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 propod for Rayleigh fading channels.The new models employ random path gain,random initial pha,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 a single-digit integer.More-over,the probability density functions of the envelope and pha, 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 ud to gen-erate multiple uncorrelated fading waveforms for frequency lec-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,cond-order sta-tistics.
I.I NTRODUCTION
M OBILE radio channel simulators are commonly ud in the laboratory becau they allow system tests and evaluations which are less expensive and more reproducible than field trials.Many different approaches have been ud 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 ud 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-lective 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 April1,2002;revid November7,2002.This work was supported in part by the University of Mis-souri(UM)-Columbia Rearch Council under Grant URC-0
2-050and in part by the UM System Rearch Board under Grant URB-02-124.This paper was prented in part at the IEEE Vehicular Technology Conference,Birmingham, AL,May6–9,2002.
The authors are with the Department of Electrical and Computer Engineering, University of Missouri,Columbia,MO65211USA(e-mail:yzheng@ee.mis-souri.edu;xiaoc@missouri.edu).
Digital Object Identifier10.1109/TCOMM.2003.813259Despite the extensive acceptance and application of Jakes’simulator,some important limitations of the simulator were determined and discusd in detail recently[22].It was shown in[22]that Jakes’simulator is wide-n nonstationary when averaged across the physical enmble of fading channels.Pop and Beaulieu[22]propod an improved simulator by intro-ducing random pha 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 cond-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 versio
ns of it,have similar problems with the cond-order statistics.
In this paper,new sum-of-sinusoids statistical simulation models are propod 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 pha,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 the properties,except for the fading pha’s PDF,is rapid and clo approximation is achieved even when the number of sinusoids is as small an integer as eight,and the number of random trials is only50.Additionally,and importantly,the new simulator can be directly ud to generate multiple uncorrelated fading waveforms,which are needed to simulate some realistic frequency-lective 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 t
he reference model and an improved Jakes simulator. Section III propos a new sum-of-sinusoids simulation model for Rayleigh fading channels,and the statistical properties of
0090-6778/03$17.00©2003IEEE
this new model are analyzed in detail.Other models with iden-tical or similar statistical properties are also briefly discusd in this ction.Section IV prents the performance evaluation of one of the new simulators by extensive numerical results.Sec-tion V draws some conclusions.
II.R EFERENCE M ODEL AND J AKES’S IMULATOR F AMILY A.Mathematical Reference Model
Consider a frequency-nonlective fading channel comprid
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of
th propagation path,
and
.
Assuming
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that
.
Assuming
that
(3a)
(3b)
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(3c)
(3d)
(3f)
where
and
(5b)
(6b)
(6c)
where
(7c)
,
for
all
eliminates the stationarity problem occurring in Jakes’original design.However,some problems with higher-order statistics remain.
Some cond-order statistics of the improved Jakes model are given by
[24]
(9b)
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(9c)
are the autocorrelations of the complex envelope
and the squared envelope,respectively.Although the autocorre-
lation of the complex
envelope approaches the desired
autocorrelation
when
(10)
where
(11a)
(11b)
,
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and
can be further simplified to
be
(13)
where,
and is in-
cluded to make the total power remain unchanged.Bad on
,we can define a new simulation model as follows.
Definition:The normalized low-pass fading process of a new
statistical sum-of-sinusoids simulation model is defined
by
(15)
where,
and
in the following theorem.
Theorem 1:The autocorrelation and cross-correlation func-tions of the quadrature components,and the autocorrelation functions of the complex envelope and the squared envelope of the fading
signal
(16a)
(16b)
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,and they match the desired
cond-order statistics exactly,irrespective of the value
of
as the number of
sinusoids
.
Theorem 2:
When
is uniformly dis-tributed
over
approaches
infinity,
is Rayleigh distributed,and the
pha is uniformly
间接测量distributed
over
,one can obtain (18).
Two other important statistical properties associated with fading envelope are the level crossing rate (LCR)and the average fade duration (AFD).The LCR is defined as the rate at which the envelope cross a specified level in the positive slope,and the AFD is the average time duration that the fading envelope remains below a specified level.Both the LCR and
AFD provide important information about the statistics of burst errors [28],[29],which facilitates the design and lection of error-correction technique.It is shown in the following theorem that the LCR and AFD of the new simulator asymptotically match tho of Clarke’s mathematical reference model [25].Theorem 3:
When
is the normalized fading envelope level given
by
approaches infinity,the fading envelope
is Rayleigh distributed,as shown in Theorem 2.Using the same procedure provided in [25],one can prove (19).Details are omitted here for brevity.
Before concluding this ction,we have three remarks on the new simulation model.
Remark 1:The initial
pha )in (15)is to randomize the radian Doppler
frequencies
.This makes the new model dif-ferent from all the existing Jakes family simulators,such as [10],[16],[18],[19],[21],[22],and [31].The random
variables
(20a)
,
and
,defined by (14).
Remark 3:The new simulation model can be directly ud to generate uncorrelated fading waveforms for frequency-lec-tive Rayleigh channels,MIMO channels,and diversity-combing
scenarios.
Let
th Rayleigh fader given
by (21)
stands for the number of statistical trials.
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