# beaulieu2005

```IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005
Efficient Nakagami-
413
Norman C. Beaulieu, Fellow, IEEE, and Christine Cheng
Abstract—An efficient method for generating correlated Nakagami- fading envelope samples is presented; this method is
applicable for arbitrary values of the fading parameter . The
new method is compared to other methods used to generate
Nakagami- random variates. An accurate approximation to the
inverse Nakagami- cumulative distribution function, valid for
all values of , is derived. Uncertainties regarding the autocorrelation of the Nakagami- fading process are discussed. The
fading envelope autocorrelation is determined by simulation and
asymptotic analysis.
Index Terms—Fading channels, probability, simulation, stochastic processes.
I. INTRODUCTION
T
[1]–[8]. Through the parameter , this distribution can model
signal fading conditions that range from severe to moderate, to
light or no fading. The primary justification for the use of the
data [9]–[11]. Much theoretical and numerical analysis of the
performances of diverse communication systems operating in
Nakagami fading has been reported in the literature [1]–[8],
[12]–[15]. Fewer results pertaining to the computer simulation
of Nakagami- fading have been reported. Recognizing that the
square root of a sum of squares of zero-mean identically distributed Gaussian random variables (RVs) has a Nakagami distribution with
[16, pp. 46–48] leads to a “brute force
method” for integer and half-integer values of . This method
is used [17], but becomes more inefficient computationally as
increases. Previous work that deals with the simulation of Nakagami- fading channels includes the following.
Yip and Ng [18] proposed a Nakagami- fading channelsimulation method based on implementing the product of a
square-root beta process and a complex Gaussian process.
This method is restricted to values of
. Dersch
and R&uacute;egg [19] developed a semi-empirical method, partially
simulation based, using a theoretical model proposed in [20]
with measured data. This method requires determining the
values of certain coefficients from measured data. Furthermore,
the generation of 100 dissimilarly distributed Gaussian RVs is
required to obtain one random envelope/phase sample pair.
In this paper, we give a procedure for simulating both the
amplitude and the phase processes of a Nakagami- fading
channel, for arbitrary values of the fading parameter . In deManuscript received January 13, 2001; revised June 14, 2004 and October 5,
2004. The review of this paper was coordinated by Prof. E. Sourour.
N. C. Beaulieu is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail:
[email protected]).
C. Cheng is with Nortel Networks, Nepean, ON K1Y 4H7, Canada.
Digital Object Identifier 10.1109/TVT.2004.841555
signing our phasor Nakagami-fading simulation procedure, we
discuss longstanding uncertainties regarding the autocorrelation
of the Nakagami-fading amplitude model appropriate to Nakagami amplitude fading. In addition, we derive a new useful approximation to the inverse cumulative distribution function (cdf)
of the Nakagami distribution, valid for arbitrary values of fading
parameter .
This paper is organized as follows. Section II presents a description of the Nakagami- fading model and a brief discussion of its uncertainties. Previous simulation methods are examined prior to giving our new method, as well as some rationales
for our design. In Section III, we derive a useful approximation
to the inverse function of the Nakagami cdf needed for our simulator design. Section IV examines some statistical properties
of our simulator. Comparisons of the new method with other
methods are given in Section V and some conclusions are given
in Section VI.
II. NAKAGAMI-
In this section, we present the Nakagami- fading model.
The properties of this model as well as some uncertainties associated with the model are examined. Also, previous simulation
methods are discussed in detail. The bases and design of our
new simulation method are given.
Nakagami [1] has reported that signal amplitude fading in
radio wave propagation can be well modeled by the probability
density function (pdf)
(1a)
where
(1b)
is the expectation operator and
(1c)
In (1a),
is a parameter that controls the severity, or depth,
of the amplitude fading. The value
results in the widespread Rayleigh-fading model [1], while values of less than
greater than one correspond to fading less severe
values of
than Rayleigh fading. Fig. 1 shows the pdf for some different
values of . Note that for integer and half-integer values of ,
the pdf is that of the amplitude of a sum of squared independent
Gaussian RVs appropriately normalized.1 That is
(2)
1The Nakagami distribution in these cases is equivalent to the generalized
Rayleigh distribution described in [16, pp. 46–48].
0018-9545/\$20.00 &copy; 2005 IEEE
414
Fig. 1. Nakagami-
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005
m pdf for various values of fading parameter m.
where
are independent and identically distributed (i.i.d.) Gaussian RVs each with zero-mean and variance
. The RV in (2) has a Nakagami- pdf with parameter
and second moment
. If the second motends to infinity,
ment is held constant, then in the limit as
the Nakagami- pdf tends to an impulse as seen in Fig. 1; this
is a consequence of the normalization chosen by Nakagami and
permits obtaining a static signal amplitude as a limiting case
).
(obtained as
While (1) gives the pdf of the amplitude of the modeled fading
signal, it is not clear what the associated phase pdf should be
[1], [18]. A clear and plausible physical or intuitive basis for
Nakagami- amplitude fading is not known and this fact has
been a basis for some criticism of the use of this distribution to
model fading [21]. Nonetheless, the distribution has been widely
used owing to empirical justifications [9]–[11], of its quality in
matching measured amplitude fading data. We are left with the
open question of what phase pdf to associate with Nakagami
amplitude fading if our simulation method is to provide phasor
(amplitude and phase) fading gains. We postpone discussing
how we choose to deal with this uncertainty to discuss a second
uncertainty associated with the Nakagami-fading model.
It is, perhaps, even less clear what the second- or higher-order
distributions should be. The fading makes samples of the signal
correlated. Gans [22] has derived the fading spectrum associated
with two-dimensional (2-D) isotropic scattering and an omnidirectional receiving antenna, for the case of Rayleigh amplitude
fading. The corresponding normalized autocorrelation of the
complex Gaussian quadrature components is given by the inverse
Fourier transform of the fading spectrum as [23, pp. 39–42]
(3)
where
is the zero-order Bessel function of the first kind,
is the maximum Doppler frequency, and represents time.
It can be shown that the normalized autocorrelation function of
the received signal envelope is then given by [23, p. 50]
(4)
is the hypergeometric function defined in
where
[24, p. 555].
To the best of the authors’ knowledge, it has neither been established by physical argument nor by empirical evidence what
the autocorrelation of the quadrature signal components or the
autocorrelation of the envelope should be for Nakagami fading
[18].
We do not rigorously answer the fundamental uncertainties
regarding the phase and the signal autocorrelations that should
be associated with Nakagami fading, noting that little explicit
discussion of these problems can be found in the literature.
However, the following plausible artifice permits us to assign
an autocorrelation to the faded Nakagami signal. It is based on
considering the cdf defined by
(5)
where is a dummy variable of integration and
is the pdf
in (1).
Consider the fading environments illustrated in Fig. 2. We
assume that the signal experiences a multipath transmission
environment such that it undergoes slow Rayleigh fading. We
assume further that the Rayleigh-faded signal experiences a
subsequent transmission environment such that the resultant
puts the Nakagami-fading signal into a correspondence with
the Rayleigh-fading signal through the cdfs of each. Thus,
if we specify that the Rayleigh-faded signal has a particular
autocorrelation function, the autocorrelation function of the
BEAULIEU AND CHENG: EFFICIENT NAKAGAMI-
Fig. 2. A cdf correspondence between Nakagami-
Fig. 3.
Block diagram of the phasor Nakagami-
415
are left with needing to assign a phase to the Nakagami fading
to have a phasor Nakagami simulator. Somewhat arbitrarily,
we assign a uniform phase to the faded Nakagami signal. This
assignment is motivated (but not justified) by the fact that a
Rayleigh-faded signal has a uniform phase distribution. This
assumption is most common in the literature.
Our phasor Nakagami simulation method is illustrated in
Fig. 3. An existing Rayleigh-fading simulator generates a
Rayleigh-faded envelope sample sequence having the specified
denote
autocorrelation and phase properties. Let the RV
an envelope sample and the RV
denote a phase sample
corresponding to . Then, the transformation
TABLE I
INVERSE NAKAGAMI CDF APPROXIMATION COEFFICIENTS
(6)
where
is the second moment of the random variable (i.e.,
) transforms the RV into a uniform RV on
.
is the cdf of a Rayleigh
This follows from the fact that
is the
RV [16, p. 45]. The subsequent transformation
inverse function of the Nakagami- cdf defined implicitly by
(7)
by
It is well known that transforming a uniform RV
results in a transformed RV
the inverse cdf function
[25, pp. 101–102]. Thus
having cdf given by
(8)
is an RV having Nakagami- distribution. Observe that our
simulator design generates a uniform random variable from a
Rayleigh random variable using the transformation (6), rather
than starting directly from a uniform random variable. This design builds in appropriate correlation properties and appropriate
higher order statistical properties into both the generated uniform random variable at the intermediate stage and into the generated Nakagami- random variable at the final (output) stage.
There is no closed-form expression for
except for
. In the next section, we derive a
the special case of
useful and accurate approximation to the Nakagami- inverse
cdf function.
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005
Fig. 4.
Inverse normalized Nakagami cdf approximation for
m = 2.
Fig. 5.
Inverse normalized Nakagami cdf approximation for
m = 10.
III. INVERSE CDF APPROXIMATION
In order to implement the direct method to generate phasor
Nakagami- distributed random variates described in Section II, the Nakagami- inverse cdf function is needed. Since
this is not available in closed-form and since approximations to
it that are useful for the problem at hand have not appeared in
the literature, we derive an accurate useful approximation in this
section. This approximation will be useful in other work, for
example, in outage probability calculations [14]. We consider a
.
normalized Nakagami cdf, defined by
Hastings [26, pp. 192–193], [27] has derived approximations
to the inverse complementary cdf of the Gaussian distribution
that are widely used [24, pp. 931–933]. Following the approach
of Hastings, we have obtained [28] the approximation
(9)
where
is an ancillary variable defined as
(10)
, and
are coefficients chosen to minimize the
. Unfortunately,
approximation error and
BEAULIEU AND CHENG: EFFICIENT NAKAGAMI-
Fig. 6. Relative approximation error of inverse Nakagami cdf approximation for
m = 2.
Fig. 7.
m = 10.
Relative approximation error of inverse Nakagami cdf approximation for
it is not possible to find one set of coefficients that results
in an accurate approximation for all values of the fading
[28]. Table I gives coefficients for values of
parameter
, and
. These coefficients were determined according to a minimum–maximum error criterion implemented by numerical
than the values in
search. Coefficients for other values of
Table I can be determined by interpolation or by replicating our
approach.
, the approximation function
,
Note that for
which gives the exact inverse Rayleigh cdf. (Thus, values of
, and are not given in Table I for
.) The ac-
417
curacy of the approximation for other values of is illustrated
in Figs. 4 and 5, which show the exact inverse cdf (computed
numerically) and the approximation obtained using the coeffiand
, respectively. Also
cients in Table I for
shown is the ancillary variable . In these figures, the abssica
of each plot has been scaled (nonlinearly) such that the true cdf
plots as a straight line; in this way, the error of the approximation is immediately obvious. The relative error defined as
Approximation-Function
(11)
Function
is shown for
and
in Figs. 6 and 7, respectively.
In deriving these approximations, we have sought accuracy
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Fig. 8. Comparison of theoretical and empirical Nakagami-
m cdfs for three values of m for small values of cdf argument.
Fig. 9. Comparison of theoretical and empirical Nakagami-
m cdfs for three values of m for large values of cdf argument.
in both “tails” of the cdf approximation, i.e., the region
and the region
. The former region is
important in outage problems where the goal is to minimize
the probability that the signal amplitude falls below a specified
minimum level. The latter region is important in co-channel
interference (CCI) problems where the probability that the CCI
exceeds a specified level is important. The approximations for
, and
and the corresponding relative errors are shown graphand is
ically in [28]. The relative error is typically
no more than
for these values of
for the region
to
.
These inverse Nakagami cdf approximations are incorporated
into the Nakagami- fading simulator given in Fig. 3. In the
BEAULIEU AND CHENG: EFFICIENT NAKAGAMI-
419
Fig. 10. Empirical and theoretical phase pdfs of the simulator output.
next section, we present some discussion of the statistical properties of the simulator output.
IV. SIMULATOR STATISTICS
In this section, we examine some statistical properties of the
simulator output sequences. The distribution, the amplitude
autocorrelation, the squared-amplitude autocorrelation, the
quadrature component autocorrelation, and the phase autocorrelation are discussed.
Figs. 8 and 9 highlight the cdf and complementary cdf, respectively, of the simulator output sequences. In both figures,
the desired theoretical values are plotted as well as the empir, and
. It
ical simulator values for
is seen in Fig. 8 that the small argument “tail” of the simulator
distribution is in excellent agreement with the theoretical model.
The theoretical and empirical values are graphically indistinin the range
.
guishable for values of the cdf
Fig. 9 gives empirical and theoretical values of the complemen, in the range
. Again,
tary cdf
the agreement between empirical and theoretical values is exin the region
cellent. (There is a small difference for
to
. This difference is expected
from the results given for the relative error of the approximation
in [28].)
Fig. 10 shows the phase pdf of the simulator output determined empirically and the theoretical phase pdf. Note that the
generated phase is uniformly distributed in the interval
and the empirical and theoretical pdfs are in excellent agreement. Fig. 11 shows a typical amplitude sequence and its corresponding phase sequence. The samples used in this figure have
. Observe that, typically, when the
signal has deep amplitude fades, it also has marked changes in
phase. This behavior is characteristic of real world channels [29]
and it is important that a simulator replicate it.
The autocorrelation of the amplitude of the envelope is widely
used as an indicator of the quality of a fading channel simulator [18], [23, ch. 2], [30, ch. 1]. The Nakagami random variates are obtained by a nonlinear transformation of the Rayleigh
(or the underlying complex Gaussian) random variates. The analytical determination of the envelope amplitude autocorrelation is complicated and cumbersome. We, therefore, determine
the amplitude autocorrelation function empirically, as discussed
further in the sequel. Another relevant statistical measure of the
fading process is provided by the squared-envelope autocorrelation function [23, ch. 2], [31]. Here, we derive the squared-envelope autocorrelation function for whole- and half-integer values
Let and denote absolute time and a time increment, respectively. Then, using (2), the squared-envelope autocorrelation function is given by
(12)
where
denotes the autocorrelation function of the
denotes the autocorrelation function
squared-envelope,
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Fig. 11.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005
(a) Typical amplitude sequence and (b) its corresponding phase sequence.
of the squared quadrature component, and the linearity property
of expectation has been used as well as the independence of
and
for
. Furthermore, one has that [25, pp.
305–306]
The autocorrelation function in (14) is not normalized such that
. For consistency and clarity, we normalize it;
gives
then, using
(13)
(15)
for a normal stationary process with zero mean and autocorre. Combining (13) with (12) and using
lation
gives
(14)
Note that equality has not been preserved in going from (14)
as given in (15) will be considered
to (15); only
for values of
further in the sequel. Fig. 12 shows
, and
. Also shown are empirically
BEAULIEU AND CHENG: EFFICIENT NAKAGAMI-
421
Fig. 12.
Normalized theoretical and empirical squared-envelope autocorrelation functions of Nakagami random sequences for five values of
Fig. 13.
Normalized empirical envelope autocorrelation functions of generated Nakagami sequences for five values of
determined values of
; these are in excellent agreement with the theoretical values. Observe that (15) predicts the
asymptotes
(16)
since
(17)
m.
m.
inas expected, since the amplitude becomes constant as
creases without limit and the Nakagami- pdf tends to an
impulse.
Fig. 13, shows the normalized amplitude autocorrelation
of the Nakagami simulator output, deterfunction
mined empirically. The normalized empirical autocorrelation
was computed according to
(19)
This asymptotic behavior is evident in Fig. 12. Note further that
(18)
where
is the th sample of the amplitude sequence, is the
total number of samples, is the discrete relative time differdenotes an empirical average of
.
ence, and
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005
Fig. 14. Normalized quadrature component autocorrelation functions of generated Nakagami sequences for five values of
The averages were computed using
random samples.
(The empirical squared-envelope autocorrelation was computed
similarly.) Observe that, as for the squared-envelope autocorrelation, there are asymptotes for large values of and these
asymptotes depend on . Again, the asymptotic, large , autocorrelation approaches the value 1 as increases without limit.
The autocorrelation of the quadrature components of the
Nakagami-fading process are also of interest. This is shown in
Fig. 14 for five finite values of fading parameter . Interestingly, there is little difference in
for different values
of . This is explained as follows. One has
(20)
where
(21)
is the autocorrelation function of the projection of the phase
process
onto a coordinate axis and where the independence
been used. Observe now, from Fig. 13, that
for the typical values of considered there. Thus,
(22)
for different values of . Furthermore, as increases, the approximate equality in (22) becomes more accurate. That this
is so can be seen from the results in Fig. 13. It also follows
from the fact that as increases, the amplitude spread becomes
narrower as the amplitude approaches a constant. Fig. 14 also
. It is seen in Fig. 14 that (22) is a very good apshows
proximation. Thus, the autocorrelation of the quadrature com-
m.
is seen to be an exponent of the Rayleigh-fading
cellent approximation to the autocorrelation of the quadrature
component of the Nakagami fading for the conditions considered in Fig. 14.
Finally, we point out that the design of our simulator (Fig. 3)
implies that the autocorrelation of the Nakagami- fading
process phase is identical to the phase autocorrelation of the
2-D, isotropic scattering Rayleigh fading process. This can be
found in, for example, [30, Ch. 1].
V. EFFICIENCY
In this section, the efficiency of the new method is compared to the efficiency of the “brute-force” method. Both
methods were programmed in the C language. The input to
each method was an autocorrelated complex Gaussian random
and Beaulieu [32]. The comparisons were based on the genNakagami- random variates for values of
eration of
, and . A comparison to the method of [18] was abandoned due to the complexity
of implementing the algorithm of [18] and because it is only
.
valid for restricted values of
The new method requires only two independent autocorrelated Gaussian random variates for the generation of each Nakagami- random variate. In contrast, the “brute-force” method
independent autocorrelated Gaussian random
requires
variates to generate one Nakagami- random variate. Table II
shows the generation times of the two methods for selected
values of . Observe that the new method has the shorter generation time for all values of . Also, the generation times are
essentially independent of the parameter . In contrast, the run
times for the brute-force method increase as increases. When
, the time taken by the brute-force method is four
times the time required by the new method. Importantly, the new
method is suited for all values of fading parameter .
BEAULIEU AND CHENG: EFFICIENT NAKAGAMI-
TABLE II
EFFICIENCY COMPARISON BETWEEN THE BRUTE FORCE METHOD AND
APPROXIMATION METHOD (GENERATION OF 2 RANDOM SAMPLES)
VI. CONCLUSION
A new efficient method for generating Nakagami- random
samples has been presented. The new method can be used to
generate Nakagami- random variates for arbitrary values of
, whereas previous generation methods are restricted to values
of between 0 and 1 and integer and half-integer values of .
The new method required less computer time than the standard
method in all cases examined.
Longstanding uncertainties associated with the autocorrelation of the Nakagami- fading model have been discussed. The
autocorrelation of Nakagami- fading sequences was specified
using a transformation mapping Rayleigh sequences into Nakagami- sequences. Simulation was then used to obtain heretofore unknown autocorrelations of Nakagami- fading for different values of . A closed-form expression for the asymptotic (large time lag) value of the envelope autocorrelation was
derived for arbitrary values of . It was seen that the autocorrelation of the envelope tends to 1 as the fading parameter
tends to infinity.
m
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Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of
British Columbia, Vancouver, Canada, in 1980, 1983,
and 1986, respectively.
He was a Queen’s National Scholar Assistant Professor from September 1986 to June 1988, an Associate Professor from July 1998 to June 1993, and a
Professor from July 1993 to August 2000 with the
Department of Electrical Engineering, Queen s University, Kingston, ON, Canada. In September 2000,
he became the iCORE Research Chair in Broadband Wireless Communications
at the University of Alberta, Edmonton, Canada, and, in January 2001, he became the Canada Research Chair in Broadband Wireless Communications. His
current research interests include broadband digital communications systems,
fading channel modeling and simulation, interference prediction and cancellation, decision-feedback equalization, and space–time coding.
Dr. Beaulieu is a Member of the IEEE Communication Theory Committee
and served as its Representative to the Technical Program Committee for the
1991 International Conference on Communication and as Co-Representative to
the Technical Program Committee for the 1993 International Conference on
Communications and the 1996 International Conference on Communications.
He was General Chair of the Sixth Communication Theory Mini-Conference in
association with GLOBECOM’97 and Co-Chair of the Canadian Workshop on
Information Theory 1999. He has been an Editor of the IEEE TRANSACTIONS
ON COMMUNICATIONS for wireless communication theory since January 1992
and has served on the Editorial Board of the PROCEEDINGS OF THE IEEE since
November 2000. He served as Editor-in-Chief of the IEEE TRANSACTIONS ON
COMMUNICATIONS from January 2000 to December 2003 and as an Associate
Editor of the IEEE COMMUNICATIONS LETTERS for wireless communication
theory from November 1996 to August 2003. He received the Natural Science
and Engineering Research Council of Canada (NSERC) E. W. R. Steacie Memorial Fellowship in 1999. He was awarded the University of British Columbia
Special University Prize in Applied Science in 1980 as the highest standing
graduate in the Faculty of Applied Science. He is a Fellow of The Royal Society of Canada.