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920 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 6, JUNE 2003

Simulation Models With Correct Statistical

Properties for Rayleigh Fading Channels

Yahong Rosa Zheng and Chengshan Xiao, Senior Member, IEEE

Abstract—In this paper, new sum-of-sinusoids statistical sim-

ulation models are proposed for Rayleigh fading channels. These

new models employ random path gain, random initial phase, and

conditional random Doppler frequency for all individual sinusoids.

It is shown that the autocorrelations and cross correlations of the

quadrature components, and the autocorrelation of the complex

envelope of the new simulators match the desired ones exactly, even

if the number of sinusoids is as small as asingle-digitinteger. More-

over, the probability density functions of the envelope and phase,

the level crossing rate, the average fade duration, and the auto-

correlation of the squared fading envelope which contains fourth-

order statistics of the new simulators, asymptotically approach the

correct ones as the number of sinusoids approaches infinity, while

good convergence is achieved even when the number of sinusoids is

as small as eight. The new simulators can be directly used to gen-

erate multiple uncorrelated fading waveforms for frequency selec-

tive fading channels, multiple-input multiple-output channels, and

diversity combining scenarios. Statistical properties of one of the

new simulators are evaluated by numerical results, finding good

agreements.

Index Terms—Channel models, fading channel simulator, fading

channels, high-order statistics, Rayleigh fading, second-order sta-

tistics.

I. INTRODUCTION

OBILE radio channel simulators are commonly used

in the laboratory because they allow system tests and

evaluations which are less expensive and more reproducible

than field trials. Many different approaches have been used for

the modeling and simulation of mobile radio channels [1]–[23],

[31], [32]. Among them, the well-known mathematical ref-

erence model due to Clarke [1] and its simplified simulation

model due to Jakes [6] have been widely used for Rayleigh

fading channels for about three decades. However, Jakes’ sim-

ulator is a deterministic model, and it has difficulty in creating

multiple uncorrelated fading waveforms for frequency-selective

fading channels and multiple-input multiple-output (MIMO)

channels, therefore, different modifications of Jakes’ simulator

have been reported in the literature [10], [16]–[19], [31].

Paper approved by R. A. Valenzuela, the Editor for Transmission Systems of

the IEEE Communications Society. Manuscript received April 1, 2002; revised

November 7, 2002. This work was supported in part by the University of Mis-

souri (UM)-Columbia Research Council under Grant URC-02-050 and in part

by the UM System Research Board under Grant URB-02-124. This paper was

presented in part at the IEEE Vehicular Technology Conference, Birmingham,

AL, May 6–9, 2002.

The authorsare with the Department of Electrical and Computer Engineering,

University of Missouri, Columbia, MO 65211 USA (e-mail: yzheng@ee.mis-

souri.edu; xiaoc@missouri.edu).

Digital Object Identifier 10.1109/TCOMM.2003.813259

Despite the extensive acceptance and application of Jakes’

simulator, some important limitations of the simulator were

determined and discussed in detail recently [22]. It was shown

in [22] that Jakes’ simulator is wide-sense nonstationary when

averaged across the physical ensemble of fading channels. Pop

and Beaulieu [22] proposed an improved simulator by intro-

ducing random phase shifts in the low-frequency oscillators to

remove the stationary problem. However, it was pointed out in

[22] that higher-order statistics of this improved simulator may

not match the desired ones. Consistent with Pop and Beaulieu’s

caution about higher-order statistics of the improved simulator,

it was further proved in [24] that second-order statistics of

the quadrature components and the envelope do not match the

desired ones. Moreover, even in the limit as the number of

sinusoids approaches infinity, the autocorrelations and cross

correlations of the quadrature components, and the autocorre-

lation of the squared envelope of the improved simulator, fail

to match the desired correlations. Jakes’ original simulator and

published modified versions of it, have similar problems with

these second-order statistics.

In this paper, new sum-of-sinusoids statistical simulation

models are proposed for Rayleigh fading channels. It is

shown that the autocorrelations and cross correlations of

the quadrature components, and the autocorrelation of the

complex envelope of the new simulators, match the desired

ones exactly even if the number of sinusoids is so small as

a single-digit integer. Furthermore, the autocorrelation of the

squared envelope which contains fourth-order statistics, the

probability density functions (PDFs) of the fading envelope

and the phase, the level-crossing rate, and the average fade

duration of our new simulators asymptotically approach the

desired ones as the number of sinusoids approaches infinity.

Moreover, convergence to the limiting (exact) values of these

properties, except for the fading phase’s PDF, is rapid and

close approximation is achieved even when the number of

sinusoids is as small an integer as eight, and the number of

random trials is only 50. Additionally, and importantly, the new

simulator can be directly used to generate multiple uncorrelated

fading waveforms, which are needed to simulate some realistic

frequency-selective fading channels, MIMO channels, and

diversity-combining scenarios.

The remainder of this paper is organized as follows. Section II

briefly reviews the mathematical reference model and the family

of Jakes’ simulators. Attention is given to the statistical proper-

ties of the reference model and an improved Jakes simulator.

Section III proposes a new sum-of-sinusoids simulation model

for Rayleigh fading channels, and the statistical properties of

ZHENG AND XIAO: SIMULATION MODELS WITH CORRECT STATISTICAL PROPERTIES FOR RAYLEIGH FADING CHANNELS 921

this new model are analyzed in detail. Other models with iden-

tical or similar statistical properties are also briefly discussed in

this section. Section IV presents the performance evaluation of

one of the new simulators by extensive numerical results. Sec-

tion V draws some conclusions.

II. R

EFERENCE MODEL AND JAKES’SIMULATOR FAMILY

A. Mathematical Reference Model

Consider a frequency-nonselective fading channel comprised

propagation paths; the low-pass fading process is given by

[1] and [6]

(1)

where

is a scaling constant, , , and are, respectively,

the random path gain, angle of incoming wave, and initial phase

associated with the

th propagation path, and is the max-

imum radian Doppler frequency occurring when

Assuming that

is real valued, (1) can be written as

(2a)

(2b)

(2c)

The central limit theorem justifies that

and can

be approximated as Gaussian random processes for large

Assuming that

and are mutually independent and uni-

formly distributed over

for all , and adopting Clarke’s

two-dimensional (2-D) isotropic scattering model, some desired

second-order statistics for fading simulators are manifested in

the autocorrelation and cross-correlation functions [1], [25]

(3a)

(3b)

(3c)

(3d)

(3e)

(3f)

where

denotes expectation, is the zero-order Bessel

function of the first kind [26], and without loss of generality,

we have set

and . The first-order

PDFs of the fading envelope,

, and the phase,

are given by

(4a)

(4b)

The function arctan(

) maps the arguments (

) into a phase in the

correct quadrant in

[

;

)

Clearly, the fading envelope is Rayleigh distributed ac-

cording to (4a), and the phase

is uniformly distributed.

B. Jakes’ Simulator Family

Based on Clarke’s reference model (1), and by selecting

(5a)

(5b)

(5c)

Jakes derived his well-known simulation model for Rayleigh

fading channels. The normalized low-pass fading process of this

model is given by

(6a)

(6b)

(6c)

where

, and

(7a)

(7b)

(7c)

(7d)

The simplifying relationships forced in (5) make this simula-

tionmodel deterministic[10],[16]and wide-sensenonstationary

[22]. Therefore, various modifications of Jakes’ simulator have

been proposed in [10], [16]–[19], [21], [22], [31] and the ref-

erences therein, which we call the family of Jakes’ simulators.

Among the Jakes simulator family, the recently improved model

proposed by Pop and Beaulieu in [22] is worthy of mention

due to its wide-sense stationarity.

The normalized low-pass fading process of the improved

Jakes’simulator proposed in [22] is given by

(8a)

(8b)

(8c)

where

, , , and are the same as those of Jakes’original

model given by (7), and

are independent random variables

uniformly distributed on

for all .

Clearly, the difference between this improved simulator

and Jakes’ original simulator lies in the random phases

, as the original Jakes simulator assumes that

for all . The introduction of these random phases

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