International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
A Novel Phase Model for Nakagami-m Fading
Channels
Jyotheeswar K M1, C.Subhas2
1
Student, 2 Professor
Department of Electronics and Communication Engineering,
Sree Vidyanikethan Engineering College, TIRUPATI – 517 102, A. P., INDIA
Abstract—The Nakagami-m distribution approximates a
Rayleigh for = , Rician for > and Hoyt for . < <
distributions. Utilizing the characteristic function (c.f.) of a
gamma Random Variable (RV) and by approximating it to
that of a Rician and Hoyt RVs, a novel Probability Density
Functions (p.d.f.) are obtained for the real and imaginary parts
of the complex Nakagami-m fading gain.
Keywords—Nakagami-m, Characteristic functions,
distribution, Rician distribution, phase model.
1.
Hoyt
INTRODUCTION
With the advent of mobile technology, there has been a
rigorous and widespread study on the wireless channels.
These studies include modeling the physical phenomena that
are characteristic of those channels and mitigating the
effects of the channel on the transmitted data. One such
important characteristic of a ‘no-wire’ channel is ‘fading’.
The fundamental cause for the fading effect is that the
transmitting antenna, generally in a mobile handset, is at the
ground level. While the receiving antenna may be at a few
tens of meters, the presence of buildings, power scatters, and
sharp edges, make the received signal fluctuate very rapidly,
which is termed as fading. There has been a significant
research to model this effect so that mitigating techniques
can be easily described and further applied to the fast
growing wireless technology. Physical modeling of such an
effect offers the flexibility of easy analysis for any user.
Modeling, generally, considers overall effect as a black box
effect, where the end user, may not have to explicitly
consider the whole phenomenon. Hence the user can easily
expect what can be the output for a given input.
Since, the received signals are totally random in nature,
the study of random variables are very much needed.
Specifically, probability density functions (p.d.f.s), which
give the probability ‘that the random variable taking a
particular value’, are required. In recent years, Nakagami-m
p.d.f gained much importance as it models well many
experimental data.
ISSN: 2231-5381
In the literature, there exists p.d.f. for the Nakagami-m
distribution [3] and also the phase distribution considered is
uniform in nature. In this paper, the real and imaginary parts
of a complex Nakagami-m fading gain are derived and from
this the phase distribution is derived. The paper is organized
as follows. After this introduction, in section 2, the actual
characterization of the real and imaginary parts of the fading
gain is given. In section 3, the required phase distribution is
obtained. In section 4, numerical results are shown to
validate the obtained results.
2. CHARACTERIZATION OF REAL AND
IMAGINARY PARTS OF COMPLEX NAKAGAMIm FADING GAIN
Let a signal transmitted through a wireless channel has
undergone Nakagami-m fading with parameter value m and
mean square value Ω and let its envelope be R, and in order
to obtain the p.d.f. of phase, let Θ be the random phase.
Then
R ℐ = X + ℐY.
(1)
where X and Y are the in phase and quadrature components
of R ℐ
If R is a Nakagami-m random variable, then the c.f. of , is
given as
Ψ (ℐω) = ℐ
=Ψ
(ℐω) =
ℐ
Ω
(2)
In order to obtain the novel model, it is instructive to
consider two necessary cases, viz., > 1 and ≤ < 1.
From [2], it is evident that the fading parameter ‘m’ is
inversely proportional to the fading severity i.e., for small m,
the p.d.f. experiences a high fading level and
=∞
corresponds to a deterministic signal. According to central
limit theorem, the p.d.f. of a sum of (large number of)
random variables approximates a Gaussian density function
of zero mean.
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For large m ( > 1), the p.d.f. of R has a non-zero mean,
since a line of sight can be expected (in order to decrease the
fading severity). This case is more predominant in rural and
in some parts of sub urban areas. In order to get the p.d.f. of
X and Y, without loss of generality, we assume that the
variance, which gives the spread of the p.d.f. of R about the
mean, gets equally divided into the in phase and quadrature
components
For small m ≤ < 1 , the p.d.f. of R has a zero mean
and the p.d.f. of X and Y experiences unequal variance
levels.
Considering the above analysis, for > 1 the characteristic
function (c.f.) of X can be approximated to that of a N
Ω
Ω
(µ , ) distribution and the c.f. of Y to that of a N (µ , )
distribution. Further assuming independency of X and Y, the
approximate c.f. of
is given by
+µ
+
,
(10)
Ω =
Ω,
(11)
Ω =
Ω,
(12)
(3)
ℐ Ω
+Ω = Ω,
ℐ
ℐ Ω
≈
Ω
ℐ
ℐ Ω
≈
Ω
ℐ
ℐ Ω(
)
/
(
ℐ Ω(
)) /
,
is valid for the corresponding parameters. Taking square
roots on both sides of (12), we obtain
≤ 1,
(6)
Ω
⎞
⎠
(15)
ℐωΩ
⎞
ℐω
⎝
− 1 , 0≤
ℐωΩ
⎝
≈
µ
⎛
=
µ
ℐω
(5)
where
(14)
ℐ Ω
⎛
(ℐω) ≈
(13)
ℐ Ω
The relation
(4)
For ≤ < 1, the c.f. of X can be approximated to that of
a N (0, Ω (1+ )/2) distribution and the c.f. of Y to that of a
N (0, Ω (1− )/2) distribution. Further assuming the
independency of X and Y, the approximate c.f. of
is
given by
=
ℐ
ℐ Ω
Case when m > 1 (m is large)
Since in this case, we can expect a direct LOS component,
the mean of the received signal will be non-zero. This can
be concluded by considering the Central limit theorem,
which derives that the p.d.f. of sum of the random variables
will be of Gaussian with zero mean. So, the below equation
is valid
ℐ
(ℐω) ≈
µ
We put the condition that
+
must have the same c.f.
as that of
+
given by (2), the conditions must hold.
⎠
Substituting
implying that R approximates a Hoyt (or Nakagami-q)
distribution with
= µ , Ω
=Ω ,
=
(16)
in (15), we obtain
=
,0≤
≤1,
(7)
Let U and V be independent Nakagami distributed random
variables with fading parameters
and , respectively,
and mean square values Ω andΩ , respectively. This
implies
and
are gamma distributed with means Ω
and Ω , respectively, and parameters
and ,
respectively, so, we get
(ℐω) =
ℐ
(ℐω) =
ℐ
Ω
,
ℐ
(
ℐ
,
(17)
)
ℐ
Similarly, substitution of
= µ ,Ω
= Ω ,
=
(16)
(8)
in (15) results in
Ω
,
(9)
ℐ
ℐ
≈
ℐ
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ℐ
≈
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(
ℐ
,
(18)
)
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
If the means
and
of the random variables
respectively (see (8) and (9)), satisfy
=
+
,
=
+
and
,
,
(19)
=
,
(20)
Then we conclude the following
If we choose the distribution of |X| to be the same as the
distribution of U and the distribution of |Y| to be the same as
the distribution of V, then, for
> 1/2,
> 1/2, and
condition (18), the c.f.s of
and
are given, respectively,
by the approximations
ℐωµ
ℐωΩ
(ℐω) ≈ (
( ) = (1 −
)g( ;
,
(21)
,
(22)
( )=
+
ℐωΩ )
(
(
=2
,
(
)
−
)
−
)
,
,
(24)
)=
| |(
)
=2
(
)
(
)
−
, x > 0
)=
| | (−
−
0,x ≥ 0
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(30)
(
−
)
1
+
2
( ;
)
)
,
(− ;
)
,
(31a)
−
, x < 0
(
)
)
( ;
)
,
(− ;
,
)
(31b)
µ =
1−
√Ω cos φ,
(32)
µ =
1−
√Ω sin φ,
(33)
(26)
Ω =
)
(
+
The parametersµ , µ , Ω , Ω ,
,and
in are
expressed in terms of the parameters m,Ω , and φ as
(25)
On the other hand (− ;
, ) denotes the p.d.f. of a
negative-sided Nakagami random variable with parameter
and mean square value , and is given by
,
)
(23)
0,x ≤ 0
(− ;
(
1
+
2 2
( )=
Let ( ;
, ) denote the p.d.f. of a positive-sided
Nakagami random variable with parameter
and mean
square value , given by
( ;
(29)
Therefore, the p.d.f. of X is given as
+
)=2
(
= −
It is clear that if eq. (21) and (22) are satisfied, then the
p.d.f. of | | and | | are given by
| |(
)
,
A similar procedure on Y gives the following p.d.f.
ℐωµ
ℐωΩ
(
|
g(− ;
ℐωΩ )
(ℐω) ≈ |( ) = 2
)+
,
To find , we use the method of mean matching. Since the
distribution of X approximates a Gaussian distribution with
mean , we obtain
Such that, from (10), (11) and (12)
=
In order to obtain, the p.d.f. of X and Y, we linearly
combine the above two functions as follows.
Ω =
Ω
Ω
1+
1−
cos 2
,
(34)
1−
1−
cos 2
,
(35)
(27)
=
1+
1−
cos 2
,
(36)
=
1−
1−
cos 2
,
(37)
(28)
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
Case when 1⁄ 2 ≤
< 1 (m is small)
In this case, the sum of sinusoids obtained at the receiver
has the p.d.f. of mean zero and of different variances. So, we
can approximate the corresponding c.f.s of Hoyt and gamma
RVs and further substituting Ω = Ω and m = m we
obtain
=
,
(38)
Ω
ℐ
(
ℐ
Ω =
Ω
Ω
,
(46a)
< ∞, ≠ 0,
A similar procedure on Y gives the following p.d.f.
1
( ) = [ ( ;
2
,
(
ℐ Ω )
1+
− 1 ,
(40)
1−
− 1 ,
(41)
1+
−1 ,
(42)
) + (− ;
,
| |
=
(39)
−∞ <
)
< ∞,
−
)]
,
,
(46b)
≠0
0, = 0
=
Further in order to extend the obtained results, we try to
obtain the magnitude distribution from the joint distribution
function, so that the results can be validated with that of the
original Nakagami –m distribution.
We have that
=
−
= 0, = 0,
If the means Ω and Ω and the parameters m and m of
the random variables
and , respectively satisfy
Ω =
)
ℐ Ω )
=
(
(
−∞ <
Similar substituting of Ω = Ω and m = m gives
Ω
| |
=
,Θ (
, )=
( cos ) ( sin )
(47)
In order to get the magnitude marginal distribution of R, we
integrate the above joint function w.r.t to i.e.,
=
1−
− 1 ,
(43)
Then we conclude the following:
If we choose the distribution of | | to be the same as the
distribution of U and the distribution of | | to be same as
the distribution of V, then the c.f.s of
and are given,
respectively, by the approximations
∞
( ) =∫ ( cos ) ( sin )d By substituting, the required equations into eq. (51), we
finally arrive at
( )=2
(
,
(
)
) (
)
(1 −
(ℐω) ≈
)(1 −
) + (1 −
)
+
ℐ Ω
]
(49)
3. DISTRIBUTION OF PHASE OF COMPLEX
NAKAGAMI-m FADING GAIN
(45)
Similar to the case of > 1, the p.d.f.s of | | and | | are
given by g (x, ;,) and g (-x,:,) functions
By mean matching, the value of is obtained as
=
Therefore, the p.d.f. of X is given as
We now find the distribution of the phase Θ corresponding
to the novel model given by the p.d.f.s (31) and (46) for the
cases of > 1 and1⁄2 ≤ < 1, respectively.
Phase distribution when > 1
From (28) and the fact that = cos Θ , =
joint p.d.f. of R and Θ is expressed as
,
)+
(ℐω) ≈
ISSN: 2231-5381
× [(1 −
(44)
ℐ Ω
( ) = [ ( ;
(48)
) + (− ;
,
)]
,Θ (
sin Θ , the
, ) = (1 − )(1 − ) ( cos ;
( sin ; , )
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,
)×
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
+ (1 −
(− cos ;
+
+(1 −
) (− cos ;
) × ( sin ;
,
) × (− sin ;
,
)( ) ( cos ;
)
,
) × ( sin ;
,
)
,
The phase distribution given by [3] can be obtained from
(49) by putting µ = µ ≪ Ω ⁄2 and
=
=
⁄2 , which results in
)
,
, ⁄
0 < < ∞, − < < Θ(
∞
) =∫ ( cos ) ( sin )d +
−
−
(
)
(
)
(
)
(
+
×
+
×
+
×
−
)
×
−
(
)
(
)
(
)
(
( )+
( )+
( )+
)
( )
(52)
where
2 (
( ) =
(
)
| cos |
)
+
) (
| sin |
<1
The p.d.f. of phase Θ is given by ( ) = ∫
which results in
( )=
(
(
)
) (
)
|cos |
− <
4.
|sin |
,Θ (
, )
(53)
<
For the case when > 1 , the original p.d.f.s of the real part
X and the imaginary part of Y of the complex Nakagami-m
fading gain R ℐ , are simply the Gaussian functions with
the corresponding means and variances. The original phase
p.d.f. is given by
1
(
)
( )≈
1 + 2√
cos( − ) × 1
2
− √2 cos( − )
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<
(55)
( )≈
(
(
(56)
))
Fig.1. illustrates the probability density functions of X and
Y for the case of large m along with that of the original
Gaussian approximation.
Fig.2. shows the p.d.f. of phase Θ of the complex
Nakagami-m fading gain. Also shown are the Rician
approximation and the
approximation.
⁄
Also, it is clear that the
has
four
modes and
⁄
only one mode matches well with the Rician approximation,
which clearly states that
⁄ is only applicable
3
for = ± 4 and = ±
4
Fig.4. gives the plots of phase Θ along with the Hoyt
approximation and the
⁄ approximation. We
find that there are singularities at = 0
= ± .These
singularities corresponds to the high probability
concentrations of the phase around = 0 and = ± ,
implying that the real and imaginary parts of the fading gain
have high probability of being either equal or antipodal.
Fig.5 shows the magnitude distribution for = 2 (for large
m case) of the novel model along with the original p.d.f.
given below
NUMERICAL RESULTS
− <
|cos sin |
Fig.3. shows the p.d.f of the novel model for X and Y for the
case of small m. Since the p.d.f. of X and Y are obtained by
linearly combining the positive sided and negative sided
p.d.f. this novel model deviates from its Hoyt approximation
at zero. The same discrepancy can be seen also for the case
of large m.
i=1 corresponding to first quadrant and similar cases
Phase distribution when 1⁄ 2 ≤
( ⁄ )
And we call this the
⁄ characterization. For
the case1⁄2 ≤ < 1, similar plots are obtained and the
original phase function is given as
(51)
Now it can be easily seen that
( )=
( )
(50)
by (30), The p.d.f. of Θ is
where g(.) is given by (26), and
given by
( )=
( ;
, Ω) = 2
Fig.6. shows the same for
Ω
( )
−
(57)
Ω
= 0.9 (for small m case).
(54)
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
Fig.1. PDFs of real part X and imaginary part Y of complex
Nakagami-m fading gain and of corresponding Rician
approximation for m=5, φ = π/6, and Ω=10.
Fig.3. PDF of real part X and imaginary part Y of complex
Nakagami-m fading gain and of corresponding Hoyt approximation
for m=0.9 and Ω=10.
Fig.2. PDF of phase Θ of complex Nakagami-m fading gain and of
corresponding Rician approximation for m=5, φ = π/6 and its
comparison with PDF of Θ obtained from Nakagami-m/2
approximation.
Fig.4. PDF of phase Θ of complex Nakagami-m fading gain and of
corresponding Hoyt approximation for m=0,9 and its comparison
with PDF of Θ obtained from Nakagami-m/2 characterization.
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
5. CONCLUSION
In this paper, a new model for Nakagami-m fading
channels has been proposed. This is obtained by
approximating the model with the distributions of Hoyt
and Rician distributed envelopes. The phase distribution
here derived is non uniform in nature and the results
confirm the validity of the model for the two cases > 1
and0.5 ≤ < 1.
REFERENCES
Fig.5. Magnitude distribution for m=2
[1].Ranjan K. Mallik,”A New Statistical Model of the complex
Nakagami-m Fading Gain”, IEEE Trans. Commun vol. 58, no. 9,
september 2010.
[2].M. A. Taneda, J. Takada, and K. Araki, ”The problem of the fading
model selection”, IEICE Trans. Commun, vol. E84-B, no.3. pp.
660.-666, Mar. 2001.
Jyotheeswar
completed
his Bachelor
B.Tech
[3]M. Nakagami,
S. Wada,
and S. Fujimura,
“Some degree
considerations
on
random
phase
problems
from
the
standpoint
fading”,
Inst.
(ECE) from Sreenivasa Institute of Technology and
Electrical Commun. Engineers Proc., vol. 36, no. 11, pp. 595-602,
Nov.
1953
Mangagement
Studies, Chittoor and is currently
[4].M.D. Yacoub, G. Fraidenraich, and J.C.S. Santos Filho,
pursuing phase-envelope
his master’s degree
at Sree Vidyanikethan
“Nakagami-m
joint distribution”,
Electronn. Lett.,
vol. 41, no.5, pp.259-261, Mar. 3, 2005.
Engineering College, Tirupathi.
[5].N. C. Sagias and G. K. Karagiannidis, “Effects of carrier phase
error on EGC receivers in correlated Nakagami-m fading”, IEEE
Commun. Lett., vol.9, no.7, pp. 580-582, Jul 2005.
[6].G. Fraidenraich,
C. Subhas M.
didD.his
Yacoub,
Ph.D.J.inR. wireless
Mendes, and
communications
J. C. S. Santos at
Filho, “Second order statistics for diversity combining of nonJawaharlal
Nehru IEEE
Technological
University
Anantapur,
identical
Hoyt signals”,
Trans. Commun.,
vol. 56,
no. 2,
pp,183-188,
Feb, 2008.
Anantapur,
A.P. He has over 26 years of Industry,
Research, Academic and Administrative experience.
Presently he is with Sree Vidyanikethan Engineering
College, Tirupati, A.P. as Professor of Electronics and
Communication Engineering and Dean (Academic). He
has eight publications in international journal with good
impact factors and ten presentations in IEEE international
conferences. His interests of research are wireless
communications and signal processing. He is a member of
IEEE and IEICE of Japan, and Fellow of IETE.
Fig.6. Magnitude distribution for m=0.6
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