CAPACITY OF DUAL BRANCH MRC SYSTEM FOR CORRELATED IMBALANCED AVERAGE SNRS OVER NAKAGAMI-M FADING CHANNELS USING OPRA SCHEME

Abstract

Wireless communication is the fastest growing segment of the communications industry. It provide services, such as wireless telephones, internet access, video teleconferencing and satellite services, etc. These services require high data rate with reliable communication which is a challenging task to achieve in wireless environment as wireless systems operate over multipath fading channels (Khatalin and Fonseka, 2006). Fading has adverse effects on wireless communication system since it affects the signal destructively leading to poor reception. Thus, the capacity of the fading channels is the most important factor to consider while designing the wireless communication systems as it determines the maximum achievable data rate of the system (Khatalin and Fonseka, 2007). Channel capacity is also known as average channel capacity as it is achieved by averaging the Shannon capacity of an Additive White Gaussian Noise (AWGN) channel.

Specification

Description:FORM 2
THE PATENTS ACT, 1970
(39 OF 1970)
&
THE PATENTS RULES, 2003 5
COMPLETE SPECIFICATION
(See section 10; rule 13)
10
Title: Capacity of dual branch MRC system for correlated imbalanced average SNRs
over Nakagami-m fading channels using OPRA scheme
15
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 20
25
PREAMBLE TO THE DESCRIPTION:
The following specification particularly describes the nature of this invention and the manner
in which it is to be performed. 30
384 A. Yengkhom et al.
1 Introduction
Wireless communication is the fastest growing segment of the
communications industry. It provide services, such as wireless
telephones, internet access, video teleconferencing and satellite
services, etc. These services require high data rate with reliable
communication which is a challenging task to achieve in
wireless environment as wireless systems operate over
multipath fading channels (Khatalin and Fonseka, 2006).
Fading has adverse effects on wireless communication system
since it affects the signal destructively leading to poor
reception. Thus, the capacity of the fading channels is the most
important factor to consider while designing the wireless
communication systems as it determines the maximum
achievable data rate of the system (Khatalin and Fonseka,
2007). Channel capacity is also known as average channel
capacity as it is achieved by averaging the Shannon capacity of
an Additive White Gaussian Noise (AWGN) channel.
Channel capacity is defined as the maximum achievable
rate at which information can be reliably transmitted over a
channel. The capacity of a band limited AWGN channel (nonfading
channel) was obtained by Shannon as (Khatalin and
Fonseka, 2006)
? ? 2 log 1 C B ? ? ? bits/s (1)
where B is the channel bandwidth (Hz), ? is the instantaneous
received signal-to-noise (SNR).
Shannon capacity of an AWGN channel represents an
upper limit for reliable information transmission (Goldsmith
and Varaiya, 1997). The channel capacity in fading
environment is always less than the channel capacity under
AWGN channel but improvement in the capacity under
fading environment can be obtained by increasing the signal
transmit power well enough to reduce the effect of fading
(Simon and Alouini, 2005). However, it results in increased
level of interference. Diversity techniques are used to
improve the performance of the fading channels. In diversity
technique, multiple transmitting and receiving antennas are
used in communication channel and the signal is transmitted
and received through multiple paths (Simon and Alouini,
2005). As a result, the probability that all replicas of signals
will fade at the same time is reduced significantly. Diversity
combining is a method of collecting uncorrelated faded
signals from diversity branches and combining them in such a
manner that can optimise received signal power or SNR and
improve performance of communication systems. Pure
diversity combining is of three types; Selection Combining
(SC), Maximal Ratio Combining (MRC), Equal Gain
Combining (EGC) (Simon and Alouini, 2005).
Diversity combining and/or adaptive transmission
schemes are another class of technique employed to improve
the capacity exclusive of the requirement of increasing
transmit power and/or bandwidth in fading environment
(Simon and Alouini, 2005). Adaptive transmission schemes
are classified into four categories: Optimal Rate Adaptation
(ORA) with Constant Transmit Power, Optimal Power and
Rate Adaptation (OPRA), Channel Inversion with Fixed Rate
(CIFR) and Truncated Channel Inversion with Fixed Rate
(TIFR) (Simon and Alouini, 2005). Under ORA scheme, the
transmit power is kept constant and data rate is adapted by the
transmitter according to the fading conditions of the channel.
In OPRA scheme, transmitter in accordance with channel
condition adapts both data rate and power. In CIFR,
transmitter adapts power for maintaining constant SNR so
that the channel is inverted into AWGN channel. CIFR
scheme is the simplest system to employ when there are good
channel estimates at both the transmitter and receiver. TIFR
scheme is a modification of the CIFR scheme in which power
can be adapted about a fixed cutoff, 0 ? (Simon and Alouini,
2005).
Previous studies (Khatalin and Fonseka, 2006, 2007;
Goldsmith and Varaiya, 1997; Simon and Alouini, 2005)
show that OPRA scheme is the most complex scheme to
employ compared to the other schemes. OPRA scheme yields
a small improvement in channel capacity over ORA, but with
increase in the average SNR and/or decrease in severity
of fading, this small improvement decreases. Also, the
difference in capacity between these two schemes becomes
negligible for all average received SNRs when diversity
combining technique is employed, even in deep fading
environments. That is if the data rate is varied according to
the conditions of the channel, varying the transmit power
gives negligible improvement.
Several works have been done on the average capacity
under various fading channels. We summarise some of the
related works. The capacity of Rayleigh and Rician fading
channels with MRC diversity has been considered in many
papers. Particularly, Simon and Alouini (2005), Alouini and
Goldsmith (1999) and Alouini and Goldsmith (1997)
discussed the capacity of MRC using different schemes of
adaptive transmission over uncorrelated Rayleigh fading
channels. The results in Alouini and Goldsmith (1999) were
extended in Shao et al. (1999), Mallik and Win (2000) and
Mallik et al. (2004) to obtain the expressions intended for the
channel capacity using MRC under Rayleigh fading taking
into account the impact of correlation in the branches for both
the cases of balanced and unbalanced branch SNRs for all
four schemes. Subadar and Sahu (2010) presented the
analysis of channel capacity of dual branch SC as well as
MRC systems using Nakagami-m fading with non-identical
and arbitrary fading parameters of correlated (? = 0, 0.3 and
0.6). In Khatalin and Fonseka (2006), capacity expression of
MRC using ORA scheme over uncorrelated Hoyt and Rician
fading channels is extended by obtaining lower- and upperbound
expressions. Channel capacity of dual MRC under
uncorrelated and correlated Nakagami-0.5 fading channels
using OPRA, TIFR, ORA and CIFR schemes is obtained in
Hasan and Kumar (2014), Hasan and Kumar (2015a, 2015b,
2015c) , respectively. In Hasan and Kumar (2016), channel
capacity of dual-SC over correlated Nakagami-0.5 fading
with imbalanced average received SNR under ORA, TIFR
and OPRA schemes was obtained. Closed formed expression
for the channel capacity of MRC for independent Nakagamiq
fading under different scheme of adaptive transmission is
obtained in Cheng and Berger (2003). Da Costa and Yacoub
(2007) presented the average channel capacity under ? ? ?
and ? ? ? fading channels. In Pena-Martin et al. (2013),
Capacity of dual branch MRC system for correlated imbalanced average SNRs 385
performance analysis of wireless system with MIMO under
Hoyt fading channel is presented. BER performance of MRC
system using Nakagami-m is analysed in Aalo (1995).
Capacity of MRC over arbitrary Rician fading channel has
been obtained in Hamdi (2008). Annavajjala and Milstein
(2004) considered the average channel capacity of dual-
MRC, SC and EGC system with unequal average SNRs over
Rayleigh fading channels. Alouini and Simon (1998)
obtained the capacity expression of Rayleigh channel with
MRC, EGC and SC taking into account level of imbalanced
in mean signal strength and correlation effect. The channel
capacity of Rayleigh fading is investigated in Lee (1990) and
Gunther (1996), respectively. Channel capacity expression
with MRC using Nakagami-m fading for the ORA scheme
assuming i.i.d. diversity branches was obtained in Yao and
Sheikh (1993). In Alouini and Goldsmith (1997) and Alouini
et al. (2001), the channel capacity with MRC diversity using
Nakagami-m fading under different adaptive transmission
schemes considering i.i.d. diversity branches was studied.
The capacity of MRC diversity systems under i.i.d. Rician
and Hoyt fading channels has been investigated in Cheng and
Berger (2001) and also for the four adaptive transmission
schemes mentioned above. In Devi et al. (2016), the channel
capacity with MRC diversity using Nakagami-m fading for
Rayleigh fading channel has been reviewed. In Pukhrambam
et al. (2017), the channel capacity with MRC diversity using
Nakagami-m fading channels with non-identical fading
parameters with first branch Rayleigh and second branch
Rician (m = 3) and imbalanced average SNRs has been
obtained. The capacity in the literature is also known as
average capacity as it is realised by averaging the Shannon
capacity of an AWGN channel.
However, an analytical study on channel capacity of
correlated imbalanced average SNRs Nakagami-m fading for
m? 1 under OPRA has not been done so far. Imbalanced
average SNRs means unequal average received SNR between
the branches. It is also known as unequal average SNRs.
Generally, Nakagami-m is used to a great extent to study the
performance of wireless mobile communication system but
Nakagami-m with imbalanced average SNRs appears to have
less attention. Also the results obtained for capacity of
Nakagami-m with imbalanced average SNRs will have huge
practical value, as in practical scenario branches may not
experience same amount of fade. For that reason it is
imperative to design a system for imbalanced average SNRs
for realistic scenario, and obtain results for different fading
correlation with imbalanced average SNRs. Therefore, this
paper fills a much needed gap by introducing the effect of
different physical constraints e.g., antenna space requirements,
correlation and level of imbalanced in average SNRs on the
capacity using OPRA scheme. In this paper, we investigate the
channel capacity over Nakagami-m fading using MRC systems
under OPRA schemes with balanced as well as imbalanced
average received SNRs. We consider slowly-varying flat
multipath fading channels. Similar to the assumption made in
Simon and Alouini (2005), an error free feedback path with
negligible time delay is assumed while considering adaptive
transmission schemes.
The rest of the paper is organised as follows: Section 2
defines the channel model. In Section 3, channel capacity
of MRC considering balanced and imbalanced average
SNRs is derived using OPRA scheme. Section 4 presents
numerically evaluated results. Concluding remarks are
given in Section 5.
2 Channel model
The Probability Distribution Function (PDF), ( ) p ? ? of the
received instantaneous SNR, ? at MRC combiner for
L-branch correlated Nakagami-m fading channels with
balanced average SNRs ? ? 1 2 ? ? ? ? ? can be presented as
Khatalin and Fonseka (2006)
? ?
? ?? ?
? ? ? ? ? ?
1
1
1
exp
1
; ;
1 1
( ) , 0
1 1
Lm
Lm m m
m m
Lm F m Lm
L
p
L Lm
m
?
? ?
? ? ?
??
? ? ? ?
? ?
? ? ? ?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ?
(2)
where ? is the average received SNR, m is the fading
parameter,1 1[...] F is the Kummer confluent hypergeometric
function (Gradshteyn and Ryzhik, 2000) and ? is the
correlation coefficient.
The PDF of ? at L-branch MRC combiner over
uncorrelated fading channels using Nakagami-m can be
obtained by first setting ? = 0 in (2), which was also given by
Simon and Alouini (2005) and Alouini and Goldsmith (1999)
as
? ?
1
exp
( ) , 0
Lm m m
p
Lm
m
?
? ?
? ?
? ?
?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ?? ? ?
? ?
(3)
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 2
1
2
( ) 1 2 '
exp ' , 0
m m
m
m p
m
I
?
? ?
?
? ? ? ?
? ? ? ? ?
?
? ? ? ? ?
? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ?
(4)
where In(.) is the n-th order Bessel function and the parameter
' ? and ' ? are the parametre (Alouini and Simon, 1998), and
is given by,
? ?
? ?
1 2
1 2 2 1
m ? ?
?
?? ?
?
? ?
?
386 A. Yengkhom et al.
and
? ? ? ? ? ?
? ?
1
2 2
1 2 1 2
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 1 2
1 2
2
1 2 1 2 1 2
1 2
2
1 2 1 2
4 1
exp
2 1
4 1
exp
2 1
4 1
p ?
? ? ? ? ? ? ?
?
? ? ?
? ? ? ? ? ? ?
?
? ? ?
?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
? ? ?
(5)
For m = 3, (4) reduces as
? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
3
4 5
1 2
2
1 9
2 1
3 exp exp
exp exp
3 exp exp
p ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ?? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?
(6)
Different values of m describe several important distributions.
For example, m = 1 corresponds to distribution of Rayleigh,
m = 1 corresponds to Rician distribution.
3 Channel capacity
In this section, we obtain the closed form expressions for the
channel capacity of uncorrelated and correlated fading channels
using Nakagami-m with MRC for optimal power and rate
adaptation (OPRA) scheme considering balanced and
imbalanced average SNRs. The corresponding expressions for
correlated Rayleigh (m=1) channels with imbalanced average
SNRs and correlated Rician (m=3) channels with both balanced
and imbalanced average SNRs are obtained. The effect of
correlation and level of imbalanced in average SNRs on the
channel capacity is also analysed.
3.1 OPRA
In OPRA scheme, transmitter in accordance with channel
condition, adapts both data rate and power. If this power
adaptation ? ? P ? is subjected to average power constraint
P (Simon and Alouini, 2005), then
0
( ) ( ) P p d P ? ? ? ?
?
? ? (7)
where ( ) P? ? indicates the PDF under Nakagami-m fading
channel. The capacity with P is the capacity in Simon and
Alouini (2005), equation (15.21) with the power optimally
distributed over time (Simon and Alouini, 2005)
? ? ? ?
? ? ? ?
0
2 max log 1
P p P
P
C B p d
P ?
?
? ?
?
? ?
? ?
?
?
? ?
? ? ? ?
? ? ? ? (8)
The power adaptation that optimises the above expression was
obtained in Simon and Alouini (2005) by using Lagrange
multipliers as
? ? 0
0
1 1,
0,
P
P
? ? ?
? ?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
(9)
where 0 ? is the optimal cut-off value below which data
transmission is suspended and 0 ? must satisfy
? ?
0 0
1 1 1 p d ?
?
? ?
? ?
?? ?
? ? ? ?
? ? ? (10)
Substituting (10) into (8) the expression for the channel
capacity of fading channels with OPRA is obtained as
Khatalin and Fonseka (2006, 2007), Goldsmith and Varaiya
(1997), Simon and Alouini (2005)
? ? 2
0 0
log OPRA C B p d ?
?
? ?
?
? ? ?
? ? ?
? ? ? (11)
OPRA is also known as water filling technique. Higher
power and data rates are offered when channel conditions is
good (large ?), and as channel condition degrades (small ?),
lower power and data rates are given. When 0 ? is greater
than the received SNR, the data transmission is suspended,
which is also known as an outage probability Pout that is
equal to the probability when there no transmission. Outage
probability is given by
? ? ? ? ? ? 0
0
0
0
1 out P P p d p d
?
? ?
?
? ? ? ? ? ?
?
? ? ? ? ? ? ? (12)
Probability of the outage is different from outage capacity
which is the probability that the instantaneous capacity, C,
drops below predetermined threshold or target capacity, Cth,
which is given as Simon and Alouini (2005) Pout (Cth) =
P[C = Cth].
3.1.1 Capacity for uncorrelated fading channels
using Nakagami-m (m = 3) with balanced
average SNRs
The capacity for uncorrelated fading channels using Nakagamim
(m = 3) under OPRA scheme can be possibly obtained by
first substituting the PDF in (3) into (11) and putting m = 3 as,
? ? 0
3 1
2
0
3 3 exp
log
G 3
3
L
OPRA C B d
L ?
? ?
? ? ?
?
? ?
?
?
? ? ? ? ?
? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? (13)
Capacity of dual branch MRC system for correlated imbalanced average SNRs 387
To get an optimal cut-off SNR 0 ? , we first set m = 3, and substitute (3) in (10) and obtained
? ? 0
3 1
0
3 3 exp
1 1 1
G 3
3
L
d
L ?
? ?
? ?
?
? ? ?
?
?
? ? ? ? ?
? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? (14)
Putting the values of L=1 in (14) and evaluating the above integral and after some mathematical transformation using
(Wolfram, 2015; Gradshteyn and Ryzhik, 2000) we obtain
0 0
0
3 2 3 1 exp 1
2
? ? ?
? ? ?
? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ?
(15)
Putting the values of m=3 and L=2 in (14) and evaluating the integral and after some mathematical transformation by Wolfram
(2015) and Gradshteyn and Ryzhik (2000), we can obtained the following cut off SNR 0 ? ,
? ? ? ?
6 5 4
0 0 0
6 5 6
0 0 0
3 5! 4! 3 1 exp 1
120 3 3 ! !
k z
k z
k z k z
? ? ?
? ? ?
? ?
? ?
? ?
? ?
? ? ? ?
? ? ? ? ? ? ?
? ?? ?
? ? ? ?
? ? (16)
After solving (15) and (16), we get a unique value of 0 ? for each ? fulfilling (15) and (16) that take values from 0 [0,1] ? ? . Result
shows that 0 ? increase with increase in ? . The cut-off SNR value, 0 ? satisfying (15) and (16) for every ? value, has been used for
calculating channel capacity.
Channel capacity without diversity under uncorrelated Rician fading (m=3) can be obtained by first putting m=3, setting
L=1 in (14) and then substituting in (13) as,
0
2 2
2 3
0
27 3 log exp
2 OPRA C B d
?
? ? ?
?
? ? ?
? ? ?? ? ? ? ?
? ? ?? ? ? ?
? ? ? ? ? ? ? (17)
? ? ? ?
0 0
2 2
0 3
27 1.443 3 3 log exp log exp
2 OPRA C B d d
? ?
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? (18)
After evaluating the above integral and after some mathematical transformation using (Wolfram, 2015; Gradshteyn and Ryzhik,
2000) we get,
? ? 2 2 2 0 0
0 1 2
3 3 1.443 exp 3 2 2
2 OPRA C B E
? ?
? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
(19)
The channel capacity under uncorrelated Rician fading (m = 3) of dual branch MRC can be obtained by first putting m = 3, setting
L=2 in (14), substituting in (13) as,
0
5
6
2
0
3 exp
3 log
120 OPRA C B d
?
?
?
? ?
?
? ?
?
? ? ?
? ? ? ?? ? ? ? ? ? ?? ?
? ? ? ? ? (20)
? ? ? ?
0 0
6
5 5
0
1.443 3 3 3 log exp log exp
120 OPRA C B d d
? ?
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? (21)
After solving the integration in (21) by partial integration and some mathematical transformation using (Gradshteyn and
Ryzhik, 2000), we get
? ? ? ? ? ?
2 1
2 1
2 1
2 3 4
0 0 0 0 0 0
1 3 4 5 3 4 5
0 0 0
2 1
3 3 3 9 20.25 48.6 1.443 exp 1.5
2 3 3 3 ! ! !
k k k
OPRA k k k
k k k
C B E
k k k
? ? ? ? ? ?
? ? ? ? ? ?
? ? ?
? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? (22)
388 A. Yengkhom et al.
3.1.2 Capacity of correlated fading channels using Nakagami-m considering balanced and imbalanced average
SNRs with dual branch MRC
The capacity of correlated Rayleigh fading channels for OPRA scheme with imbalanced average SNRs using dual branch
MRC, can be found by first substituting the PDF expression in (5) into (11) as,
? ? ? ?
2
0 0
exp exp
log OPRA
a b
C B d
x
? ? ?
?
?
? ? ? ? ? ?
? ? ?
? ? ? (23)
where
? ? ? ?
? ?
? ? ? ?
? ? ? ? ? ?
2 2
2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 1 2
1 2 1 2
4 1 4 1
4 1
2 1 2 1
a b x
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ?
After evaluating the integral of (23) and some mathematical transformation using (Wolfram, 2015; Gradshteyn and Ryzhik,
2000) we get,
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 1 0 0 0 1 0 0 0
0
log exp log exp exp exp 1.443 log OPRA
a E a b E b a b B C
x a b a b
? ? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
(24)
Putting (5) in (10) and after evaluating the integral using partial integral and after some mathematical transformation using
(Wolfram, 2015; Gradshteyn and Ryzhik, 2000), we can obtained the following cut off SNR 0 ? ,
? ? ? ? ? ? ? ? ? ? 0 0
1 0 1 0
0
exp exp 1 1 1
a b
E a E b
x a b x
? ?
? ?
?
? ? ? ?
? ? ? ? ? ?
? ?
(25)
Similarly, capacity of correlated Rician (m = 3) fading channels for OPRA scheme with balanced and imbalanced average
SNRs using dual branch MRC, can be found by first substituting the PDF expression in (6) into (11) as,
? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
3
2
2 4 5
0 12 0
3 exp exp
1 9 log exp exp
2 1
3 exp exp
OPRA C B d
? ? ? ? ? ?
?
?? ? ? ? ? ? ? ?
? ? ?? ?
? ? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?
? (26)
After evaluating the integral of (26) and some mathematical transformation using (Wolfram, 2015; Gradshteyn and Ryzhik,
2000) we get,
? ? ? ?
3
1 2 3 4 4 5
1 2
1.443 9
2 1 OPRA
B C II I I
? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ? ?
(27)
where
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ?
0 0 1 0
0 0 1 0
1
log exp ' '
log exp ' ' '
'
3
'
E
E
I
? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ?
2 2
0
0 0 1
0
0 0 0 1 0
3
2
2
0 0 0 1 0
3
2 2
0
0 0 1
0
2 log exp ' '
2 ! ' '
exp ' ' 1 ' ' 2exp ' ' 2 ' '
' '
'
exp ' ' 1 ' ' 2exp ' ' 2 ' '
' '
2 log exp ' '
2 ! ' '
k
k
k
l
l
l
k
E
I
E
l
?
? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ?
?
? ? ? ? ? ? ? ? ? ? ? ?
? ?
?
? ? ? ?
? ?
?
?
?
?
?
?
?
? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ?
? ? ?
?
?
??????
? ?
? ?
? ?
? ?
? ?
? ?
? ?
?
Capacity of dual branch MRC system for correlated imbalanced average SNRs 389
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
0 0 0 0 1 0
2
3
0 0 0 0 1 0
2
log exp ' ' 1 ' ' exp ' ' ' '
' '
3 '
log exp ' ' 1 ' ' exp ' ' ' '
' '
E
I
E
? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
and
? ?
? ? ? ? ? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
0 0
2 2
2 0 0
4 0 0 0 3 3
0 0
0 0 0 0
2
exp ' ' exp ' '
3
' '
2 2 log ' exp ' ' exp ' '
! ' ' ! ' '
exp ' ' 1 ' ' exp ' ' 1 ' '
3 '
' '
y s
y s
y s
I
y s
? ? ? ? ? ?
? ?
? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ?
?
?
?
??
Putting (6) in (10) and after evaluating the integral using partial integral and after some mathematical transformation using
(Wolfram, 2015; Gradshteyn and Ryzhik, 2000), we can obtained the following cut off SNR 0 ? ,
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ?
0 0
2
0
0 3
0
2
2 0
0
0 3
0
0 0
2
0 0
exp exp
3
2 exp
!
2 exp
!
exp 1
3
exp 1
k
k
k
z
z
z
k A
z
? ? ? ? ? ?
? ? ? ?
?
? ? ?
? ?
?
? ?
? ? ?
? ?
? ? ? ? ? ?
? ?
?
? ? ? ? ? ?
?
?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ?
? ? ? ? ? ? ?
?
?
?
?
? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ?
2
1 0 1 0
0 0
2
2
0 0
2
0 0
3
exp 1
exp 1
exp exp
3
E E
A
? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ?
?
? ? ? ? ? ?
? ?
? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ?
1
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ??
? ?
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
(28)
The corresponding expressions for capacity with balanced SNR can be obtained by setting ? ? 1 2 ? ? ? ? ? . Also capacity over
uncorrelated Rayleigh and Rician (m = 3) with imbalanced average can be obtained by putting ? = 0 in (24) and (26),
respectively.
390 A. Yengkhom et al.
4 Numerical results and analysis
Results for the capacity of MRC over uncorrelated and
correlated fading channels using Nakagami-m (m = 1) with
imbalanced average SNRs and Nakagami-m (m = 3) with
both balanced and imbalanced average SNRs under OPRA
scheme are presented in this section. For comparison,
capacity of fading channels using correlated Nakagami-m
(m = 1) fading with and without diversity (Goldsmith and
Varaiya, 19997; Shao et al., 1999; Mallik and Win, 2000)
considering balanced average received SNRs is presented in
the figures.
Figure 1 shows the channel capacity over correlated
fading channels using Nakagami-m under OPRA scheme of
dual branch MRC for m=1 (Rayleigh) with balanced
average SNRs which was obtained in Khatalin and Fonseka
(2006) and Simon and Alouini (2005). Figure 2 shows the
channel capacity over correlated fading channels using
Nakagami-m under OPRA scheme of dual branch MRC for
m = 1 (Rayleigh) with imbalanced average SNRs and is
compared with already published results of Shao et al.
(1999). In both the figures, employing diversity using MRC
and/or increasing average received SNR ? ? 1 or ? ? gives
improvement in channel capacity. It is also seen that with
dual branch MRC, the capacity is almost identical for
1 , 1dB ? ? ? with m = 1 even when branch correlation
increases. So for low value of SNR the appropriate spacing
essential for creating uncorrelated diversity path at the
receiver end may be ignored particularly for 1 , 1dB ? ? ?
with m = 1.
Figure 1 Capacity of MRC for correlated fading channels using
Nakagami-m (m = 1) under OPRA scheme
0 5 10 15 20 25 30
1
2
3
4
5
6
7
8
9
10
11
Average received SNR[dB] per branch
Channel capacity per Unit Bandwidth [Bits/Sec/Hz]
dual-branch,?=0,equal SNR
dual-branch,?=0.1,equal SNR
dual-branch,?=0.3,equal SNR
dual-branch,?=0.8,equal SNR
no-diversity
Figure 3 shows the capacity comparison for correlated
fading channels using Nakagami-m under OPRA scheme of
dual branch MRC of balanced with imbalanced average
SNRs for m = 1. It is seen that the channel capacity
improvement considering imbalanced average SNRs gives
the better capacity than those with balanced SNRs as
expected, because the overall SNR of imbalanced average
SNRs is greater than that of balanced average SNRs
between the branches as we have considered 2 1 2 ? ? ? . From
this figure also it was observed that MRC diversity and
increasing average SNR increase the capacity.
Figure 2 Channel capacity of MRC for correlated fading
channels using Nakagami-m (m = 1) under OPRA
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/Sec/Hz]
dual-branch,?=0,imbalanced SNR
dual-branch,?=0.1,imbalanced SNR
dual-branch,?=0.3,imbalanced SNR
dual-branch,?=0.8,imbalanced SNR
no-diversity
Figure 3 Capacity comparison of MRC for correlated fading
channels using Nakagami-m (m = 1)
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/Sec/Hz]
dual-branch,?=0
dual-branch,?=0.1
dual-branch,?=0.3
dual-branch,?=0.8
No diversity
Imbalanced SNRs
equal SNRs
Figure 4 shows the channel capacity over correlated fading
channels using Nakagami-m under OPRA scheme of dual
branch MRC for m = 3 (Rician) with balanced average
SNRs. Figure 5 shows the channel capacity over correlated
fading channels using Nakagami-m under OPRA scheme of
dual branch MRC for m = 3 (Rician) with imbalanced
average SNRs. By increasing average receive SNR ? ? 1 ?
and/or employing diversity using MRC, channel capacity
should improve, the same has been observed. It is seen that
with dual branch MRC, the capacity is almost identical for
1 4 dB ? ? with m = 3 even when branch correlation
increases. So for low values of SNR the appropriate
Capacity of dual branch MRC system for correlated imbalanced average SNRs 391
spacing essential for creating uncorrelated diversity path at
the receiver end may be ignored particularly for 1 4 dB ? ?
with m = 3.
Figure 4 Capacity of MRC for correlated fading channels using
Nakagami-m (m = 3) under OPRA scheme
0 5 10 15 20 25 30
1
2
3
4
5
6
7
8
9
10
11
Average received SNR[dB] per branch
Channel capacity per Unit Bandwidth [Bits/Sec/Hz]
dual-branch,?=0,equal SNR
dual-branch,?=0.1,equal SNR
dual-branch,?=0.3,equal SNR
dual-branch,?=0.8,equal SNR
no-diversity
Figure 5 Channel capacity of MRC for correlated fading
channels using Nakagami-m (m = 3) under OPRA
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/Sec/Hz]
dual-branch,?=0,imbalanced SNR
dual-branch,?=0.1,imbalanced SNR
dual-branch,?=0.3,imbalanced SNR
dual-branch,?=0.8,imbalanced SNR
no-diversity
Figure 6 shows the capacity comparison for correlated fading
channels using Nakagami-m under OPRA scheme of dual
branch MRC of balanced with imbalanced average SNRs for
m = 3 and is compared with the already published results
(Shao et al., 1999) which are also shown in Figures 1, 2 and 3.
It can be observed the capacity of MRC over Nakagami-m
(m = 3) is greater than the capacity of MRC over Nakagami-m
(m = 1) for both the cases of balanced and imbalanced SNRs. It
is seen that the channel capacity improvement considering
imbalanced average SNRs gives the better capacity than
those with balanced SNRs as expected, because the overall
SNR of imbalanced average SNRs is greater than that of
balanced average SNRs between the branches as we have
considered 2 1 2 ? ? ? . From this figure also it was observed that
MRC diversity and increasing average SNR increase the
capacity.
Figure 6 Capacity comparison of MRC for correlated fading
channels using Nakagami-m (m = 3)
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/Sec/Hz]
dual-branch,?=0
dual-branch,?=0.1
dual-branch,?=0.3
dual-branch,?=0.8
no diversity
imbalanced SNR
equal SNR
5 Conclusion
We have obtained expressions for the capacity of MRC for
correlated and uncorrelated fading channels using Nakagami-m
under OPRA scheme with balanced and imbalanced average
received SNRs. As expected, employing diversity using MRC
and/or increasing average received SNRs gives improvement in
channel capacity. By numerical evaluation we also observed
that, channel capacity improvement considering imbalanced
average received SNRs gives the better capacity than those
with balanced average received SNRs as expected. Further,
with dual branch MRC, the channel capacity is almost identical
for smaller values of average received SNRs even when branch
correlation increases. Therefore, for low values of SNR, the
appropriate spacing required for creating uncorrelated diversity
path at the receiver end may be ignored.
References
Aalo, V.A. (1995) 'Performance of maximal-ratio diversity systems in
a correlated Nakagami fading environment', IEEE Transactions
on Communications, Vol. 43, No. 8, pp.2360-2369.
Alouini, M-S. and Goldsmith, A. (1997) 'Capacity of Nakagami
multipath fading channels', Proceedings of the IEEE 47th
Vehicular Technology Conference Technology in Motion, IEEE,
USA, Vol. 1.
Alouini, M-S. and Goldsmith, A.J. (1999) 'Capacity of Rayleigh
fading channels under different adaptive transmission and
diversity-combining techniques," IEEE Transactions on
Vehicular Technology, Vol. 48, pp.1165-1181.
Alouini, M-S. and Simon, M.K. (1998) 'Multichannel reception
of digital signals over correlated Nakagami fading channels',
Proceedings of the Annual Allerton Conference on Communication
Control and Computing', Vol. 36, pp.146-155.
392 A. Yengkhom et al.
Alouini, M-S., Abdi, A. and Kaveh, M. (2001) 'Sum of gamma
variates and performance of wireless communication systems
over Nakagami-fading channels', IEEE Transactions on
Vehicular Technology, Vol. 50, No. 6, pp.1471-1480.
Annavajjala, R. and Milstein, L.B. (2004) 'On the capacity of dual
diversity combining schemes on correlated Rayleigh fading
channels with unequal branch gains', IEEE Wireless
Communications and Networking Conference, IEEE, USA,
Vol. 1.
Cheng, J. and Berger, T. (2003) 'Capacity of Nakagami-q (Hoyt)
fading channels with channel side information', International
Conference on Communication Technology (ICCT'03), IEEE,
China, Vol. 2.
Cheng, J. and Berger, T.E. (2001) 'Capacity of a class of fading
channels with channel state information (CSI) feedback',
Proceedings of the 39th Annual Allerton Conference
on Communication, Control, and Computing (Allerton'01),
3-5 October, Monticello, IL, USA.
Da Costa, D.B. and Yacoub, M.D. (2007) 'Average channel capacity
for generalized fading scenarios", IEEE Communications Letters,
Vol. 11, No. 12, pp.949-951.
Devi, P.B., Yengkhom, A., Hasan, M.I. and Chandwani, G. (2016)
'Capacity of dual branch MRC system over correlated
Nakagami-m fading channels: a review', Wireless Personal
Communications, pp.1-18. Doi 10.1007/s11577-017-4942-4.
Goldsmith, A.J. and Varaiya, P.P. (1997) 'Capacity of fading
channels with channel side information', IEEE Transactions
on Information Theory, Vol. 43, No. 6, pp.1986-1992.
Gradshteyn, S. and Ryzhik, I.M. (2000) Table of Integrals, Series,
and Products, 6th ed., Academic Press, New York.
Gunther, C.G. (1996) 'Comment on Estimate of channel capacity
in Rayleigh fading environment', IEEE Transactions on
Vehicular Technology, Vol. 45, No. 2, pp.401-403.
Hamdi, K.A. (2008) 'Capacity of MRC on correlated Rician fading
channels', IEEE Transactions on Communications, Vol. 56,
No. 5, pp.708-711.
Hasan, M.I. and Kumar, S. (2014) 'Channel capacity of dualbranch
maximal ratio combining under worst case of fading
scenario', WSEAS Transactions on Communications, Vol. 13,
pp.162-170.
Hasan, M.I. and Kumar, S. (2015a) 'Average channel capacity of
correlated dual-branch maximal ratio combining under worst
case of fading scenario', Wireless Personal Communications,
Springer, Vol. 83, pp.2624-2646.
Hasan, M.I. and Kumar, S. (2015b) 'Average channel capacity
evaluation for selection combining diversity schemes over
nakagami-0.5 fading channels', International Journal of
Communication Networks and Information Security (IJCNIS),
Vol. 7, No. 2, pp.69-79.
Hasan, M.I. and Kumar, S. (2015c) 'Spectral efficiency evaluation
for selection combining diversity scheme under worst case of
fading scenario', International Journal of Communication
Networks and Information Security (IJCNIS), Vol. 7, No. 3,
pp.123-130.
Hasan, M.I. and Kumar, S. (2016) 'Spectral efficiency of dual
diversity selection combining schemes under correlated
Nakagami-0.5 fading with unequal average received SNR',
Telecommunication Systems, pp.1-14.
Khatalin, S. and Fonseka, J.P. (2006) 'Capacity of correlated
Nakagami-m fading channels with diversity combining
techniques', IEEE Transactions on Vehicular Technology,
Vol. 55, No. 1, pp.142-150.
Khatalin, S. and Fonseka, J.P. (2006) 'On the channel capacity in
Rician and Hoyt fading environments with MRC diversity',
IEEE Transactions on Vehicular Technology, Vol. 55, No. 1,
pp.137-141.
Khatalin, S. and Fonseka, J.P. (2007) 'Channel capacity of
dual-branch diversity systems over correlated Nakagami-m
fading with channel inversion and fixed rate transmission
scheme', IET Communications Proceedings, Vol. 1, No. 6,
pp.1161-1169.
Lee, W.C.Y. (1990) 'Estimate of channel capacity in Rayleigh
fading environment', IEEE Transactions on Vehicular
Technology, Vol. 39, No. 3, pp.187-189.
Mallik, R.K. and Win, M.Z. (2000) 'Channel capacity in evenly
correlated Rayleigh fading with different adaptive transmission
schemes and maximal ratio combining', IEEE International
Symposium on Information Theory, IEEE, Italy.
Mallik, R.K. et al. (2004) 'Channel capacity of adaptive transmission
with maximal ratio combining in correlated Rayleigh fading',
IEEE Transactions on Wireless Communications, Vol. 3, No. 4,
pp.1124-1133.
Pena-Martin, J.P., Romero-Jerez, J.M. and Tellez-Labao, C. (2013)
'Performance of TAS/MRC wireless systems under Hoyt fading
channels', IEEE Transactions on Wireless Communications,
Vol. 12, No. 7, pp.3350-3359.
Pukhrambam, B.D., Yengkhom, A., Hasan, M.I. and Chandwani,
G. (2017) 'Capacity of dual-branch MRC system over
correlated Nakagami-m fading channels with non-identical
fading parameters and imbalanced average SNRs',
International Journal on Wireless and Mobile Computing,
Vol. 13, No. 2, pp.151-163.
Shao, J.W., Alouini, M-S. and Goldsmith, A.J. (1999) 'Impact of
fading correlation and unequal branch gains on the capacity of
diversity systems', IEEE Transactions on Vehicular
Technology, Vol. 3, No. 1, pp.142-150.
Simon, M.K. and Alouini, M-S. (2005) Digital Communication
over Fading Channels, John Wiley & Sons, Inc.
Subadar, R. and Sahu, P.R. (2010) 'Capacity analysis of Dual-SC
and-MRC systems over correlated Nakagami-m fading
channels with non-identical and arbitrary fading parameters',
National Conference on Communications (NCC), IEEE,
India.
Wolfram (2015) The Wolfram function site. Available online at:
http://functions.wolfram.com
Yao, Y-D. and Sheikh, A.U.H. (1993) 'Evaluation of channel
capacity in a generalized fading channel', Proceedings of the
43rd IEEE Vehicular Technology Conference, IEEE, USA. , Claims:FORM 2
THE PATENTS ACT, 1970
(39 OF 1970)
&
THE PATENTS RULES, 2003 5
COMPLETE SPECIFICATION
(See section 10; rule 13)
10
Title: Capacity of dual branch MRC system for correlated imbalanced average SNRs
over Nakagami-m fading channels using OPRA scheme
15
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 20
25
PREAMBLE TO THE DESCRIPTION:
The following specification particularly describes the nature of this invention and the manner
in which it is to be performed. 30
384 A. Yengkhom et al.
1 Introduction
Wireless communication is the fastest growing segment of the
communications industry. It provide services, such as wireless
telephones, internet access, video teleconferencing and satellite
services, etc. These services require high data rate with reliable
communication which is a challenging task to achieve in
wireless environment as wireless systems operate over
multipath fading channels (Khatalin and Fonseka, 2006).
Fading has adverse effects on wireless communication system
since it affects the signal destructively leading to poor
reception. Thus, the capacity of the fading channels is the most
important factor to consider while designing the wireless
communication systems as it determines the maximum
achievable data rate of the system (Khatalin and Fonseka,
2007). Channel capacity is also known as average channel
capacity as it is achieved by averaging the Shannon capacity of
an Additive White Gaussian Noise (AWGN) channel.
Channel capacity is defined as the maximum achievable
rate at which information can be reliably transmitted over a
channel. The capacity of a band limited AWGN channel (nonfading
channel) was obtained by Shannon as (Khatalin and
Fonseka, 2006)
? ? 2 log 1 C B ? ? ? bits/s (1)
where B is the channel bandwidth (Hz), ? is the instantaneous
received signal-to-noise (SNR).
Shannon capacity of an AWGN channel represents an
upper limit for reliable information transmission (Goldsmith
and Varaiya, 1997). The channel capacity in fading
environment is always less than the channel capacity under
AWGN channel but improvement in the capacity under
fading environment can be obtained by increasing the signal
transmit power well enough to reduce the effect of fading
(Simon and Alouini, 2005). However, it results in increased
level of interference. Diversity techniques are used to
improve the performance of the fading channels. In diversity
technique, multiple transmitting and receiving antennas are
used in communication channel and the signal is transmitted
and received through multiple paths (Simon and Alouini,
2005). As a result, the probability that all replicas of signals
will fade at the same time is reduced significantly. Diversity
combining is a method of collecting uncorrelated faded
signals from diversity branches and combining them in such a
manner that can optimise received signal power or SNR and
improve performance of communication systems. Pure
diversity combining is of three types; Selection Combining
(SC), Maximal Ratio Combining (MRC), Equal Gain
Combining (EGC) (Simon and Alouini, 2005).
Diversity combining and/or adaptive transmission
schemes are another class of technique employed to improve
the capacity exclusive of the requirement of increasing
transmit power and/or bandwidth in fading environment
(Simon and Alouini, 2005). Adaptive transmission schemes
are classified into four categories: Optimal Rate Adaptation
(ORA) with Constant Transmit Power, Optimal Power and
Rate Adaptation (OPRA), Channel Inversion with Fixed Rate
(CIFR) and Truncated Channel Inversion with Fixed Rate
(TIFR) (Simon and Alouini, 2005). Under ORA scheme, the
transmit power is kept constant and data rate is adapted by the
transmitter according to the fading conditions of the channel.
In OPRA scheme, transmitter in accordance with channel
condition adapts both data rate and power. In CIFR,
transmitter adapts power for maintaining constant SNR so
that the channel is inverted into AWGN channel. CIFR
scheme is the simplest system to employ when there are good
channel estimates at both the transmitter and receiver. TIFR
scheme is a modification of the CIFR scheme in which power
can be adapted about a fixed cutoff, 0 ? (Simon and Alouini,
2005).
Previous studies (Khatalin and Fonseka, 2006, 2007;
Goldsmith and Varaiya, 1997; Simon and Alouini, 2005)
show that OPRA scheme is the most complex scheme to
employ compared to the other schemes. OPRA scheme yields
a small improvement in channel capacity over ORA, but with
increase in the average SNR and/or decrease in severity
of fading, this small improvement decreases. Also, the
difference in capacity between these two schemes becomes
negligible for all average received SNRs when diversity
combining technique is employed, even in deep fading
environments. That is if the data rate is varied according to
the conditions of the channel, varying the transmit power
gives negligible improvement.
Several works have been done on the average capacity
under various fading channels. We summarise some of the
related works. The capacity of Rayleigh and Rician fading
channels with MRC diversity has been considered in many
papers. Particularly, Simon and Alouini (2005), Alouini and
Goldsmith (1999) and Alouini and Goldsmith (1997)
discussed the capacity of MRC using different schemes of
adaptive transmission over uncorrelated Rayleigh fading
channels. The results in Alouini and Goldsmith (1999) were
extended in Shao et al. (1999), Mallik and Win (2000) and
Mallik et al. (2004) to obtain the expressions intended for the
channel capacity using MRC under Rayleigh fading taking
into account the impact of correlation in the branches for both
the cases of balanced and unbalanced branch SNRs for all
four schemes. Subadar and Sahu (2010) presented the
analysis of channel capacity of dual branch SC as well as
MRC systems using Nakagami-m fading with non-identical
and arbitrary fading parameters of correlated (? = 0, 0.3 and
0.6). In Khatalin and Fonseka (2006), capacity expression of
MRC using ORA scheme over uncorrelated Hoyt and Rician
fading channels is extended by obtaining lower- and upperbound
expressions. Channel capacity of dual MRC under
uncorrelated and correlated Nakagami-0.5 fading channels
using OPRA, TIFR, ORA and CIFR schemes is obtained in
Hasan and Kumar (2014), Hasan and Kumar (2015a, 2015b,
2015c) , respectively. In Hasan and Kumar (2016), channel
capacity of dual-SC over correlated Nakagami-0.5 fading
with imbalanced average received SNR under ORA, TIFR
and OPRA schemes was obtained. Closed formed expression
for the channel capacity of MRC for independent Nakagamiq
fading under different scheme of adaptive transmission is
obtained in Cheng and Berger (2003). Da Costa and Yacoub
(2007) presented the average channel capacity under ? ? ?
and ? ? ? fading channels. In Pena-Martin et al. (2013),
Capacity of dual branch MRC system for correlated imbalanced average SNRs 385
performance analysis of wireless system with MIMO under
Hoyt fading channel is presented. BER performance of MRC
system using Nakagami-m is analysed in Aalo (1995).
Capacity of MRC over arbitrary Rician fading channel has
been obtained in Hamdi (2008). Annavajjala and Milstein
(2004) considered the average channel capacity of dual-
MRC, SC and EGC system with unequal average SNRs over
Rayleigh fading channels. Alouini and Simon (1998)
obtained the capacity expression of Rayleigh channel with
MRC, EGC and SC taking into account level of imbalanced
in mean signal strength and correlation effect. The channel
capacity of Rayleigh fading is investigated in Lee (1990) and
Gunther (1996), respectively. Channel capacity expression
with MRC using Nakagami-m fading for the ORA scheme
assuming i.i.d. diversity branches was obtained in Yao and
Sheikh (1993). In Alouini and Goldsmith (1997) and Alouini
et al. (2001), the channel capacity with MRC diversity using
Nakagami-m fading under different adaptive transmission
schemes considering i.i.d. diversity branches was studied.
The capacity of MRC diversity systems under i.i.d. Rician
and Hoyt fading channels has been investigated in Cheng and
Berger (2001) and also for the four adaptive transmission
schemes mentioned above. In Devi et al. (2016), the channel
capacity with MRC diversity using Nakagami-m fading for
Rayleigh fading channel has been reviewed. In Pukhrambam
et al. (2017), the channel capacity with MRC diversity using
Nakagami-m fading channels with non-identical fading
parameters with first branch Rayleigh and second branch
Rician (m = 3) and imbalanced average SNRs has been
obtained. The capacity in the literature is also known as
average capacity as it is realised by averaging the Shannon
capacity of an AWGN channel.
However, an analytical study on channel capacity of
correlated imbalanced average SNRs Nakagami-m fading for
m? 1 under OPRA has not been done so far. Imbalanced
average SNRs means unequal average received SNR between
the branches. It is also known as unequal average SNRs.
Generally, Nakagami-m is used to a great extent to study the
performance of wireless mobile communication system but
Nakagami-m with imbalanced average SNRs appears to have
less attention. Also the results obtained for capacity of
Nakagami-m with imbalanced average SNRs will have huge
practical value, as in practical scenario branches may not
experience same amount of fade. For that reason it is
imperative to design a system for imbalanced average SNRs
for realistic scenario, and obtain results for different fading
correlation with imbalanced average SNRs. Therefore, this
paper fills a much needed gap by introducing the effect of
different physical constraints e.g., antenna space requirements,
correlation and level of imbalanced in average SNRs on the
capacity using OPRA scheme. In this paper, we investigate the
channel capacity over Nakagami-m fading using MRC systems
under OPRA schemes with balanced as well as imbalanced
average received SNRs. We consider slowly-varying flat
multipath fading channels. Similar to the assumption made in
Simon and Alouini (2005), an error free feedback path with
negligible time delay is assumed while considering adaptive
transmission schemes.
The rest of the paper is organised as follows: Section 2
defines the channel model. In Section 3, channel capacity
of MRC considering balanced and imbalanced average
SNRs is derived using OPRA scheme. Section 4 presents
numerically evaluated results. Concluding remarks are
given in Section 5.
2 Channel model
The Probability Distribution Function (PDF), ( ) p ? ? of the
received instantaneous SNR, ? at MRC combiner for
L-branch correlated Nakagami-m fading channels with
balanced average SNRs ? ? 1 2 ? ? ? ? ? can be presented as
Khatalin and Fonseka (2006)
? ?
? ?? ?
? ? ? ? ? ?
1
1
1
exp
1
; ;
1 1
( ) , 0
1 1
Lm
Lm m m
m m
Lm F m Lm
L
p
L Lm
m
?
? ?
? ? ?
??
? ? ? ?
? ?
? ? ? ?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ?
(2)
where ? is the average received SNR, m is the fading
parameter,1 1[...] F is the Kummer confluent hypergeometric
function (Gradshteyn and Ryzhik, 2000) and ? is the
correlation coefficient.
The PDF of ? at L-branch MRC combiner over
uncorrelated fading channels using Nakagami-m can be
obtained by first setting ? = 0 in (2), which was also given by
Simon and Alouini (2005) and Alouini and Goldsmith (1999)
as
? ?
1
exp
( ) , 0
Lm m m
p
Lm
m
?
? ?
? ?
? ?
?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ?
? ?? ? ?
? ?
(3)
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 2
1
2
( ) 1 2 '
exp ' , 0
m m
m
m p
m
I
?
? ?
?
? ? ? ?
? ? ? ? ?
?
? ? ? ? ?
? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ?
(4)
where In(.) is the n-th order Bessel function and the parameter
' ? and ' ? are the parametre (Alouini and Simon, 1998), and
is given by,
? ?
? ?
1 2
1 2 2 1
m ? ?
?
?? ?
?
? ?
?
386 A. Yengkhom et al.
and
? ? ? ? ? ?
? ?
1
2 2
1 2 1 2
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 1 2
1 2
2
1 2 1 2 1 2
1 2
2
1 2 1 2
4 1
exp
2 1
4 1
exp
2 1
4 1
p ?
? ? ? ? ? ? ?
?
? ? ?
? ? ? ? ? ? ?
?
? ? ?
?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ?
? ? ?
? ? ?
(5)
For m = 3, (4) reduces as
? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
3
4 5
1 2
2
1 9
2 1
3 exp exp
exp exp
3 exp exp
p ? ?
? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ?? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?
(6)
Different values of m describe several important distributions.
For example, m = 1 corresponds to distribution of Rayleigh,
m = 1 corresponds to Rician distribution.
3 Channel capacity
In this section, we obtain the closed form expressions for the
channel capacity of uncorrelated and correlated fading channels
using Nakagami-m with MRC for optimal power and rate
adaptation (OPRA) scheme considering balanced and
imbalanced average SNRs. The corresponding expressions for
correlated Rayleigh (m=1) channels with imbalanced average
SNRs and correlated Rician (m=3) channels with both balanced
and imbalanced average SNRs are obtained. The effect of
correlation and level of imbalanced in average SNRs on the
channel capacity is also analysed.
3.1 OPRA
In OPRA scheme, transmitter in accordance with channel
condition, adapts both data rate and power. If this power
adaptation ? ? P ? is subjected to average power constraint
P (Simon and Alouini, 2005), then
0
( ) ( ) P p d P ? ? ? ?
?
? ? (7)
where ( ) P? ? indicates the PDF under Nakagami-m fading
channel. The capacity with P is the capacity in Simon and
Alouini (2005), equation (15.21) with the power optimally
distributed over time (Simon and Alouini, 2005)
? ? ? ?
? ? ? ?
0
2 max log 1
P p P
P
C B p d
P ?
?
? ?
?
? ?
? ?
?
?
? ?
? ? ? ?
? ? ? ? (8)
The power adaptation that optimises the above expression was
obtained in Simon and Alouini (2005) by using Lagrange
multipliers as
? ? 0
0
1 1,
0,
P
P
? ? ?
? ?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
(9)
where 0 ? is the optimal cut-off value below which data
transmission is suspended and 0 ? must satisfy
? ?
0 0
1 1 1 p d ?
?
? ?
? ?
?? ?
? ? ? ?
? ? ? (10)
Substituting (10) into (8) the expression for the channel
capacity of fading channels with OPRA is obtained as
Khatalin and Fonseka (2006, 2007), Goldsmith and Varaiya
(1997), Simon and Alouini (2005)
? ? 2
0 0
log OPRA C B p d ?
?
? ?
?
? ? ?
? ? ?
? ? ? (11)
OPRA is also known as water filling technique. Higher
power and data rates are offered when channel conditions is
good (large ?), and as channel condition degrades (small ?),
lower power and data rates are given. When 0 ? is greater
than the received SNR, the data transmission is suspended,
which is also known as an outage probability Pout that is
equal to the probability when there no transmission. Outage
probability is given by
? ? ? ? ? ? 0
0
0
0
1 out P P p d p d
?
? ?
?
? ? ? ? ? ?
?
? ? ? ? ? ? ? (12)
Probability of the outage is different from outage capacity
which is the probability that the instantaneous capacity, C,
drops below predetermined threshold or target capacity, Cth,
which is given as Simon and Alouini (2005) Pout (Cth) =
P[C = Cth].
3.1.1 Capacity for uncorrelated fading channels
using Nakagami-m (m = 3) with balanced
average SNRs
The capacity for uncorrelated fading channels using Nakagamim
(m = 3) under OPRA scheme can be possibly obtained by
first substituting the PDF in (3) into (11) and putting m = 3 as,
? ? 0
3 1
2
0
3 3 exp
log
G 3
3
L
OPRA C B d
L ?
? ?
? ? ?
?
? ?
?
?
? ? ? ? ?
? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? (13)
Capacity of dual branch MRC system for correlated imbalanced average SNRs 387
To get an optimal cut-off SNR 0 ? , we first set m = 3, and substitute (3) in (10) and obtained
? ? 0
3 1
0
3 3 exp
1 1 1
G 3
3
L
d
L ?
? ?
? ?
?
? ? ?
?
?
? ? ? ? ?
? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? (14)
Putting the values of L=1 in (14) and evaluating the above integral and after some mathematical transformation using
(Wolfram, 2015; Gradshteyn and Ryzhik, 2000) we obtain
0 0
0
3 2 3 1 exp 1
2
? ? ?
? ? ?
? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ?
(15)
Putting the values of m=3 and L=2 in (14) and evaluating the integral and after some mathematical transformation by Wolfram
(2015) and Gradshteyn and Ryzhik (2000), we can obtained the following cut off SNR 0 ? ,
? ? ? ?
6 5 4
0 0 0
6 5 6
0 0 0
3 5! 4! 3 1 exp 1
120 3 3 ! !
k z
k z
k z k z
? ? ?
? ? ?
? ?
? ?
? ?
? ?
? ? ? ?
? ? ? ? ? ? ?
? ?? ?
? ? ? ?
? ? (16)
After solving (15) and (16), we get a unique value of 0 ? for each ? fulfilling (15) and (16) that take values from 0 [0,1] ? ? . Result
shows that 0 ? increase with increase in ? . The cut-off SNR value, 0 ? satisfying (15) and (16) for every ? value, has been used for
calculating channel capacity.
Channel capacity without diversity under uncorrelated Rician fading (m=3) can be obtained by first putting m=3, setting
L=1 in (14) and then substituting in (13) as,
0
2 2
2 3
0
27 3 log exp
2 OPRA C B d
?
? ? ?
?
? ? ?
? ? ?? ? ? ? ?
? ? ?? ? ? ?
? ? ? ? ? ? ? (17)
? ? ? ?
0 0
2 2
0 3
27 1.443 3 3 log exp log exp
2 OPRA C B d d
? ?
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? (18)
After evaluating the above integral and after some mathematical transformation using (Wolfram, 2015; Gradshteyn and Ryzhik,
2000) we get,
? ? 2 2 2 0 0
0 1 2
3 3 1.443 exp 3 2 2
2 OPRA C B E
? ?
? ? ? ? ?
? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
(19)
The channel capacity under uncorrelated Rician fading (m = 3) of dual branch MRC can be obtained by first putting m = 3, setting
L=2 in (14), substituting in (13) as,
0
5
6
2
0
3 exp
3 log
120 OPRA C B d
?
?
?
? ?
?
? ?
?
? ? ?
? ? ? ?? ? ? ? ? ? ?? ?
? ? ? ? ? (20)
? ? ? ?
0 0
6
5 5
0
1.443 3 3 3 log exp log exp
120 OPRA C B d d
? ?
? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? (21)
After solving the integration in (21) by partial integration and some mathematical transformation using (Gradshteyn and
Ryzhik, 2000), we get
? ? ? ? ? ?
2 1
2 1
2 1
2 3 4
0 0 0 0 0 0
1 3 4 5 3 4 5
0 0 0
2 1
3 3 3 9 20.25 48.6 1.443 exp 1.5
2 3 3 3 ! ! !
k k k
OPRA k k k
k k k
C B E
k k k
? ? ? ? ? ?
? ? ? ? ? ?
? ? ?
? ? ?
? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? (22)
388 A. Yengkhom et al.
3.1.2 Capacity of correlated fading channels using Nakagami-m considering balanced and imbalanced average
SNRs with dual branch MRC
The capacity of correlated Rayleigh fading channels for OPRA scheme with imbalanced average SNRs using dual branch
MRC, can be found by first substituting the PDF expression in (5) into (11) as,
? ? ? ?
2
0 0
exp exp
log OPRA
a b
C B d
x
? ? ?
?
?
? ? ? ? ? ?
? ? ?
? ? ? (23)
where
? ? ? ?
? ?
? ? ? ?
? ? ? ? ? ?
2 2
2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 1 2
1 2 1 2
4 1 4 1
4 1
2 1 2 1
a b x
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ?
After evaluating the integral of (23) and some mathematical transformation using (Wolfram, 2015; Gradshteyn and Ryzhik,
2000) we get,
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 0 1 0 0 0 1 0 0 0
0
log exp log exp exp exp 1.443 log OPRA
a E a b E b a b B C
x a b a b
? ? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
(24)
Putting (5) in (10) and after evaluating the integral using partial integral and after some mathematical transformation using
(Wolfram, 2015; Gradshteyn and Ryzhik, 2000), we can obtained the following cut off SNR 0 ? ,
? ? ? ? ? ? ? ? ? ? 0 0
1 0 1 0
0
exp exp 1 1 1
a b
E a E b
x a b x
? ?
? ?
?
? ? ? ?
? ? ? ? ? ?
? ?
(25)
Similarly, capacity of correlated Rician (m = 3) fading channels for OPRA scheme with balanced and imbalanced average
SNRs using dual branch MRC, can be found by first substituting the PDF expression in (6) into (11) as,
? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
3
2
2 4 5
0 12 0
3 exp exp
1 9 log exp exp
2 1
3 exp exp
OPRA C B d
? ? ? ? ? ?
?
?? ? ? ? ? ? ? ?
? ? ?? ?
? ? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?
? (26)
After evaluating the integral of (26) and some mathematical transformation using (Wolfram, 2015; Gradshteyn and Ryzhik,
2000) we get,
? ? ? ?
3
1 2 3 4 4 5
1 2
1.443 9
2 1 OPRA
B C II I I
? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ? ?
(27)
where
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ?
0 0 1 0
0 0 1 0
1
log exp ' '
log exp ' ' '
'
3
'
E
E
I
? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ?
2 2
0
0 0 1
0
0 0 0 1 0
3
2
2
0 0 0 1 0
3
2 2
0
0 0 1
0
2 log exp ' '
2 ! ' '
exp ' ' 1 ' ' 2exp ' ' 2 ' '
' '
'
exp ' ' 1 ' ' 2exp ' ' 2 ' '
' '
2 log exp ' '
2 ! ' '
k
k
k
l
l
l
k
E
I
E
l
?
? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ?
?
? ? ? ? ? ? ? ? ? ? ? ?
? ?
?
? ? ? ?
? ?
?
?
?
?
?
?
?
? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ?
? ? ?
?
?
??????
? ?
? ?
? ?
? ?
? ?
? ?
? ?
?
Capacity of dual branch MRC system for correlated imbalanced average SNRs 389
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
0 0 0 0 1 0
2
3
0 0 0 0 1 0
2
log exp ' ' 1 ' ' exp ' ' ' '
' '
3 '
log exp ' ' 1 ' ' exp ' ' ' '
' '
E
I
E
? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
and
? ?
? ? ? ? ? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
0 0
2 2
2 0 0
4 0 0 0 3 3
0 0
0 0 0 0
2
exp ' ' exp ' '
3
' '
2 2 log ' exp ' ' exp ' '
! ' ' ! ' '
exp ' ' 1 ' ' exp ' ' 1 ' '
3 '
' '
y s
y s
y s
I
y s
? ? ? ? ? ?
? ?
? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ?
?
?
?
??
Putting (6) in (10) and after evaluating the integral using partial integral and after some mathematical transformation using
(Wolfram, 2015; Gradshteyn and Ryzhik, 2000), we can obtained the following cut off SNR 0 ? ,
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ?
0 0
2
0
0 3
0
2
2 0
0
0 3
0
0 0
2
0 0
exp exp
3
2 exp
!
2 exp
!
exp 1
3
exp 1
k
k
k
z
z
z
k A
z
? ? ? ? ? ?
? ? ? ?
?
? ? ?
? ?
?
? ?
? ? ?
? ?
? ? ? ? ? ?
? ?
?
? ? ? ? ? ?
?
?
?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ?
? ? ? ? ? ? ?
?
?
?
?
? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ? ? ?
2
1 0 1 0
0 0
2
2
0 0
2
0 0
3
exp 1
exp 1
exp exp
3
E E
A
? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ?
?
? ? ? ? ? ?
? ?
? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ?
1
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ??
? ?
? ?
? ?
? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
(28)
The corresponding expressions for capacity with balanced SNR can be obtained by setting ? ? 1 2 ? ? ? ? ? . Also capacity over
uncorrelated Rayleigh and Rician (m = 3) with imbalanced average can be obtained by putting ? = 0 in (24) and (26),
respectively.
390 A. Yengkhom et al.
4 Numerical results and analysis
Results for the capacity of MRC over uncorrelated and
correlated fading channels using Nakagami-m (m = 1) with
imbalanced average SNRs and Nakagami-m (m = 3) with
both balanced and imbalanced average SNRs under OPRA
scheme are presented in this section. For comparison,
capacity of fading channels using correlated Nakagami-m
(m = 1) fading with and without diversity (Goldsmith and
Varaiya, 19997; Shao et al., 1999; Mallik and Win, 2000)
considering balanced average received SNRs is presented in
the figures.
Figure 1 shows the channel capacity over correlated
fading channels using Nakagami-m under OPRA scheme of
dual branch MRC for m=1 (Rayleigh) with balanced
average SNRs which was obtained in Khatalin and Fonseka
(2006) and Simon and Alouini (2005). Figure 2 shows the
channel capacity over correlated fading channels using
Nakagami-m under OPRA scheme of dual branch MRC for
m = 1 (Rayleigh) with imbalanced average SNRs and is
compared with already published results of Shao et al.
(1999). In both the figures, employing diversity using MRC
and/or increasing average received SNR ? ? 1 or ? ? gives
improvement in channel capacity. It is also seen that with
dual branch MRC, the capacity is almost identical for
1 , 1dB ? ? ? with m = 1 even when branch correlation
increases. So for low value of SNR the appropriate spacing
essential for creating uncorrelated diversity path at the
receiver end may be ignored particularly for 1 , 1dB ? ? ?
with m = 1.
Figure 1 Capacity of MRC for correlated fading channels using
Nakagami-m (m = 1) under OPRA scheme
0 5 10 15 20 25 30
1
2
3
4
5
6
7
8
9
10
11
Average received SNR[dB] per branch
Channel capacity per Unit Bandwidth [Bits/Sec/Hz]
dual-branch,?=0,equal SNR
dual-branch,?=0.1,equal SNR
dual-branch,?=0.3,equal SNR
dual-branch,?=0.8,equal SNR
no-diversity
Figure 3 shows the capacity comparison for correlated
fading channels using Nakagami-m under OPRA scheme of
dual branch MRC of balanced with imbalanced average
SNRs for m = 1. It is seen that the channel capacity
improvement considering imbalanced average SNRs gives
the better capacity than those with balanced SNRs as
expected, because the overall SNR of imbalanced average
SNRs is greater than that of balanced average SNRs
between the branches as we have considered 2 1 2 ? ? ? . From
this figure also it was observed that MRC diversity and
increasing average SNR increase the capacity.
Figure 2 Channel capacity of MRC for correlated fading
channels using Nakagami-m (m = 1) under OPRA
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/Sec/Hz]
dual-branch,?=0,imbalanced SNR
dual-branch,?=0.1,imbalanced SNR
dual-branch,?=0.3,imbalanced SNR
dual-branch,?=0.8,imbalanced SNR
no-diversity
Figure 3 Capacity comparison of MRC for correlated fading
channels using Nakagami-m (m = 1)
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/Sec/Hz]
dual-branch,?=0
dual-branch,?=0.1
dual-branch,?=0.3
dual-branch,?=0.8
No diversity
Imbalanced SNRs
equal SNRs
Figure 4 shows the channel capacity over correlated fading
channels using Nakagami-m under OPRA scheme of dual
branch MRC for m = 3 (Rician) with balanced average
SNRs. Figure 5 shows the channel capacity over correlated
fading channels using Nakagami-m under OPRA scheme of
dual branch MRC for m = 3 (Rician) with imbalanced
average SNRs. By increasing average receive SNR ? ? 1 ?
and/or employing diversity using MRC, channel capacity
should improve, the same has been observed. It is seen that
with dual branch MRC, the capacity is almost identical for
1 4 dB ? ? with m = 3 even when branch correlation
increases. So for low values of SNR the appropriate
Capacity of dual branch MRC system for correlated imbalanced average SNRs 391
spacing essential for creating uncorrelated diversity path at
the receiver end may be ignored particularly for 1 4 dB ? ?
with m = 3.
Figure 4 Capacity of MRC for correlated fading channels using
Nakagami-m (m = 3) under OPRA scheme
0 5 10 15 20 25 30
1
2
3
4
5
6
7
8
9
10
11
Average received SNR[dB] per branch
Channel capacity per Unit Bandwidth [Bits/Sec/Hz]
dual-branch,?=0,equal SNR
dual-branch,?=0.1,equal SNR
dual-branch,?=0.3,equal SNR
dual-branch,?=0.8,equal SNR
no-diversity
Figure 5 Channel capacity of MRC for correlated fading
channels using Nakagami-m (m = 3) under OPRA
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/Sec/Hz]
dual-branch,?=0,imbalanced SNR
dual-branch,?=0.1,imbalanced SNR
dual-branch,?=0.3,imbalanced SNR
dual-branch,?=0.8,imbalanced SNR
no-diversity
Figure 6 shows the capacity comparison for correlated fading
channels using Nakagami-m under OPRA scheme of dual
branch MRC of balanced with imbalanced average SNRs for
m = 3 and is compared with the already published results
(Shao et al., 1999) which are also shown in Figures 1, 2 and 3.
It can be observed the capacity of MRC over Nakagami-m
(m = 3) is greater than the capacity of MRC over Nakagami-m
(m = 1) for both the cases of balanced and imbalanced SNRs. It
is seen that the channel capacity improvement considering
imbalanced average SNRs gives the better capacity than
those with balanced SNRs as expected, because the overall
SNR of imbalanced average SNRs is greater than that of
balanced average SNRs between the branches as we have
considered 2 1 2 ? ? ? . From this figure also it was observed that
MRC diversity and increasing average SNR increase the
capacity.
Figure 6 Capacity comparison of MRC for correlated fading
channels using Nakagami-m (m = 3)
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/Sec/Hz]
dual-branch,?=0
dual-branch,?=0.1
dual-branch,?=0.3
dual-branch,?=0.8
no diversity
imbalanced SNR
equal SNR
5 Conclusion
We have obtained expressions for the capacity of MRC for
correlated and uncorrelated fading channels using Nakagami-m
under OPRA scheme with balanced and imbalanced average
received SNRs. As expected, employing diversity using MRC
and/or increasing average received SNRs gives improvement in
channel capacity. By numerical evaluation we also observed
that, channel capacity improvement considering imbalanced
average received SNRs gives the better capacity than those
with balanced average received SNRs as expected. Further,
with dual branch MRC, the channel capacity is almost identical
for smaller values of average received SNRs even when branch
correlation increases. Therefore, for low values of SNR, the
appropriate spacing required for creating uncorrelated diversity
path at the receiver end may be ignored.
References
Aalo, V.A. (1995) 'Performance of maximal-ratio diversity systems in
a correlated Nakagami fading environment', IEEE Transactions
on Communications, Vol. 43, No. 8, pp.2360-2369.
Alouini, M-S. and Goldsmith, A. (1997) 'Capacity of Nakagami
multipath fading channels', Proceedings of the IEEE 47th
Vehicular Technology Conference Technology in Motion, IEEE,
USA, Vol. 1.
Alouini, M-S. and Goldsmith, A.J. (1999) 'Capacity of Rayleigh
fading channels under different adaptive transmission and
diversity-combining techniques," IEEE Transactions on
Vehicular Technology, Vol. 48, pp.1165-1181.
Alouini, M-S. and Simon, M.K. (1998) 'Multichannel reception
of digital signals over correlated Nakagami fading channels',
Proceedings of the Annual Allerton Conference on Communication
Control and Computing', Vol. 36, pp.146-155.
392 A. Yengkhom et al.
Alouini, M-S., Abdi, A. and Kaveh, M. (2001) 'Sum of gamma
variates and performance of wireless communication systems
over Nakagami-fading channels', IEEE Transactions on
Vehicular Technology, Vol. 50, No. 6, pp.1471-1480.
Annavajjala, R. and Milstein, L.B. (2004) 'On the capacity of dual
diversity combining schemes on correlated Rayleigh fading
channels with unequal branch gains', IEEE Wireless
Communications and Networking Conference, IEEE, USA,
Vol. 1.
Cheng, J. and Berger, T. (2003) 'Capacity of Nakagami-q (Hoyt)
fading channels with channel side information', International
Conference on Communication Technology (ICCT'03), IEEE,
China, Vol. 2.
Cheng, J. and Berger, T.E. (2001) 'Capacity of a class of fading
channels with channel state information (CSI) feedback',
Proceedings of the 39th Annual Allerton Conference
on Communication, Control, and Computing (Allerton'01),
3-5 October, Monticello, IL, USA.
Da Costa, D.B. and Yacoub, M.D. (2007) 'Average channel capacity
for generalized fading scenarios", IEEE Communications Letters,
Vol. 11, No. 12, pp.949-951.
Devi, P.B., Yengkhom, A., Hasan, M.I. and Chandwani, G. (2016)
'Capacity of dual branch MRC system over correlated
Nakagami-m fading channels: a review', Wireless Personal
Communications, pp.1-18. Doi 10.1007/s11577-017-4942-4.
Goldsmith, A.J. and Varaiya, P.P. (1997) 'Capacity of fading
channels with channel side information', IEEE Transactions
on Information Theory, Vol. 43, No. 6, pp.1986-1992.
Gradshteyn, S. and Ryzhik, I.M. (2000) Table of Integrals, Series,
and Products, 6th ed., Academic Press, New York.
Gunther, C.G. (1996) 'Comment on Estimate of channel capacity
in Rayleigh fading environment', IEEE Transactions on
Vehicular Technology, Vol. 45, No. 2, pp.401-403.
Hamdi, K.A. (2008) 'Capacity of MRC on correlated Rician fading
channels', IEEE Transactions on Communications, Vol. 56,
No. 5, pp.708-711.
Hasan, M.I. and Kumar, S. (2014) 'Channel capacity of dualbranch
maximal ratio combining under worst case of fading
scenario', WSEAS Transactions on Communications, Vol. 13,
pp.162-170.
Hasan, M.I. and Kumar, S. (2015a) 'Average channel capacity of
correlated dual-branch maximal ratio combining under worst
case of fading scenario', Wireless Personal Communications,
Springer, Vol. 83, pp.2624-2646.
Hasan, M.I. and Kumar, S. (2015b) 'Average channel capacity
evaluation for selection combining diversity schemes over
nakagami-0.5 fading channels', International Journal of
Communication Networks and Information Security (IJCNIS),
Vol. 7, No. 2, pp.69-79.
Hasan, M.I. and Kumar, S. (2015c) 'Spectral efficiency evaluation
for selection combining diversity scheme under worst case of
fading scenario', International Journal of Communication
Networks and Information Security (IJCNIS), Vol. 7, No. 3,
pp.123-130.
Hasan, M.I. and Kumar, S. (2016) 'Spectral efficiency of dual
diversity selection combining schemes under correlated
Nakagami-0.5 fading with unequal average received SNR',
Telecommunication Systems, pp.1-14.
Khatalin, S. and Fonseka, J.P. (2006) 'Capacity of correlated
Nakagami-m fading channels with diversity combining
techniques', IEEE Transactions on Vehicular Technology,
Vol. 55, No. 1, pp.142-150.
Khatalin, S. and Fonseka, J.P. (2006) 'On the channel capacity in
Rician and Hoyt fading environments with MRC diversity',
IEEE Transactions on Vehicular Technology, Vol. 55, No. 1,
pp.137-141.
Khatalin, S. and Fonseka, J.P. (2007) 'Channel capacity of
dual-branch diversity systems over correlated Nakagami-m
fading with channel inversion and fixed rate transmission
scheme', IET Communications Proceedings, Vol. 1, No. 6,
pp.1161-1169.
Lee, W.C.Y. (1990) 'Estimate of channel capacity in Rayleigh
fading environment', IEEE Transactions on Vehicular
Technology, Vol. 39, No. 3, pp.187-189.
Mallik, R.K. and Win, M.Z. (2000) 'Channel capacity in evenly
correlated Rayleigh fading with different adaptive transmission
schemes and maximal ratio combining', IEEE International
Symposium on Information Theory, IEEE, Italy.
Mallik, R.K. et al. (2004) 'Channel capacity of adaptive transmission
with maximal ratio combining in correlated Rayleigh fading',
IEEE Transactions on Wireless Communications, Vol. 3, No. 4,
pp.1124-1133.
Pena-Martin, J.P., Romero-Jerez, J.M. and Tellez-Labao, C. (2013)
'Performance of TAS/MRC wireless systems under Hoyt fading
channels', IEEE Transactions on Wireless Communications,
Vol. 12, No. 7, pp.3350-3359.
Pukhrambam, B.D., Yengkhom, A., Hasan, M.I. and Chandwani,
G. (2017) 'Capacity of dual-branch MRC system over
correlated Nakagami-m fading channels with non-identical
fading parameters and imbalanced average SNRs',
International Journal on Wireless and Mobile Computing,
Vol. 13, No. 2, pp.151-163.
Shao, J.W., Alouini, M-S. and Goldsmith, A.J. (1999) 'Impact of
fading correlation and unequal branch gains on the capacity of
diversity systems', IEEE Transactions on Vehicular
Technology, Vol. 3, No. 1, pp.142-150.
Simon, M.K. and Alouini, M-S. (2005) Digital Communication
over Fading Channels, John Wiley & Sons, Inc.
Subadar, R. and Sahu, P.R. (2010) 'Capacity analysis of Dual-SC
and-MRC systems over correlated Nakagami-m fading
channels with non-identical and arbitrary fading parameters',
National Conference on Communications (NCC), IEEE,
India.
Wolfram (2015) The Wolfram function site. Available online at:
http://functions.wolfram.com
Yao, Y-D. and Sheikh, A.U.H. (1993) 'Evaluation of channel
capacity in a generalized fading channel', Proceedings of the
43rd IEEE Vehicular Technology Conference, IEEE, USA.

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