n,k, n,k and 0,k are mutually inde-pendent and uniformly distributed over [, ) 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 ofn,k, n,k, 0,k, n,l, n,l and 0,l when k = l.

IV. EMPIRICAL TESTINGVerification of the proposed fading simulator is carried out

by 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 StatisticsWe 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 PDFsFigs. 6 and 7 show that the PDFs of the fading envelope and

phase of the simulator with N = 8 are in very good agreementwith the theoretical ones. It is noted that when N > 8, thesePDFs 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.


0 2 4 6 8 101

0.5

0

0.5

1

Real part of RZZ(), N = 8

Re[

RZZ

()]

Normalized time: fd

SimulationTheory

K = 1 K = 3

K = 0 (Rayleigh)

Fig. 3. The real part of the autocorrelation of the complex envelope Z(t);0 = /4 for K = 1 and K = 3 Rician cases.

0 2 4 6 8 101

0.5

0

0.5

1

Imaginary part of RZZ(), N = 8

Im[R

ZZ(

)]

Normalized time: fd

SimulationTheoryK = 1 K = 3

K = 0 (Rayleigh)

Fig. 4. The imaginary part of the autocorrelation of the complex envelopeZ(t); 0 = /4 for K = 1 and K = 3 Rician cases.

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

Normalized time: fd

SimulationTheory

K = 3

K = 1

K = 0 (Rayleigh)

Fig. 5. The autocorrelation of the squared envelope |Z(t)|2 with 0 = /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 0 = /2, theDoppler frequency of the specular component is zero, for eachsingle trial and the AFD becomes larger when the value of Kis increased.

V. CONCLUSIONIn this paper, it was shown that Clarkes model with a

finite 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 Clarkes 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


1 0.5 0 0.5 10.14

0.15

0.16

0.17

0.18

f (

)

, ( )

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 (t).

25 20 15 10 5 0 5 10103

102

101

100

Nor

mal

ized

leve

l cro

ssin

g ra

te

Normalized fading envelope level (dB)

LCR of the fading, 0 = /4


K = 0

K = 1

K = 3

K = 5 K = 10

Fig. 8. The normalized LCR of the fading envelope |Z(t)|, where 0 = /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 [, ).

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 PDFs 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

101

100

101AFD of the fading, 0 = 0

Nor

mal

ized

ave

rage

fade

dur

atio

n

Normalized fading envelope level (dB)


K = 0

K = 1

K = 3

K = 5

K = 10

Fig. 9. The normalized AFD of the fading envelope |Z(t)|, where 0 = 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() = E, [Yc(t)Yc(t + )]

=1N

Nn=1

Ni=1

E {cos(wdt cosn + n)

cos[wd(t + ) cosi + i]}

=1

2N

{N

n=1

E[cos(wd cosn)]

}

=1

2N

{N

n=1

cos[wd cos

(2n + n

N

)]dn2

}

=1

2N

{N

n=1

2n+N

2nN

cos (wd cos n)N

2dn

}

=14

2+ NN

cos (wd cos ) d

=14

20

cos(wd cos )d

=12J0(wd).

Similarly, we can obtain the autocorrelation of the imagi-


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

RYsYs() = E [Ys(t)Ys(t + )]

=14

20

cos(wd cos )d

=12J0(wd).

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

RYcYs() = E [Yc(t)Ys(t + )]

=1N

Nn=1

Ni=1


sin[wd(t + ) cosi + i]}

=1

2N

Nn=1

E [sin(wd cosn)]

=14

20

sin(wd cos )d = 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() = E[Y 2c (t)Y

2c (t + )

]+ E

[Y 2s (t)Y

2s (t + )

]+E

[Y 2c (t)Y

2s (t + )

]+ E

[Y 2s (t)Y

2c (t + )

].

(19)

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

E[Y 2c (t)Y

2c (t + )

]

=1

N2 E

{N

n=1

cos(wdt cosn + n)

N

i=1

cos(wdt cosi + i)

N

p=1

cos[wd(t + ) cosp + p]

N

q=1

cos[wd(t + ) cosq + q]

}. (20)

Since the random phases k and l 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 2c (t)Y 2c (t + )] is derived for eachof the four cases.

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

E[Y 2c (t)Y

2c (t + )

]1st

=1

N2

Nn=1

E{cos2(wdt cosn + n)

cos2[wd(t + ) cosn + n]}

=1

N2

{N

n=1

E

[1 + cos(2wdt cosn + 2n)

2

1 + cos[2wd(t + ) cosn + 2n]2

]}

=1

N2

{N

4+

18

Nn=1

E[cos(2wd cosn)]

}

=1

4N+

18N

J0(2wd).

For the second case, n = i, p = q, and n = p, we haveE[Y 2c (t)Y 2c (t + )

]2nd

=1

N2

Nn=1

Np=1

p=n

E[cos2(wdt cosn + n)

]

E (cos2[wd(t + ) cosp + p])}=

1N2

[N2 N

4

]=

14 1

4N.

For the third case, n = p, i = q, and n = i, we haveE[Y 2c (t)Y 2c (t + )

]3rd

=1

N2

Nn=1

Ni=1i=n


cos[wd(t + ) cosn + n]}E {cos(wdt cosi + i) cos[wd(t + ) cosi + i]}

=1

N2

{N

n=1

12E[cos(wd cosn)]

}2

1N2

Nn=1

{12E[cos(wd cosn)]

}2

=14J20 (wd)

fc (wd,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 2c (t)Y

2c (t + )

]4th

=14J20 (wd)

fc (wd,N)4

.

Since these four cases are the exclusive and exhaustivepossibilities for E

[Y 2c (t)Y

2c (t + )

]being non-zero, adding

them together we haveE[Y 2c (t)Y 2c (t + )

]= E

[Y 2c (t)Y

2c (t + )

]1st

+ E[Y 2c (t)Y

2c (t + )

]2nd

+E[Y 2c (t)Y

2c (t + )

]3rd

+ E[Y 2c (t)Y

2c (t + )

]4th

=14

+12J20 (wd) +

18N

J0(2wd) fc (wd,N)2 .


This completes the derivation of E[Y 2c (t)Y

2c (t + )

].

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

E[Y 2s (t)Y

2s (t + )

]=

14

+12J20 (wd) +

18N

J0(2wd)

fs (wd,N)2

E[Y 2c (t)Y

2s (t + )

]=

14 1

8NJ0(2wd) fc (wd,N)2

E[Y 2s (t)Y

2c (t + )

]=

14 1

8NJ0(2wd) fs (wd,N)2 .

Therefore,

R|Y |2|Y |2() = 1 + J20 (wd) fc (wd,N) fs (wd,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 [, ),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 hasR|Z|2|Z|2() = E

[Z2c (t)Z

2c (t + )

]+ E

[Z2s (t)Z

2s (t + )

]+E

[Z2c (t)Z

2s (t + )

]+ E

[Z2s (t)Z

2c (t + )

].

Then,E[Z2c (t)Z

2c (t + )

]=

1(1 + K)2

E

{[Yc(t) +

K cos (wdt cos 0 + 0)

]2(Yc(t + ) +

K cos [wd(t + ) cos 0 + 0]

)2}

=E[Y 2c (t)Y

2c (t + )

](1 + K)2

+K E [Y 2c (t)] E {cos2 [wd(t + ) cos 0 + 0]}

(1 + K)2

+K E [Y 2c (t + )] E [cos2 (wdt cos 0 + 0)]

(1 + K)2

+4K E [Yc(t)Yc(t + )]

(1 + K)2E {cos (wdt cos 0+0)

cos [wd(t + ) cos 0 + 0]}+

K2

(1 + K)2E {cos2 (wdt cos 0 + 0) cos2 [wd(t + ) cos 0 + 0]

}

=E[Y 2c (t)Y 2c (t + )

](1 + K)2

+K

2(1 + K)2[1 + 2J0(wd) cos(wd cos 0)]

+K2

4(1 + K)2

[1 +

cos(2wd cos 0)2

].

Similarly, we have

E[Z2s (t)Z

2s (t + )

]=

1(1 + K)2

E

{[Ys(t) +

K sin (wdt cos 0 + 0)

]2(Ys(t + ) +

K sin [wd(t + ) cos 0 + 0]

)2}

=E[Y 2s (t)Y 2s (t + )

](1 + K)2

+K

2(1 + K)2[1 + 2J0(wd) cos(wd cos 0)]

+K2

4(1 + K)2

[1 +

cos(2wd cos 0)2

]

andE[Z2c (t)Z

2s (t + )

]=

1(1 + K)2

E

{[Yc(t) +

K cos (wdt cos 0 + 0)

]2(Ys(t + ) +

K sin [wd(t + ) cos 0 + 0]

)2}

=E[Y 2c (t)Y

2s (t + )

](1 + K)2

+K E [Y 2c (t)] E {sin2 [wd(t + ) cos 0 + 0]}

(1 + K)2

+K E [Y 2s (t + )] E [cos2 (wdt cos 0 + 0)]

(1 + K)2

+K2

(1 + K)2 E {cos2 (wdt cos 0 + 0)

sin2 [wd(t + ) cos 0 + 0]}

=E[Y 2c (t)Y

2s (t + )

](1 + K)2

+K

2(1 + K)2

+K2

4(1 + K)2

[1 cos(2wd cos 0)

2

]

andE[Z2s (t)Z2c (t + )

]=

1(1 + K)2

E

{[Ys(t) +

K sin (wdt cos 0 + 0)

]2(Yc(t + ) +

K cos [wd(t + ) cos 0 + 0]

)2}

=E[Y 2c (t)Y

2s (t + )

](1 + K)2

+K

2(1 + K)2

+K2

4(1 + K)2

[1 cos(2wd cos 0)

2

].

Therefore,R|Z|2|Z|2()

=R|Y |2|Y |2()+K2+2K [1+J0(wd) cos(wd cos 0)]

(1 + K)2

=1+J20 (wd)+K

2+2K [1+J0(wd) cos(wd cos 0)](1 + K)2

fc(wd,N) + fs(wd,N)(1 + K)2

This completes the proof.


REFERENCES

[1] R. H. Clarke, A statistical theory of mobile-radio reception, Bell Syst.Tech. J., pp. 957-1000, Jul.-Aug. 1968.

[2] M. J. Gans, A power-spectral theory of propagation in the mobile-radioenvironment, IEEE Trans. Veh. Technol., vol. 21, pp. 27-38, Feb. 1972.

[3] W. C. Jakes, Microwave Mobile Communications. Wiley, 1974; re-issuedby IEEE Press, 1994.

[4] T. Aulin, A modified model for the fading signal at a mobile radiochannel, IEEE Trans. Veh. Technol., vol. 28, pp. 182-203, Aug. 1979.

[5] A. A. M. Saleh and R. A. Valenzuela, A statistical model for indoormultipath propagation, IEEE J. Select. Areas Commun., vol. 5, pp. 128-137, Feb. 1987.

[6] W. R. Braun and U. Dersch, A physical mobile radio channel model,IEEE Trans. Veh. Technol., vol. 40, pp. 472-482, May 1991.

[7] P. Hoeher, A statistical discrete-time model for the WSSUS multipathchannel, IEEE Trans. Veh. Technol., vol. 41, pp. 461-468, Nov. 1992.

[8] S. A. Fechtel, A novel approach to modeling and efficient simulationof frequency-selective fading radio channels, IEEE J. Select. AreasCommun., vol. 11, pp. 422-431, Apr. 1993.

[9] P. Dent, G. E. Bottomley, and T. Croft, Jakes fading model revisited,IEEE Electron. Lett., vol. 29, pp. 1162-1163, June 1993.

[10] H. Hashemi, The indoor radio propagation channel, Proc. IEEE, vol.81, pp. 943-968, July 1993.

[11] U. Dersch and R. J. Ruegg, Simulations of the time and frequencyselective outdoor mobile radio channel, IEEE Trans. Veh. Technol., vol.42, pp. 338-344, Aug. 1993.

[12] P. M. Crespo and J. Jimenez, Computer simulation of radio channelsusing a harmonic decomposition technique, IEEE Trans. Veh. Technol.,vol. 44, pp. 414-419, Aug. 1995.

[13] K.-W. Yip and T.-S. Ng, Discrete-time model for digital communica-tions over a frequency-selective Rician fading WSSUS channel, IEEProc. Commun., vol. 143, pp. 37-42, Feb. 1996.

[14] K.-W. Yip and T.-S. Ng, Karhunen-Loeve expansion of the WSSUSchannel output and its application to efficient simulation, IEEE J.Select. Areas Commun., vol. 15, pp. 640-646, May 1997.

[15] A. Anastasopoulos and K. M. Chugg, An efficient method for simu-lation of frequency selective isotropic Rayleigh fading, in Proc. IEEEVTC, vol. 3, pp. 2084-2088, May 1997.

[16] M. Patzold, U. Killat, F. Laue, and Y. Li, On the statistical propertiesof deterministic simulation models for mobile fading channels, IEEETrans. Veh. Technol., vol. 47, pp. 254-269, Feb. 1998.

[17] Y. X. Li and X. Huang, The generation of independent Rayleighfaders, in Proc. IEEE ICC, June 2000, vol. 1, pp. 41-45; also IEEETrans. Commun., vol. 50, pp. 1503-1514, Sep. 2002.

[18] K.-W. Yip and T.-S. Ng, A simulation model for Nakagami-m fadingchannels, m < 1, IEEE Trans. Commun., vol. 48, pp. 214-221, Feb.2000.

[19] M. F. Pop and N. C. Beaulieu, Limitations of sum-of-sinusoids fadingchannel simulators, IEEE Trans. Commun., vol. 49, pp. 699-708,Apr. 2001.

[20] E. Chiavaccini and G. M. Vitetta, GQR models for multipath Rayleighfading channels, IEEE J. Select. Areas Commun., vol. 19, pp. 1009-1018, June 2001.

[21] M. F. Pop and N. C. Beaulieu, Design of wide-sense stationary sum-of-sinusoids fading channel simulators, in Proc. IEEE ICC, Apr. 2002,pp. 709-716.

[22] C. Xiao and Y. R. Zheng, A generalized simulation model for Rayleighfading channels with accurate second-order statistics, in Proc. IEEEVTC-Spring, May 2002, pp. 170-174.

[23] C. Xiao, Y. R. Zheng, and N. C. Beaulieu, Second-order statisticalproperties of the WSS Jakes fading channel simulator, IEEE Trans.Commun., vol. 50, pp. 888-891, June 2002.

[24] Y. R. Zheng and C. Xiao, Improved models for the generation ofmultiple uncorrelated Rayleigh fading waveforms, IEEE Commun.Lett., vol. 6, pp. 256-258, June 2002.

[25] Y. R. Zheng and C. Xiao, Simulation models with correct statisticalproperties for Rayleigh fading channels, IEEE Trans. Commun., vol.51, pp. 920-928, June 2003.

[26] F. Vatta, G. Montorsi, and F. Babich, Achievable performance ofturbo codes over the correlated Rician channel, IEEE Trans. Commun.,vol. 51, pp. 1-4, Jan. 2003.

[27] S. Gazor and H. S. Rad, Space-time coding ambiguities in joint adap-tive channel estimation and detection, IEEE Trans. Signal Processing,vol. 52, pp. 372-384, Feb. 2004.

[28] F. Babich, On the performance of efficient coding techniques overfading channels, IEEE Trans. Wireless Commun., vol. 3, pp. 290-299,Jan. 2004.

[29] J. Liu, Y. Yuan, L. Xu, R. Wu, Y. Dai, Y. Li, L. Zhang, M. Shi,and Y. Du, Research on smart antenna technology for terminals forthe TD-SCDMA system, IEEE Commun. Mag., vol. 41, pp. 116-119,June 2003.

[30] G. L. Stuber, Principles of Mobile Communication, 2nd ed. Norwell,MA: Kluwer Academic Publishers, 2001.

[31] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, andProducts, 6th ed. A. Jeffrey, ed. (San Diego: Academic Press, 2000).

[32] Qinfang Sun, Huanchun Ye, and Won-Joon Choi, Private Communica-tions, Feb. 9, 2004.

[33] C. S. Patel, G. L. Stuber, and T. G. Pratt, Comparative analysis ofstatistical models for the simulation of Rayleigh faded cellular channels,IEEE Trans. Commun., vol. 53, pp. 1017-1026, June 2005.

[34] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw Hill,2001.

[35] M. Slack, The probability distributions of sinusoidal oscillations com-bined in random phase, IEE Proceedings, vol. 93, pp. 76-86, 1946.

[36] W. R. Bennett, Distribution of the sum of randomly phased compo-nents, Quart. Appl. Math., vol. 5, pp. 385-393, Jan. 1948.

[37] K. Ohtani, K. Daikoku, and H. Omori, Burst error performanceencountered in digital land mobile radio channel, IEEE Trans. Veh.Technol., vol. 30, no. 4, pp. 156-160, Nov. 1981.

[38] J. M. Morris, Burst error statistics of simulated Viterbi decoded BPSKon fading and scintillating channels, IEEE Trans. Commun., vol. 40,pp. 34-41, 1992.

[39] M. K. Simon and M.-S. Alouini, Digital Communication over FadingChannels, 2nd ed. New York: John Wiley & Sons, 2005.

[40] S. O. Rice, Statistical properties of a sine wave plus random noise,Bell Syst. Tech. J., vol. 27, pp. 109-157, Jan. 1948.

Chengshan Xiao (M99-SM02) 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 intoNortels 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.


Yahong Rosa Zheng (S99-M03) 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 (S82-M86-SM89-F99) 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 Queens National Scholar Assistant Pro-fessor with the Department of Electrical Engineer-ing, Queens 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.

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure true /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /LeaveUntagged /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice

Novel Sum-of-Sinusoids Simulation Models for Rayleigh and Rician Fading Channels

Documents

Transcript of Novel Sum-of-Sinusoids Simulation Models for Rayleigh and Rician Fading Channels