精确统计模型：模拟瑞利衰落信道的新方法

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本文主要探讨了"Simulation models with correct statistical properties for Rayleigh fading channels"这一主题，作者 Yahong Rosa Zheng 和 Chengshan Xiao 提出了新的瑞利信道统计仿真模型。瑞利信道是无线通信中常见的随机衰落模型，其特点是频率选择性衰落，信号在不同频率成分上的衰落是独立且服从瑞利分布的。新提出的仿真模型的核心在于使用随机的路径损耗、初始相位以及条件随机的多普勒频移来模拟每个正弦波分量。这种设计的关键优势在于即使使用较少数量（如个位数）的正弦波，也能精确地匹配实际信道的自相关和交叉相关特性。这意味着即使在低复杂度的情况下，模型也能提供高度逼真的信道特性，这对于快速而准确的系统仿真至关重要。文章深入研究了这些模型的统计特性，包括幅度（envelope）和相位的概率密度函数，它们在正弦波数量趋近于无限大时能够达到理想状态。同时，即使只有8个正弦波，模型也能表现出良好的收敛性，这对于计算诸如级联衰落的统计量（如幅度的平方的自相关，这包含了信道的四阶统计特性）非常有用。此外，论文还考察了模型在频率选择性衰落、多输入多输出（MIMO）通道以及多样性合并场景中的应用，证明了其通用性和有效性。通过数值结果的评估，新模型与真实瑞利信道的统计特性有良好的一致性，这表明该模型不仅理论上有严谨性，而且在实际应用中具有很高的实用价值。因此，对于无线通信系统的设计、分析和测试来说，这些新的统计仿真模型提供了一个强大的工具，可以节省大量的实验资源并提高研究的精度。

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

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