Microsoft Word UK VN TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ, ĐẠI HỌC ĐÀ NẴNG SỐ ���� 1 ĐẶC TRƯNG THỐNG KÊ CỦA CÁC MÔ HÌNH FADING RAYLEIGH TRONG THÔNG TIN VÔ TUYẾN STATISTICAL PROPERTIES OF RAYLEIGH FADING MODE[.]
Trang 1ĐẶC TRƯNG THỐNG KÊ CỦA CÁC MÔ HÌNH FADING RAYLEIGH
TRONG THÔNG TIN VÔ TUYẾN
STATISTICAL PROPERTIES OF RAYLEIGH FADING MODELS IN WIRELESS
COMMUNICATIONS Vien Nguyen-Duy-Nhat, Hung Nguyen-Le, and Chien Tang-Tan
Department of Electronics and Telecommunications Engineering, Danang University of Technology
E-mail: ndnvien@dut.edu.vn, nlhung@dut.udn.vn, ttchien@ac.udn.vn
ABSTRACT
The paper studies statistical properties of existing fading channel models in wireless communications Several statistical characteristics of various fading channel models are numerically verified with related theoretical values The comparison results can be used as guidelines in selecting suitable fading channel models for a specific wireless system design
Keywords: Rayleigh fading channels, fading simulator, sum-of-sinusoids channel simulator, correlation properties.
The past decade has witnessed explosive
growth in wireless data traffic of ubiquitous
multimedia services for both business and
residential customers worldwide To meet the
development trends, numerous advanced
transmission techniques have been proposed
for wireless communications Unlike wireline
transmissions, wireless communications lends
itself to many technical challenges in signal
processing due to the randomness of radio
channel responses As a result, understanding
the nature of wireless channels is of paramount
importance in optimizing the wireless system
performance under limited radio resources and
power constraints To characterize the complex
mathematical models have been proposed in
the literature [1-8][12] Among them, an early
fading channel model was proposed by Clarke
[1] Jakes [2] has proposed a simplified version
of Clarke’s model for generating multiple
Rayleigh fading waveforms deterministically
by superimposing multiple sinusoidal, are
called sum-of-sinusoids (SoS), with different
frequencies and initial phases The simplified model has been widely accepted for about three decades Later, Dent [3] modified Jakes’ model by using orthogonal Walsh-Hadamard
uncorrelated faded envelopes
Pop and Beaulieu [4] showed that there are some undesirable properties in the Jakes' model More specifically, the autocorrelation functions (ACF) of the in-phase differs from that of quadrature components, and the cross-correlation function (XCF) between the in-phase and quadrature components is not always zero For any pair of faders, they are not mutually independent because the XCF between them is generally not zero To alleviate the drawbacks, Li and Huang proposed a model [5] based on Jakes’ model, this model overcomes some undesirable properties in the Jakes and Dent models The in-phase and quadrature components in any single fader are independent and have the same autocorrelation functions However, the model
of Li and Huang possesses a highly
computation complexity, a novel model is proposed by Wu [6] The model’s correlation properties are as good as those of Li and
Trang 22
Huang model, yet computational complexity is
reduced by a half Zheng and Xiao [7-9]
proposed a new model and this model have
been widely used for Rayleigh fading channels
in recent years
The SoS models can be classified into
statistical and deterministic Deterministic
models have fixed phases, amplitudes, and
simulation trials In contrast, the statistical
models have at least one of above parameters
as a random variable The statistical properties
of the models will vary for each simulation
trial, but converge to the desired properties
with a large number of simulation trials An
ergodic statistical model converges to the
desired properties in only a simulation trial
The simulation model must ensure
accurately evaluation under realistic fading
conditions In Rayleigh fading simulation
components of the complex Gaussian
mean Gaussian with equal variance The
the Rayleigh distribution || = ,
the fading process at the receiver The ideal
auto-correlation functions of the in-phase or
quadrature parts and the complex envelope are
scale with the zeroth-order Bessel function of
the maximum Doppler frequency and the time
lag The ideal cross-correlation function
components is zero
In this paper, some important statistical
properties of the SoS fading simulation models
are analyzed and compared with each others
The numerical results show that the model
with correct statistical properties for Rayleigh
fading channel simulation
The rest of this article is organized as follows Section 2 reviews the Clarke’s reference SoS fading models with its desired statistical properties The SoS based Rayleigh fading simulation models are described in Section 3 The statistical properties of SoS Rayleigh models are simulated and evaluated
in Section 4 Finally, Section 5 concludes the paper
Clarke [1] showed that the complex
expressed by
= ∑ & !"#$ " ,
with /* is the
path gain, the angle of arival, and the phase
components of the complex faded envelope are Gaussian random processes for large N Thus,
distributed
The statistical properties of the Clarke’s model can be consulted in [2] and [10] for the autocorrelations and cross-correlations of the reference model are summarized by
E F,G,F,GH = IJ , + H , K = L M ) * H (2)
Trang 3E F,GH = IJ + H K = 0, (3)
E O,PH = I6 Q + H R S = T2L M ) * H, U = V
0 , U ≠ V 8,
(4)
E |O| ,|O| H = I6| Q + H| | Q | S
= 4 + 4L M )*H (5)
where E[.] is the expectation operator,
first kind
Models
From (1) and selecting = &(, Z =
[
& , and 3= 0 for 4 = 0, , 1, Jake [2]
derived deterministic simulation model for
Rayleigh fading channels The complex faded
envelope as
= + (6)
=√& ∑ a#('(] cos ) * + 3 (7)
=√& ∑ a#( b cos ) * + 3
] = T2cde , 4 = 1,2, , g
√2cde , 4 = g + 1 8 (9)
b = T2dh4e√2dh4e , 4 = 1,2, , g
, 4 = g + 1 8 (10)
e = i
[
& , 4 = 1,2, , g
[
j , 4 = g + 1 8 (11) ) = k) * cd2[& , 4 = 1,2, , g
) * , 4 = g + 1 8 (12)
cross-correlation functions of the quadrature
components, and the autocorrelation functions
of the envelope and the squared envelope of
fading signal are given by [11]
E FFH =&jl∑ m"
cos ) H
a#(
E GGH =j&l∑ o"
cos) H
a#(
E F,GG,FH =&jl∑ m" o"
cos ) H
a#(
E H =&j62 ∑ a#( cos) H + cos ) * H
(16)
E || || H = 4 + 2E FFH + 2E GGH +4E FGH +&pL M 2) * H +(q&(& (17)
Pop and Beaulieu improved Jakes’ fading channel simulator to eliminate the stationary problem occurring in Jakes’ original design Pop and Beaulieu modified Jakes’
for all n = 1, 2, , N
When approaching infinity, the ACF and
components, the envelope and the squared envelope of fading signal are given by [11]
E FFH = L ) s H + L j ) * H (18)
E GGH = L ) s H − L j ) * H (19)
E F,GG,FH = [t sin4Z cos )M[/ * H cos θ wZ
(20)
E H = 2L M ) * H (21)
E || || H = 4LM)*H + 4L j )*H +4 l [t sin4Z cos )M[/ * H cosZ wZ n (22)
shortcoming of the Jake’s model, Dent [3]
can be generated by using
Trang 44
Q = & y∑&y z Q 46cdecd) * cd2 +
'(
Z 8 8+dh4e cd) * cd2 + Z S (23)
where U = 1,2, , 1M; 2= [[& ; e=[&
y;
codeword to decor relate the multiple faded
envelopes in Jake’s model
properties of Dent’s model are given by [5]
E FFH =&(
y ∑&y cos e cos) cos2 H
E GGH =&(y∑&y sin e cos) cos2 H
E F,GG,FH = &(
y ∑&y sin2e cos) cos2 H
'(
(26)
E OPH =
(
&y∑|y z Q 4zR4cd)cd2 H
The independence between different
faded envelopes in the Dent’s model is still not
so good, so Li and Huang [5] have proposed a
novel model as the following
Q = ~11
M 6cd) a cd2 Q + Z Q 8
&y(
'M
8+dh4) a dh42 Q + ZQ S (28)
random phases uniformly distributed on the
maximum angular Doppler frequency The
angles of arrivals are 2Q = [& + Q[& for
4 = 0, , g, where 2MM is an initial angle of
arrival, chosen to be 0 < 2MM < [& and
The autocorrelation function of the quadrature component of the faded envelope, the cross-correlation function between the in-phase and the quadrature components and the cross-correlation function between the in-phase and the quadrature components are then derived as [5]
E FFH = 2 ∑&y ( cos) cos2 Q H
E GGH = 2 ∑&y ( cos) sin2 Q H
E F,GG,FH = 0 (31)
E PPH = 2 ∑&y ( cd)cd2 Q H +
'M
cd) dh42 Q H (32)
Zheng and Xiao proposed several novel statistical models [7-9] by allowing all three parameter sets (amplitudes, phases, and Doppler frequencies) to be random variables
Q =
a ∑ a'(cd Q cd) a cd2 Q + 3 Q 8 8+ ∑ a dh4 Q cd) a cd2 Q + 3 Q
variables uniformly distributed on the interval
608, 82, for all n and k
cross-correlation functions of the in-phase and quadrature components, the envelope and the squared envelope of faded envelopes are given
E F,G,F,GH = IJ , + H , K = L M ) * H(34)
E F,GH = IJ + H K = 0, (35)
E O , P H = I6Q + HRS = T2L M ) * H, U = V
0 , U ≠ V 8, (36)
E |O| ,| O | H = I6| Q + H| | Q | S
Trang 5= 4 + 4L M ) * H (37)
Zajic and Stuber proposed an new
Q =√& ∑ a 2 cose Q cos) * cd2 Q +
'(
3 Q , (38)
Q =√& ∑ a 2 sine Q sin)*cd2 Q +
'(
3 Q (39)
It is assumed that P independent
each having g =&j sinusoidal terms in the I
The angles of arrivals are chosen as follows:
2Q= [& + [Q
& +[& , for 4 = 1, , g; U =
0, , − 1
auto-correlation function of the in-phase and
function of the squared envelope is presented
by [12]
E F,G,F,GH = L M ) * H (40)
E F,GH = 0, (41)
E O,PH = T2L M ) * H, U = V
0 , U ≠ V 8, (42)
E |O| ,|O| H = 4 + 4LM) * H, g → ∞ (43)
The performance evaluation of the SoS
fading simulation models was carried out by
comparing the statistical properties with each
implemented to generate SoS complex faded
for all the simulation results are based on 10,
50, and 100 random trials as indicated in the figures
and Phase Fig 1 show the PDFs of the generated SoS Rayleigh fading models which are plotted and compared with PDF of Rayleigh distribution (with variance=1) when the number of random trials is 10 We can see that the models of Dent [3], Li [5] and Zheng [7] are in good agreement with the PDF of theoretical values of Rayleigh distribution while the others do not
Figure 1 The PDFs of faded waveforms
Figure 2 Envelope PDF of the waveform generated by the Zheng model [7] for various numbers of trial simulation
Moreover, as observed from Fig 2, an increase in the number of trials results in a
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
c(t)
PDFs of the faded envelope
Theoretic Jake Dent Li Pop Zheng Zajic
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
c(t)
PDFs of the faded envelope
Theoretic N=10 N=50 N=100
Trang 66
better agreement with the theoretical ones
autocorrelations of the quadrature components,
the cross correlations of the quadrature
components, and the autocorrelations of the
complex envelope and squared envelope of the
simulator output are shown in Figs 3–6,
respectively The second-order statistics of the
mathematical ideal model, which are analyzed
above, are also included in the figures for
comparison purposes Figs 1-3 show the
autocorrelations of the complex envelope, the
cross-correlation function between the
in-phase and the quadrature components, and the
autocorrelations of squared envelope for these
models with the number of trials is 10
Figure 3 The auto-correlation of faded waveforms
From Fig 3, we can observe that the
auto-correlation of the complex envelope of
Pop’s [4] and Zheng’s [7] models are identical
with the theoretical ones of the reference
model In contrast, the models of Dent and Li
are different from the theoretical ones
Other comparisons are summarized in
Fig 4, where we plot the cross-correlation of
the real and imaginary parts of the faded
waveforms The models of Zheng [7] and Zajic
[12] agree very well with the theoretical
autocorrelation given by (3)
Figure 4 The cross-correlation function between the in-phase and the quadrature components
Figure 5 The auto-correlation of I/Q-components
In Fig 5 we compare the ACFs of the in-phase component for the reference model with the other simulation models The models
of Zheng [7] and Zajic [12] have the best uncorrelation property
Compared with the others models, the Zheng’s [7] and Zajic’s [12] models provide similar approximations to the theoretical ACFs
of the squared envelope as in Fig.6, where the squared envelope correlation is plotted for different time values
-1
-0.5
0
0.5
1
1.5
2
t(Second)
Theoretic Jake Dent Li Pop Zheng Zajic
-0.4 -0.2 0 0.2 0.4 0.6
t(Second)
Theoretic Jake Dent Li Pop Zheng Zajic
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
t(Second)
Theoretic Jake Dent Li Pop Zheng Zajic
Trang 7Figure 6 The auto-correlation of faded squared
waveforms
The accuracy of the SoS fading
simulation models can be measured by the
mean-square-error (MSE) Figs 7, 8 and Table
I summarize MSEs of PDF, ACF of the
quardrature components, the squared envelope,
and the XCFs of the intra and inter waveforms
The Fig.7 and Table 1 show numerical results
when the number of simulation trials is 10 The
MSE of statistical properties using 100 trials
are presented in the Fig 8
Figure 7 The MSE of statistical properties with
M=8, N=10
Figure 8 The MSE of statistical properties with M=8, N=100
We can observe that the Zheng’s [7] and Zajic’s [12] models have the best statistical properties But when increase the number of trials, the Zajic’s [12] model achieves a larger de-correlation than the Zheng’s [7] and the other models
Table1 Mean-Square-Error of Correlation
Functions SoS model
of XCF Complex
Envelope
In-phase component
Quadrature component
Li and Huang 0.0135 0.0191 0.0217 0.0154 Pop and
Beaulieu 0.0091 0.0185 0.0237 0.0126 Zheng and
Zajic and Stuber 0.0022 0.0029 0.0049 0.0030
The paper presented an analysis of various SoS fading simulation models in terms
of statistical properties Based on the numerical results, we can conclude that the statistical properties of the fading models proposed by Zheng [7] and Zajic [12] well coincide with the theoretical values We can select one of these models to generate multiple uncorrelated fading waveforms for mobile channels
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
t(Second)
Theoretic Jake Dent Li Pop Zheng Zajic
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Models: 1 Jake; 2 Dent; 3 Li; 4 Pop; 5 Zheng; 6.Zajic
MSE of PDF MSE of R
c(t)
i (t)
q (t)
i (t)c
q (t)
l (t)c
k (t)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Models: 1 Jake; 2 Dent; 3 Li; 4 Pop; 5 Zheng; 6.Zajic
MSE of PDF MSE of R
c(t)
MSE of R
ci(t)
q (t)
MSE of R
ci(t)cq(t)
MSE of R
cl(t)ck(t)
Trang 88
References [1] R H Clarke, “A statïstical theory of mobile radio reception,” Bell Systems Technical Journal,
1968
[2] W C Jakes, Microwave Mobile Communications New York: Wiley, 1974
[3] P Dent, G E Bottomley, and T Croft, “Jakes fading model revisited ” Electron Letter, vol 29,
no 13, pp 1162–1163, June 1993
[4] M F Pop, and N C Beaulieu, “Limitations of sum-of-sinusoids fading channel simulators,” IEEE Trans Commun, vol 49, pp 699–708, 2011
[5] Y Li, and X Huang, “The simulation of independent Rayleigh faders,” IEEE Trans Commun., vol 50, pp 1503–1514, 2002
[6] Z Wu, “Model of independent Rayleigh faders,” Electronics Letters vol 40, no 15, 2004
[7] Y R Zheng, and C Xiao, “Simulation models with correct statistical properties for Rayleigh fading channels,” IEEE Trans Commun., vol 51, no 6, pp 920-928, Jun 2003
[8] Y R Zheng, and C Xiao, “Improved models for the generation of multiple uncorrelated Rayleigh fading waveforms,” Communications Letters, vol 6, no 6, pp 256–258, Jun 2002
[9] Y R Zheng, and C Xiao, “A statistical simulation model for mobile radio fading channels,” Proc IEEE WCNC’03, New Orleans, USA, pp 144-149, March 2003
[10] G L Stuber, Principles of Mobile Communication, 2 ed.: Norwell, MA: Kluwer, 2001
[11] C Xiao, Y R Zheng, and N C Beaulieu, “Second-Order Statistical Properties of the WSS Jakes’ Fading Channel Simulator,” IEEE Transactions on communications, vol 50, no 6, Jun 2002 [12] A G Zajic, and G L Stuber, “Efficient simulation of Rayleigh fading with enhanced de-correlation properties,” IEEE Transactions on Wireless Communications, vol 5, pp 1866-1875, Jul
2006
... Zajic-0 .4 -0 .2 0.2 0.4 0.6
t(Second)
Theoretic Jake Dent Li Pop Zheng Zajic
-0 .8 -0 .6 -0 .4 -0 .2 0.2 0.4 0.6 0.8...
Figure The cross-correlation function between the in-phase and the quadrature components
Figure The auto-correlation of I/Q-components
In... component of the faded envelope, the cross-correlation function between the in-phase and the quadrature components and the cross-correlation function between the in-phase and the quadrature components