CAPACITY OF DUAL-BRANCH MRC SYSTEM OVER CORRELATED NAKAGAMI-M FADING CHANNELS WITH NON-IDENTICAL FADING PARAMETERS AND IMBALANCED AVERAGE SNRS

Abstract

Abstract: We present the closed-form expressions for the capacity with correlated, non-identical fading parameters and imbalanced average SNRs (Signal-to-Noise Ratios) of dual-branch MRC (Maximal Ratio Combining) diversity system over Nakagami-m fading channels. This capacity is evaluated for ORA (Optimum Rate Adaptation with constant transmit power) and CIFR (Channel Inversion with Fixed Rate) schemes. Numerical results have been presented and compared with the available capacity results of ORA and CIFR schemes in the literature. The effect of different practical constraints, e.g. non-identical fading parameters, fade correlation and level of imbalanced in average SNRs on the channel capacity of the systems, is analysed.

Specification

Description:FORM 2
THE PATENTS ACT, 1970
(39 OF 1970)
&
THE PATENTS RULES, 2003
COMPLETE SPECIFICATION
(See section 10; rule 13)
Title: Capacity of dual-branch MRC system over correlated Nakagami-m fading
channels with non-identical fading parameters and imbalanced average SNRs
APPLICANT DETAILS:
(a) NAME: GRAPHIC ERA DEEMED TO BE UNIVERSITY
(b) NATIONALITY: Indian
(c) ADDRESS: 566/6, Bell Road, Society Area, Clement Town, Dehradun - 248002,
Uttarakhand, India
PREAMBLE TO THE DESCRIPTION:
The following specification particularly describes the nature of this invention and the manner
in which it is to be performed.
152 B.D. Pukhrambam et al.
1 Introduction
Demand for wireless communication services is growing
rapidly and requires high data rate and reliability which is
difficult to achieve owing to multipath fading in channels
(Khatalin and Fonseka, 2006a). Channel capacity is a
significant parameter in designing wireless communication
systems as it determines the maximum achievable data rate
(Khatalin and Fonseka, 2006a; Khatalin and Fonseka, 2007).
The channel capacity in the literature is also known as
average channel capacity as it is achieved by averaging the
Shannon capacity of an AWGN channel. Improvement in
channel capacity can be achieved by increasing transmit
power of the signal well enough to reduce the effect of fading
but it results in increase in level of interference (Khatalin and
Fonseka, 2006a). Diversity combining and/or adaptive
transmission schemes are other techniques employed in
fading environment to improve the channel capacity without
the need of increasing transmit power and/or bandwidth
(Khatalin and Fonseka, 2006a). Pure diversity combining
techniques are of three types: Maximal Ratio Combining
(MRC), Equal Gain Combining (EGC), and Selection
Combining (SC) (Khatalin and Fonseka, 2006a). Adaptive
transmission schemes already available in literature are
Optimum Rate Adaptation with constant transmit power
(ORA), Optimum Power and Rate Adaptation (OPRA),
Channel Inversion with Fixed Rate (CIFR), and Truncated
Channel Inversion with Fixed Rate (TIFR) (Khatalin and
Fonseka, 2006a; Khatalin and Fonseka, 2007). Under ORA
scheme, data rate is adapted by the transmitter according to
the channel fading conditions while the transmit power
remains constant (Khatalin and Fonseka, 2006a). Under
OPRA scheme, transmitter can realised optimum capacity by
transmitting appropriate power and data rate in accordance
with the channel conditions (Khatalin and Fonseka, 2006a;
Khatalin and Fonseka, 2007; Goldsmith and Varaiya, (1997).
In CIFR scheme, power is adapted by the transmitter to
maintain constant Signal-to-Noise Ratio (SNR) in order to
invert the channel into additive white Gaussian noise
(AWGN) channel (Simon and Alouini, 2005). TIFR is the
modified version of CIFR. In case of CIFR, there is power
penalty when the channel is in deep fade condition.
Therefore, we use a modified version of CIFR, which is
called TIFR scheme, in which power can be adapted around a
cut-off value below which data transmission is suspended
(Simon and Alouini, 2005; Alouini and Goldsmith, 1999).
Average channel capacity under various fading channels
is a well-researched topic. Khatalin and Fonseka (2006a,
2007) and Goldsmith and Varaiya, (1997) discuss the average
channel capacity under correlated Nakagami-m (m = 1, 2)
fading with equal average SNRs under ORA and CIFR
schemes for different combining techniques. In Simon and
Alouini (2005), capacity expressions of MRC using different
scheme of adaptive transmission over uncorrelated Rayleigh
fading have been obtained. Capacity using different schemes
of adaptive transmission and techniques of combining over
uncorrelated Rayleigh fading is presented in Alouini and
Goldsmith (1999, 1997). Subadar and Sahu (2010) present
the analysis of capacity of dual-SC and MRC systems over
fading channels using correlated (? = 0, 0.3 and 0.6)
Nakagami-m with non-identical (m1 = 1, m2 = 2) and arbitrary
fading parameters. In Khatalin and Fonseka (2006b), capacity
expression of MRC over uncorrelated Hoyt and Rician fading
channels using ORA scheme is extended by obtaining lower
and upper bound expressions. Average channel capacity of
dual-branch MRC using OPRA and TIFR schemes under
uncorrelated and correlated fading channels using Nakagami0.5 were obtained in Hasan and Kumar (2014, 2015a),
respectively. In Hasan and Kumar (2015b, 2015c), average
channel capacity of dual-branch SC with ORA and CIFR
schemes over fading channels using correlated and
uncorrelated Nakagami-0.5 was obtained. In Hasan and
Kumar (2017), spectral efficiency of dual-branch SC with
ORA, TIFR and OPRA schemes over correlated Nakagami0.5 with unequal average received SNR was obtained. In
Shao et al. (1999), results of Goldsmith and Varaiya (1997)
have been extended by obtaining the expression for channel
capacity of MRC diversity systems with the fade correlation
and imbalanced average SNRs between the branches. Closedformed expression for the channel capacity of MRC over
independent Nakagami-q fading under different scheme of
adaptive transmission is obtained in Cheng and Berger
(2003). Da Costa and Yacoub (2007) present the average
channel capacity under ? ? ? and k ? ? fading channel.
Performance analysis of wireless system with MIMO under
Hoyt fading channel is presented in Pena-Martin et al. (2013).
BER performance of MRC over Nakagami-m is analysed in
Aalo (1995). Capacity of single branch MRC with Rayleigh
fading channel under ORA, CIFR and OPRA with
consideration of effect of correlation on capacity has been
analysed in Mallik et al. (2004). Capacity of MRC over
arbitrary Rician fading channel has been obtained in Hamdi
(2008). Annavajjala and Milstein (2004) obtained the
expression of capacity of Rayleigh channel with MRC, EGC
and SC taking into account level of imbalanced in mean
signal strength and effect of correlation. However, an
analytical study on capacity over correlated Nakagami-m
fading channels with identical fading parameter (m > 1) and
non-identical fading parameters with first branch Rayleigh
and second branch Rician (m = 3) considering imbalanced
average SNRs under ORA and CIFR schemes using MRC
has not been considered so far.
Generally, Nakagami-m is used to a great extent to study
the performance of wireless mobile communication system
(Khatalin and Fonseka, 2006a), less concentration appears
to have been focused on non-identical fading parameters
and fade correlation under Nakagami-m channels. In a
number of real-life scenarios, physical space constraints
may not permit antenna spacing that is required for
constantly uncorrelated fading across diversity branches
(Subadar and Sahu, 2010). At the same time, branches may
experience different fading severity and imbalanced SNRs.
So, in cases where quality of service with high data rate
requirements mandate designing for real-life fading
Capacity of dual-branch MRC system 153
scenario, results obtained for the same fading conditions
will have great practical usefulness in wireless mobile
environments. That allows the researchers or system
designers to perform comparison and trade-off studies
among the various adaptive transmission schemes with
optimal combining technique MRC, so as to determine the
optimal choice in the face of their available constraints.
Therefore, this paper fills this gap by presenting the
impact of various practical constraints, e.g. physical space
requirements, effect of fade correlation, non-identical fading
parameters and level of imbalance in the branch average
received SNRs on the channel capacity under Nakagami-m
fading channels.
The remainder of this paper is organised as follows:
Section 2 defined the channel model. In Section 3, channel
capacity of MRC with identical and non-identical fading
parameters considering equal and imbalanced average SNRs
has been derived using ORA and CIFR schemes. Section 4
presents numerically evaluated results. Concluding remarks
are given in Section 5.
2 Channel model
The Probability Distribution Function (PDF), p ? ? ? ? , of the
output SNR, ? at MRC combiner for L-branch correlated
Nakagami-m fading channels with identical (m1 = m2 = 1)
fading parameters and equal average SNRs ? ? 1 2 ? ? ? ? ?
can be presented as (Khatalin and Fonseka, 2006a):
? ?
? ?
? ? ? ?
? ? ? ? ? ?
1
1 1
exp
1
; ;
1 1
, 0
1 1
L m
Lm m m
m m
Lm F m Lm
L
p
L Lm m
?
? ?
? ? ?
??
? ? ??
? ?
? ? ??
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ?
? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ?
(1)
where ? is the average received SNR, m is the fading
parameter, 1F1[.;.;.] is the Kummer confluent hypergeometric
function and ? is the correlation coefficient.
The PDF of ? at dual-branch MRC combiner output over
correlated and imbalanced average SNRs ? ? 1 2 ? ? ? between
the branch is obtained in Alouini and Simon (1998) as:
? ? ? ? ? ?
?? ? ?
1
2 2
1
1 2 2 1 2
exp , 0
m m
m
m p I
m ?
? ? ? ?? ? ?
?? ?? ?
? ? ? ? ? ? ? ?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
(2)
where In(.) is the nth-order Bessel Function and the
parameter ?? and ?? are the normalised form of ? and ?
(Alouini and Simon, 1998), and given by:
? ?
? ?
1 2
1 2 2 1
m ? ?
?
? ? ?
? ? ? ?
and
? ? ? ? ? ?
? ?
1
2 2
1 2 12
1 2
4 1
2 1
m ? ? ?? ?
?
?? ?
?? ? ? ? ?
For m = 1 (Rayleigh), expression of PDF was obtained in
[Alouini and Simon, 1998, equation (12)] as,
? ?
? ? ? ?
? ?
? ? ? ?
2
1 2 1 2 12
1 2
2
1 2 12
4 1
exp 2 1
4 1
p?
? ? ? ? ?? ?
?
?? ?
?
? ? ?? ?
? ? ?? ? ? ? ? ? ? ? ? ? ?
?? ?

? ? ? ?
? ?
? ? ? ?
2
1 2 1 2 12
1 2
2
1 2 12
4 1
exp 2 1
4 1
? ? ? ? ?? ?
?
?? ?
? ? ?? ?
? ? ?? ? ? ? ? ? ? ? ? ? ?
?? ?
(3)
Putting m = 3 and expanding the Bessel function using
(Wolfram, 2015), we can reduce the PDF of equation (2) as
follows:
? ? ? ?
? ? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
3
4 5
1 2
2
1 9
2 1
3 exp exp
exp exp
3 exp exp
p? ? ? ?? ?
? ?? ? ??
?? ? ? ? ? ? ?
?? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ?? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? ? ?? ?? ? ? ? ? ? ?? ? ? ? ?? ?? ?? ??
(4)
The PDF of ? at dual-branch MRC (L = 2) combiner output
with correlated, non-identical (m1 ? m2) fading parameters
and imbalanced average SNRs over fading channels using
Nakagami-m is obtained in Subadar and Sahu (2010) and
Subadar (2011) as:
? ? ? ? ? ?
? ?
? ? ? ?
? ?
1 2
1
1 2 1 2
2
1 2 0 1 2
1
1
2
1
2
12 21
11 1 1
1 2
1 1
! exp 1
, ; 1
m m k
k
k
m m mmr
p r r
m
m k r
m m Fmk r
?
?
?
? ? ? ?
?
? ?
?
? ? ? ?
? ?
?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
(5)
where 112 ? ? mm k ? ? 2 and ? ?k a is the Pochhammer
function, 2
1
m r m ? ? and 1
2
m
m
? ? with m m 2 1 ? .
Different values of m1 and m2 describe several important
distributions. Like m1 = m2 = 1 distribution reduces to
Rayleigh, ?m m 1 2 ? ? ? 1 corresponds to Rician distribution
and m1 = 1 and m2 > 1 describe the non-identical nature of
the fading parameter with first branch experiencing
Rayleigh distribution and second branch Rician distribution.
154 B.D. Pukhrambam et al.
Replacing 1F1 [.;.;.] in its infinite series form as given in
Wolfram (2015), equation (5) can be written as:
? ? ? ? ? ?
? ?
? ?? ?
? ?? ? ? ?
1 2
1
1 2 12
2
1 2 0 0 1 2
1
12 21 1 1
1 2 2
1 1
2
1 1
1
! ! exp 1
m k m
k t
t t
k t
t
m m mmr p r r
m m m mk
r m k t r
?
?
?
? ? ? ?
? ? ?
? ? ?? ?
?
? ?
? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
(6)
Expanding the Pochhammer function using Wolfram (2015),
equation (6) can be reduced as:
? ? ? ? ? ?
? ?
? ?
? ?? ? ? ?
1 2
1
1 2 12
2
1 2 0 0 1 2
1
12 21 1
1 2 2
1 1
2
1 1
1
! ! exp 1
m k m
k t
t t
m m mmr p r r
m m m kt
r m kt m t r
?
?
?
? ? ? ?
? ? ?
? ? ? ?
?
? ?
? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ??? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?
? ?
(7)
3 Channel capacity
In this section, closed-form expressions are derived for the
channel capacity of dual-branch MRC under correlated,
identical and non-identical fading parameters for Nakagamim fading considering equal and imbalanced average SNRs
between the branches using ORA and CIFR schemes. Similar
to the assumption made in Alouini and Goldsmith (1999), an
error-free feedback path with negligible time delay is
assumed while considering adaptive transmission schemes.
3.1 ORA
The channel capacity of fading channels under ORA scheme
with received SNR distribution p? ?? ? is defined in Simon
and Alouini (2005) as:
2 ? ? ?? 0
log 1 C B pd ORA ? ? ??
?
? ? ? (8)
where B[Hz] is the bandwidth of the channel and p? ?? ? is
the PDF of the output SNR, ?.
Substituting equation (7) in equation (8), channel capacity
of dual-branch MRC with correlated, non-identical and
imbalanced average SNRs over Nakagami-m fading can be
obtained as:
? ?
? ? ? ?
? ?
? ?? ? ? ?
? ?
1 2
1
1 2
1 2
12 12 21
2 2
0 0 12 12
1
1
2
1 1 0 2
2
1
1 1
log 1 ! !
exp 1
ORA
m m
k t
k t
t
m m C B r
mmr m m
r r
m kt
d
kt m t m
r
?
? ?
? ?
?? ??
? ? ? ?
?
?
? ?
? ?
? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?? ? ? ? ?? ? ? ? ? ?? ?
? ??
?
? ?? ? ? ? ? ? ? ?
? ?
?
(9)
The integral part in equation (9) can be solved using partial
integration as follows:
?? ?? 0 0 0
u dv u v u v vdu lim lim ? ?
? ?
?? ?
??? ? ? (10)
Let
u ? ln 1? ? ? ? (11)
Then
1
d du ?
? ? ?
(12)
Now let
? ? 1 1 2
2
exp 1
t m dv d r
? ? ? ?
?
? ? ? ? ? ?? ? ? ? ?

Integrating the above expression, we obtain:
? ? 1 1 2
2
exp 1
t m v d r
? ? ? ?
?
? ? ? ? ? ?? ? ? ? ? ? (13)
Putting the values of equations (11), (12) and (13) in
equation (10), we get:
? ?
? ?
? ?
1
1
1
2
0 2
2
1 2
0 2
log 1 1.443
exp 1
1
exp 1 1
t
t
d
m
r
m d d
r
?
?
? ? ?
?
?
? ? ??
? ?
? ? ?
?
? ?
? ?? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ?

Integrating the above expression using Gradshteyn and
Ryzhik (2000) we get:
? ?
? ?
? ?
? ? ? ? ? ?
1
1 1
1
2 1
0 2
2
1
22 2
22 2 0
log 1 1.443 G
exp 1
exp G , 11 1
t
z t t
z
d t
m
r
mm m z rr r
?
? ?
? ? ??
?
?
?? ?
? ? ?
? ? ? ?
?
? ? ?? ? ? ? ? ? ? ?
? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ?
?
?
(14)
where ?[.,.] is the incomplete gamma function.
Substituting equation (14) in equation (9), we get:
? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ? ? ?
1 2
1 1
1 12 2
2
1 2 0 0 1 2
1 12 21 2
1 12 2
2 22
2 22 0
1.443
1 1
G
exp ! !G 1 1
G , 1 11
ORA
m k m
k t
t
t z t
z
m mmr m
C B
r r
m kt mm m
kt m r r
m mm z r rr
? ?
? ? ? ?
? ?
?? ?
? ??
? ?
? ?
? ? ?
?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?? ? ? ? ?? ? ? ? ? ?? ?
? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ?? ? ? ?
??
?1
?
(15)
Channel capacity of dual-branch MRC with correlated and
identical fading parameter for Nakagami-m fading can be
Capacity of dual-branch MRC system 155
obtained by putting L = 2, replacing 1F1[.;.;.] with its series
representation (Wolfram, 2015), and then substituting
equation (1) into equation (8) as:
? ? ??
? ? ? ?
? ?
? ? ? ?
2
0
2 1
2
0
1
2
2 !
log 1 exp
1
ORA
m
m
k
k
k k
k m
B m C
m
m b
m k
m d
? ?
?
? ?? ?
? ?
?
?
?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ?? ? ? ? ?
?
?

After integrating above expression using Gradshteyn and
Ryzhik (2000) we obtained:
? ?
? ?
? ? ? ?
? ? ? ?
? ?
? ? ? ?
2
0
2 1
0
1 1.443
exp 1 1
2
! 1
, 1 1
ORA
m
n
n
z
n m
z
B m C
m
m n
n
m m z
?
? ? ?
? ?
?
?? ??
?
?
? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ?
? ?? ? ? ?? ? ? ? ? ? ? ?? ?
?
?
(16)
Further, for no diversity ? ? L ?1 ,
? ? ? ? ? ?
? ? ? ?
0
1
0
1.443 1
1 !
() !
, 1 1
m k
ORA
k
z
m k
z
B
C mk
m k
m m z
? ?
?? ??
?
?
? ?
?
? ?? ? ??
?
? ?? ? ? ?? ? ? ? ? ? ? ?? ?
?
?
(17)
Similarly, after some mathematical transformation using
Wolfram (2015) and Gradshteyn and Ryzhik (2000), channel
capacity of dual-branch MRC with correlated, identical fading
parameters and imbalanced average SNRs for Nakagami-m
(m = 1 and m = 3) fading is obtained by:
Substituting equation (3) in equation (8) for m = 1 as:
? ?
? ?
? ? ? ?
? ?
? ? ? ?
1 2
1
1
1.443
2 1
exp
exp
ORA
B C
E
E
?? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
?? ? ? ? ?? ? ? ? ? ? ?? ? ???
? ? ? ?? ? ?
? ?? ? ? ? ? ?
? ? ? ? ? ??
(18)
Substituting equation (4) in equation (8) for m = 3 we get
equation (19).
? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ? ? ?? ?
? ? ? ?? ?
3
4 5
1 2
1
1
2 3
2 0
2 3
0
1.443 9
2 1
3
exp
3
exp
exp ,
2
exp ,
ORA
z
z
k
k
B C
E
E
z
k
? ?? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ????
?
?? ?? ? ?
?
?
?
?
? ? ? ? ? ? ? ? ? ? ? ?
??? ? ? ?? ? ?? ? ? ? ? ? ?? ? ??? ?
? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ??
?? ? ? ?? ? ?? ???? ? ?? ? ? ? ?? ?? ? ? ? ? ? ? ? ?? ? ? ?? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ?
?
?
?
? ? ? ?? ?
? ? ? ?? ?
1 2
0
1 2
0
exp ,
3
exp ,
e
e
f
f
e
f
?? ? ???
?
?? ? ? ? ?
?
?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ???? ? ?? ? ? ?? ? ? ? ?? ? ?? ? ?? ?? ? ? ? ? ?? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?
?
?
(19)
When m1 = m2 = m in equation (15), it reduces to equation (17).
Also for m = 1, equation (16) is comparable to [Shao et al.,
1999, equation (15)] and when ? = 0, equation (16) reduces
to [Alouini and Goldsmith, 1999, equation (40)] and
equation (18) reduces to [Shao et al., 1999, equation (15)] for
Rayleigh with imbalanced average SNRs.
3.2 CIFR
The channel capacity (bits/s) of fading channels under CIFR
scheme with distribution of received SNR p? ?? ? is defined
in Simon and Alouini (2005) as:
? ? 2
0
1 log 1 C B CIFR p d ? ?
?
?
?
? ? ? ?
? ? ? ? ? ?
? ?
? ? ?
(20)
Substituting equation (5) in ? ?
0
p d ? ?
?
?
?
? we obtain:
? ?
? ?
? ?
? ?
? ? ? ?
? ?
1 2
1
1 2
0 0 1 2
1
1 2 1
2
0 1 2 2
1
2
12 21
11 1 1
1 2
1
1
1 ! exp 1
, ; 1
m m
k
k
k
p m m d r
mmr m
r m k r
m m Fm k d r
?
?
?
?
? ?? ?
?
? ? ? ?
?
? ? ? ??
? ?
? ?
? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
?
156 B.D. Pukhrambam et al.
Using [Gradshteyn and Ryzhik, 2000, (7.621 (4))], the
above integration can be transformed as follows:
? ?
? ?
? ? ? ?
? ?? ?
? ?? ?
1 2
1
1 2
0 1 2
1
12 2
2
0 2 1 2
1 1 12 21
21 1 1 1
1 1 12
1
1 1
1
, 1; ; !
m m
k
k
p m m d r
mmr m
r r
m k m m Fm k
km m
?
?
?
?
?? ?
? ? ?
? ? ? ? ?
? ?
?
? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?? ? ? ? ? ? ? ? ?
? ? ? ?
?
? (21)
where 2F1 (a, b, c, z) is the Gauss hypergeometric function.
Channel capacity of dual-branch MRC diversity system
with non-identical fading parameters and imbalanced
average SNRs under CIFR scheme can be obtained by
substituting equation (21) into equation (20) as follows:
? ?
? ? ? ?
? ?? ?
? ?? ?
1 2
1
1 2
1 2
1
2 12
2
2 1 2
2
0 1 1
1 1
1 2 21
21 1 1 1
1 2
1
1 1
log 1 1
! !
, 1; ;
m m
k
CIFR
k
m m
r
m mmr
r r C B
m k
kt m
m m
Fm k
m
?
? ?
? ? ?
?
?
? ? ? ?
?
? ?
? ?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?? ?
? ?
? ? ? ? ? ? ?
? ?
? ?
? (22)
Again substituting equation (1) in ? ?
0
p d ? ?
?
?
?
? , and after
integrating using Gradshteyn and Ryzhik (2000), we get:
? ?
? ? ??
? ? ? ? ? ?
2
0
1 2
0
1
2
2 1
!
m
m
k
k m
k
m
p d m
mk b
m ka
k
? ? ?
?
? ?
?
?
? ? ?
?
? ? ? ? ? ? ? ? ?
? ??
?? ?
?
?
(23)
where ? ? 1
m
a
? ? ? ? and ? ? 1
m b
? ? ? ? .
Channel capacity of dual-branch MRC with identical fading
parameter and equal average SNRs under CIFR scheme can
be obtained by substituting equation (23) into equation (20)
as follows:
? ? ??
? ? ? ? ? ?
2 2
2 1
0
1
log 1
2
2 1
!
CIFR
m
k m
m k
k
C
m B
m mk b
m ka
k
?
?
?
? ? ??
?
? ? ? ?
? ? ? ?
? ?? ? ? ? ? ?? ?
? ? ? ? ?
(24)
Further for no diversity ?L ?1? ,
? ?
? ?
? ? ? ?
0
1
1
1
!
m
k
m k
k o
p m d m
a m ka
k
? ?
?
? ?
?
?
? ? ?
?
? ? ? ? ? ? ? ?
? ?? ?
?
?
(25)
Then substituting equation (25) in equation (20) we get:
? ?
? ? ? ?
2
1
log 1
1
!
CIFR
k m
m k
k o
C
m B
m a m ka
k
?
?
? ? ?
?
? ? ? ?
?
?? ?
? ? ? ?? ? ? ? ? ? ? ?
(26)
Substituting equation (3) in ? ?
0
p d ? ?
?
?
?
? and solving the
integration using Gradshteyn and Ryzhik (2000) we get:
? ?
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ?
0
2
1 2 1 2 12
1
1 2
0 2
1 2 1 2 12
1
1 2
2
1 2 12
4 1
2 1
lim
4 1
2 1
4 1
u
p d
E u
E u
? ?
?
?
? ? ? ? ?? ?
?? ?
? ? ? ? ?? ?
?? ?
? ? ?? ?
?
?
?
? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?
?? ?
?
(27)
Channel capacity of dual-branch MRC with identical fading parameter (with m1 = m2 = m = 1) and imbalanced average SNRs
under CIFR scheme can be obtained by substituting equation (27) into equation (20) as follows:
? ? ? ?
? ? ? ?
? ?
? ? ? ?
? ?
2
1 2 12
2 2 2
1 2 1 2 12 1 2 1 2 12
1 1 0 12 12
4 1
C log 1
41 41
lim
21 21
CIFR
u
B
E uE u
? ? ?? ?
? ? ? ? ?? ? ? ? ? ? ?? ?
?? ? ?? ? ? ?
? ? ? ? ? ? ? ?? ? ?
? ? ? ? ? ? ?? ?? ? ?? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ?? ??? ?
(28)
Capacity of dual-branch MRC system 157
Similarly for ? ? 1 2 mmm ? ?? 3 , substituting equation (4) in ? ?
0
p d ? ?
?
?
?
? and solving the integration using Wolfram (2015)
and Gradshteyn and Ryzhik (2000) we get:
? ?
? ?
? ? ?? ? ? ? ? ? ? ? ? ? ?
3
4 5
0 1 2
2
1 1 2 2 0
1 9
2 1
1 1 11 3 lim 3 u
p d
E E
? ?
?
? ?? ? ?
?? ?? ? ?
?? ?? ?? ?? ?
?
?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ?? ? ? ? ? ? ?? ?? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ?? ? ? ?? ?? ? ? ?? ? ? ? ? ???
?
(29)
Channel capacity of dual-branch MRC with identical fading parameter (with m = 3) and imbalanced average SNRs under
CIFR scheme, can be obtained by substituting equation (29) into equation (20) as follows:
? ?
? ? ?? ? ? ? ? ? ? ? ? ? ?
3
4 5
1 2
2
2
1 1 2 2 0
9 2
1
log 1
1 1 11 3 lim 3
CIFR
u
C B
E E
?
?? ?
?? ?? ? ?
?? ?? ?? ?? ?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???? ? ? ? ?? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ???? ? ? ? ? ? ? ?? ?? ? ? ?? ?? ? ? ??? ? ? ? ? ? ????
(30)
4 Numerical results and analysis
In this section, the effect of non-identical fading parameters,
level of imbalanced in average SNRs and correlation
coefficient on the capacity under Nakagami-m fading
channels using ORA and CIFR schemes with MRC
diversity has been presented. Channel capacity over fading
channels of correlated Nakagami-m fading for MRC with no
diversity for Rayleigh channel which was obtained in
Khatalin and Fonseka (2006a) and Simon and Alouini
(2005) is also presented for comparison. We have
considered imbalance in average SNRs as 2 1 ? ? 2? and
m1 = 1, m2 = 3 for non-identical fading parameters.
Figures 1 and 2 show the channel capacity over fading
channels of correlated Nakagami-m considering equal average
SNRs and imbalanced average SNRs ? ? 2 1 ? ? 2? respectively
using ORA scheme for m = 3. As expected, by employing
diversity using MRC and/or increasing average received SNR
? ? 1 ? or ? gives improvement in channel capacity in both
Figures 1 and 2. It is observed that the channel capacity with
dual-branch MRC gives almost identical performance for
1 ? , 2dB ? ? even when branch correlation increases. So for
low value of SNR ? ? 1 i e. . , 2dB ? ? ? the appropriate antenna
spacing required for uncorrelated diversity path at the receiver
end may be ignored.
Figure 3 shows the channel capacity with non-identical
fading parameters over correlated Nakagami-m fading using
ORA scheme. As expected, the channel capacity with no
diversity for m = 3 gives better channel capacity than no
diversity for m = 1. Since the channel capacity improves with
increase in m (Simon and Alouini, 2005). It is seen that under
low SNR value up to 2 dB, channel capacity with no diversity
for m = 3 and channel capacity of non-identical fading
parameters ? ? 1 2 m= m= 1, 3 for lower correlation ? ? i e. . 0.1 ? ?
are almost same. And at higher SNR value ? ? i e. . 8dB ? ? ,
channel capacity of non-identical fading parameters is superior
to the channel capacity of no diversity with m = 1 as expected.
Theoretically, employing diversity improves channel
capacity but in contrary channel capacity of non-identical
fading parameters is inferior to the channel capacity with no
diversity for m = 3 and m = 3 at higher value of SNR
?i e. . 2dB and 8dB ? ? ? ? ? , respectively. It is also observed
that for higher correlation ?i e. . 0.1 ? ? ? , the channel capacity
for m = 1 and m = 3 with no diversity is better than channel
capacity with non-identical fading parameters. This is because
the effect of branch correlation is more on the branch having
less amount of fade (m2 = 3) (Simon and Alouini, 2005) and as
the correlation in the branch increases, the weighting problem
arises at the MRC combiner resulting in decrease in capacity.
So for dual-branch MRC with non-identical fading parameters
and higher correlation ?i e. . 0.1 ? ? ? , it is not necessary to
create diversity as it does not provide any improvement in
capacity under ORA scheme.
Figure 1 Channel capacity over Nakagami-m (m = 3) fading
under ORA scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR, ?=0
dual branch M RC, equal SNR, ?=0.1
dual branch M RC, equal SNR, ?=0.3
dual branch M RC, equal SNR, ?=0.5
no diversity
158 B.D. Pukhrambam et al.
Figure 2 Channel capacity over Nakagami-m (m = 3) fading
under ORA scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, unequal SNR,?=0
dual branch M RC, unequal SNR,?=0.1
dual branch M RC, unequal SNR,?=0.3
dual branch M RC, unequal SNR,?=0.5
no diversity
Figure 3 Channel capacity of non-identical fading parameters
using Nakagami-m fading under ORA scheme
0 5 10 15 20 25 30 0
1
2
3
4
5
6
7
8
9
10
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
no diversity, m=1 (Rayleigh)
no diversity, m=3 (Rician)
dual branch M RC, equal SNR, m1
=1, m2
=3,?=0.1
dual branch M RC, equal SNR, m1
=1, m2
=3, ?=0.3
dual branch M RC, equal SNR, m1
=1, m2
=3, ?=0.5
Similarly, in Figure 4 considering MRC with non-identical
fading parameters and imbalanced average SNRs, the
channel capacity with no diversity for m = 3 gives better
channel capacity than no diversity for m = 3 as expected.
For lower value of first branch average received SNR
? ? 1 i e. . 9dB ? ? the channel capacity with non-identical fading
parameters ? ? 1 2 m= m= 1, 3 for low correlation ? ? i e. . 0.1 ? ?
is better than capacity for m = 1 and m = 3 with no diversity
as expected and at higher value of first branch average
received SNR ? ? 1 1 i e. . 9dB and 14dB ? ? ? ? , even
employing diversity fails to provide better channel capacity
than the capacity of m = 3 and m = 1 with no diversity,
respectively. It is also observed that at higher correlation
? ? i e. . 0.1 ? ? , the channel capacity for m = 1 and m = 3
with no diversity is better than channel capacity with nonidentical fading parameters. Since branch correlation is
more effective when the fading is less severe (m2 = 3)
(Simon and Alouini, 2005), and as the branch correlation
increases it becomes difficult to weight the branch at the
MRC combiner that results in capacity to decrease. So under
ORA scheme, for dual-branch MRC with non-identical
fading parameters and higher correlation ? ? i e. . 0.1 ? ? , it is
not necessary to create diversity as it does not provide any
improvement in capacity from single branch.
Figure 4 Channel capacity as a function of first branch average
SNR under correlated and non-identical fading parameters
for Nakagami-m fading using ORA scheme
0 5 10 15 20 25 30 0
1
2
3
4
5
6
7
8
9
10
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
no diversity,m=1
no diversity,m=3
dual branch M RC, unequal SNR, m1
=1,m2
=3, ?=0.1
dual branch M RC, unequal SNR, m1
=1,m2
=3, ?=0.3
dual branch M RC, unequal SNR, m1
=1,m2
=3, ?=0.5
Figure 5 shows the comparison of channel capacity with
identical and non-identical fading parameters over
Nakagami-m fading using ORA scheme considering equal
average SNRs for ? = 0.1. As expected, identical fading
parameters case for m = 3 gives the largest capacity. Under
ORA scheme, capacity of correlated and non-identical
fading parameters may seem to provide greater capacity
than m = 1 as in non-identical case one path experiences
Rayleigh channel (m1 = 1) and other path experiences
Rician channel (m2 = 3), but in contrary the capacity of nonidentical fading parameter is inferior to Rayleigh channel.
This is because the effect of branch correlation is more on
the second branch having less amount of fading (m2 = 3)
(Simon and Alouini, 2005), so the branch correlation
increases resulting in decrease in capacity.
Figure 6 shows the comparison of channel capacity with
identical and non-identical fading parameters considering
imbalanced average SNRs between the branches over
Nakagami-m fading using ORA scheme with ? = 0.1.
Identical fading parameters with m = 3 and m = 1, equally
gives the largest capacity up to 3 dB point after that m = 3
gives better capacity than m = 1 as expected. The difference
in capacity between m = 3 and m = 1 is less in imbalanced
average SNRs than in equal average SNRs case as the
overall SNR is greater in imbalanced average SNRs as we
consider 2 1 ? ? 2? .
Capacity of dual-branch MRC system 159
Figure 5 Channel capacity over Nakagami-m fading for ? = 0.1
with identical and non-identical fading parameter under
ORA scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,m=1,?=0.1
dual branch M RC, equal SNR,m=3,?=0.1
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.1
no diversity,m=1 (Rayleigh)
no diversity,m=3 (Rician)
Figure 6 Channel capacity with identical and non-identical
fading parameters over Nakagami-m fading for ? = 0.1
under ORA scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, unequal SNR,m=1,?=0.1
dual branch M RC, unequal SNR,m=3,?=0.1
dual branch M RC, unequal SNR,m1
=1,m2
=3,?=0.1
no diversity,m=1 (Rayleigh)
no diversity,m=3 (Rician)
Figure 7 shows the channel capacity with correlated and
equal average SNRs over Nakagami-m fading using CIFR
scheme for m = 3. It is seen that for ? ? 1dB , channel
capacity is almost identical for every ? value. And as
expected for ? ? 1dB , channel capacity decreases as
correlation increases. But channel capacity for ? = 0 and ? = 0.1
gives almost same for each value of average received
SNR. So for ? ? 0.1 and ? ? 1dB required antenna spacing
cannot be avoided.
Figure 7 Channel capacity over Nakagami-m (m = 3) fading
under CIFR scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,?=0
dual branch M RC, equal SNR,?=0.1
dual branch M RC, equal SNR,?=0.3
dual branch M RC, equal SNR,?=0.5
no diversity
Figure 8 shows the channel capacity with correlated and
imbalanced average SNRs over Nakagami-m fading for m = 3
using CIFR scheme. It is seen that for 1 ? ? 0dB , channel
capacity is almost same for all ? value and as expected with
increase in 1 ? and decrease in ?, capacity improves. But ? = 0
and ? = 0.1 gives almost same channel capacity for each
value of 1 ? . 2 1 ? ? 2?
Figure 8 Channel capacity over Nakagami-m (m = 3) fading
under CIFR scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC,unequal SNR,?=0
dual branch M RC,unequal SNR,?=0.1
dual branch M RC,unequal SNR,?=0.3
dual branch M RC,unequal SNR,?=0.5
no diversity
Figures 9 and 10 show the channel capacity with correlated
and non-identical fading parameters over Nakagami-m
fading using CIFR scheme considering equal and
imbalanced average SNRs, respectively. It is seen that CIFR
160 B.D. Pukhrambam et al.
scheme suffers large capacity penalty in deep fading
environments such as m = 1 with no diversity. However, this
penalty diminishes as m increases and/or when diversity
combining is employed. It is also seen that for non-identical
fading parameters (m1 ? m2) channel capacity increases as ?
increases unlike in ORA scheme. Because fading in channel
increases with increase in ? that let the transmit power to
increase so that constant received SNR is maintained. This
increase in transmit power results channel capacity to
increase due to non-identical fading parameters (Subadar
and Sahu, 2010).
Figure 9 Channel capacity with correlated and non-identical
fading parameters over Nakagami-m fading under
CIFR scheme
0 5 10 15 20 25 30 0
5
10
15
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.1
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.3
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.5
no diversity,m=3(Rician)
no diversity,m=1(Rayleigh)
Figure 10 Channel capacity with correlated and non-identical
fading parameters using Nakagami-m fading under
CIFR scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
14
16
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, unequal SNR,m1
=1,m2
=3,?=0.1
dual branch M RC, unequal SNR,m1
=1,m2
=3,?=0.3
dual branch M RC, unqual SNR,m1
=1,m2
=3,?=0.5
no diversity,m=1(Rayleigh)
no diversity,m=3(Rician)
Figures 11 and 12 show comparison of the channel capacity
for identical with non-identical fading parameters over
Nakagami-m fading using CIFR for ? = 0.1 considering
both equal and imbalanced average SNRs, respectively. As
expected, there is improvement in capacity in both the figures
by increasing ? and m and/or by employing diversity. But
the channel capacity of dual-branch MRC with non-identical
fading parameters gives the largest capacity in both the
figures. It is because capacity with non-identical fading
parameter under CIFR scheme increases with increase in ?
(Subadar and Sahu, 2010), whereas capacity with identical
fading parameter under CIFR scheme decreases with increase
in ?.
Figure 11 Channel capacity over Nakagami-m fading for ? = 0.1
with identical and non-identical fading parameters
under CIFR scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,m=1,?=0.1
dual branch M RC, equal SNR,m=3,?=0.1
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.1
no diversity,m=1 (Rayleigh)
no diversity,m=3 (Rician)
Figure 12 Channel capacity using Nakagami-m fading for ? = 0.1
with identical and non-identical fading parameter under
CIFR scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,m=1,?=0.1
dual branch M RC, equal SNR,m=3,?=0.1
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.1
no diversity, m=1 (Rayleigh)
no diversity, m=3 (Rician)
In Figures 13, 14, 15, and 16, comparison of channel capacity
for equal and imbalance average SNRs over correlated
Nakagami-m fading are shown. It is seen that the channel
Capacity of dual-branch MRC system 161
capacity improvement considering imbalanced average SNRs
gives the better capacity than those with equal SNRs as
expected, because the overall SNR of imbalanced average
SNRs is greater than that of equal average SNRs between the
branches as we have considered 2 1 ? ? 2? .
Figure 17 compares the channel capacity of identical
and non-identical fading parameters for ORA with CIFR
schemes considering imbalanced average SNRs between the
branches when ? = 0.1. As expected, in identical cases
channel capacity of dual-branch MRC under ORA schemes
gives greater capacity than CIFR scheme. It is also observed
that for 1 ? ? 3dB the capacity for m = 1 using ORA scheme
is similar to the capacity dual-branch MRC with m = 3 using
CIFR scheme.
Figure 13 Channel capacity over Nakagami-m (m = 3) fading
under ORA scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC,?=0
dual branch M RC,?=0.1
dual branch M RC,?=0.3
dual branch M RC,?=0.5
no diversity
equal SNR
unequal SNR
Figure 14 Channel capacity with correlated and non-identical
fading parameters using Nakagami-m fading under
ORA scheme
0 5 10 15 20 25 30 0
1
2
3
4
5
6
7
8
9
10
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC,m1
=1, m2
=3, ?=0.1
dual branch M RC,m1
=1, m2
=3, ?=0.3
dual branch M RC,m1
=1, m2
=3, ?=0.5
no diversity, m=1 (Rayleigh)
no diversity, m=3 (Rician)
equal SNR
unequal SNR
Figure 15 Channel capacity using Nakagami-m (m = 3) fading
under CIFR scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, ?=0
dual branch M RC, ?=0.1
dual branch M RC, ?=0.3
dual branch M RC, ?=0.5
no diversity
unequal SNR
equal SNR
Figure 16 Channel capacity with correlated and non-identical
fading parameters using Nakagami-m fading under
CIFR scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
14
16
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC,m1
=1,m2
=3,?=0.1
dual branch M RC,m1
=1,m2
=3,?=0.3
dual branch M RC,m1
=1,m2
=3,?=0.5
no diversity, m=1(Rayleigh)
no diversity, m=3(Rician)
unequal SNR
equal SNR
It is because severity of the fading in m = 1 is higher, and
there is less improvement in capacity in higher severity
environment. But if the fading parameter is non-identical,
the capacity of dual-branch MRC using CIFR scheme gives
better performance than that of ORA. It is because capacity
under ORA gets reduced as branch becomes correlated
(? = 0.1), whereas in case of CIFR scheme in non-identical
fading environment as the correlation increases capacity
also increases (Subadar and Sahu, 2010). It has also been
observed that for 1 ? ? 13dB only, the channel capacity of
non-identical fading parameters under ORA scheme
provides better channel capacity than the channel capacity
162 B.D. Pukhrambam et al.
of m = 3 with no diversity under CIFR scheme. That is if
1 ? ? 13dB and the channel is Rician (m = 3), even single
branch gives better performance under CIFR scheme than
dual-branch MRC with non-identical fading parameter
under ORA scheme.
Figure 17 Channel capacity using Nakagami-m fading for ? = 0.1
with identical and non-identical fading parameters
under ORA and CIFR schemes
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, unequal SNR,m=1,?=0.1
dual branch M RC, unequal SNR,m=3,?=0.1
dual branch M RC, unqual SNR,m1
=1,m2
=3,?=0.1
no diversity, m=1(Rayleigh)
no diversity, m=3(Rician)
ORA
CIFR
5 Conclusion
In this paper, we have derived closed-form expressions for
the channel capacity of MRC over dual correlated
Nakagami-m fading for identical and/or non-identical fading
parameters with equal and/or imbalanced average SNRs
under ORA and CIFR schemes. By numerical evaluation it
has been found that for identical fading parameters, there is
an improvement in channel capacity under ORA and CIFR
schemes by employing diversity using MRC and/or
increasing average received SNR and decreasing the
correlation between branches. However, as the fading
severity reduces, capacity decreases due to decreased
correlation between branches. It has also been observed that
for ORA and CIFR schemes, capacity with dual-branch
MRC is almost same for 1 ? , 2dB ? ? and 1 ? , 1dB ? ? ,
respectively, even though correlation increases. Therefore, it
is recommended that under less fading severity conditions
(higher m), the appropriate antenna spacing required for
uncorrelated diversity path at the receiver end may be
ignored particularly for 1 ? , 2dB ? ? under ORA scheme
and 1 ? , 1dB ? ? under CIFR scheme. However, for nonidentical fading parameters, it is seen that at low value of
SNR only creating diversity gives the improvement in
channel capacity. From this observation, it can be
recommended that at higher value of SNR instead of
implementing dual-branch, it is better to go with no
diversity as it gives better performance. It has also been
observed that under ORA scheme for dual-branch MRC
with non-identical fading parameters and higher correlation
(i.e. ? > 0.1), it is not necessary to create diversity as it does
not provide any improvement in capacity. The channel
capacity of dual-branch MRC with correlated and nonidentical fading parameters over Nakagami-m fading under
CIFR scheme increases with an increase in correlation
coefficient (?). Because fading in channel increases with
increase in correlation coefficient, ? that let the transmit
power to increase so that constant received SNR is
maintained. This increase in transmit power results channel
capacity to increase due to non-identical fading parameters.
For identical fading parameters as expected, increase in
correlation coefficient (?) decreases the channel capacity.
This paper also concludes that at lower correlation
coefficient (? = 0.1), for identical fading parameters
capacity under ORA scheme gives greater capacity than
CIFR scheme and for non-identical fading parameters,
capacity under CIFR gives greater capacity than ORA.
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Wolfram (2015) The Wolfram Function Site. Available online at:
http://functions.wolfram.com , Claims:Demand for wireless communication services is growing
rapidly and requires high data rate and reliability which is
difficult to achieve owing to multipath fading in channels
(Khatalin and Fonseka, 2006a). Channel capacity is a
significant parameter in designing wireless communication
systems as it determines the maximum achievable data rate
(Khatalin and Fonseka, 2006a; Khatalin and Fonseka, 2007).
The channel capacity in the literature is also known as
average channel capacity as it is achieved by averaging the
Shannon capacity of an AWGN channel. Improvement in
channel capacity can be achieved by increasing transmit
power of the signal well enough to reduce the effect of fading
but it results in increase in level of interference (Khatalin and
Fonseka, 2006a). Diversity combining and/or adaptive
transmission schemes are other techniques employed in
fading environment to improve the channel capacity without
the need of increasing transmit power and/or bandwidth
(Khatalin and Fonseka, 2006a). Pure diversity combining
techniques are of three types: Maximal Ratio Combining
(MRC), Equal Gain Combining (EGC), and Selection
Combining (SC) (Khatalin and Fonseka, 2006a). Adaptive
transmission schemes already available in literature are
Optimum Rate Adaptation with constant transmit power
(ORA), Optimum Power and Rate Adaptation (OPRA),
Channel Inversion with Fixed Rate (CIFR), and Truncated
Channel Inversion with Fixed Rate (TIFR) (Khatalin and
Fonseka, 2006a; Khatalin and Fonseka, 2007). Under ORA
scheme, data rate is adapted by the transmitter according to
the channel fading conditions while the transmit power
remains constant (Khatalin and Fonseka, 2006a). Under
OPRA scheme, transmitter can realised optimum capacity by
transmitting appropriate power and data rate in accordance
with the channel conditions (Khatalin and Fonseka, 2006a;
Khatalin and Fonseka, 2007; Goldsmith and Varaiya, (1997).
In CIFR scheme, power is adapted by the transmitter to
maintain constant Signal-to-Noise Ratio (SNR) in order to
invert the channel into additive white Gaussian noise
(AWGN) channel (Simon and Alouini, 2005). TIFR is the
modified version of CIFR. In case of CIFR, there is power
penalty when the channel is in deep fade condition.
Therefore, we use a modified version of CIFR, which is
called TIFR scheme, in which power can be adapted around a
cut-off value below which data transmission is suspended
(Simon and Alouini, 2005; Alouini and Goldsmith, 1999).
Average channel capacity under various fading channels
is a well-researched topic. Khatalin and Fonseka (2006a,
2007) and Goldsmith and Varaiya, (1997) discuss the average
channel capacity under correlated Nakagami-m (m = 1, 2)
fading with equal average SNRs under ORA and CIFR
schemes for different combining techniques. In Simon and
Alouini (2005), capacity expressions of MRC using different
scheme of adaptive transmission over uncorrelated Rayleigh
fading have been obtained. Capacity using different schemes
of adaptive transmission and techniques of combining over
uncorrelated Rayleigh fading is presented in Alouini and
Goldsmith (1999, 1997). Subadar and Sahu (2010) present
the analysis of capacity of dual-SC and MRC systems over
fading channels using correlated (? = 0, 0.3 and 0.6)
Nakagami-m with non-identical (m1 = 1, m2 = 2) and arbitrary
fading parameters. In Khatalin and Fonseka (2006b), capacity
expression of MRC over uncorrelated Hoyt and Rician fading
channels using ORA scheme is extended by obtaining lower
and upper bound expressions. Average channel capacity of
dual-branch MRC using OPRA and TIFR schemes under
uncorrelated and correlated fading channels using Nakagami0.5 were obtained in Hasan and Kumar (2014, 2015a),
respectively. In Hasan and Kumar (2015b, 2015c), average
channel capacity of dual-branch SC with ORA and CIFR
schemes over fading channels using correlated and
uncorrelated Nakagami-0.5 was obtained. In Hasan and
Kumar (2017), spectral efficiency of dual-branch SC with
ORA, TIFR and OPRA schemes over correlated Nakagami0.5 with unequal average received SNR was obtained. In
Shao et al. (1999), results of Goldsmith and Varaiya (1997)
have been extended by obtaining the expression for channel
capacity of MRC diversity systems with the fade correlation
and imbalanced average SNRs between the branches. Closedformed expression for the channel capacity of MRC over
independent Nakagami-q fading under different scheme of
adaptive transmission is obtained in Cheng and Berger
(2003). Da Costa and Yacoub (2007) present the average
channel capacity under ? ? ? and k ? ? fading channel.
Performance analysis of wireless system with MIMO under
Hoyt fading channel is presented in Pena-Martin et al. (2013).
BER performance of MRC over Nakagami-m is analysed in
Aalo (1995). Capacity of single branch MRC with Rayleigh
fading channel under ORA, CIFR and OPRA with
consideration of effect of correlation on capacity has been
analysed in Mallik et al. (2004). Capacity of MRC over
arbitrary Rician fading channel has been obtained in Hamdi
(2008). Annavajjala and Milstein (2004) obtained the
expression of capacity of Rayleigh channel with MRC, EGC
and SC taking into account level of imbalanced in mean
signal strength and effect of correlation. However, an
analytical study on capacity over correlated Nakagami-m
fading channels with identical fading parameter (m > 1) and
non-identical fading parameters with first branch Rayleigh
and second branch Rician (m = 3) considering imbalanced
average SNRs under ORA and CIFR schemes using MRC
has not been considered so far.
Generally, Nakagami-m is used to a great extent to study
the performance of wireless mobile communication system
(Khatalin and Fonseka, 2006a), less concentration appears
to have been focused on non-identical fading parameters
and fade correlation under Nakagami-m channels. In a
number of real-life scenarios, physical space constraints
may not permit antenna spacing that is required for
constantly uncorrelated fading across diversity branches
(Subadar and Sahu, 2010). At the same time, branches may
experience different fading severity and imbalanced SNRs.
So, in cases where quality of service with high data rate
requirements mandate designing for real-life fading
Capacity of dual-branch MRC system 153
scenario, results obtained for the same fading conditions
will have great practical usefulness in wireless mobile
environments. That allows the researchers or system
designers to perform comparison and trade-off studies
among the various adaptive transmission schemes with
optimal combining technique MRC, so as to determine the
optimal choice in the face of their available constraints.
Therefore, this paper fills this gap by presenting the
impact of various practical constraints, e.g. physical space
requirements, effect of fade correlation, non-identical fading
parameters and level of imbalance in the branch average
received SNRs on the channel capacity under Nakagami-m
fading channels.
The remainder of this paper is organised as follows:
Section 2 defined the channel model. In Section 3, channel
capacity of MRC with identical and non-identical fading
parameters considering equal and imbalanced average SNRs
has been derived using ORA and CIFR schemes. Section 4
presents numerically evaluated results. Concluding remarks
are given in Section 5.
2 Channel model
The Probability Distribution Function (PDF), p ? ? ? ? , of the
output SNR, ? at MRC combiner for L-branch correlated
Nakagami-m fading channels with identical (m1 = m2 = 1)
fading parameters and equal average SNRs ? ? 1 2 ? ? ? ? ?
can be presented as (Khatalin and Fonseka, 2006a):
? ?
? ?
? ? ? ?
? ? ? ? ? ?
1
1 1
exp
1
; ;
1 1
, 0
1 1
L m
Lm m m
m m
Lm F m Lm
L
p
L Lm m
?
? ?
? ? ?
??
? ? ??
? ?
? ? ??
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ?
? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ?
(1)
where ? is the average received SNR, m is the fading
parameter, 1F1[.;.;.] is the Kummer confluent hypergeometric
function and ? is the correlation coefficient.
The PDF of ? at dual-branch MRC combiner output over
correlated and imbalanced average SNRs ? ? 1 2 ? ? ? between
the branch is obtained in Alouini and Simon (1998) as:
? ? ? ? ? ?
?? ? ?
1
2 2
1
1 2 2 1 2
exp , 0
m m
m
m p I
m ?
? ? ? ?? ? ?
?? ?? ?
? ? ? ? ? ? ? ?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
(2)
where In(.) is the nth-order Bessel Function and the
parameter ?? and ?? are the normalised form of ? and ?
(Alouini and Simon, 1998), and given by:
? ?
? ?
1 2
1 2 2 1
m ? ?
?
? ? ?
? ? ? ?
and
? ? ? ? ? ?
? ?
1
2 2
1 2 12
1 2
4 1
2 1
m ? ? ?? ?
?
?? ?
?? ? ? ? ?
For m = 1 (Rayleigh), expression of PDF was obtained in
[Alouini and Simon, 1998, equation (12)] as,
? ?
? ? ? ?
? ?
? ? ? ?
2
1 2 1 2 12
1 2
2
1 2 12
4 1
exp 2 1
4 1
p?
? ? ? ? ?? ?
?
?? ?
?
? ? ?? ?
? ? ?? ? ? ? ? ? ? ? ? ? ?
?? ?

? ? ? ?
? ?
? ? ? ?
2
1 2 1 2 12
1 2
2
1 2 12
4 1
exp 2 1
4 1
? ? ? ? ?? ?
?
?? ?
? ? ?? ?
? ? ?? ? ? ? ? ? ? ? ? ? ?
?? ?
(3)
Putting m = 3 and expanding the Bessel function using
(Wolfram, 2015), we can reduce the PDF of equation (2) as
follows:
? ? ? ?
? ? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
3
4 5
1 2
2
1 9
2 1
3 exp exp
exp exp
3 exp exp
p? ? ? ?? ?
? ?? ? ??
?? ? ? ? ? ? ?
?? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ?? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? ? ?? ?? ? ? ? ? ? ?? ? ? ? ?? ?? ?? ??
(4)
The PDF of ? at dual-branch MRC (L = 2) combiner output
with correlated, non-identical (m1 ? m2) fading parameters
and imbalanced average SNRs over fading channels using
Nakagami-m is obtained in Subadar and Sahu (2010) and
Subadar (2011) as:
? ? ? ? ? ?
? ?
? ? ? ?
? ?
1 2
1
1 2 1 2
2
1 2 0 1 2
1
1
2
1
2
12 21
11 1 1
1 2
1 1
! exp 1
, ; 1
m m k
k
k
m m mmr
p r r
m
m k r
m m Fmk r
?
?
?
? ? ? ?
?
? ?
?
? ? ? ?
? ?
?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
(5)
where 112 ? ? mm k ? ? 2 and ? ?k a is the Pochhammer
function, 2
1
m r m ? ? and 1
2
m
m
? ? with m m 2 1 ? .
Different values of m1 and m2 describe several important
distributions. Like m1 = m2 = 1 distribution reduces to
Rayleigh, ?m m 1 2 ? ? ? 1 corresponds to Rician distribution
and m1 = 1 and m2 > 1 describe the non-identical nature of
the fading parameter with first branch experiencing
Rayleigh distribution and second branch Rician distribution.
154 B.D. Pukhrambam et al.
Replacing 1F1 [.;.;.] in its infinite series form as given in
Wolfram (2015), equation (5) can be written as:
? ? ? ? ? ?
? ?
? ?? ?
? ?? ? ? ?
1 2
1
1 2 12
2
1 2 0 0 1 2
1
12 21 1 1
1 2 2
1 1
2
1 1
1
! ! exp 1
m k m
k t
t t
k t
t
m m mmr p r r
m m m mk
r m k t r
?
?
?
? ? ? ?
? ? ?
? ? ?? ?
?
? ?
? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
(6)
Expanding the Pochhammer function using Wolfram (2015),
equation (6) can be reduced as:
? ? ? ? ? ?
? ?
? ?
? ?? ? ? ?
1 2
1
1 2 12
2
1 2 0 0 1 2
1
12 21 1
1 2 2
1 1
2
1 1
1
! ! exp 1
m k m
k t
t t
m m mmr p r r
m m m kt
r m kt m t r
?
?
?
? ? ? ?
? ? ?
? ? ? ?
?
? ?
? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ??? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?
? ?
(7)
3 Channel capacity
In this section, closed-form expressions are derived for the
channel capacity of dual-branch MRC under correlated,
identical and non-identical fading parameters for Nakagamim fading considering equal and imbalanced average SNRs
between the branches using ORA and CIFR schemes. Similar
to the assumption made in Alouini and Goldsmith (1999), an
error-free feedback path with negligible time delay is
assumed while considering adaptive transmission schemes.
3.1 ORA
The channel capacity of fading channels under ORA scheme
with received SNR distribution p? ?? ? is defined in Simon
and Alouini (2005) as:
2 ? ? ?? 0
log 1 C B pd ORA ? ? ??
?
? ? ? (8)
where B[Hz] is the bandwidth of the channel and p? ?? ? is
the PDF of the output SNR, ?.
Substituting equation (7) in equation (8), channel capacity
of dual-branch MRC with correlated, non-identical and
imbalanced average SNRs over Nakagami-m fading can be
obtained as:
? ?
? ? ? ?
? ?
? ?? ? ? ?
? ?
1 2
1
1 2
1 2
12 12 21
2 2
0 0 12 12
1
1
2
1 1 0 2
2
1
1 1
log 1 ! !
exp 1
ORA
m m
k t
k t
t
m m C B r
mmr m m
r r
m kt
d
kt m t m
r
?
? ?
? ?
?? ??
? ? ? ?
?
?
? ?
? ?
? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?? ? ? ? ?? ? ? ? ? ?? ?
? ??
?
? ?? ? ? ? ? ? ? ?
? ?
?
(9)
The integral part in equation (9) can be solved using partial
integration as follows:
?? ?? 0 0 0
u dv u v u v vdu lim lim ? ?
? ?
?? ?
??? ? ? (10)
Let
u ? ln 1? ? ? ? (11)
Then
1
d du ?
? ? ?
(12)
Now let
? ? 1 1 2
2
exp 1
t m dv d r
? ? ? ?
?
? ? ? ? ? ?? ? ? ? ?

Integrating the above expression, we obtain:
? ? 1 1 2
2
exp 1
t m v d r
? ? ? ?
?
? ? ? ? ? ?? ? ? ? ? ? (13)
Putting the values of equations (11), (12) and (13) in
equation (10), we get:
? ?
? ?
? ?
1
1
1
2
0 2
2
1 2
0 2
log 1 1.443
exp 1
1
exp 1 1
t
t
d
m
r
m d d
r
?
?
? ? ?
?
?
? ? ??
? ?
? ? ?
?
? ?
? ?? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ?

Integrating the above expression using Gradshteyn and
Ryzhik (2000) we get:
? ?
? ?
? ?
? ? ? ? ? ?
1
1 1
1
2 1
0 2
2
1
22 2
22 2 0
log 1 1.443 G
exp 1
exp G , 11 1
t
z t t
z
d t
m
r
mm m z rr r
?
? ?
? ? ??
?
?
?? ?
? ? ?
? ? ? ?
?
? ? ?? ? ? ? ? ? ? ?
? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ?
?
?
(14)
where ?[.,.] is the incomplete gamma function.
Substituting equation (14) in equation (9), we get:
? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ? ? ?
1 2
1 1
1 12 2
2
1 2 0 0 1 2
1 12 21 2
1 12 2
2 22
2 22 0
1.443
1 1
G
exp ! !G 1 1
G , 1 11
ORA
m k m
k t
t
t z t
z
m mmr m
C B
r r
m kt mm m
kt m r r
m mm z r rr
? ?
? ? ? ?
? ?
?? ?
? ??
? ?
? ?
? ? ?
?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?? ? ? ? ?? ? ? ? ? ?? ?
? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ?? ? ? ?
??
?1
?
(15)
Channel capacity of dual-branch MRC with correlated and
identical fading parameter for Nakagami-m fading can be
Capacity of dual-branch MRC system 155
obtained by putting L = 2, replacing 1F1[.;.;.] with its series
representation (Wolfram, 2015), and then substituting
equation (1) into equation (8) as:
? ? ??
? ? ? ?
? ?
? ? ? ?
2
0
2 1
2
0
1
2
2 !
log 1 exp
1
ORA
m
m
k
k
k k
k m
B m C
m
m b
m k
m d
? ?
?
? ?? ?
? ?
?
?
?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ?? ? ? ? ?
?
?

After integrating above expression using Gradshteyn and
Ryzhik (2000) we obtained:
? ?
? ?
? ? ? ?
? ? ? ?
? ?
? ? ? ?
2
0
2 1
0
1 1.443
exp 1 1
2
! 1
, 1 1
ORA
m
n
n
z
n m
z
B m C
m
m n
n
m m z
?
? ? ?
? ?
?
?? ??
?
?
? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ?
? ?? ? ? ?? ? ? ? ? ? ? ?? ?
?
?
(16)
Further, for no diversity ? ? L ?1 ,
? ? ? ? ? ?
? ? ? ?
0
1
0
1.443 1
1 !
() !
, 1 1
m k
ORA
k
z
m k
z
B
C mk
m k
m m z
? ?
?? ??
?
?
? ?
?
? ?? ? ??
?
? ?? ? ? ?? ? ? ? ? ? ? ?? ?
?
?
(17)
Similarly, after some mathematical transformation using
Wolfram (2015) and Gradshteyn and Ryzhik (2000), channel
capacity of dual-branch MRC with correlated, identical fading
parameters and imbalanced average SNRs for Nakagami-m
(m = 1 and m = 3) fading is obtained by:
Substituting equation (3) in equation (8) for m = 1 as:
? ?
? ?
? ? ? ?
? ?
? ? ? ?
1 2
1
1
1.443
2 1
exp
exp
ORA
B C
E
E
?? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ?
?? ? ? ? ?? ? ? ? ? ? ?? ? ???
? ? ? ?? ? ?
? ?? ? ? ? ? ?
? ? ? ? ? ??
(18)
Substituting equation (4) in equation (8) for m = 3 we get
equation (19).
? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ? ? ?? ?
? ? ? ?? ?
3
4 5
1 2
1
1
2 3
2 0
2 3
0
1.443 9
2 1
3
exp
3
exp
exp ,
2
exp ,
ORA
z
z
k
k
B C
E
E
z
k
? ?? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ????
?
?? ?? ? ?
?
?
?
?
? ? ? ? ? ? ? ? ? ? ? ?
??? ? ? ?? ? ?? ? ? ? ? ? ?? ? ??? ?
? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ??
?? ? ? ?? ? ?? ???? ? ?? ? ? ? ?? ?? ? ? ? ? ? ? ? ?? ? ? ?? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ?
?
?
?
? ? ? ?? ?
? ? ? ?? ?
1 2
0
1 2
0
exp ,
3
exp ,
e
e
f
f
e
f
?? ? ???
?
?? ? ? ? ?
?
?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ???? ? ?? ? ? ?? ? ? ? ?? ? ?? ? ?? ?? ? ? ? ? ?? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?
?
?
(19)
When m1 = m2 = m in equation (15), it reduces to equation (17).
Also for m = 1, equation (16) is comparable to [Shao et al.,
1999, equation (15)] and when ? = 0, equation (16) reduces
to [Alouini and Goldsmith, 1999, equation (40)] and
equation (18) reduces to [Shao et al., 1999, equation (15)] for
Rayleigh with imbalanced average SNRs.
3.2 CIFR
The channel capacity (bits/s) of fading channels under CIFR
scheme with distribution of received SNR p? ?? ? is defined
in Simon and Alouini (2005) as:
? ? 2
0
1 log 1 C B CIFR p d ? ?
?
?
?
? ? ? ?
? ? ? ? ? ?
? ?
? ? ?
(20)
Substituting equation (5) in ? ?
0
p d ? ?
?
?
?
? we obtain:
? ?
? ?
? ?
? ?
? ? ? ?
? ?
1 2
1
1 2
0 0 1 2
1
1 2 1
2
0 1 2 2
1
2
12 21
11 1 1
1 2
1
1
1 ! exp 1
, ; 1
m m
k
k
k
p m m d r
mmr m
r m k r
m m Fm k d r
?
?
?
?
? ?? ?
?
? ? ? ?
?
? ? ? ??
? ?
? ?
? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
?
156 B.D. Pukhrambam et al.
Using [Gradshteyn and Ryzhik, 2000, (7.621 (4))], the
above integration can be transformed as follows:
? ?
? ?
? ? ? ?
? ?? ?
? ?? ?
1 2
1
1 2
0 1 2
1
12 2
2
0 2 1 2
1 1 12 21
21 1 1 1
1 1 12
1
1 1
1
, 1; ; !
m m
k
k
p m m d r
mmr m
r r
m k m m Fm k
km m
?
?
?
?
?? ?
? ? ?
? ? ? ? ?
? ?
?
? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?? ? ? ? ? ? ? ? ?
? ? ? ?
?
? (21)
where 2F1 (a, b, c, z) is the Gauss hypergeometric function.
Channel capacity of dual-branch MRC diversity system
with non-identical fading parameters and imbalanced
average SNRs under CIFR scheme can be obtained by
substituting equation (21) into equation (20) as follows:
? ?
? ? ? ?
? ?? ?
? ?? ?
1 2
1
1 2
1 2
1
2 12
2
2 1 2
2
0 1 1
1 1
1 2 21
21 1 1 1
1 2
1
1 1
log 1 1
! !
, 1; ;
m m
k
CIFR
k
m m
r
m mmr
r r C B
m k
kt m
m m
Fm k
m
?
? ?
? ? ?
?
?
? ? ? ?
?
? ?
? ?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?? ?
? ?
? ? ? ? ? ? ?
? ?
? ?
? (22)
Again substituting equation (1) in ? ?
0
p d ? ?
?
?
?
? , and after
integrating using Gradshteyn and Ryzhik (2000), we get:
? ?
? ? ??
? ? ? ? ? ?
2
0
1 2
0
1
2
2 1
!
m
m
k
k m
k
m
p d m
mk b
m ka
k
? ? ?
?
? ?
?
?
? ? ?
?
? ? ? ? ? ? ? ? ?
? ??
?? ?
?
?
(23)
where ? ? 1
m
a
? ? ? ? and ? ? 1
m b
? ? ? ? .
Channel capacity of dual-branch MRC with identical fading
parameter and equal average SNRs under CIFR scheme can
be obtained by substituting equation (23) into equation (20)
as follows:
? ? ??
? ? ? ? ? ?
2 2
2 1
0
1
log 1
2
2 1
!
CIFR
m
k m
m k
k
C
m B
m mk b
m ka
k
?
?
?
? ? ??
?
? ? ? ?
? ? ? ?
? ?? ? ? ? ? ?? ?
? ? ? ? ?
(24)
Further for no diversity ?L ?1? ,
? ?
? ?
? ? ? ?
0
1
1
1
!
m
k
m k
k o
p m d m
a m ka
k
? ?
?
? ?
?
?
? ? ?
?
? ? ? ? ? ? ? ?
? ?? ?
?
?
(25)
Then substituting equation (25) in equation (20) we get:
? ?
? ? ? ?
2
1
log 1
1
!
CIFR
k m
m k
k o
C
m B
m a m ka
k
?
?
? ? ?
?
? ? ? ?
?
?? ?
? ? ? ?? ? ? ? ? ? ? ?
(26)
Substituting equation (3) in ? ?
0
p d ? ?
?
?
?
? and solving the
integration using Gradshteyn and Ryzhik (2000) we get:
? ?
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ?
0
2
1 2 1 2 12
1
1 2
0 2
1 2 1 2 12
1
1 2
2
1 2 12
4 1
2 1
lim
4 1
2 1
4 1
u
p d
E u
E u
? ?
?
?
? ? ? ? ?? ?
?? ?
? ? ? ? ?? ?
?? ?
? ? ?? ?
?
?
?
? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?
?? ?
?
(27)
Channel capacity of dual-branch MRC with identical fading parameter (with m1 = m2 = m = 1) and imbalanced average SNRs
under CIFR scheme can be obtained by substituting equation (27) into equation (20) as follows:
? ? ? ?
? ? ? ?
? ?
? ? ? ?
? ?
2
1 2 12
2 2 2
1 2 1 2 12 1 2 1 2 12
1 1 0 12 12
4 1
C log 1
41 41
lim
21 21
CIFR
u
B
E uE u
? ? ?? ?
? ? ? ? ?? ? ? ? ? ? ?? ?
?? ? ?? ? ? ?
? ? ? ? ? ? ? ?? ? ?
? ? ? ? ? ? ?? ?? ? ?? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ?? ??? ?
(28)
Capacity of dual-branch MRC system 157
Similarly for ? ? 1 2 mmm ? ?? 3 , substituting equation (4) in ? ?
0
p d ? ?
?
?
?
? and solving the integration using Wolfram (2015)
and Gradshteyn and Ryzhik (2000) we get:
? ?
? ?
? ? ?? ? ? ? ? ? ? ? ? ? ?
3
4 5
0 1 2
2
1 1 2 2 0
1 9
2 1
1 1 11 3 lim 3 u
p d
E E
? ?
?
? ?? ? ?
?? ?? ? ?
?? ?? ?? ?? ?
?
?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ?? ? ? ? ? ? ?? ?? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ?? ? ? ?? ?? ? ? ?? ? ? ? ? ???
?
(29)
Channel capacity of dual-branch MRC with identical fading parameter (with m = 3) and imbalanced average SNRs under
CIFR scheme, can be obtained by substituting equation (29) into equation (20) as follows:
? ?
? ? ?? ? ? ? ? ? ? ? ? ? ?
3
4 5
1 2
2
2
1 1 2 2 0
9 2
1
log 1
1 1 11 3 lim 3
CIFR
u
C B
E E
?
?? ?
?? ?? ? ?
?? ?? ?? ?? ?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???? ? ? ? ?? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ???? ? ? ? ? ? ? ?? ?? ? ? ?? ?? ? ? ??? ? ? ? ? ? ????
(30)
4 Numerical results and analysis
In this section, the effect of non-identical fading parameters,
level of imbalanced in average SNRs and correlation
coefficient on the capacity under Nakagami-m fading
channels using ORA and CIFR schemes with MRC
diversity has been presented. Channel capacity over fading
channels of correlated Nakagami-m fading for MRC with no
diversity for Rayleigh channel which was obtained in
Khatalin and Fonseka (2006a) and Simon and Alouini
(2005) is also presented for comparison. We have
considered imbalance in average SNRs as 2 1 ? ? 2? and
m1 = 1, m2 = 3 for non-identical fading parameters.
Figures 1 and 2 show the channel capacity over fading
channels of correlated Nakagami-m considering equal average
SNRs and imbalanced average SNRs ? ? 2 1 ? ? 2? respectively
using ORA scheme for m = 3. As expected, by employing
diversity using MRC and/or increasing average received SNR
? ? 1 ? or ? gives improvement in channel capacity in both
Figures 1 and 2. It is observed that the channel capacity with
dual-branch MRC gives almost identical performance for
1 ? , 2dB ? ? even when branch correlation increases. So for
low value of SNR ? ? 1 i e. . , 2dB ? ? ? the appropriate antenna
spacing required for uncorrelated diversity path at the receiver
end may be ignored.
Figure 3 shows the channel capacity with non-identical
fading parameters over correlated Nakagami-m fading using
ORA scheme. As expected, the channel capacity with no
diversity for m = 3 gives better channel capacity than no
diversity for m = 1. Since the channel capacity improves with
increase in m (Simon and Alouini, 2005). It is seen that under
low SNR value up to 2 dB, channel capacity with no diversity
for m = 3 and channel capacity of non-identical fading
parameters ? ? 1 2 m= m= 1, 3 for lower correlation ? ? i e. . 0.1 ? ?
are almost same. And at higher SNR value ? ? i e. . 8dB ? ? ,
channel capacity of non-identical fading parameters is superior
to the channel capacity of no diversity with m = 1 as expected.
Theoretically, employing diversity improves channel
capacity but in contrary channel capacity of non-identical
fading parameters is inferior to the channel capacity with no
diversity for m = 3 and m = 3 at higher value of SNR
?i e. . 2dB and 8dB ? ? ? ? ? , respectively. It is also observed
that for higher correlation ?i e. . 0.1 ? ? ? , the channel capacity
for m = 1 and m = 3 with no diversity is better than channel
capacity with non-identical fading parameters. This is because
the effect of branch correlation is more on the branch having
less amount of fade (m2 = 3) (Simon and Alouini, 2005) and as
the correlation in the branch increases, the weighting problem
arises at the MRC combiner resulting in decrease in capacity.
So for dual-branch MRC with non-identical fading parameters
and higher correlation ?i e. . 0.1 ? ? ? , it is not necessary to
create diversity as it does not provide any improvement in
capacity under ORA scheme.
Figure 1 Channel capacity over Nakagami-m (m = 3) fading
under ORA scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR, ?=0
dual branch M RC, equal SNR, ?=0.1
dual branch M RC, equal SNR, ?=0.3
dual branch M RC, equal SNR, ?=0.5
no diversity
158 B.D. Pukhrambam et al.
Figure 2 Channel capacity over Nakagami-m (m = 3) fading
under ORA scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, unequal SNR,?=0
dual branch M RC, unequal SNR,?=0.1
dual branch M RC, unequal SNR,?=0.3
dual branch M RC, unequal SNR,?=0.5
no diversity
Figure 3 Channel capacity of non-identical fading parameters
using Nakagami-m fading under ORA scheme
0 5 10 15 20 25 30 0
1
2
3
4
5
6
7
8
9
10
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
no diversity, m=1 (Rayleigh)
no diversity, m=3 (Rician)
dual branch M RC, equal SNR, m1
=1, m2
=3,?=0.1
dual branch M RC, equal SNR, m1
=1, m2
=3, ?=0.3
dual branch M RC, equal SNR, m1
=1, m2
=3, ?=0.5
Similarly, in Figure 4 considering MRC with non-identical
fading parameters and imbalanced average SNRs, the
channel capacity with no diversity for m = 3 gives better
channel capacity than no diversity for m = 3 as expected.
For lower value of first branch average received SNR
? ? 1 i e. . 9dB ? ? the channel capacity with non-identical fading
parameters ? ? 1 2 m= m= 1, 3 for low correlation ? ? i e. . 0.1 ? ?
is better than capacity for m = 1 and m = 3 with no diversity
as expected and at higher value of first branch average
received SNR ? ? 1 1 i e. . 9dB and 14dB ? ? ? ? , even
employing diversity fails to provide better channel capacity
than the capacity of m = 3 and m = 1 with no diversity,
respectively. It is also observed that at higher correlation
? ? i e. . 0.1 ? ? , the channel capacity for m = 1 and m = 3
with no diversity is better than channel capacity with nonidentical fading parameters. Since branch correlation is
more effective when the fading is less severe (m2 = 3)
(Simon and Alouini, 2005), and as the branch correlation
increases it becomes difficult to weight the branch at the
MRC combiner that results in capacity to decrease. So under
ORA scheme, for dual-branch MRC with non-identical
fading parameters and higher correlation ? ? i e. . 0.1 ? ? , it is
not necessary to create diversity as it does not provide any
improvement in capacity from single branch.
Figure 4 Channel capacity as a function of first branch average
SNR under correlated and non-identical fading parameters
for Nakagami-m fading using ORA scheme
0 5 10 15 20 25 30 0
1
2
3
4
5
6
7
8
9
10
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
no diversity,m=1
no diversity,m=3
dual branch M RC, unequal SNR, m1
=1,m2
=3, ?=0.1
dual branch M RC, unequal SNR, m1
=1,m2
=3, ?=0.3
dual branch M RC, unequal SNR, m1
=1,m2
=3, ?=0.5
Figure 5 shows the comparison of channel capacity with
identical and non-identical fading parameters over
Nakagami-m fading using ORA scheme considering equal
average SNRs for ? = 0.1. As expected, identical fading
parameters case for m = 3 gives the largest capacity. Under
ORA scheme, capacity of correlated and non-identical
fading parameters may seem to provide greater capacity
than m = 1 as in non-identical case one path experiences
Rayleigh channel (m1 = 1) and other path experiences
Rician channel (m2 = 3), but in contrary the capacity of nonidentical fading parameter is inferior to Rayleigh channel.
This is because the effect of branch correlation is more on
the second branch having less amount of fading (m2 = 3)
(Simon and Alouini, 2005), so the branch correlation
increases resulting in decrease in capacity.
Figure 6 shows the comparison of channel capacity with
identical and non-identical fading parameters considering
imbalanced average SNRs between the branches over
Nakagami-m fading using ORA scheme with ? = 0.1.
Identical fading parameters with m = 3 and m = 1, equally
gives the largest capacity up to 3 dB point after that m = 3
gives better capacity than m = 1 as expected. The difference
in capacity between m = 3 and m = 1 is less in imbalanced
average SNRs than in equal average SNRs case as the
overall SNR is greater in imbalanced average SNRs as we
consider 2 1 ? ? 2? .
Capacity of dual-branch MRC system 159
Figure 5 Channel capacity over Nakagami-m fading for ? = 0.1
with identical and non-identical fading parameter under
ORA scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,m=1,?=0.1
dual branch M RC, equal SNR,m=3,?=0.1
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.1
no diversity,m=1 (Rayleigh)
no diversity,m=3 (Rician)
Figure 6 Channel capacity with identical and non-identical
fading parameters over Nakagami-m fading for ? = 0.1
under ORA scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, unequal SNR,m=1,?=0.1
dual branch M RC, unequal SNR,m=3,?=0.1
dual branch M RC, unequal SNR,m1
=1,m2
=3,?=0.1
no diversity,m=1 (Rayleigh)
no diversity,m=3 (Rician)
Figure 7 shows the channel capacity with correlated and
equal average SNRs over Nakagami-m fading using CIFR
scheme for m = 3. It is seen that for ? ? 1dB , channel
capacity is almost identical for every ? value. And as
expected for ? ? 1dB , channel capacity decreases as
correlation increases. But channel capacity for ? = 0 and ? = 0.1
gives almost same for each value of average received
SNR. So for ? ? 0.1 and ? ? 1dB required antenna spacing
cannot be avoided.
Figure 7 Channel capacity over Nakagami-m (m = 3) fading
under CIFR scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,?=0
dual branch M RC, equal SNR,?=0.1
dual branch M RC, equal SNR,?=0.3
dual branch M RC, equal SNR,?=0.5
no diversity
Figure 8 shows the channel capacity with correlated and
imbalanced average SNRs over Nakagami-m fading for m = 3
using CIFR scheme. It is seen that for 1 ? ? 0dB , channel
capacity is almost same for all ? value and as expected with
increase in 1 ? and decrease in ?, capacity improves. But ? = 0
and ? = 0.1 gives almost same channel capacity for each
value of 1 ? . 2 1 ? ? 2?
Figure 8 Channel capacity over Nakagami-m (m = 3) fading
under CIFR scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC,unequal SNR,?=0
dual branch M RC,unequal SNR,?=0.1
dual branch M RC,unequal SNR,?=0.3
dual branch M RC,unequal SNR,?=0.5
no diversity
Figures 9 and 10 show the channel capacity with correlated
and non-identical fading parameters over Nakagami-m
fading using CIFR scheme considering equal and
imbalanced average SNRs, respectively. It is seen that CIFR
160 B.D. Pukhrambam et al.
scheme suffers large capacity penalty in deep fading
environments such as m = 1 with no diversity. However, this
penalty diminishes as m increases and/or when diversity
combining is employed. It is also seen that for non-identical
fading parameters (m1 ? m2) channel capacity increases as ?
increases unlike in ORA scheme. Because fading in channel
increases with increase in ? that let the transmit power to
increase so that constant received SNR is maintained. This
increase in transmit power results channel capacity to
increase due to non-identical fading parameters (Subadar
and Sahu, 2010).
Figure 9 Channel capacity with correlated and non-identical
fading parameters over Nakagami-m fading under
CIFR scheme
0 5 10 15 20 25 30 0
5
10
15
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.1
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.3
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.5
no diversity,m=3(Rician)
no diversity,m=1(Rayleigh)
Figure 10 Channel capacity with correlated and non-identical
fading parameters using Nakagami-m fading under
CIFR scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
14
16
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, unequal SNR,m1
=1,m2
=3,?=0.1
dual branch M RC, unequal SNR,m1
=1,m2
=3,?=0.3
dual branch M RC, unqual SNR,m1
=1,m2
=3,?=0.5
no diversity,m=1(Rayleigh)
no diversity,m=3(Rician)
Figures 11 and 12 show comparison of the channel capacity
for identical with non-identical fading parameters over
Nakagami-m fading using CIFR for ? = 0.1 considering
both equal and imbalanced average SNRs, respectively. As
expected, there is improvement in capacity in both the figures
by increasing ? and m and/or by employing diversity. But
the channel capacity of dual-branch MRC with non-identical
fading parameters gives the largest capacity in both the
figures. It is because capacity with non-identical fading
parameter under CIFR scheme increases with increase in ?
(Subadar and Sahu, 2010), whereas capacity with identical
fading parameter under CIFR scheme decreases with increase
in ?.
Figure 11 Channel capacity over Nakagami-m fading for ? = 0.1
with identical and non-identical fading parameters
under CIFR scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,m=1,?=0.1
dual branch M RC, equal SNR,m=3,?=0.1
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.1
no diversity,m=1 (Rayleigh)
no diversity,m=3 (Rician)
Figure 12 Channel capacity using Nakagami-m fading for ? = 0.1
with identical and non-identical fading parameter under
CIFR scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
Average received SNR [dB] per branch
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, equal SNR,m=1,?=0.1
dual branch M RC, equal SNR,m=3,?=0.1
dual branch M RC, equal SNR,m1
=1,m2
=3,?=0.1
no diversity, m=1 (Rayleigh)
no diversity, m=3 (Rician)
In Figures 13, 14, 15, and 16, comparison of channel capacity
for equal and imbalance average SNRs over correlated
Nakagami-m fading are shown. It is seen that the channel
Capacity of dual-branch MRC system 161
capacity improvement considering imbalanced average SNRs
gives the better capacity than those with equal SNRs as
expected, because the overall SNR of imbalanced average
SNRs is greater than that of equal average SNRs between the
branches as we have considered 2 1 ? ? 2? .
Figure 17 compares the channel capacity of identical
and non-identical fading parameters for ORA with CIFR
schemes considering imbalanced average SNRs between the
branches when ? = 0.1. As expected, in identical cases
channel capacity of dual-branch MRC under ORA schemes
gives greater capacity than CIFR scheme. It is also observed
that for 1 ? ? 3dB the capacity for m = 1 using ORA scheme
is similar to the capacity dual-branch MRC with m = 3 using
CIFR scheme.
Figure 13 Channel capacity over Nakagami-m (m = 3) fading
under ORA scheme
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC,?=0
dual branch M RC,?=0.1
dual branch M RC,?=0.3
dual branch M RC,?=0.5
no diversity
equal SNR
unequal SNR
Figure 14 Channel capacity with correlated and non-identical
fading parameters using Nakagami-m fading under
ORA scheme
0 5 10 15 20 25 30 0
1
2
3
4
5
6
7
8
9
10
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC,m1
=1, m2
=3, ?=0.1
dual branch M RC,m1
=1, m2
=3, ?=0.3
dual branch M RC,m1
=1, m2
=3, ?=0.5
no diversity, m=1 (Rayleigh)
no diversity, m=3 (Rician)
equal SNR
unequal SNR
Figure 15 Channel capacity using Nakagami-m (m = 3) fading
under CIFR scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, ?=0
dual branch M RC, ?=0.1
dual branch M RC, ?=0.3
dual branch M RC, ?=0.5
no diversity
unequal SNR
equal SNR
Figure 16 Channel capacity with correlated and non-identical
fading parameters using Nakagami-m fading under
CIFR scheme
0 5 10 15 20 25 30
0
2
4
6
8
10
12
14
16
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC,m1
=1,m2
=3,?=0.1
dual branch M RC,m1
=1,m2
=3,?=0.3
dual branch M RC,m1
=1,m2
=3,?=0.5
no diversity, m=1(Rayleigh)
no diversity, m=3(Rician)
unequal SNR
equal SNR
It is because severity of the fading in m = 1 is higher, and
there is less improvement in capacity in higher severity
environment. But if the fading parameter is non-identical,
the capacity of dual-branch MRC using CIFR scheme gives
better performance than that of ORA. It is because capacity
under ORA gets reduced as branch becomes correlated
(? = 0.1), whereas in case of CIFR scheme in non-identical
fading environment as the correlation increases capacity
also increases (Subadar and Sahu, 2010). It has also been
observed that for 1 ? ? 13dB only, the channel capacity of
non-identical fading parameters under ORA scheme
provides better channel capacity than the channel capacity
162 B.D. Pukhrambam et al.
of m = 3 with no diversity under CIFR scheme. That is if
1 ? ? 13dB and the channel is Rician (m = 3), even single
branch gives better performance under CIFR scheme than
dual-branch MRC with non-identical fading parameter
under ORA scheme.
Figure 17 Channel capacity using Nakagami-m fading for ? = 0.1
with identical and non-identical fading parameters
under ORA and CIFR schemes
0 5 10 15 20 25 30 0
2
4
6
8
10
12
First branch average received SNR [dB]
Channel capacity per unit bandwidth [bits/s/Hz]
dual branch M RC, unequal SNR,m=1,?=0.1
dual branch M RC, unequal SNR,m=3,?=0.1
dual branch M RC, unqual SNR,m1
=1,m2
=3,?=0.1
no diversity, m=1(Rayleigh)
no diversity, m=3(Rician)
ORA
CIFR
5 Conclusion
In this paper, we have derived closed-form expressions for
the channel capacity of MRC over dual correlated
Nakagami-m fading for identical and/or non-identical fading
parameters with equal and/or imbalanced average SNRs
under ORA and CIFR schemes. By numerical evaluation it
has been found that for identical fading parameters, there is
an improvement in channel capacity under ORA and CIFR
schemes by employing diversity using MRC and/or
increasing average received SNR and decreasing the
correlation between branches. However, as the fading
severity reduces, capacity decreases due to decreased
correlation between branches. It has also been observed that
for ORA and CIFR schemes, capacity with dual-branch
MRC is almost same for 1 ? , 2dB ? ? and 1 ? , 1dB ? ? ,
respectively, even though correlation increases. Therefore, it
is recommended that under less fading severity conditions
(higher m), the appropriate antenna spacing required for
uncorrelated diversity path at the receiver end may be
ignored particularly for 1 ? , 2dB ? ? under ORA scheme
and 1 ? , 1dB ? ? under CIFR scheme. However, for nonidentical fading parameters, it is seen that at low value of
SNR only creating diversity gives the improvement in
channel capacity. From this observation, it can be
recommended that at higher value of SNR instead of
implementing dual-branch, it is better to go with no
diversity as it gives better performance. It has also been
observed that under ORA scheme for dual-branch MRC
with non-identical fading parameters and higher correlation
(i.e. ? > 0.1), it is not necessary to create diversity as it does
not provide any improvement in capacity. The channel
capacity of dual-branch MRC with correlated and nonidentical fading parameters over Nakagami-m fading under
CIFR scheme increases with an increase in correlation
coefficient (?). Because fading in channel increases with
increase in correlation coefficient, ? that let the transmit
power to increase so that constant received SNR is
maintained. This increase in transmit power results channel
capacity to increase due to non-identical fading parameters.
For identical fading parameters as expected, increase in
correlation coefficient (?) decreases the channel capacity.
This paper also concludes that at lower correlation
coefficient (? = 0.1), for identical fading parameters
capacity under ORA scheme gives greater capacity than
CIFR scheme and for non-identical fading parameters,
capacity under CIFR gives greater capacity than ORA.
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