-
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
-
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() = 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
-
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(
) 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)
-
3672 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO.
12, DECEMBER 2006
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
-
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)
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.
-
3674 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO.
12, DECEMBER 2006
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
-
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 (
)
, ( )
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
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
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)
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
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-
-
3676 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO.
12, DECEMBER 2006
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
E {cos(wdt cosn + n)
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
E {cos(wdt cosn + 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 .
-
XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR
RAYLEIGH AND RICIAN FADING CHANNELS 3677
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.
-
3678 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO.
12, DECEMBER 2006
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.
-
XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR
RAYLEIGH AND RICIAN FADING CHANNELS 3679
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