Download Complex Probability Density Function

A Complex Probability Density Function: An
Application to Multipath Fading Channels in
Wireless Communication Systems
Chirasil Chayawan, Ph.D.
Department of Electronics and Telecommunication Engineering King Mongkut’s University of
Technology Thonburi, Bangkok, 10140 Thailand
Phone: (662) 470-9063, Fax: (662) 470-9070
Email: [email protected]
ABSTRACT
An argument of common probability
density function (pdf) is generally a real
number, which describes the density, or mass
in case of discrete random variable, of a
random variable.
However, in some
applications, the pdf can be a complex
function, so called complex probability
density function. This kind of pdf is then, as
general complex function, characterized by
the amplitude and phase and is usually
extensively used in quantum physics. In this
paper the properties of complex pdf are
investigated.
The application to the
multipath fading channels in wireless
communication systems is also studied.
Some models of multipath fading channels
are given for supporting the validity of
complex pdf in wireless communication
systems.
Keywords: Complex Probability Density
Function, Superposition, Multipath Fading
Channel
1.
INTRODUCTION
The probability density function (pdf) in
general is a real function. This kind of pdf is
widely used in many applications including
the modeling of multipath fading channels in
wireless communication systems. When the
attribute of a pdf is real, the statistical
interpretation is not complicated and the
statistical property is then characterized in
term of the amplitude or absolute value of
that pdf. However, the notation of complex
probability is not common in the literature, the
concept and its considerations have been
discussed in physics pertinent to the physical
characteristics of tunneling probability amplitude
functions [1], complex diffusion wave function
[2], wave-matter interactions [3], etc.
For the best of our knowledge, the
application of complex pdf in multipath fading
channels has been limited and need to be
investigated in details. In this paper, therefore,
the applications of complex pdf in multipath
fading channels of wireless communication
systems are examined. Some examples of valid
conditions in wireless communications are given
to support the usefulness of complex pdf, which
turn out to be the general form of commonly
used models such as Rice or Rayleigh distributed
multipath fading channels. The organization of
this paper is as follow. The concepts of complex
pdf are investigated in section II. In section III,
the application of complex in multipath fading
channels is given. The analysis and conclusion
are then provided in section IV and V,
respectively.
2. COMPLEX
FUNCTION
PROBABILIT
DENSITY
The complex pdf in specific aspects does
not signify as a pdf of a complex random
variable, which is, in general, defined as a joint
pdf of real and imaginary parts of corresponding
complex random variable. In addition, the
complex pdf is described as a probability density
of any random variables; however, that density
function takes a possibility to be complex. At
this point, some confusion might be raised, since
the word “density” or mass, in general, is a real
measurable quantity. When the statistical
property of a random process is characterized
by a complex pdf, two questions are needed
to answer properly [2]. i) What is the
physical inference of complex pdf? ii) Does
the complex pdf provide any measurable
quantity?
In general, the pdf of a random
variable is, by definition, a properly
normalized function that assigns a probability
"density" to each possible outcome within
some interval. For example, a pdf of a
Gaussian random variable A with mean m
and variance σ2 can be written as
 a − m 2 
) 
1
− (
f A (a ) =
exp
 ∈ R ; − ∞ ≤ a ≤ ∞,

2πσ 2
2σ 2 






under the pdf curve cannot be calculated as a
volume (unit of cube), due to the density
function being a function of one variable a
(comparing with the pdf of a complex random
variable). In addition, the volume in figure 1
also includes a complex probability, which is
impractical for probability value taking from
zero to one in real axis. Therefore, in the case of
a random variable having a complex density, the
cdf can be evaluated by find the area under the
real part of the corresponding pdf to satisfy the
definition and relationship of cdf and pdf [5].
Im[fA(a)]
Re[fA(a)]
(1)
which is always a positive real number and
(1) is the one (many)-to-one transformation
from a real plane to a real plane. In contrast,
if f A (a ) is complex, i.e.
f A ( a ) ∈ C;
-∞ ≤ a ≤∞ ,
(2)
The transformation is now an one (many)-toone mapping from a real plane to a complex
plane, but all statistical properties are needed
to maintain in the complex plane as that of in
a real plane. Note that the relationship of a
density function and a cumulative
distribution function (cdf) should be
primarily studied. An cdf of a random
variable A in [(2)] is defined as the
probability of a random variable that is less
than or equal to a specific value of A, and can
be obtained by integrating the density
function of A, i.e.,
a
FA (a ) = P ( A ≤ a ) =
∫
f A (α ) d α . (3)
−∞
Also note that the cdf is nothing but is the
area under the pdf curve that less than or
equal to an upper limit a of the integration in
(3). Returning to the complex pdf, the
corresponding cdf should also be, by
definition, the area under the complex pdf
curve. Since the density is complex having
both real and imaginary quantities, the area
under the pdf curve does not satisfied the
definition. Moreover, for the case of a
complex pdf as shown in figure 2, the area
a
Fig.1: An example of a complex pdf of a random
variable a, showing the real and imaginary parts
of the complex pdf.
However, what the significance property is that
the imaginary value of the complex pdf contains.
Note that the complex pdf of a real random
variable especially in wireless communication
systems is generated by a superposition of finite
random variables and usually involves the
movement, scattering, diffusion or diffraction
[6], [7]. It can also be shown by [5] that the
imaginary part is proportional to the degree of
the correlation. The imaginary part is then a
function of a correlation coefficient or other
parameters that state the degree of the
relationship of each individual random variable
of the superposition of the random variable
having a complex pdf. At this point, one can
see that the envelope of the complex pdf might
be insignificant to interpret since the real and
imaginary part have diverse properties, i.e. one
for pdf and the other for elementary correlation,
respectively. Finally, the phase of the complex
pdf needs to be revealed. It can be shown by
Barkay and Moiseyev [2] that the phase of the
complex probability amplitudes plays an
important role in the interference among
resonance states during scattering experiments.
In other words, it associated with the phase of
the resonance channel probability amplitudes.
3. AN APPLICATION OF COMPLEX
PDF
IN
MULTIPATH
FADING
CHANNELS
In this section, the application of
complex pdf will be studied for multipath
fading
environment
in
wireless
communication systems. The complex pdf
will be used to analyze the exacted envelope
pdf for multipath fading channels. First the
theory of multipath fading is summarized.
Then the individual phase of each channel,
which lies beneath the probability
assumptions of the complex random signals,
is investigated and shows that for some
specified phase distributions resulting the
complex envelope pdf, which turns out to be
more general than a classical assumption of
uniform distribution.
In wireless communication systems, a
multipath fading channel can be modeled as a
superposition of sinusoidal signals. The inphase and quadrature-phase components of
the resultant vector are given by [6]
N
X c + jX s =
∑R
n
(cos ϕn + j sin ϕn ) (4)
n =1
where there are N random vectors
(representing the number of scatterers in the
channel), each with corresponding amplitude
Rn and phase ϕn. When the amplitude Rn
and phase
ϕn
of each individual vector are
independent, and it is known that the pdf of
phase ϕn is either i) uniformly distributed in
any interval of 2π or, ii) nonuniformly
distributed but its pdf is symmetrical about
zero, the resultant envelope and phase, which
can be given by
R=
X c + X s θ = tan
2
2
−1
Xs
Xc
(5)
are independent, and also the quadrature
components X c and X s are uncorrelated
regardless of the distribution of Rn [1], [7].
When the number of random vectors is
sufficiently large ( N → ∞) , the quadrature
components are Gaussian distributed based
on the central limit theorem (CLT). Then
since Xc and Xs are jointly uncorrelated, they
are also independent and the resultant
envelope is Rayleigh distributed. There are
many pdf’s that are usually applied as the
individual phase pdf, fϕn (ϕn ) such as uniform,
Simpson, Cardiod and von Mises distribution
[9].
The significant phase distribution in this
literature is the von Mises distribution that is
commonly used in modeling angle distributions
in man applications.
This distribution is
generally used in an application of directional
statistics, and first studied for measuring the
atomic weights from integral value [10]. It has
also been widely used as a noise model in
oscillators [11], [12], and the angle of arrival in
antenna array systems [10]. The pdf of phase ϕn
when it is distributed by the von Mises
distribution is given by [9]
fϕ (ϕ ) =
1
exp κ cos (α (ϕ − µ)) ,
2π I 0 (κ )
(
)
µ − π ≤ ϕ ≤ µ + π,
(6)
Where the parameter µ is the mean direction, the
parameter κ is known as the concentration
parameter, I n ( x ) is an n-th order modified
Bessel function of the second kind [13], and α
is the number of modes of the von Mises
distribution, which, in this case, represents the
number of concentrated direction of signals at
the receiver. Figure 2 compares the von Mises
with Cardiod phase distribution with uniform
distribution as a special case. It can be seen that
Cardiod distribution can be approximated, for a
small value of κ, by von Mises pdf. The Cardiod
distribution then may be obtained by [9]
 κ
C  µ ,  ~ M ( µ , κ ) ,
 2 
(7)
where C (*,*) and M (*,*) stand for Cardiod and
von Mises, respectively.
Therefore, it is
reasonable to use the von Mises distribution to
model the elementary phase of the fading
channel in the remainder of this paper.
When the von Mises distribution is used to
modeled the individual phase of multipath fading
channel in wireless communication systems, the
exact envelope distribution can be evaluated by
[1]
f R (r ) = r
×e
∞ N

∫
∫0 ∏
n=1 
jrn q cos(ϕn −ϑ )
∞
0
f Rn (rn ) ∫
2π
0
d ϕn drn  J 0 (qr )qdq ,

fϕn (ϕn )
(8)
0.4
Von Mises
Cardiod
---------
0.35
κ=1
4.
0.3
0.5
fϕ(ϕ )
0.25
0.2
0.2
0.15
0
0.5
0.1
0.05
0
-3
-2
-1
0
1
2
3
ϕ
Fig.2:: Density of the von Mises distribution
π
for µ = and κ = 0,1, 3, 8,12 .
2
ϑ = tan−1 (ωc , ω s )
where
is the angle
between Xc and Xs in frequency domain,
f Rn (rn ) is the pdf of the elementary
envelope and J 0 ( ⋅) is the Bessel function of
the first kind of order zero. Substituting (6)
in (7), and evaluating the inner integral,
yields
J=
2π
∫
0
∫
2π
×
0
=
pϕn (ϕn ) e
jrn q cos(ϕn −ϑ)
d ϕn
=
1
2π I 0 (κ )
exp (κ cos (ϕn − µ ) + jrn q cos (ϕn − ϑ)) d ϕn


2
I 0  κ 2 − ( rn q ) − j 2rn q cos (µ − ϑ)


I 0 (κ )
. (9)
Substituting (9) in (8), the exact pdf of the
envelope with the elementary phase being the
von Mises distribution is given by
f R (r ) =
×I 0
(
∞ N
r
I0

∏  ∫
(κ ) ∫
0
n =1
∞
0
f Rn (rn )
)

2
κ 2 − (rn q ) − j 2rn q cos (µ − ϑ) drn  J 0 (qr )qdq

(10)
Note that (10) shows that the fading channel
under some certain conditions may have a
complex envelope probability. Therefore, the
combining signals in diversity detection
should have a mechanism to take the
statistics of the imaginary part and phase of
the pdf into consideration so as to optimally
devise the detection outcome.
NUMERICAL ANALYSIS
In this section, some examples of the complex
pdf are provided to validate the results. Figure 3
shows the comparison of the exact envelope pdf
by plotting the real part of (10) the concentration
parameter κ as a parameter, mean direction
µ = π 4 and the number of fading channels
N = 8 , respectively. As expect, when the
concentration parameter decreases to zero (i.e.,
uniform distribution), the exact envelope pdf
(real part of complex pdf in (10)) is exact
Rayleigh regardless of the mean direction or
center of the von Mises pdf. Also for the case of
symmetrical pdf, the figure 4 is plotted with the
mean direction µ as a parameter, concentration
parameter κ = 0.2 and the number of fading
channels N = 8 , respectively. As the mean
direction µ reduces to zero (i.e., symmetrical
about zero), the exact envelope pdf is
characterized as the Rayleigh distribution
regardless of the density of the elementary phase
and envelope f Rn (rn ) .
5.
CONCLUSION
The complex pdf that is generally used in
quantum physics is investigated. It can be
shown that for some specific phenomena, the
complex pdf can be occurred to model the
envelope distribution. This complex pdf has
some useful meanings in explaining the
characteristics of multipath fading channels in
terms of envelope pdf and correlation. The
example of elementary phase that is widely used
in characterizing the angle of arrival and also has
other distribution as special cases is given. Then
the exact envelope pdf are derived. To validate
the results, some graphical presentations are
provided and compared with some specific cases
of Rayleigh distribution. Numerical results
presented showed that the complex pdf can
provide a useful tool for performance analysis.
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= 0.3
= 0.2
= 0.1
= 0, Rayleigh
fR(r)
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κ
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
r
Fig.3: Comparison of the envelope pdf of the
multipath fading channel when the elementary
phase is von Mises distributed with
concentration parameter κ as a parameter.
κ = 0.2 N = 8 κ = 0.2 N = 8
0.8
[7] P. Beckmann and A. Spizzichino, The
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N =8
0.7
0.6
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=π 3
[8]
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fR(r)
0.5
=π 4
= 0, Rayleigh
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
r
Fig.4: Comparison of the envelope pdf of the
multipath fading channel when the elementary
phase is von Mises distributed with mean
direction µ as a parameter.
4