IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
The Simulation of Independent Rayleigh Faders
Yunxin Li, Senior Member, IEEE, and Xiaojing Huang, Member, IEEE
Abstract—Multiple independent Rayleigh fading waveforms
are often required for the simulation of wireless communications
channels. Jakes Rayleigh fading model and its derivatives based
on sum-of-sinusoids provide simple simulators, but they have
major shortcomings in their simulated correlation functions. In
this paper, a novel sum-of-sinusoids fading model is proposed and
verified, which generates Rayleigh fading processes satisfying
the theoretical independence requirements and providing desired
power spectral densities with ideal second-order moment. The
effects of replacing sinusoids in the proposed model by their
approximate waveforms are also analyzed and tested. Performance
evaluation and comparison are provided, using the quality
measures of the mean-square-error of autocorrelation function
and the second-order moment of power spectral density.
Index Terms—Fading channels, Rayleigh channels, simulation
and mobile communications.
I. INTRODUCTION
S
IMULATING a multipath fading channel for digital
communications often requires the generation of multiple
independent random processes [1]. For example, when the
bandwidth of the signal to be transmitted is greater than the
coherence bandwidth of the channel, the channel becomes
frequency selective. In addition, if the channel transfer function
is also time variant, further signal distortion, i.e., fading in the
received signal strength, will be observed. This frequency-se-
lective fading channel can be modeled as a tapped delay line
transversal filter with time-variant tap coefficients [2]. In the
case of Rayleigh fading, the tap coefficients are complex-valued
Gaussian random processes and they are statistically independent.
Ideally, the set of independent complex Gaussian
processes should conform to the following criteria [2], [3]:
1) the real (or in-phase) and imaginary (or quadrature) components
of each complex process are zero-mean independent
Gaussian processes with identical power spectral density or
autocorrelation function. As a result, the envelope is Rayleigh
distributed and the phase is uniformly distributed; and 2) the
cross correlation between any pair of complex processes should
be zero. We refer to the complex-valued random processes
satisfying the above conditions and the Rayleigh fading spectral
requirements [3] as independent Rayleigh faders.
A number of different models have been proposed for the
simulation of Rayleigh fading channels over the past decades.
Paper approved by R. A. Valenzuela, the Editor for Transmission Systems
of the IEEE Communications Society. Manuscript received July 27, 2001; revis
ed
December 7, 2001. This paper was presented in part at the 2000 International
Conference on Communications (ICC’2000), New Orleans, LA, USA,
June 18–22, 2000, and the 6th Asia-Pacific Conference on Communications
(APCC’2000), Seoul, Korea, October 30–November 2, 2000.
The authors are with the Motorola Australian Research Center, Botany, NSW
1455, Australia.
Publisher Item Identifier 10.1109/TCOMM.2002.802562.
Generally, these models can be classified as either being
statistical or deterministic. The statistical models are based on
the shaping of the power spectral densities of white Gaussian
random processes by either time-domain or frequency-domain
filtering [4]–[7], whereas the deterministic models approximate
the Gaussian processes by the superposition of finite properly
selected sinusoids [3], [8]–[11]. The statistical models are
suitable for the generation of independent Rayleigh fading
waveforms because the above criteria are easily satisfied.
However, a set of independent white Gaussian processes must
be generated first, and extra effort is needed for the design and
implementation of the digital shaping filters with small bandwidth
and special impulse response (or frequency response).
Deterministic models are more popular because of their simplicity,
even though they may have limitations in their statistical
stationariness and ergodicity [8]. In [8], a random phase shift
was introduced to each sinusoidal oscillator of the Jakes model
[3]so that the Rayleigh fader became wide-sense stationary.
To generate multiple Rayleigh faders deterministically, Jakes
proposed a simple method by introducing an additional phase
shift in each oscillator used to simulate an incident ray in his
fading model [3], and this method had been widely accepted. A
modified Jakes model was later proposed in which orthogonal
functions were used to weight each oscillator’s output, and the
incident wave arrival angles were rearranged to enable equal
oscillator power [9]. However, the brief review and simulation
results of these two models [3], [9], provided in Appendix A,
reveal some of their undesirable properties. For example, for
any single fader, the autocorrelation function of the in-phase
component differs from that of the quadrature component, and
the cross-correlation function between the in-phase and quadrature
components is not always zero. For any pair of faders,
they are not independent because the cross-correlation function
between them are generally not zero, although the modified
Jakes model [9] does guarantee a zero cross correlation at zero
time shift.
In this paper, a novel sum-of-sinusoids fading model is proposed
which deterministically generates multiple independent
Rayleigh faders satisfying the above outlined conditions. By
use of asymmetrical arrival angle arrangement and appropriately
chosen incident wave phases, the in-phase and quadrature
components in any single fader are independent and have almost
identical autocorrelation functions. The independence between
different faders is also guaranteed. To reduce the computational
complexity and increase the speed of the proposed
simulator, the sinusoids can be replaced by their approximate
waveforms. However, it is necessary to consider the impact of
such approximation on simulator performance. Our analysis and
simulation show that the performance is still acceptable, even
if the sinusoids are replaced by periodic triangular waveforms
0090-6778/02$17.00 . 2002 IEEE
with the same frequencies. The resultant model of sum of triangular
waveforms is much more computationally efficient and
is particularly useful in a real-time channel simulator. Two different
quality criteria based on the mean-square-errors of the
autocorrelation functions and the second-order moments of the
power spectral densities are also proposed to evaluate the performances
of the proposed fading model and its approximate
derivatives. These criteria can be used in conjunction with other
performance evaluation methods such as those proposed in [4]
and [5].
The rest of this paper is structured as follows. In Section II,
the proposed model for the generation of multiple independent
Rayleigh faders is derived. The autocorrelation and cross-corre-
lation functions of the generated Rayleigh fading waveforms are
analyzed and tested to show their conformation to the theoretical
requirements. Other important properties of this fading model
are also given. The effects of replacing sinusoids by their approximate
waveforms are analyzed in Section III. Performance
evaluations for the proposed fading model and its approximate
derivatives are presented in Section IV. Finally, a summary and
conclusions are given in Section V.
II. INDEPENDENT RAYLEIGH FADERS
As a consequence of the central limit theorem, a Gaussian
process can be approximated by the superposition of a large
number of properly weighted sinusoids with uniformly
distributed phases. This fact is exploited by most of the
deterministic fading models and also serves as the basis on
which our proposed model is established. We suppose that
independent fading waveforms are required, each of which
is composed of
sinusoids. With the
th fading waveform
denoted by
,wehave
(1)
where
,
, and
represent the amplitude, frequency,
and uniformly distributed random phase of the
th complex
sinusoid in the
th fader, respectively. Referring to the Jakes
fading model, which is characterized by
incident rays with
equally spaced arrival angles around a moving receiver of velocity
,
is interpreted as the Doppler frequency shift introduced
by the
th incident wave and can be expressed as
(2)
where
is the maximum Doppler frequency
shift with
being the wavelength of the transmitted carrier
frequency, and
denotes the
th arrival angle in the
th
fader. Equation (2) also means that
is the projection of a
vector with polar coordinates (
) on the direction in
which the moving receiver is traveling.
By assuming a uniform antenna gain pattern and uniformly
distributed incident power,
can be reduced to a constant
value
. Also, by choosing
in the form of
(3)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
Fig. 1. Arrival angle arrangement on the x-. plane for fader 0 (solid dots)
and
fader 1 (blank dots). The moving receiver is located at the origin and is tr
aveling
in the . axis direction.
with being an initial arrival angle, the th fading waveform
can be rewritten as shown in (4) at the bottom of the next page,
where is the number of sinusoids in one quadrant.
If are so chosen that the following conditions are always
satisfied
(5)
(6)
where denotes new phase variables related to and
for ; then, (4) can be simplified
as
(7)
where
(8)
(9)
are the in-phase and quadrature components of , respectively,
and , for and
, are 2 independent random phases, each
of which is uniformly distributed in .
With the arrival angle arrangement formulated by (3), the
th fader’s arrival angles can be obtained by rotating the
th fader’s arrival angles counterclockwise by as illustrated
in Fig. 1, where each dot represents a vector with polar
coordinates ( ).
If the initial arrival angle is properly chosen so that an
asymmetrical arrival angle arrangement is ensured, the Doppler
frequency shifts and in (8) and (9) will
always be different from each other. As illustrated in Fig. 2,
where are represented by the projections of vectors
( ) in quadrant 1 on the axis, whereas are
represented by the projections of vectors ( )in
quadrant 4 on the axis, if satisfies
and (10)
then (or ) different Doppler frequency shifts will
be introduced (see Appendix B).
The autocorrelation function of the
in-phase component of , where denotes the ensemble
average of a process, is evaluated as shown in (11) at the bottom
LI AND HUANG: THE SIMULATION OF INDEPENDENT RAYLEIGH FADERS
(a) (b)
(c) (d)
=2. . )on . axis with
=MN, the
arrival angle pattern is symmetrical and the Doppler frequency shift set
. co. . coincides with the Doppler frequency shift set . si. . .
=MN, the arrival angle
pattern is asymmetrical and different Doppler frequency shifts are introduce
d.
of the next page. In deriving (11), we have used the fact that the
ensemble average
is always zero, and the ensemble average
is
zero for , since and are uniformly distributed
in and independent when [13].
Similarly, the autocorrelation function of the quadrature
component of
, the cross-correlation function between
the in-phase and the quadrature components of
and the
cross-correlation function between
and
are then
derived as (12), (13), and in (14) as shown at the bottom of the
next page
(12)
(13)
denotes complex conjugation. Equations
where the superscript
(13) and (14) imply that the in-phase and the quadrature components
of any single fading process are independent, and the
fading processes of different faders are also independent of each
other.
The time-averaged autocorrelation function
of
, where
denotes the time average of a process, can
also be shown to be the same as the ensemble-averaged one expressed
in (14), which is a function of the time difference
only. In addition, the mean value of
can be easily found,
using either ensemble or time averaging, to be zero, which is
a constant independent of time. Consequently,
is at least
wide-sense stationary and ergodic for both mean and autocorrelation
function.
Letting
and simply choosing
,
can be normalized to unity energy, and the new
fading model is finally formulated as
(15)
The random phases
and
can be considered as the seeds
used to generate different realizations of
. Also note that
the sinusoids in (15) can be evaluated using either cosine or sine
(4)
function without affecting the statistical properties of the fading
model.
According to (14), the normalized autocorrelation function of
becomes
(16)
and the corresponding power spectral density function is given
by
(17)
where
denotes the Dirac delta function, and
.
The
th moments, for 0, 1, 2, of
are found to be exactly
the same as those of the ideal Jakes power spectral density
function [3], i.e.
(18)
(19)
(20)
where
and
denote the first-and second-order
, respectively.
derivatives of
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
Fig. 3 shows some examples of simulated correlation functions
of this new fading model with parameters
, from which the independence between the in-phase and the
quadrature components of any single fading waveform and the
independence between different faders are confirmed. While the
autocorrelation function of a single fader is almost the same
as the theoretical Bessel function
, the autocorrelation
functions of the in-phase and quadrature components are
much closer to their theoretical values. The residual errors in
the in-phase and the quadrature autocorrelation functions can
be further reduced by the use of a larger .
III. COMPUTATIONAL REDUCTION
In the previous section, the new model for generating
multiple independent Rayleigh faders is proposed, which
conforms to the theoretical requirements. However, great
computational effort is also needed, because different sets
of Doppler frequency shifts are used in different faders. As
indicated in (15), the generation of
independent Rayleigh
distributed waveforms, each using
equal-strength incident
rays, requires the computation of
sinusoids at each time
instant. It is obvious that avoiding the use of standard sinusoidal
function, which is usually evaluated in a general-purpose
computer by power series expansion, will save considerable
computer execution time. In fact, as a fundamental building
block of digital signal-processing systems, digital sinusoidal
wave generator usually uses some kind of approximation to
provide efficient sinusoid evaluation [12]. Therefore, replacing
the sinusoids in the fading model with their approximate
waveforms is useful to both software channel simulation and
(11)
(14)
LI AND HUANG: THE SIMULATION OF INDEPENDENT RAYLEIGH FADERS
(a) (b)
Fig. 3. Example correlation functions of the proposed fading model. (a) In-p
hase and quadrature autocorrelation functions and cross-correlation function
for
fader 0 (0: in-phase autocorrelation; 1: quadrature autocorrelation; 2: theo
retical autocorrelation; 3: cross correlation). (b) Autocorrelation function
of fader 0 and
cross-correlation function of fader 0 and fader 4 (0: autocorrelation; 1: th
eoretical autocorrelation; 2: cross correlation). The normalized time delay
is defined as
.
. .
digital real-time channel simulator implementation. The issue
is how such approximations affect the simulation performance.
In this section, we generate approximate sinusoid by interpolation,
which makes use of known or tabulated sinusoidal
values of limited size, and analyze the autocorrelation function,
the power spectral density, and its
th moments ( 0, 1, 2)
of the resulting reduced-complexity fading model. Suppose that
the cosine values for
angles uniformly spaced around the unit
circle are known and stored in a lookup table. Then, the sinusoidal
function
can be approximated by its interpolated
version
generally expressed as
(21)
where
choosing , (21) can to
and
(22)
is any even interpolation function. By properly
be reduced very simple form,
and thus the evaluation of is greatly simplified (see
Table I). For example, the simple table lookup approximation
(or the zero-order interpolation) only uses one addition and
one modulo to generate one sample of the approximated
sinusoidal wave. For the linear interpolation (or the first-order
interpolation), only one more addition and multiplication are
Suppose that the lookup table size is any even number
larger than two. By replacing in (8) and in (9) with
, respectively, the approximated Rayleigh
waveform , after normalizing to unity energy, becomes
needed.
where
is defined by
(23)
The normalized autocorrelation function and corresponding
power spectral density are derived as
(24)
(25)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
TABLE I
SUMMARY OF FORMULAS FOR SINUSOID APPROXIMATION BY INTERPOLATION
where
(26)
The th moments, for
and its derivatives, are
0, 1, 2, of , derived from
(27)
(28)
(29)
is the second-order derivative of . Note that the
second-order moment is no longer the same as its theoretical
value. For the cases of table lookup approximation and linear
interpolation,
where
is a function of
as listed in Table I.
IV. PERFORMANCE EVALUATION
The performances of the independent Rayleigh faders are
evaluated using two different quality criteria in this section. The
first criterion is to compare the measured autocorrelation function
to the theoretical Bessel function
. The second is
to compare the second-order moment of the measured power
spectral density with the ideal value given in (20). Other quantitative
measures such as those proposed in [4] and [5] could also
be applied.
As the first criterion, the quality of the fader’s autocorrelation
function can be measured by the mean-square-error defined by
(30)
where denotes the time interval over which the mean-
square-error is evaluated. As suggested in [10], in order to adapt
to the number of sinusoids used in the generation of the
Rayleigh waveforms, we choose according to the relation
(31)
The relationships between and for both table lookup approximation
and linear interpolation are given in Fig. 4. The
curve for the exact model (i.e., without sinusoid approximation)
is also shown for comparison purposes. Generally,
becomes smaller when is larger. Also we see that, for
agiven , the larger the lookup table size is, the smaller the
mean-square-error will be. When the table lookup approximation
is used, at is nearly the same as that obtained
using exact sinusoids. For the linear interpolation method, very
good approximation is already achieved at . Fig. 5 shows
the autocorrelation functions of the Rayleigh waveforms after
using the above computational reduction methods, which confirm
the results shown in Fig. 4.
The second-order moment of the Rayleigh fader’s power
spectral density is another important quality measure because
it is directly related to other properties of the fader such as the
level crossing rate and the average duration of fading time [3].
For the proposed fading model without sinusoid approximation,
this moment is always the same as its theoretical value (i.e.,
). However, when interpolation approximation is used,
LI AND HUANG: THE SIMULATION OF INDEPENDENT RAYLEIGH FADERS
(a) (b)
Fig. 4. Mean-square-error .
as reference. Other parameters are . =. and . =0.
Nat different . (a) Table lookup approximation. (b) Linear interpolation. Th
e performance obtained using the exact model is shown
(a) (b)
Fig. 5. Autocorrelation functions of Rayleigh waveforms compared with that o
btained by the exact model. (a) Generated using table lookup approximation.
(b) Generated using linear interpolation. Other parameters are .
=8, . =8, and . =0. The normalized time delay is defined as .
. .
depends on
and its second-order derivative as indicated
in (29).
For table lookup approximation,
will be theoretically infinite
for a finite lookup table size
as seen in Table I. The interpretation
for this infinite
is that the power spectral density
of the approximated Rayleigh waveform rolls off in the order
of
, so that the integral of
overall frequency range
becomes infinite. In practice, the fading waveform is sampled at
frequency
, so that the maximum frequency is limited to half
of the sampling frequency. Therefore, the actual second-order
moment can be evaluated by
(32)
where [ ] denotes the nearest integer larger than . Apparently,
depends on the lookup table size as well as the sampling
frequency . For linear interpolation, is finite and only depends
on as seen in Table I. In this case, rolls off in
the order of and the integral of overall frequency
range converges.
Fig. 6 shows the normalized second-order moments
of the reduced-complexity independent Rayleigh faders using
interpolation approximation. We see that for table lookup approximation
approaches the ideal value only when
very large lookup table size is used, and this moment increases
as becomes higher. For the linear interpolation method, however,
is very close to even at very small . Fig. 7
shows how the simulated level crossing rate (defined as the
rate at which the envelope of the Rayleigh waveform crosses a
specified level in the positive direction) is affected by the large
value. When for table lookup approximation, the
normalized level crossing rates are always higher than that obtained
by the exact model (which is almost identical to the the
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
(a) (b)
Fig. 6.
Normalized second-order moments ~b
=!
as a function of lookup table size L. (a) Table lookup approximation with di
fferent sampling frequencies.
(b) Linear interpolation method. Other parameters are .
=2. 2 5. rad/s, .
=8, . =8, and . =0.
(a) (b)
Fig. 7. Normalized level crossing rates .
=.
as a function of the normalized fading envelope. (a) Table lookup approximat
ion with different sampling
frequencies. (b) Linear interpolation method. Other parameters are .
=2. 2 50rad/s, .
=8, . =8, and . =0.
oretical value
, where the normalized fading envelope
is defined as the ratio of the fading envelope over its root
mean-square value). To secure similar performance,
has to be
as large as 512. For the linear interpolation method, very good
approximation is already approached at .
From the above performance evaluation results, we note that
the linear interpolation method with
already provides
very good approximation for the generation of multiple independent
Rayleigh fading waveforms. Also note that when
, the linear interpolated sinusoid
becomes a triangular
wave
. This implies that, according to (22), the reducedcomplexity
fading model with acceptable performance can be finally
formulated as the superposition of finite triangular waves,
i.e.,
Compared to the exact model (15), the sinusoid
(33)
is now
replaced by the triangular wave
with the same average
power 1/2. Fig. 8 shows the comparison of these two
waveforms, illustrating how the sinusoid is approximated by the
triangular wave.
V. CONCLUSIONS
We have proposed a novel method for generating multiple
independent Rayleigh fading waveforms, which overcomes
some undesirable properties in conventional deterministic
models. By use of asymmetrical arrival angle arrangement and
appropriately chosen incident wave phases, the in-phase and
quadrature components in any single fader are independent
and have almost identical autocorrelation functions. The independence
between different faders is also guaranteed, because
different sets of Doppler frequency shifts are used. Moreover,
the second-order moment of the power spectral density of the
fading waveform generated by the proposed model is always
the same as its theoretical value. To reduce the computational
effort involved in evaluating the sinusoids related to these
Doppler frequency shifts, different computational reduction
methods are analyzed and tested. The table lookup method
LI AND HUANG: THE SIMULATION OF INDEPENDENT RAYLEIGH FADERS
Fig. 8. Approximation of sinusoid cos(x. (curve 1) by triangular wave
3=2c(x)(curve 0).
dramatically increases the second-order moment of the power
spectral density of the simulated fading waveform, which
degrades the performance such as the level crossing rate. To
secure better performance, sufficiently large lookup table size
should be used. The linear interpolation method provides good
approximation to the ideal autocorrelation function as well as
the second-order moment of the simulated power spectral density,
even if the size of the lookup table is small. As a result, the
fading model can be even approximated as the superposition of
finite triangular waves. Consequently, the reduced-complexity
independent Rayleigh faders with linear interpolation provide
both computational efficiency and performance acceptance for
the simulation of multipath Rayleigh fading channels.
APPENDIX A
JAKES FADING MODEL AND ITS MODIFIED VERSION
By assuming uniform antenna gain pattern and uniform distributed
incident power, the
th fading waveform of the Jakes
Fig. 9. Symmetrical arrival angle arrangement of Jakes fading model with
. =10.
fading models [3] with
equal-strength incident rays can be
expressed as
(A1)
,
,
, and
where
. Fig. 9
shows its symmetrical arrival angle arrangement where the dots
represent vectors with polar coordinates (
).
Using time averaging, the autocorrelation functions of the
in-phase and quadrature components of
, the cross-correla-
tion function between the in-phase and quadrature components
of
, and the cross-correlation function between any
and
are derived as shown in (A2)–(A5) at the bottom of
the page.
Fig. 10 shows some examples of measured correlation functions
with , rad/s, and a sampling
rate of 9.6 kHz. These results clearly show that the autocorrelation
function of the in-phase component of any fader is different
from that of the quadrature component, the cross correlation between
the in-phase and quadrature components in any fader is
(A2)
(A3)
(A4)
(A5)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
(a) (b)
Fig. 10. Example correlation functions of Jakes fading model. (a) In-phase a
nd quadrature autocorrelation functions and cross-correlation function for f
ader
1 (0: in-phase autocorrelation; 1: quadrature autocorrelation; 2: theoretica
l autocorrelation; 3: cross correlation). (b) Autocorrelation function of fa
der 1 and
cross-correlation function of fader 1 and fader 5 (0: autocorrelation; 1: th
eoretical autocorrelation; 2: cross correlation). The normalized time delay
is defined as
.
. .
Fig. 11. Symmetrical arrival angle arrangement of modified Jakes fading
model with . =8.
fader is not always zero, and the cross-correlation function between
any pair of faders is not always zero, either.
The modified Jakes fading model proposed in [9] can be expressed
as
,
(A6)
where
,
,
are random seeds, and
. Fig. 11 shows
are any orthogonal functions
satisfying
the rearranged arrival angle pattern, which is still symmetrical.
Similarly, when
is chosen as the Walsh–Hadamard
function set, we have
(A9)
(A10)
Fig. 12 shows some examples of measured correlation functions
with the same parameters as shown in Fig. 10. These results
show in the same way that the independence requirements are
still not satisfied, although this model does guarantee a zero
cross correlation between any pair of faders at zero time delay.
APPENDIX B
CONDITION FOR PROVIDING DIFFERENT DOPPLER
FREQUENCY SHIFTS
We denote the Doppler frequency shifts in (8) and (9) as
(B1)
(A7)
(B2)
(A8)
respectively. When
,wehave
LI AND HUANG: THE SIMULATION OF INDEPENDENT RAYLEIGH FADERS
(a) (b)
Fig. 12. Example correlation functions of modified Jakes fading model. (a) I
n-phase and quadrature autocorrelation functions and cross-correlation funct
ion for
fader 1 (0: in-phase autocorrelation; 1: quadrature autocorrelation; 2: theo
retical autocorrelation; 3: cross correlation). (b) Autocorrelation function
of fader 1 and
cross-correlation function of fader 1 and fader 5 (0: autocorrelation; 1: th
eoretical autocorrelation; 2: cross correlation). The normalized time delay
is defined as
.
. .
,
where ,
, .
REFERENCES
for
(B3)
[1] “Radio transmission and reception,”, GSM 05.05 Version 5.2.0, European
Telecommunications Standards Institute (ETSI), 1996.
[2] J. G. Proakis, Digital Communications, 3rd ed. New York: McGraw-
Hill, 1995.
[3] W. C. Jakes, Ed., Microwave Mobile Communications. New York:
Wiley, 1974.
[4] D. J. Young and N. C. Beaulieu, “A quantitative evaluation of generatio
n
发表于 2004-12-3 23:03:00