SPECTRAL EFFICIENCY OF DUAL DIVERSITY SELECTION COMBINING SCHEMES UNDER CORRELATED NAKAGAMI-0.5 FADING WITH UNEQUAL AVERAGE RECEIVED SNR

Abstract

The spectral efficiency results for different adaptive transmission schemes over correlated diversity branches with unequal average signal to noise ratio (SNR) obtained so far in literature are not applicable for Nakagami-0.5 fading channels. In this paper, we investigate the effect of fade correlation and level of imbalance in the branch average received SNR on the spectral efficiency of Nakagami-0.5 fading channels in conjunction with dual-branch selection combining (SC). This paper derived the expressions for the spectral efficiency over correlated Nakagami-0.5 fading channels with unequal average received SNR. This spectral efficiency is evaluated under different adaptive transmission schemes using dual-branch SC diversity scheme. The corresponding expressions for Nakagami-0.5 fading are considered to be the expressions under worst fading conditions. Finally, numerical results are provided to illustrate the spectral efficiency degradation due to channel correlation and unequal average received SNR between the different combined branches under different adaptive transmission schemes. It has been observed that optimal simultaneous power and rate adaptation (OPRA) scheme provides improved spectral efficiency as compared to truncated channel inversion with fixed rate (TIFR) and optimal rate adaptation with constant transmit power (ORA) schemes under worst case fading scenario. It is very interesting to observe that TIFR scheme is always a B Mohammad Irfanul Hasan irfanhasan25@rediffmail.com Sanjay Kumar skumar@bitmesra.ac.in 1 Department of Electronics and Communication Engineering, Birla Institute of Technology, Mesra, Ranchi 835215, India 2 Department of Electronics and Communication Engineering, Graphic Era University, Dehradun 248002, India better choice over ORA scheme under correlated Nakagami- 0.5 fading channels with unequal average received SNR. Keywords Dual-branch · Nakagami-0.5 fading channels · Optimal simultaneous power and rate adaptation · Optimal rate adaptation with constant transmit power · Selection combining · Spectral efficiency · Truncated channel inversion with fixed rate

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Title: Spectral efficiency of dual diversity selection combining schemes under correlated
Nakagami-0.5 fading with unequal average received SNR
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Telecommun Syst
DOI 10.1007/s11235-016-0152-8
Spectral efficiency of dual diversity selection combining schemes
under correlated Nakagami-0.5 fading with unequal average
received SNR
Mohammad Irfanul Hasan1,2 · Sanjay Kumar1
© Springer Science+Business Media New York 2016
Abstract The spectral efficiency results for different adaptive
transmission schemes over correlated diversity branches
with unequal average signal to noise ratio (SNR) obtained so
far in literature are not applicable for Nakagami-0.5 fading
channels. In this paper, we investigate the effect of fade correlation
and level of imbalance in the branch average received
SNR on the spectral efficiency of Nakagami-0.5 fading channels
in conjunction with dual-branch selection combining
(SC). This paper derived the expressions for the spectral
efficiency over correlated Nakagami-0.5 fading channels
with unequal average received SNR. This spectral efficiency
is evaluated under different adaptive transmission schemes
using dual-branch SC diversity scheme. The corresponding
expressions for Nakagami-0.5 fading are considered to be the
expressions under worst fading conditions. Finally, numerical
results are provided to illustrate the spectral efficiency
degradation due to channel correlation and unequal average
received SNR between the different combined branches
under different adaptive transmission schemes. It has been
observed that optimal simultaneous power and rate adaptation
(OPRA) scheme provides improved spectral efficiency
as compared to truncated channel inversion with fixed rate
(TIFR) and optimal rate adaptation with constant transmit
power (ORA) schemes under worst case fading scenario. It
is very interesting to observe that TIFR scheme is always a
B Mohammad Irfanul Hasan
irfanhasan25@rediffmail.com
Sanjay Kumar
skumar@bitmesra.ac.in
1 Department of Electronics and Communication Engineering,
Birla Institute of Technology, Mesra, Ranchi 835215, India
2 Department of Electronics and Communication Engineering,
Graphic Era University, Dehradun 248002, India
better choice over ORA scheme under correlated Nakagami-
0.5 fading channels with unequal average received SNR.
Keywords Dual-branch · Nakagami-0.5 fading channels · Optimal simultaneous power and rate adaptation · Optimal
rate adaptation with constant transmit power · Selection
combining · Spectral efficiency · Truncated channel inversion
with fixed rate
1 Introduction
The channel capacity is of fundamental importance in the
design ofwirelessmobile communication systems as it determines
the maximum achievable data rate of the wireless
communication systems [1-4]. Since wireless mobile environment
is subjected to fading, which degrades the data rate
performance. The channel capacity in fading environment,
which is less than that of anAWGNchannel, can be improved
by increasing the transmit power. Increasing transmit power
to mitigate the fading effect on the signal results in increasing
the interference level, which is undesirable. Another way
to improve channel capacity in fading environment is by
employing diversity combining techniques and/ or adaptive
transmission schemes. The biggest advantage of such mechanism
is that there is no requirement to increase the transmit
power and/ or bandwidth to attain improvements in channel
capacity. However, the improvement in channel capacity
comes at the expense of an added system complexity [1-4].
The adaptive transmission schemes with diversity combining
techniques are currently receiving a great deal of
attention as very promising techniques to achieve high channel
capacity in thewirelessmobile environment. The primary
objective behind adaptive transmission schemes is to use CSI
(channel state information) at receiver or at transmitter in
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order to maximize the channel capacity and lower the probability
of outage [3-5]. Moreover, correlation at the receive
antennas can potentially lead to degradation in the channel
capacity in fading environment [3-21]. In number of
real-life scenarios, physical restrictions may not allow using
antenna spacing that is required for independent fading across
diversity branches. Therefore, investigations on the effect of
correlation in adaptive transmission schemes using diversity
combining techniques becomes essential in gaining a better
understanding of the trade-off involved in systems design.
Diversity combining is known to be a powerful technique
that can be used to combat fading in wireless mobile
environment. Maximal ratio combining (MRC), equal gain
combining (EGC) and SC aremost prevalent diversity combing
techniques [3,4].
Adaptive transmission schemes, which requires accurate
channel estimation at the receiver and a reliable feedback
path between the estimator and the transmitter, provides
great improvement to the channel capacity [3,4].The capacity
of flat fading channels was derived in [5] for four different
adaptive transmission schemes such as OPRA, ORA,
channel inversion with fixed rate transmission (CIFR) and
TIFR. In studying the channel capacity performances of different
adaptive transmission schemes, the usual assumption
made is that the combined branches are independent to one
another and have the same average received SNR. However,
independent fading is not always realized in practice due to
insufficient antenna spacing in case of small-size terminals
equipments. In addition, the diversity branches in practical
system may have unequal average received SNR's due to
different noise figures or feeding lengths [6,7].
Numerous researchers [3-25] haveworked on the study of
channel capacity over different fading channels. We discuss
here some representative examples. Specifically [3,4] discuss
the channel capacity over correlated Nakagami-m (m = 1 and m < 1) fading channels underORAand CIFR schemes
using different diversity combining techniques. In [8], the
channel capacity over uncorrelated Nakagami-m (m = 1)
fading channels with MRC and without diversity under
different adaptive transmissions schemes was analyzed.
Expressions for the capacity over uncorrelated Rayleigh fading
channels with MRC and SC under different adaptive
transmission schemes were obtained in [9]. An analytical
performance study of the channel capacity for correlated generalized
gamma fading channels with dual-branch SC under
different adaptive transmission schemes was introduced in
[10]. The channel capacity under Nakagami-m (m = 1) fading
channel without diversitywas derived in [11] for different
adaptive transmission schemes. In [12], channel capacity of
dual-branch SC and MRC systems over correlated Hoyt fading
channels using different adaptive transmission schemes
was presented. In [13], expression for the ergodic capacity of
MRC over arbitrarily correlated Rician fading channels was
derived. In [14], an expression for lower and upper bounds
in the channel capacity expression for uncorrelated Rician
and Hoyt fading channels with MRC using ORA scheme
were obtained. In [22], an analytical performance study of
the channel capacity for uncorrelated Nakagami-0.5 with
dual-branch MRC using OPRA and TIFR was obtained. In
[23], the channel capacity over correlated Nakagami-0.5 fading
channels under OPRA and TIFR schemes with MRC
was discussed. An analytical performance study of the channel
capacity for uncorrelated Nakagami-0.5 fading channels
with dual-branch SC under OPRA, TIFR, ORA, and CIFR
schemes was introduced in [24,25]. However, an analytical
study of channel capacity over correlated Nakagami-0.5
fading channels with unequal average received SNR under
different adaptive transmission schemes using SC has not
been considered so far.
Nakagami-m model has been extensively used in general
to study wireless mobile communication system performance,
less concentration appears to have been focused on
the particular case of Nakagami-0.5 fading. At the same time
that results obtained for Nakagami-0.5 will have immense
practical value as a worst case fading scenario. Hence, in
cases where quality of service with high data rate requirements
mandate designing for worst case fading scenario,
results obtained for the same fading condition will have
great practical applications in wireless mobile environment
[26,27]. In this paper, correlated dual-branch SC under worst
case of fading conditions has been considered to investigate
the effect of fade correlation (?) and average received SNR
imbalance between the different combined branches on the
link spectral efficiency. The link spectral efficiency, defined
as the ratio of the average channel capacity to the given
bandwidth [9]. The dual-branch diversity has been considered
since it offers themaximum SNR improvement, besides
offering minimum complexity and physical space requirements
[26].
Therefore, this paper fills this gap by presenting the impact
of various practical constraints, e.g. physical space requirements,
effect of fade correlation and level of imbalance in
the branch average received SNR on channel capacity under
worst case fading scenario.
Finally, some recommendation is given for proper antenna
spacing required for uncorrelated diversity branch and the
choice for adaptive transmission schemes under average
received SNR imbalance. That allows the researchers or system
designers to perform comparison and tradeoff studies
among the various adaptive transmission schemes with simple
combining technique SC, so as to determine the optimal
choice in the face of their available constraints.
The remainder of this paper is organized as follows: In
Sect. 2, the channel model is defined. In Sect. 3, spectral
efficiency of dual-branch SC over correlated Nakagami-
0.5 fading channels with unequal average received SNR
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Spectral efficiency of dual diversity selection...
are derived for different adaptive transmission schemes. In
Sect. 4, several numerical results are presented and analyzed,
whereas in Sect. 5, concluding remarks are given.
2 Channel model
The probability density function (pdf) of the instantaneous
received SNR,? , at the output of a correlated dual-branch SC
combiner, p? (? ), for the Nakagami-m fading channels with
unequal average received SNR is obtained in [27,28]
p? (? ) =
? m-1
(m) m
¯ ?1 m
exp-
m?
¯ ?1 1
- Qm  2m??
¯ ?1(1 - ?)
, 2m?
¯ ?2(1 - ?)
+
? m-1
(m) m
¯ ?2 m
exp-
m?
¯ ?2 1
-Qm  2m??
¯ ?2(1 - ?)
, 2m?
¯ ?1(1 - ?), ? = 0
(1)
where ¯ ?l is the average received SNR of the l th branch
(l = 1, 2) represents level of imbalance in the branch
average received SNR, m (m = 0.5) is the fading parameter,
(.) is the gamma function, Qm(., .) is the generalized
(mth-order ) Marcum Q-function, and ? is the correlation
coefficient between two fading envelopes represents effect
of fade correlation between diversity branches [27,28].
The fading parameter, m measures the amount of fading
(amount of fading = 1/m). For different values of m,
this expression simplifies to several important distributions
describing fading models. Like m = 0.5 corresponds to one
sided Gaussian distribution said to be worst case fading scenario,
m = 1 corresponds to Rayleigh distribution,m = 1
corresponds to Rician distribution, and as m ?8, the distribution
converges to a non fading AWGN [27].
Replacing the Qm(., .) function in (1) with its series
representation as given in [29], and then the pdf under
Nakagami-m fading using (1) becomes
p? (? ) =
? m-1
(m) m
¯ ?1 m
exp-
m?
¯ ?1  2m?
¯ ?1(1 - ?)
m
×
8k=0
(-1)k  2m?
¯ ?1(1-?)
k
1F1 -k;m;- m?2?
¯ ?1(1-?)

2me
m??
¯ ?1(1-?) (m)
×
? m-1
(m) m
¯ ?2 m
exp-
m?
¯ ?2  2m?
¯ ?2(1 - ?)
m
×
8
k=0
(-1)k  2m?
¯ ?2(1-?)
k
1F1 -k;m;- m?2?
¯ ?2(1-?)

2me
m??
¯ ?2(1-?) (m)
(2)
Putting m = 0.5 and simplified this pdf using [29], we
obtained
p? (? ) = (1 - ?)0.5 8
k=0
?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×?
??
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r (3)
3 Spectral efficiency
In this section, we present expressions for the spectral effi-
ciency of correlated Nakagami-0.5 fading channels with
unequal average received SNR using dual-branch SC under
OPRA, TIFR and ORA schemes. It is assumed that, for the
considered adaptation schemes, there exist perfect channel
estimation and an error-free delayless feedback path, similar
to the assumption made in [9].
3.1 OPRA
The average channel capacity of fading channels with
received SNR distribution p? (? ) and optimal simultaneous
power and rate adaptation (COPRA[bit/sec]) is defined in
[5,9] as
COPRA = B
8
?
0
log2  ?
?0  p? (? )d? (4)
where B (Hz) is the channel bandwidth, and ?0 is the optimal
cutoff SNR level below which no data is transmitted. To
obtain the optimal cutoff SNR, ?0 must satisfy the equation
given by [5,9] as
8
?
0
 1
?0 -
1
? p? (? )d? =1 (5)
To achieve the average channel capacity (4), the channel
fade level must be tracked at both the receiver and transmitter,
and the transmitter has to adapt its power and rate
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M. I. Hasan, S. Kumar
accordingly, allocating high power levels and data rates for
good channel conditions (? large), and lower power levels
and data rates for poor channel conditions (? small).
Since data transmission is suspended when the received
SNR, ? is less than ?0, this optimal adaptation scheme suffers
an outage probability, which is given by [5,9]
Pout = P ? = ?0 =
?0

0
p? (? )d? = 1 -
8
?
0
p? (? )d? (6)
Substituting (3) in (5) for optimal cutoff SNR?0, and evaluating
the integral using some mathematical transformation
by [29], we obtain
1 - ?
8
k=0
?k0.54k+1
k!(k + 0.5) ×
8
r=0
(2k + r + 1)(0.5)r
(2k + r + 1.5)(r !)
×
??????
 2k + r + 1, ?0
¯ ?1(1-?)
-
2k+r,
?0
¯ ?1(1-?)

¯ ?1(1-?) 
+ 2k + r + 1, ?0
¯ ?2(1-?)
-
2k+r,
?0
¯ ?2(1-?)

¯ ?2(1-?) 
??????
= 1
(7)
The numerical evaluation techniques have confirmed that, by
solving (7), there is a unique positive value of ?0 satisfying
(7) that takes values from ?0 ? [0, 1]. Result shows that ?0
increases as ¯ ?l (l = 1, 2) increases.The value of cutoff SNR
?0 that satisfies (7) by either assuming equal average received
SNR( ¯ ?1 = ¯?2 = ¯? ) or unequal average received SNR ( ¯ ?1 = 2 ¯ ?2)is used for finding the average channel capacity in each of
the case. Substituting (3) in (4), the average channel capacity
of dual-branch SC under Nakagami-0.5 fading channels is
COPRA
= B
8
?
0
log2  ?
?0  8
k=0
(1 - ?)0.5?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×?
??
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
COPRA
= 1.443B
8
k=0
(1 - ?)0.5?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×
???????????
8
?0
log(? )???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
-
8
?0
log(?0)???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
???????????
(8)
Following can be taken from the first part of above integral
is
8
?
0
log(? )???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
This can be solved using partial integration as follows
8
?
0
udv = lim
??8
(uv) - lim
???0
(uv) -
8
?
0
vdu
Let u = log ?
then du = d?
?
Now let
dv = ???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
Integrating above expression using [29], we obtain
v = - {0.5 × (1 - ?)}2k+r+1  2k + r + 1,
?
¯ ?1 (1 - ?)

+ {0.5 × (1 - ?)}2k+r+1  2k + r + 1,
?
¯ ?2 (1 - ?)


Evaluating first part of above integral (8) using this partial
integration and some mathematical transformation [29,30],
we obtain
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Spectral efficiency of dual diversity selection...
8
?
0
log(? )???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
= 0.5
¯ ?1 2k+r+1 ??????
2k+r
t=0
2k+r ! 2k+r-t!
log(?0) × exp  -?0
¯ ?1(1-?)
×{ ¯ ?1(1 - ?)}t+1 (?0)2k+r-t+ 8
t=0
2k+r ! 2k+r-t! { ¯ ?1(1 - ?)}2k+r+1  2k + r - t, ?0
¯ ?1(1-?)

??????
+0.5
¯ ?2 2k+r+1 ??????
2k+r
t=0
2k+r ! 2k+r-t!
log(?0) × exp  -?0
¯ ?2(1-?)
×{ ¯ ?2(1 - ?)}t+1 (?0)2k+r-t+ 8
t=0
2k+r ! 2k+r-t! { ¯ ?2(1 - ?)}2k+r+1  2k + r - t, ?0
¯ ?2(1-?)

??????
(9)
Second part of above integral (8) can be solved by using
[29], we obtain
8
?
0
log(?0)???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
= +log(?0)???
0.5
¯ ?1 2k+r+1 { ¯ ?1(1 - ?)}2k+r+1  2k + r + 1, ?0
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1 { ¯ ?2(1 - ?)}2k+r+1  2k + r + 1, ?0
¯ ?2(1-?)

???
(10)
Substituting (9) and (10) in (8), the average channel capacity
of dual-branch SC under Nakagami-0.5 fading channels
is
COPRA
= 1.443B
8
k=0
(1 - ?)0.5?k
k!(k + 0.5)(2)2k
8
r=0
×
(2k + r + 1)  1
(1-?) 2k+r+1
(2k + r + 1.5)(r !)
×
??????????
0.5
¯ ?1 2k+r+1
??????????
2k+r
t=0
2k+r ! 2k+r-t! log(?0)×
exp  -?0
¯ ?1(1-?) { ¯ ?1(1 - ?)}t+1 (?0)2k+r-t
+8t=0
2k+r ! 2k+r-t! { ¯ ?1(1 - ?)}2k+r+1
× 2k + r - t,
?0
¯ ?1(1-?)

??????????
+0.5
¯ ?2 2k+r+1 ??????
2k+r
t=0
2k+r ! 2k+r-t! log(?0)×
exp  -?0
¯ ?2(1-?) { ¯ ?2(1 - ?)}t+1 (?0)2k+r-t+
8t=0
2k+r ! 2k+r-t! { ¯ ?2(1 - ?)}2k+r+1
× 2k + r - t,
?0
¯ ?2(1-?)

??????
-log(?0)
?????? 
0.5
¯ ?1 2k+r+1 { ¯ ?1(1 - ?)}2k+r+1
× 2k + r + 1,
?0
¯ ?1(1-?)

+
0.5
¯ ?2 2k+r+1 { ¯ ?2(1 - ?)}2k+r+1
× 2k + r + 1,
?0
¯ ?2(1-?)


??????
??????
Using that result we obtain spectral efficiency i.e. COPRA
B
[bit/sec/Hz] as
?OPRA
= 1.443
8
k=0
(1 - ?)0.5?k
k!(k + 0.5)(2)2k-1
×
8
r=0
(2k + r + 1)  1
(1-?) 2k+r+1
(2k + r + 1.5)(r !) 0.5
¯ ?1 2k+r+1
×
??????
2k+r
t=0
2k+r ! 2k+r-t! log(?0)×
exp  -?0
¯ ?1(1-?) { ¯ ?1(1 - ?)}t+1 (?0)2k+r-t+
8t=0
2k+r ! 2k+r-t! { ¯ ?1(1 - ?)}2k+r+1
× 2k + r - t,
?0
¯ ?1(1-?)

??????
+0.5
¯ ?2 2k+r+1 ??????
2k+r
t=0
2k+r ! 2k+r-t! log(?0)×
exp  -?0
¯ ?2(1-?) { ¯ ?2(1 - ?)}t+1 (?0)2k+r-t+
8t =0
2k+r ! 2k+r-t! { ¯ ?2(1 - ?)}2k+r+1
× 2k + r - t,
?0
¯ ?2(1-?)

??????
-log(?0)
??????
0.5
¯ ?1 2k+r+1  { ¯ ?1(1 - ?)}2k+r+1
× 2k + r+,
?0
¯ ?1(1-?)

+
0.5
¯ ?2 2k+r+1  { ¯ ?2(1 - ?)}2k+r+1
× 2k + r+,
?0
¯ ?2(1-?)


??????
??????
(11)
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M. I. Hasan, S. Kumar
We compare (11) with [24, Eq. (27)] and spectral effi-
ciency without diversity in [22, Eq. (8)].
3.2 TIFR
The average channel capacity of fading channels with
received SNRdistribution p? (? ) under TIFR scheme (CT IFR
[bit/sec]) is defined in [5,9] as
CT IFR = B log2
?????
1 +
1
8
?0  p? (? )
? d?
?????
(1 - Pout ), ? = 0
(12)
The cutoff level ?0, can be selected to achieve a speci-
fied probability of outage,Pout , or, to maximize the average
channel capacity (12).
Now, we evaluate the probability of outage using (3) and
(6) is
Pout =
?0

0
(1 - ?)0.5 8
k=0
?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×
????
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

????
? 2k+r d?
Evaluating the above integral using some mathematical
transformation [29,30], we obtain
Pout = (1 - ?)0.5 8
k=0
(?)k (0.5)4k+1
k!(k + 0.5)
×
8
r=0
(2k + r + 1)
(2k + r + 1.5)
(0.5)r
r !
×?
?
(2k + r + 1) -  2k + r + 1, ?0
¯ ?1(1-?)+
(2k + r + 1) -  2k + r + 1, ?0
¯ ?2(1-?)
??
(13)
Using the pdf of dual-branch SC under correlated
Nakagami-0.5 fading channels in (3), we obtain
p? (? )
? = (1 - ?)0.5 8
k=0
?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×?
??
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r-1
Integrating the above expression using [29,30], we obtain
8
?
0
p? (? )
?
d? = (1 - ?)0.5 8
k=0
?k (0.5)4k+1
k!(k + 0.5)(1 - ?)
×
8
r=0
(2k + r + 1)(0.5)r
(2k + r + 1.5)(r !)
??
 1
¯ ?1  2k + r,
?0
¯ ?1(1-?)
+
 1
¯ ?2  2k + r,
?0
¯ ?2(1-?)

??
(14)
Putting the value of (13) and (14) in (12), we get
CT IFR = 1.443 × B log
??????????
1 +
1
(1 - ?)0.5 8
k=0
?k (0.5)4k+1
k!(k+0.5)(1-?)
8
r=0
(2k+r+1)(0.5)r
(2k+r+1.5)(r !)
???
 1
¯ ?1  2k + r,
?0
¯ ?1(1-?)
+
 1
¯ ?2  2k + r,
?0
¯ ?2(1-?)

???
??????????
×
?????
1 - (1 - ?)0.5 8
k=0
(?)k (0.5)4k+1
k!(k + 0.5)
8
r=0
(2k + r + 1)
(2k + r + 1.5)
(0.5)r
r !
???
(2k + r + 1) -  2k + r + 1,
?0
¯ ?1(1-?)+
(2k + r + 1) -  2k + r + 1,
?0
¯ ?2(1-?)
???
??
???
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Spectral efficiency of dual diversity selection...
Using that result we obtain spectral efficiency i.e. CT IFR
B
[bit/sec/Hz] as
?T IFR = 1.443 × log
????????
1 +
1
(1 - ?)0.5 8
k=0
?k (0.5)4k+1
k!(k+0.5)(1-?)
8
r=0
(2k+r+1)(0.5)r
(2k+r+1.5)(r !) ??
 1
¯ ?1  2k + r, ?0
¯ ?1(1-?)
+
 1
¯ ?2  2k + r, ?0
¯ ?2(1-?)

??
????????
×?
??
1 - (1 - ?)0.5 8
k=0
(?)k (0.5)4k+1
k!(k + 0.5)
8
r=0
(2k + r + 1)
(2k + r + 1.5)
(0.5)r
r !
??
(2k + r + 1) -  2k + r + 1, ?0
¯ ?1(1-?)+
(2k + r + 1) -  2k + r + 1, ?0
¯ ?2(1-?)
??
???(15)
The computation of spectral efficiency according to (15)
requires the evaluation of infinite series. It is difficult but not
impossible to compute the spectral efficiency under TIFR
scheme. To efficiently compute the series, we truncate the
series using numerical evaluation techniques.
Finally we compare (15) with [24, Eq. (35)] and spectral
efficiency without diversity in [22, Eq. (18)].
3.3 ORA
With optimal rate adaptation to channel fading and a constant
transmit power, the average channel capacity (CORA[bit/sec])
with received SNR distribution p? (? ) is defined in [5,9] as
CORA = B
8

0
log2(1 + ? )p? (? )d? (16)
where B (Hz) is the channel bandwidth.
In fact, (16) represents the average capacity of the fading
channel without transmitter feedback (i.e., with the channel
fade level known at the receiver only).
Substituting (3) into (16), the average channel capacity of
dual-branch SC over correlated Nakagami-0.5 fading channels
is
CORA = B
8

0
log2(1 + ? )(1 - ?)0.5 8
k=0
?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×??? 0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
CORA = 1.443×B
8

0
log(1+? )(1-?)0.5 8
k=0
?k
k!(k+0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×?
??
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d? (17)
Following can be taken from the first part of above integral
is
 8
0
log(1 + ? ) exp -
?
¯ ?1(1 - ?)
? 2k+r d?
This can be solved using partial integration as follows
 8
0
udv = lim
??8
(uv) - lim
?? 0
(uv) -  8
0
vdu
Let u = log(1 + ? )
then du = d?
1+?
Now let
dv = exp -
?
¯ ?1(1 - ?)
? 2k+r d?
Integrating above expression using [29], we obtain
v = -exp -
?
¯ ?1(1 - ?)

×
2k+r
t=0
2k + r !
2k + r - t!
(?1(1 - ?))t+1 ? 2k+r-t
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M. I. Hasan, S. Kumar
Evaluating integral by using partial integral and some
mathematical transformation using [29,30], we obtain
8

0
log(1 + ? ) exp -
?
¯ ?1(1 - ?)
? 2k+r d?
=
2k+r
z=0
(2k + r )!( ¯ ?1(1 - ?))2k+r+1-z
×e
1
¯ ?1(1-?)  -z,
1
¯ ?1(1 - ?)
(18)
Similarly second part of above integral (17) can be solved
using [29,30], we obtain
8

0
log(1 + ? ) exp -
?
¯ ?2(1 - ?)
? 2k+r d?
=
2k+r
z=0
(2k + r )!( ¯ ?2(1 - ?))2k+r+1-z
×e
1
¯ ?2(1-?)  -z,
1
¯ ?2(1 - ?)
(19)
Substituting (18) and (19) in (17), the spectral efficiency
of dual-branch SC under Nakagami-0.5 fading channels is
CORA
= B
8
k=0
1.443 × ?k (1 - ?)0.5
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×
?????????
0.5
¯ ?1 2k+r+1
2k+r
z=0 (2k+r )!( ¯ ?1(1-?))2k+r+1-z
×e
1
¯ ?1(1-?)  -z, 1
¯ ? 1(1-?)
+
0.5?
¯ ?2 2k+r+1
2k+r
z=0 (2k+r )!( ¯ ?2(1-?))2k+r+1-z
×e
1
¯ ?2(1-?)  -z, 1
¯ ?2(1-?)

?????????
Using that result, we obtain spectral efficiency i.e. CORA
B
[bit/sec/Hz] as
?ORA
=
8
k=0
1.443 × ?k (1 - ?)0.5
k!(k + 0.5)(2)2k
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [ dB] per Branch
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0 .6
No diversity
???
Fig. 1 Spectral efficiency over correlated Nakagami-0.5 fading channels
as a function of equal average received SNR ( ¯ ?1 = ¯?2 = ¯? )using
OPRA
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×
?????????
0.5
¯ ?1 2k+r+1
2k+r
z=0 (2k+r )!( ¯ ?1(1 - ?))2k+r+1-z
×e
1
¯ ?1(1-?)  -z, 1
¯ ? 1(1-?)
+
0.5?
¯ ?2 2k+r+1
2k+r
z=0 (2k+r )!( ¯ ?2(1 - ?))2k+r+1-z
×e
1
¯ ?2(1-?)  -z, 1
¯ ?2(1-?)

?????????
(20)
To efficiently compute the series, we truncate the series
using numerical evaluation techniques. Finally, we compare
[25, Eqs. 6 and 23] with (20).
4 Numerical results and analysis
In this section, various performance evaluation results for
the spectral efficiency using dual-branch SC operating over
correlated Nakagami-0.5 fading channels with unequal average
received SNR between the different combined branches
has been presented and analyzed. We then discuss that the
generic results can be further simplified for Nakagami-0.5
fading, correlated and uncorrelated branches with equal average
received SNR. These results also compare the different
adaptive transmission schemes under worst fading channel
condition.
Figure 1 shows the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under OPRA scheme as a function of the equal average
received SNR per branch ( ¯ ?1 = ¯?2 = ¯? ) for ? = 0.2 and
123
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Table 1 Comparison of ?OPRA
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with equal SNR ( ¯ ?1 = ¯?2 = ¯? )
¯ ? (dB) ?OPRAfor ? = 0 ?OPRA for ? = 0.2 ?OPRA for ? = 0.6 ?OPRA for no diversity
-10 0.340007 0.339989 0.320618 0.2722
-5 0.703605 0.68981 0.65502 0.54773394
0 1.33269 1.29941 1.23285 1.015772433
5 2.28202 2.22688 2.1166 1.72595787
10 3.52795 3.45287 3.30271 2.69064666
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
First Branch Average Received SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 2 Spectral efficiency over correlated Nakagami-0.5 fading channels
as a function of first branch average received SNR( ¯ ?1) usingOPRA
? = 0.6. For comparison, the spectral efficiency of uncorrelated
Nakagami-0.5 fading channels with dual-branch SC
and without diversity, which was obtained in [24, Eq. (27)]
and [22, Eq. (8)] respectively, is also presented in Fig. 1. As
expected, by increasing ¯ ? and/or employing diversity, spectral
efficiency improves. It is also observed in Table 1, that the
spectral efficiencywith dual-branchSCis largestwhen ? = 0
and decreases as ? increases except for ¯ ? = -8.75 dB, gives
almost identical performance even when correlation coeffi-
cient increased to ? = 0.2, and same is shown in Fig. 1.
Figure 2 depicts the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under OPRA scheme with unequal average received SNR
( ¯ ?1 = 2 ¯ ?2) for ? = 0,? = 0.2 and ? = 0.6. For comparison,
the spectral efficiency of uncorrelated Nakagami-0.5
fading channels without diversity, which was obtained in
[22, Eq. (8)], is also presented in Fig. 2. Similarly, the spectral
efficiency with dual-branch SC as a function of the first
branch average received SNR ( ¯ ?1) is largest when ? = 0. It
is very interesting to observe in Table 2, that the spectral effi-
ciency without diversity gives almost identical performance
for ¯ ? = 0 dB even when employing diversity as a function
of first branch average received SNR ( ¯ ?1) with correlation
coefficient decrease to zero (? = 0), and same is shown in
Fig. 2.
In Fig. 3, the spectral efficiency of dual-branch SC with
unequal average received SNR ( ¯ ?1 = 2 ¯ ?2) over uncorrelated
Nakagami-0.5 fading channels using TIFR scheme is
plotted as a function of the cutoff SNR ?0 for several values
of first branch average received SNR ( ¯ ?1). As expected, by
increasing ¯ ?1 spectral efficiency improves.
In Fig. 4, the spectral efficiency of dual-branch SC for
? = 0.2 with equal average received SNR per branch ( ¯ ?1 =
¯ ?2 = ¯? ) using TIFR scheme is plotted as a function of cutoff
SNR ?0 for several values of the average received SNR per
branch ¯ ? . As expected, by increasing ¯ ? spectral efficiency
improves.
In Fig. 5, the spectral efficiency of dual-branch SC for
? = 0.2 with unequal average received SNR ( ¯ ?1 = 2 ¯ ?2)
using TIFR scheme is plotted as a function of cutoff SNR
?0 for several values of the first branch average received
SNR ¯ ?1. As expected, by increasing ¯ ?1 spectral efficiency
improves. It is also observed in Figs. 4 and 5 that the spectral
efficiency is decreases as we go from branches with equal
average received SNR ( ¯ ?1 = ¯?2 = ¯? ) to unequal average
received SNR ( ¯ ?1 = 2 ¯ ?2).
Table 2 Comparison of ?OPRA for different values of ? under worst case fading scenario using dual-branch SC and no diversity with unequal
SNR ( ¯ ?1 = 2 ¯ ?2)
¯ ?1 (dB) ?OPRA for ? = 0 ?OPRA for ? = 0.2 ?OPRA for ? = 0.6 ?OPRA for no diversity
-10 0.2722 0.2722 0.2722 0.2722
-5 0.54773399 0.54773397 0.54773395 0.54773394
0 1.015789433 1.015783932 1.015781983 1.015772433
5 1.870663 1.825 1.81645 1.72595787
10 3.06567 2.94 2.89 2.69064666
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M. I. Hasan, S. Kumar
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
First branch average received SNR = -10 dB
First branch average received SNR = - 5 dB
First branch average received SNR = 0 dB
First branch average received SNR = 5 dB
First branch average received SNR = 10 dB
Fig. 3 Spectral efficiency of an unequal SNR ( ¯ ?1 = 2 ¯ ?2) dual-branch
SC versus the cutoff SNR over uncorrelated Nakagami-0.5 fading channels
using TIFR
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Average received SNR = - 10 dB
Average received SNR = - 5 dB
Average received SNR = 0 dB
Average received SNR = 5 dB
Average received SNR = 10 dB
Fig. 4 Spectral efficiency of an equal SNR ( ¯ ?1 = ¯?2) dual-branch SC
versus the cutoff SNR over Nakagami-0.5 fading channels using TIFR
for ? = 0.2
In Fig. 6, the spectral efficiency of dual-branch SC for
? = 0.6 with equal average received SNR per branch ( ¯ ?1 =
¯ ?2 = ¯? ) using TIFR scheme is plotted as a function of cutoff
SNR ?0 for several values of the average received SNR per
branch ¯ ? . As expected, by increasing ¯ ? spectral efficiency
improves.
In Fig. 7, the spectral efficiency of dual-branch SC for
? = 0.6 under unequal average received SNR ( ¯ ?1 = 2 ¯ ?2)
using TIFR scheme is plotted as a function of cutoff SNR
?0 for several values of the first branch average received
SNR ¯ ?1. As expected, by increasing ¯ ?1 spectral efficiency
improves. It is again observed in Figs. 6 and 7 that the spectral
efficiency is decreases as we go from branches with equal
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
First branch average received SNR = -10 dB
First branch average received SNR = - 5 dB
First branch average received SNR = 0 dB
First branch average received SNR = 5 dB
First branch average received SNR = 10 dB
Fig. 5 Spectral efficiency of an unequal SNR ( ¯ ?1 = 2 ¯ ?2) dual-branch
SC versus the cutoff SNR over Nakagami-0.5 fading channels using
TIFR for ? = 0.2
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Average received SNR = - 10 dB
Average received SNR = - 5 dB
Average received SNR = 0 dB
Average received SNR = 5 dB
Average received SNR = 10 dB
Fig. 6 Spectral efficiency of an equal SNR ( ¯ ?1 = ¯?2) dual-branch SC
versus the cutoff SNR over Nakagami-0.5 fading channels using TIFR
for ? = 0.6
average received SNR ( ¯ ?1 = ¯?2 = ¯? ) to unequal average
received SNR ( ¯ ?1 = 2 ¯ ?2).
Figure 8 shows the spectral efficiency of a dual-branch SC
system over correlated Nakagami-0.5 fading channels under
TIFR scheme as a function of the equal average received SNR
per branch ( ¯ ?1 = ¯?2 = ¯? ) for ? = 0.2 and ? = 0.6. For comparison,
the spectral efficiency of uncorrelatedNakagami-0.5
fading channels with dual-branch SC and without diversity,
which was obtained in [24, Eq. (35)] and [22, Eq. (22)]
respectively, is also presented in Fig. 8. As expected, by
increasing ¯ ? and/or employing diversity, spectral efficiency
improves. It is also observed in Table 3, that the spectral
efficiency with dual-branch SC is largest when ? = 0 and
123
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Spectral efficiency of dual diversity selection...
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
First branch average received SNR = - 10 dB
First branch average received SNR = - 5 dB
First branch average received SNR = 0 dB
First branch average received SNR = 5 dB
First branch average received SNR = 10 dB
Fig. 7 Spectral efficiency of an unequal SNR ( ¯ ?1 = 2 ¯ ?2) dual-branch
SC versus the cutoff SNR over Nakagami-0.5 fading channels using
TIFR for ? = 0.6
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Average Received SNR [ dB] per Branch
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 8 Spectral efficiency over correlated Nakagami-0.5 fading channels
as a function of equal average received SNR ( ¯ ?1 = ¯?2 = ¯? ) using
TIFR
decreases as correlation coefficient ? increases except for
¯ ? = -7.5 dB, gives almost identical performance even when
correlation coefficient ? increased to ? = 0.2, and same is
shown in Fig. 8.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
First Branch Average Received SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 9 Spectral efficiency over correlated Nakagami-0.5 fading channels
as a function of first branch average received SNR ( ¯ ?1) using TIFR
Figure 9 depicts the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under TIFR scheme with unequal average received SNR
( ¯ ?1 = 2 ¯ ?2) for ? = 0, ? = 0.2 and ? = 0.6. For comparison,
the spectral efficiency of Nakagami-0.5 fading channels
without diversity, which was obtained in [22, Eq. (22)], is
also presented in Fig. 9. Similarly, the spectral efficiency
with dual-branch SC as a function of first branch average
received ( ¯ ?1) is largest when ? = 0. It is again very interesting
to observe in Table 4, that the spectral efficiency without
diversity under TIFR scheme gives almost identical performance
for ¯ ? = 2.5 dB even when employing diversity as
a function of first branch average received SNR ( ¯ ?1) with
correlation coefficient decrease to zero (? = 0), and same is
shown in Fig. 9.
Figure 10 shows the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under ORA scheme as a function of the equal average
received SNR per branch ( ¯ ?1 = ¯?2 = ¯? ) for ? = 0.2 and
? = 0.6. For comparison, the spectral efficiency without
diversity and uncorrelated dual-branch SC with equal average
received SNRper branch ( ¯ ?1 = ¯?2 = ¯? ) for ? = 0 under
Nakagami-0.5 fading channel, which was obtained in [25,
Table 3 Comparison of ?T IFR
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with equal SNR ( ¯ ?1 = ¯?2 = ¯? )
¯ ? (dB) ?T IFR for ? = 0 ?T IFR for ? = 0.2 ?T IFR for ? = 0.6 ?T IFR for no diversity
-10 0.313607599 0.307224921 0.294457017 0.2491
-5 0.641415299 0.628441118 0.59348451 0.4945
0 1.200999905 1.168007708 1.10200389 0.9039
5 2.038088171 1.972422736 1.871163546 1.5144
10 3.136902858 3.058132377 2.901971304 2.3323
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M. I. Hasan, S. Kumar
Table 4 Comparison of ?T IFR
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with unequal SNR ( ¯ ?1 = 2 ¯ ?2)
¯ ?1 (dB) ?T IFR for ? = 0 ?T IFR for ? = 0.2 ?T IFR for ? = 0.6 ?T IFR for no diversity
-10 0.2491 0.2491 0.2491 0.2491
-5 0.494502 0.494501 0.4945008 0.4945
0 0.90395 0.90393 0.90392 0.9039
5 1.60619677 1.56838859 1.552782 1.5144
10 2.77435225 2.64525 2.58713565 2.3323
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Average Received SNR [ dB] per Branch
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 10 Spectral efficiency over correlatedNakagami-0.5 fading channels
as a function of equal average received SNR ( ¯ ?1 = ¯?2 = ¯? ) using
ORA
Eqs. 6 and 23], is also presented in Fig. 10. As expected, by
increasing ¯ ? and/or employing diversity, spectral efficiency
improves. It is also observed in Table 5 that the spectral
efficiency with dual-branch SC is largest when ? = 0 and
decreases as ? increases, and same is shown in Fig. 10.
Figure 11 shows the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under ORA scheme as a function of the unequal average
received SNR ( ¯ ?1 = 2 ¯ ?2) for ? = 0,? = 0.2 and ? = 0.6.
For comparison, the spectral efficiency without diversity
under Nakagami-0.5 fading channel using [25] is also presented
in Fig. 11. Similarly, the spectral efficiency with
dual-branch SC as a function of first branch average received
SNR ( ¯ ?1) is largest when ? = 0. But the spectral effi-
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
First Branch Average Received SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 11 Spectral efficiency over correlatedNakagami-0.5 fading channels
as a function of first branch average received SNR ( ¯ ?1) using ORA
ciency without diversity gives almost identical performance
for ¯ ? = 1.2 dB even when employing diversity as a function
of first branch average received SNR( ¯ ?1) with correlation
coefficient decrease to zero (? = 0) as observe in Table 6
and shown in Fig. 11.
5 Conclusions
This research paper derives and analyzes the spectral effi-
ciency expressions over correlated Nakagami-0.5 fading
channels under different adaptive transmission schemes with
unequal average received SNRof dual-branch SC.By numerical
evaluations it has been found that the spectral efficiency
improves by increasing ¯ ? , decreasing correlation coefficient
Table 5 Comparison of ?ORA
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with equal SNR ( ¯ ?1 = ¯?2 = ¯? )
¯ ? (dB) ?ORA for ? = 0 ?ORA for ? = 0.2 ?ORA for ? = 0.6 ?ORA for no diversity
-10 0.205587 0.197396 0.177011 0.127571
-5 0.532337 0.512258 0.461776 0.337317
0 1.1733 1.13373 1.036258 0.769773
5 2.17481 2.11313 1.936045 1.49328
10 3.47351 3.39399 3.177 2.50593
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Table 6 Comparison of ?ORA
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with unequal SNR ( ¯ ?1 = 2 ¯ ?2)
¯ ?1 (dB) ?ORA for ? = 0 ?ORA for ? = 0.2 ?ORA for ? = 0.6 ?ORA for no diversity
-10 0.127571 0.127571 0.127571 0.127571
-5 0.337318 0.33731707 0.33731703 0.337317
0 0.769798 0.769783 0.769777 0.769773
5 1.558 1.5222 1.50 1.49328
10 2.678 2.567 2.532 2.50593
?, and/ or employing diversity with unequal to equal average
received SNRin all the cases of considered adaptive transmission
schemes, OPRA, tifr and ORA. However the magnitude
of improvement is slightly higher in case of OPRA. It has
been observed that for OPRA scheme, spectral efficiency
without diversity is almost same for ¯ ?1 = 0 dB, even though
employing uncorrelated dual-branch diversity with unequal
average received SNR ( ¯ ?1 = 2 ¯ ?2). It has also been observed
that for TIFR scheme, spectral efficiency without diversity
is almost same for ¯ ?1 = 2.5 dB, even though employing
uncorrelated dual-branch diversity with unequal average
received SNR ( ¯ ?1 = 2 ¯ ?2). It has also been observed that
for ORA scheme, spectral efficiency without diversity is
almost same for ¯ ?1 = 1.2 dB, even though correlation coef-
ficient decreases and employing dual-branch diversity with
unequal average received SNR ( ¯ ?1 = 2 ¯ ?2). Therefore it is
very important to recommend that under worst fading condition,
the proper antenna spacing at the receiver end, required
for uncorrelated diversity path for obtaining the optimum
spectral efficiency in each of the scheme, OPRA, TIFR, or
ORA scheme is not an important issue for low value of average
received SNR, particularly, ¯ ?1 = 0 dB in case of OPRA,
¯ ?1 = 2.5 dB in case of TIFR, or ¯ ?1 = 1.2 dB in case of
ORA This paper also states that under Nakagami-0.5 fading
channels with equal average received SNR, TIFR scheme
is a better choice over ORA for low average received SNR
and ORA scheme is for high average received SNR even
employing diversity. It is very interesting to finally conclude
that TIFR scheme with unequal average received SNR under
worst condition of fading is always a better option overORA.
References
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29. Gradshteyn, I. S.,&Ryzhik, I.M.(2000). Tables of integrals, series,
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http://functions.wolfram.com.
MohammadIrfanul Hasan was
born in Samastipur, India on
March 01, 1977. He received
his B.Tech. degree in Electronics
& Communications Engineering
from Magadh University, Bihar,
and M.Tech. degree from Institute
of Advanced Studies in
Education University, Rajasthan.
Since February 2012, he has
been working towards his Ph.D.
degree in Electronics and Communication
Engineering at Birla
Institute of Technology, Mesra,
Ranchi, India. He is currently
working as an Assistant Professor in the Department of Electronics and
Communications Engineering atGraphic Era University,Dehradun. His
research interests are in the field of Wireless Communication Technology
primarily in the area of diversity techniques.
Sanjay Kumar was born in
Ranchi, India on January 18,
1967. In 1994 he received
MBA degree from Pune University,
and M.Tech. in Electronics
and Communication Engineering
from Guru Nanak Dev Engineering
College, Ludhiana, India
in the year 2000. Subsequently
he obtained his Ph.D. degree in
Wireless Communication from
Aalborg University, Denmark in
the year 2009. He served the
Indian Air Force from the 1985 to
2000, where he was involved in
the technical supervision and maintenance activities of telecommunications
and radar systems. Hewas a guest researcher at AalborgUniversity
during 2006 to 2009, where he worked in close cooperation with Nokia
Siemens Networks and Centre for TeleInFrastruktur. He also worked
as a guest lecturer in the department of Electronics Systems at Aalborg
University during the years 2007 to 2009. Presently he is working as an
Associate Professor in the Department of Electronics and Communications
Engineering at Birla Institute of Technology, Mesra, Ranchi. His
current interest lies in the field ofWireless Communication Technology.
123
Author's personal copy , Claims:FORM 2
THE PATENTS ACT, 1970
(39 OF 1970)
&
THE PATENTS RULES, 2003 5
COMPLETE SPECIFICATION
(See section 10; rule 13)
10
Title: Spectral efficiency of dual diversity selection combining schemes under correlated
Nakagami-0.5 fading with unequal average received SNR
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, 20
Uttarakhand, India
25
PREAMBLE TO THE DESCRIPTION:
The following specification particularly describes the nature of this invention and the manner 30
in which it is to be performed.
Telecommun Syst
DOI 10.1007/s11235-016-0152-8
Spectral efficiency of dual diversity selection combining schemes
under correlated Nakagami-0.5 fading with unequal average
received SNR
Mohammad Irfanul Hasan1,2 · Sanjay Kumar1
© Springer Science+Business Media New York 2016
Abstract The spectral efficiency results for different adaptive
transmission schemes over correlated diversity branches
with unequal average signal to noise ratio (SNR) obtained so
far in literature are not applicable for Nakagami-0.5 fading
channels. In this paper, we investigate the effect of fade correlation
and level of imbalance in the branch average received
SNR on the spectral efficiency of Nakagami-0.5 fading channels
in conjunction with dual-branch selection combining
(SC). This paper derived the expressions for the spectral
efficiency over correlated Nakagami-0.5 fading channels
with unequal average received SNR. This spectral efficiency
is evaluated under different adaptive transmission schemes
using dual-branch SC diversity scheme. The corresponding
expressions for Nakagami-0.5 fading are considered to be the
expressions under worst fading conditions. Finally, numerical
results are provided to illustrate the spectral efficiency
degradation due to channel correlation and unequal average
received SNR between the different combined branches
under different adaptive transmission schemes. It has been
observed that optimal simultaneous power and rate adaptation
(OPRA) scheme provides improved spectral efficiency
as compared to truncated channel inversion with fixed rate
(TIFR) and optimal rate adaptation with constant transmit
power (ORA) schemes under worst case fading scenario. It
is very interesting to observe that TIFR scheme is always a
B Mohammad Irfanul Hasan
irfanhasan25@rediffmail.com
Sanjay Kumar
skumar@bitmesra.ac.in
1 Department of Electronics and Communication Engineering,
Birla Institute of Technology, Mesra, Ranchi 835215, India
2 Department of Electronics and Communication Engineering,
Graphic Era University, Dehradun 248002, India
better choice over ORA scheme under correlated Nakagami-
0.5 fading channels with unequal average received SNR.
Keywords Dual-branch · Nakagami-0.5 fading channels · Optimal simultaneous power and rate adaptation · Optimal
rate adaptation with constant transmit power · Selection
combining · Spectral efficiency · Truncated channel inversion
with fixed rate
1 Introduction
The channel capacity is of fundamental importance in the
design ofwirelessmobile communication systems as it determines
the maximum achievable data rate of the wireless
communication systems [1-4]. Since wireless mobile environment
is subjected to fading, which degrades the data rate
performance. The channel capacity in fading environment,
which is less than that of anAWGNchannel, can be improved
by increasing the transmit power. Increasing transmit power
to mitigate the fading effect on the signal results in increasing
the interference level, which is undesirable. Another way
to improve channel capacity in fading environment is by
employing diversity combining techniques and/ or adaptive
transmission schemes. The biggest advantage of such mechanism
is that there is no requirement to increase the transmit
power and/ or bandwidth to attain improvements in channel
capacity. However, the improvement in channel capacity
comes at the expense of an added system complexity [1-4].
The adaptive transmission schemes with diversity combining
techniques are currently receiving a great deal of
attention as very promising techniques to achieve high channel
capacity in thewirelessmobile environment. The primary
objective behind adaptive transmission schemes is to use CSI
(channel state information) at receiver or at transmitter in
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M. I. Hasan, S. Kumar
order to maximize the channel capacity and lower the probability
of outage [3-5]. Moreover, correlation at the receive
antennas can potentially lead to degradation in the channel
capacity in fading environment [3-21]. In number of
real-life scenarios, physical restrictions may not allow using
antenna spacing that is required for independent fading across
diversity branches. Therefore, investigations on the effect of
correlation in adaptive transmission schemes using diversity
combining techniques becomes essential in gaining a better
understanding of the trade-off involved in systems design.
Diversity combining is known to be a powerful technique
that can be used to combat fading in wireless mobile
environment. Maximal ratio combining (MRC), equal gain
combining (EGC) and SC aremost prevalent diversity combing
techniques [3,4].
Adaptive transmission schemes, which requires accurate
channel estimation at the receiver and a reliable feedback
path between the estimator and the transmitter, provides
great improvement to the channel capacity [3,4].The capacity
of flat fading channels was derived in [5] for four different
adaptive transmission schemes such as OPRA, ORA,
channel inversion with fixed rate transmission (CIFR) and
TIFR. In studying the channel capacity performances of different
adaptive transmission schemes, the usual assumption
made is that the combined branches are independent to one
another and have the same average received SNR. However,
independent fading is not always realized in practice due to
insufficient antenna spacing in case of small-size terminals
equipments. In addition, the diversity branches in practical
system may have unequal average received SNR's due to
different noise figures or feeding lengths [6,7].
Numerous researchers [3-25] haveworked on the study of
channel capacity over different fading channels. We discuss
here some representative examples. Specifically [3,4] discuss
the channel capacity over correlated Nakagami-m (m = 1 and m < 1) fading channels underORAand CIFR schemes
using different diversity combining techniques. In [8], the
channel capacity over uncorrelated Nakagami-m (m = 1)
fading channels with MRC and without diversity under
different adaptive transmissions schemes was analyzed.
Expressions for the capacity over uncorrelated Rayleigh fading
channels with MRC and SC under different adaptive
transmission schemes were obtained in [9]. An analytical
performance study of the channel capacity for correlated generalized
gamma fading channels with dual-branch SC under
different adaptive transmission schemes was introduced in
[10]. The channel capacity under Nakagami-m (m = 1) fading
channel without diversitywas derived in [11] for different
adaptive transmission schemes. In [12], channel capacity of
dual-branch SC and MRC systems over correlated Hoyt fading
channels using different adaptive transmission schemes
was presented. In [13], expression for the ergodic capacity of
MRC over arbitrarily correlated Rician fading channels was
derived. In [14], an expression for lower and upper bounds
in the channel capacity expression for uncorrelated Rician
and Hoyt fading channels with MRC using ORA scheme
were obtained. In [22], an analytical performance study of
the channel capacity for uncorrelated Nakagami-0.5 with
dual-branch MRC using OPRA and TIFR was obtained. In
[23], the channel capacity over correlated Nakagami-0.5 fading
channels under OPRA and TIFR schemes with MRC
was discussed. An analytical performance study of the channel
capacity for uncorrelated Nakagami-0.5 fading channels
with dual-branch SC under OPRA, TIFR, ORA, and CIFR
schemes was introduced in [24,25]. However, an analytical
study of channel capacity over correlated Nakagami-0.5
fading channels with unequal average received SNR under
different adaptive transmission schemes using SC has not
been considered so far.
Nakagami-m model has been extensively used in general
to study wireless mobile communication system performance,
less concentration appears to have been focused on
the particular case of Nakagami-0.5 fading. At the same time
that results obtained for Nakagami-0.5 will have immense
practical value as a worst case fading scenario. Hence, in
cases where quality of service with high data rate requirements
mandate designing for worst case fading scenario,
results obtained for the same fading condition will have
great practical applications in wireless mobile environment
[26,27]. In this paper, correlated dual-branch SC under worst
case of fading conditions has been considered to investigate
the effect of fade correlation (?) and average received SNR
imbalance between the different combined branches on the
link spectral efficiency. The link spectral efficiency, defined
as the ratio of the average channel capacity to the given
bandwidth [9]. The dual-branch diversity has been considered
since it offers themaximum SNR improvement, besides
offering minimum complexity and physical space requirements
[26].
Therefore, this paper fills this gap by presenting the impact
of various practical constraints, e.g. physical space requirements,
effect of fade correlation and level of imbalance in
the branch average received SNR on channel capacity under
worst case fading scenario.
Finally, some recommendation is given for proper antenna
spacing required for uncorrelated diversity branch and the
choice for adaptive transmission schemes under average
received SNR imbalance. That allows the researchers or system
designers to perform comparison and tradeoff studies
among the various adaptive transmission schemes with simple
combining technique SC, so as to determine the optimal
choice in the face of their available constraints.
The remainder of this paper is organized as follows: In
Sect. 2, the channel model is defined. In Sect. 3, spectral
efficiency of dual-branch SC over correlated Nakagami-
0.5 fading channels with unequal average received SNR
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Spectral efficiency of dual diversity selection...
are derived for different adaptive transmission schemes. In
Sect. 4, several numerical results are presented and analyzed,
whereas in Sect. 5, concluding remarks are given.
2 Channel model
The probability density function (pdf) of the instantaneous
received SNR,? , at the output of a correlated dual-branch SC
combiner, p? (? ), for the Nakagami-m fading channels with
unequal average received SNR is obtained in [27,28]
p? (? ) =
? m-1
(m) m
¯ ?1 m
exp-
m?
¯ ?1 1
- Qm  2m??
¯ ?1(1 - ?)
, 2m?
¯ ?2(1 - ?)
+
? m-1
(m) m
¯ ?2 m
exp-
m?
¯ ?2 1
-Qm  2m??
¯ ?2(1 - ?)
, 2m?
¯ ?1(1 - ?), ? = 0
(1)
where ¯ ?l is the average received SNR of the l th branch
(l = 1, 2) represents level of imbalance in the branch
average received SNR, m (m = 0.5) is the fading parameter,
(.) is the gamma function, Qm(., .) is the generalized
(mth-order ) Marcum Q-function, and ? is the correlation
coefficient between two fading envelopes represents effect
of fade correlation between diversity branches [27,28].
The fading parameter, m measures the amount of fading
(amount of fading = 1/m). For different values of m,
this expression simplifies to several important distributions
describing fading models. Like m = 0.5 corresponds to one
sided Gaussian distribution said to be worst case fading scenario,
m = 1 corresponds to Rayleigh distribution,m = 1
corresponds to Rician distribution, and as m ?8, the distribution
converges to a non fading AWGN [27].
Replacing the Qm(., .) function in (1) with its series
representation as given in [29], and then the pdf under
Nakagami-m fading using (1) becomes
p? (? ) =
? m-1
(m) m
¯ ?1 m
exp-
m?
¯ ?1  2m?
¯ ?1(1 - ?)
m
×
8k=0
(-1)k  2m?
¯ ?1(1-?)
k
1F1 -k;m;- m?2?
¯ ?1(1-?)

2me
m??
¯ ?1(1-?) (m)
×
? m-1
(m) m
¯ ?2 m
exp-
m?
¯ ?2  2m?
¯ ?2(1 - ?)
m
×
8
k=0
(-1)k  2m?
¯ ?2(1-?)
k
1F1 -k;m;- m?2?
¯ ?2(1-?)

2me
m??
¯ ?2(1-?) (m)
(2)
Putting m = 0.5 and simplified this pdf using [29], we
obtained
p? (? ) = (1 - ?)0.5 8
k=0
?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×?
??
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r (3)
3 Spectral efficiency
In this section, we present expressions for the spectral effi-
ciency of correlated Nakagami-0.5 fading channels with
unequal average received SNR using dual-branch SC under
OPRA, TIFR and ORA schemes. It is assumed that, for the
considered adaptation schemes, there exist perfect channel
estimation and an error-free delayless feedback path, similar
to the assumption made in [9].
3.1 OPRA
The average channel capacity of fading channels with
received SNR distribution p? (? ) and optimal simultaneous
power and rate adaptation (COPRA[bit/sec]) is defined in
[5,9] as
COPRA = B
8
?
0
log2  ?
?0  p? (? )d? (4)
where B (Hz) is the channel bandwidth, and ?0 is the optimal
cutoff SNR level below which no data is transmitted. To
obtain the optimal cutoff SNR, ?0 must satisfy the equation
given by [5,9] as
8
?
0
 1
?0 -
1
? p? (? )d? =1 (5)
To achieve the average channel capacity (4), the channel
fade level must be tracked at both the receiver and transmitter,
and the transmitter has to adapt its power and rate
123
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M. I. Hasan, S. Kumar
accordingly, allocating high power levels and data rates for
good channel conditions (? large), and lower power levels
and data rates for poor channel conditions (? small).
Since data transmission is suspended when the received
SNR, ? is less than ?0, this optimal adaptation scheme suffers
an outage probability, which is given by [5,9]
Pout = P ? = ?0 =
?0

0
p? (? )d? = 1 -
8
?
0
p? (? )d? (6)
Substituting (3) in (5) for optimal cutoff SNR?0, and evaluating
the integral using some mathematical transformation
by [29], we obtain
1 - ?
8
k=0
?k0.54k+1
k!(k + 0.5) ×
8
r=0
(2k + r + 1)(0.5)r
(2k + r + 1.5)(r !)
×
??????
 2k + r + 1, ?0
¯ ?1(1-?)
-
2k+r,
?0
¯ ?1(1-?)

¯ ?1(1-?) 
+ 2k + r + 1, ?0
¯ ?2(1-?)
-
2k+r,
?0
¯ ?2(1-?)

¯ ?2(1-?) 
??????
= 1
(7)
The numerical evaluation techniques have confirmed that, by
solving (7), there is a unique positive value of ?0 satisfying
(7) that takes values from ?0 ? [0, 1]. Result shows that ?0
increases as ¯ ?l (l = 1, 2) increases.The value of cutoff SNR
?0 that satisfies (7) by either assuming equal average received
SNR( ¯ ?1 = ¯?2 = ¯? ) or unequal average received SNR ( ¯ ?1 = 2 ¯ ?2)is used for finding the average channel capacity in each of
the case. Substituting (3) in (4), the average channel capacity
of dual-branch SC under Nakagami-0.5 fading channels is
COPRA
= B
8
?
0
log2  ?
?0  8
k=0
(1 - ?)0.5?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×?
??
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
COPRA
= 1.443B
8
k=0
(1 - ?)0.5?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×
???????????
8
?0
log(? )???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
-
8
?0
log(?0)???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
???????????
(8)
Following can be taken from the first part of above integral
is
8
?
0
log(? )???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
This can be solved using partial integration as follows
8
?
0
udv = lim
??8
(uv) - lim
???0
(uv) -
8
?
0
vdu
Let u = log ?
then du = d?
?
Now let
dv = ???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
Integrating above expression using [29], we obtain
v = - {0.5 × (1 - ?)}2k+r+1  2k + r + 1,
?
¯ ?1 (1 - ?)

+ {0.5 × (1 - ?)}2k+r+1  2k + r + 1,
?
¯ ?2 (1 - ?)


Evaluating first part of above integral (8) using this partial
integration and some mathematical transformation [29,30],
we obtain
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Spectral efficiency of dual diversity selection...
8
?
0
log(? )???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
= 0.5
¯ ?1 2k+r+1 ??????
2k+r
t=0
2k+r ! 2k+r-t!
log(?0) × exp  -?0
¯ ?1(1-?)
×{ ¯ ?1(1 - ?)}t+1 (?0)2k+r-t+ 8
t=0
2k+r ! 2k+r-t! { ¯ ?1(1 - ?)}2k+r+1  2k + r - t, ?0
¯ ?1(1-?)

??????
+0.5
¯ ?2 2k+r+1 ??????
2k+r
t=0
2k+r ! 2k+r-t!
log(?0) × exp  -?0
¯ ?2(1-?)
×{ ¯ ?2(1 - ?)}t+1 (?0)2k+r-t+ 8
t=0
2k+r ! 2k+r-t! { ¯ ?2(1 - ?)}2k+r+1  2k + r - t, ?0
¯ ?2(1-?)

??????
(9)
Second part of above integral (8) can be solved by using
[29], we obtain
8
?
0
log(?0)???
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
= +log(?0)???
0.5
¯ ?1 2k+r+1 { ¯ ?1(1 - ?)}2k+r+1  2k + r + 1, ?0
¯ ?1(1-?)
+
0.5
¯ ?2 2k+r+1 { ¯ ?2(1 - ?)}2k+r+1  2k + r + 1, ?0
¯ ?2(1-?)

???
(10)
Substituting (9) and (10) in (8), the average channel capacity
of dual-branch SC under Nakagami-0.5 fading channels
is
COPRA
= 1.443B
8
k=0
(1 - ?)0.5?k
k!(k + 0.5)(2)2k
8
r=0
×
(2k + r + 1)  1
(1-?) 2k+r+1
(2k + r + 1.5)(r !)
×
??????????
0.5
¯ ?1 2k+r+1
??????????
2k+r
t=0
2k+r ! 2k+r-t! log(?0)×
exp  -?0
¯ ?1(1-?) { ¯ ?1(1 - ?)}t+1 (?0)2k+r-t
+8t=0
2k+r ! 2k+r-t! { ¯ ?1(1 - ?)}2k+r+1
× 2k + r - t,
?0
¯ ?1(1-?)

??????????
+0.5
¯ ?2 2k+r+1 ??????
2k+r
t=0
2k+r ! 2k+r-t! log(?0)×
exp  -?0
¯ ?2(1-?) { ¯ ?2(1 - ?)}t+1 (?0)2k+r-t+
8t=0
2k+r ! 2k+r-t! { ¯ ?2(1 - ?)}2k+r+1
× 2k + r - t,
?0
¯ ?2(1-?)

??????
-log(?0)
?????? 
0.5
¯ ?1 2k+r+1 { ¯ ?1(1 - ?)}2k+r+1
× 2k + r + 1,
?0
¯ ?1(1-?)

+
0.5
¯ ?2 2k+r+1 { ¯ ?2(1 - ?)}2k+r+1
× 2k + r + 1,
?0
¯ ?2(1-?)


??????
??????
Using that result we obtain spectral efficiency i.e. COPRA
B
[bit/sec/Hz] as
?OPRA
= 1.443
8
k=0
(1 - ?)0.5?k
k!(k + 0.5)(2)2k-1
×
8
r=0
(2k + r + 1)  1
(1-?) 2k+r+1
(2k + r + 1.5)(r !) 0.5
¯ ?1 2k+r+1
×
??????
2k+r
t=0
2k+r ! 2k+r-t! log(?0)×
exp  -?0
¯ ?1(1-?) { ¯ ?1(1 - ?)}t+1 (?0)2k+r-t+
8t=0
2k+r ! 2k+r-t! { ¯ ?1(1 - ?)}2k+r+1
× 2k + r - t,
?0
¯ ?1(1-?)

??????
+0.5
¯ ?2 2k+r+1 ??????
2k+r
t=0
2k+r ! 2k+r-t! log(?0)×
exp  -?0
¯ ?2(1-?) { ¯ ?2(1 - ?)}t+1 (?0)2k+r-t+
8t =0
2k+r ! 2k+r-t! { ¯ ?2(1 - ?)}2k+r+1
× 2k + r - t,
?0
¯ ?2(1-?)

??????
-log(?0)
??????
0.5
¯ ?1 2k+r+1  { ¯ ?1(1 - ?)}2k+r+1
× 2k + r+,
?0
¯ ?1(1-?)

+
0.5
¯ ?2 2k+r+1  { ¯ ?2(1 - ?)}2k+r+1
× 2k + r+,
?0
¯ ?2(1-?)


??????
??????
(11)
123
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M. I. Hasan, S. Kumar
We compare (11) with [24, Eq. (27)] and spectral effi-
ciency without diversity in [22, Eq. (8)].
3.2 TIFR
The average channel capacity of fading channels with
received SNRdistribution p? (? ) under TIFR scheme (CT IFR
[bit/sec]) is defined in [5,9] as
CT IFR = B log2
?????
1 +
1
8
?0  p? (? )
? d?
?????
(1 - Pout ), ? = 0
(12)
The cutoff level ?0, can be selected to achieve a speci-
fied probability of outage,Pout , or, to maximize the average
channel capacity (12).
Now, we evaluate the probability of outage using (3) and
(6) is
Pout =
?0

0
(1 - ?)0.5 8
k=0
?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×
????
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

????
? 2k+r d?
Evaluating the above integral using some mathematical
transformation [29,30], we obtain
Pout = (1 - ?)0.5 8
k=0
(?)k (0.5)4k+1
k!(k + 0.5)
×
8
r=0
(2k + r + 1)
(2k + r + 1.5)
(0.5)r
r !
×?
?
(2k + r + 1) -  2k + r + 1, ?0
¯ ?1(1-?)+
(2k + r + 1) -  2k + r + 1, ?0
¯ ?2(1-?)
??
(13)
Using the pdf of dual-branch SC under correlated
Nakagami-0.5 fading channels in (3), we obtain
p? (? )
? = (1 - ?)0.5 8
k=0
?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×?
??
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r-1
Integrating the above expression using [29,30], we obtain
8
?
0
p? (? )
?
d? = (1 - ?)0.5 8
k=0
?k (0.5)4k+1
k!(k + 0.5)(1 - ?)
×
8
r=0
(2k + r + 1)(0.5)r
(2k + r + 1.5)(r !)
??
 1
¯ ?1  2k + r,
?0
¯ ?1(1-?)
+
 1
¯ ?2  2k + r,
?0
¯ ?2(1-?)

??
(14)
Putting the value of (13) and (14) in (12), we get
CT IFR = 1.443 × B log
??????????
1 +
1
(1 - ?)0.5 8
k=0
?k (0.5)4k+1
k!(k+0.5)(1-?)
8
r=0
(2k+r+1)(0.5)r
(2k+r+1.5)(r !)
???
 1
¯ ?1  2k + r,
?0
¯ ?1(1-?)
+
 1
¯ ?2  2k + r,
?0
¯ ?2(1-?)

???
??????????
×
?????
1 - (1 - ?)0.5 8
k=0
(?)k (0.5)4k+1
k!(k + 0.5)
8
r=0
(2k + r + 1)
(2k + r + 1.5)
(0.5)r
r !
???
(2k + r + 1) -  2k + r + 1,
?0
¯ ?1(1-?)+
(2k + r + 1) -  2k + r + 1,
?0
¯ ?2(1-?)
???
??
???
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Spectral efficiency of dual diversity selection...
Using that result we obtain spectral efficiency i.e. CT IFR
B
[bit/sec/Hz] as
?T IFR = 1.443 × log
????????
1 +
1
(1 - ?)0.5 8
k=0
?k (0.5)4k+1
k!(k+0.5)(1-?)
8
r=0
(2k+r+1)(0.5)r
(2k+r+1.5)(r !) ??
 1
¯ ?1  2k + r, ?0
¯ ?1(1-?)
+
 1
¯ ?2  2k + r, ?0
¯ ?2(1-?)

??
????????
×?
??
1 - (1 - ?)0.5 8
k=0
(?)k (0.5)4k+1
k!(k + 0.5)
8
r=0
(2k + r + 1)
(2k + r + 1.5)
(0.5)r
r !
??
(2k + r + 1) -  2k + r + 1, ?0
¯ ?1(1-?)+
(2k + r + 1) -  2k + r + 1, ?0
¯ ?2(1-?)
??
???(15)
The computation of spectral efficiency according to (15)
requires the evaluation of infinite series. It is difficult but not
impossible to compute the spectral efficiency under TIFR
scheme. To efficiently compute the series, we truncate the
series using numerical evaluation techniques.
Finally we compare (15) with [24, Eq. (35)] and spectral
efficiency without diversity in [22, Eq. (18)].
3.3 ORA
With optimal rate adaptation to channel fading and a constant
transmit power, the average channel capacity (CORA[bit/sec])
with received SNR distribution p? (? ) is defined in [5,9] as
CORA = B
8

0
log2(1 + ? )p? (? )d? (16)
where B (Hz) is the channel bandwidth.
In fact, (16) represents the average capacity of the fading
channel without transmitter feedback (i.e., with the channel
fade level known at the receiver only).
Substituting (3) into (16), the average channel capacity of
dual-branch SC over correlated Nakagami-0.5 fading channels
is
CORA = B
8

0
log2(1 + ? )(1 - ?)0.5 8
k=0
?k
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×??? 0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d?
CORA = 1.443×B
8

0
log(1+? )(1-?)0.5 8
k=0
?k
k!(k+0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×?
??
0.5
¯ ?1 2k+r+1
exp - ?
¯ ?1(1-?)
+
0.5?
¯ ?2 2k+r+1
exp - ?
¯ ?2(1-?)

???
? 2k+r d? (17)
Following can be taken from the first part of above integral
is
 8
0
log(1 + ? ) exp -
?
¯ ?1(1 - ?)
? 2k+r d?
This can be solved using partial integration as follows
 8
0
udv = lim
??8
(uv) - lim
?? 0
(uv) -  8
0
vdu
Let u = log(1 + ? )
then du = d?
1+?
Now let
dv = exp -
?
¯ ?1(1 - ?)
? 2k+r d?
Integrating above expression using [29], we obtain
v = -exp -
?
¯ ?1(1 - ?)

×
2k+r
t=0
2k + r !
2k + r - t!
(?1(1 - ?))t+1 ? 2k+r-t
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M. I. Hasan, S. Kumar
Evaluating integral by using partial integral and some
mathematical transformation using [29,30], we obtain
8

0
log(1 + ? ) exp -
?
¯ ?1(1 - ?)
? 2k+r d?
=
2k+r
z=0
(2k + r )!( ¯ ?1(1 - ?))2k+r+1-z
×e
1
¯ ?1(1-?)  -z,
1
¯ ?1(1 - ?)
(18)
Similarly second part of above integral (17) can be solved
using [29,30], we obtain
8

0
log(1 + ? ) exp -
?
¯ ?2(1 - ?)
? 2k+r d?
=
2k+r
z=0
(2k + r )!( ¯ ?2(1 - ?))2k+r+1-z
×e
1
¯ ?2(1-?)  -z,
1
¯ ?2(1 - ?)
(19)
Substituting (18) and (19) in (17), the spectral efficiency
of dual-branch SC under Nakagami-0.5 fading channels is
CORA
= B
8
k=0
1.443 × ?k (1 - ?)0.5
k!(k + 0.5)(2)2k
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×
?????????
0.5
¯ ?1 2k+r+1
2k+r
z=0 (2k+r )!( ¯ ?1(1-?))2k+r+1-z
×e
1
¯ ?1(1-?)  -z, 1
¯ ? 1(1-?)
+
0.5?
¯ ?2 2k+r+1
2k+r
z=0 (2k+r )!( ¯ ?2(1-?))2k+r+1-z
×e
1
¯ ?2(1-?)  -z, 1
¯ ?2(1-?)

?????????
Using that result, we obtain spectral efficiency i.e. CORA
B
[bit/sec/Hz] as
?ORA
=
8
k=0
1.443 × ?k (1 - ?)0.5
k!(k + 0.5)(2)2k
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [ dB] per Branch
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0 .6
No diversity
???
Fig. 1 Spectral efficiency over correlated Nakagami-0.5 fading channels
as a function of equal average received SNR ( ¯ ?1 = ¯?2 = ¯? )using
OPRA
×
8
r=0
(2k + r + 1)  1
(1-?)2k+r+1
(2k + r + 1.5)(r !)
×
?????????
0.5
¯ ?1 2k+r+1
2k+r
z=0 (2k+r )!( ¯ ?1(1 - ?))2k+r+1-z
×e
1
¯ ?1(1-?)  -z, 1
¯ ? 1(1-?)
+
0.5?
¯ ?2 2k+r+1
2k+r
z=0 (2k+r )!( ¯ ?2(1 - ?))2k+r+1-z
×e
1
¯ ?2(1-?)  -z, 1
¯ ?2(1-?)

?????????
(20)
To efficiently compute the series, we truncate the series
using numerical evaluation techniques. Finally, we compare
[25, Eqs. 6 and 23] with (20).
4 Numerical results and analysis
In this section, various performance evaluation results for
the spectral efficiency using dual-branch SC operating over
correlated Nakagami-0.5 fading channels with unequal average
received SNR between the different combined branches
has been presented and analyzed. We then discuss that the
generic results can be further simplified for Nakagami-0.5
fading, correlated and uncorrelated branches with equal average
received SNR. These results also compare the different
adaptive transmission schemes under worst fading channel
condition.
Figure 1 shows the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under OPRA scheme as a function of the equal average
received SNR per branch ( ¯ ?1 = ¯?2 = ¯? ) for ? = 0.2 and
123
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Spectral efficiency of dual diversity selection...
Table 1 Comparison of ?OPRA
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with equal SNR ( ¯ ?1 = ¯?2 = ¯? )
¯ ? (dB) ?OPRAfor ? = 0 ?OPRA for ? = 0.2 ?OPRA for ? = 0.6 ?OPRA for no diversity
-10 0.340007 0.339989 0.320618 0.2722
-5 0.703605 0.68981 0.65502 0.54773394
0 1.33269 1.29941 1.23285 1.015772433
5 2.28202 2.22688 2.1166 1.72595787
10 3.52795 3.45287 3.30271 2.69064666
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
First Branch Average Received SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 2 Spectral efficiency over correlated Nakagami-0.5 fading channels
as a function of first branch average received SNR( ¯ ?1) usingOPRA
? = 0.6. For comparison, the spectral efficiency of uncorrelated
Nakagami-0.5 fading channels with dual-branch SC
and without diversity, which was obtained in [24, Eq. (27)]
and [22, Eq. (8)] respectively, is also presented in Fig. 1. As
expected, by increasing ¯ ? and/or employing diversity, spectral
efficiency improves. It is also observed in Table 1, that the
spectral efficiencywith dual-branchSCis largestwhen ? = 0
and decreases as ? increases except for ¯ ? = -8.75 dB, gives
almost identical performance even when correlation coeffi-
cient increased to ? = 0.2, and same is shown in Fig. 1.
Figure 2 depicts the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under OPRA scheme with unequal average received SNR
( ¯ ?1 = 2 ¯ ?2) for ? = 0,? = 0.2 and ? = 0.6. For comparison,
the spectral efficiency of uncorrelated Nakagami-0.5
fading channels without diversity, which was obtained in
[22, Eq. (8)], is also presented in Fig. 2. Similarly, the spectral
efficiency with dual-branch SC as a function of the first
branch average received SNR ( ¯ ?1) is largest when ? = 0. It
is very interesting to observe in Table 2, that the spectral effi-
ciency without diversity gives almost identical performance
for ¯ ? = 0 dB even when employing diversity as a function
of first branch average received SNR ( ¯ ?1) with correlation
coefficient decrease to zero (? = 0), and same is shown in
Fig. 2.
In Fig. 3, the spectral efficiency of dual-branch SC with
unequal average received SNR ( ¯ ?1 = 2 ¯ ?2) over uncorrelated
Nakagami-0.5 fading channels using TIFR scheme is
plotted as a function of the cutoff SNR ?0 for several values
of first branch average received SNR ( ¯ ?1). As expected, by
increasing ¯ ?1 spectral efficiency improves.
In Fig. 4, the spectral efficiency of dual-branch SC for
? = 0.2 with equal average received SNR per branch ( ¯ ?1 =
¯ ?2 = ¯? ) using TIFR scheme is plotted as a function of cutoff
SNR ?0 for several values of the average received SNR per
branch ¯ ? . As expected, by increasing ¯ ? spectral efficiency
improves.
In Fig. 5, the spectral efficiency of dual-branch SC for
? = 0.2 with unequal average received SNR ( ¯ ?1 = 2 ¯ ?2)
using TIFR scheme is plotted as a function of cutoff SNR
?0 for several values of the first branch average received
SNR ¯ ?1. As expected, by increasing ¯ ?1 spectral efficiency
improves. It is also observed in Figs. 4 and 5 that the spectral
efficiency is decreases as we go from branches with equal
average received SNR ( ¯ ?1 = ¯?2 = ¯? ) to unequal average
received SNR ( ¯ ?1 = 2 ¯ ?2).
Table 2 Comparison of ?OPRA for different values of ? under worst case fading scenario using dual-branch SC and no diversity with unequal
SNR ( ¯ ?1 = 2 ¯ ?2)
¯ ?1 (dB) ?OPRA for ? = 0 ?OPRA for ? = 0.2 ?OPRA for ? = 0.6 ?OPRA for no diversity
-10 0.2722 0.2722 0.2722 0.2722
-5 0.54773399 0.54773397 0.54773395 0.54773394
0 1.015789433 1.015783932 1.015781983 1.015772433
5 1.870663 1.825 1.81645 1.72595787
10 3.06567 2.94 2.89 2.69064666
123
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M. I. Hasan, S. Kumar
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
First branch average received SNR = -10 dB
First branch average received SNR = - 5 dB
First branch average received SNR = 0 dB
First branch average received SNR = 5 dB
First branch average received SNR = 10 dB
Fig. 3 Spectral efficiency of an unequal SNR ( ¯ ?1 = 2 ¯ ?2) dual-branch
SC versus the cutoff SNR over uncorrelated Nakagami-0.5 fading channels
using TIFR
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Average received SNR = - 10 dB
Average received SNR = - 5 dB
Average received SNR = 0 dB
Average received SNR = 5 dB
Average received SNR = 10 dB
Fig. 4 Spectral efficiency of an equal SNR ( ¯ ?1 = ¯?2) dual-branch SC
versus the cutoff SNR over Nakagami-0.5 fading channels using TIFR
for ? = 0.2
In Fig. 6, the spectral efficiency of dual-branch SC for
? = 0.6 with equal average received SNR per branch ( ¯ ?1 =
¯ ?2 = ¯? ) using TIFR scheme is plotted as a function of cutoff
SNR ?0 for several values of the average received SNR per
branch ¯ ? . As expected, by increasing ¯ ? spectral efficiency
improves.
In Fig. 7, the spectral efficiency of dual-branch SC for
? = 0.6 under unequal average received SNR ( ¯ ?1 = 2 ¯ ?2)
using TIFR scheme is plotted as a function of cutoff SNR
?0 for several values of the first branch average received
SNR ¯ ?1. As expected, by increasing ¯ ?1 spectral efficiency
improves. It is again observed in Figs. 6 and 7 that the spectral
efficiency is decreases as we go from branches with equal
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
First branch average received SNR = -10 dB
First branch average received SNR = - 5 dB
First branch average received SNR = 0 dB
First branch average received SNR = 5 dB
First branch average received SNR = 10 dB
Fig. 5 Spectral efficiency of an unequal SNR ( ¯ ?1 = 2 ¯ ?2) dual-branch
SC versus the cutoff SNR over Nakagami-0.5 fading channels using
TIFR for ? = 0.2
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Average received SNR = - 10 dB
Average received SNR = - 5 dB
Average received SNR = 0 dB
Average received SNR = 5 dB
Average received SNR = 10 dB
Fig. 6 Spectral efficiency of an equal SNR ( ¯ ?1 = ¯?2) dual-branch SC
versus the cutoff SNR over Nakagami-0.5 fading channels using TIFR
for ? = 0.6
average received SNR ( ¯ ?1 = ¯?2 = ¯? ) to unequal average
received SNR ( ¯ ?1 = 2 ¯ ?2).
Figure 8 shows the spectral efficiency of a dual-branch SC
system over correlated Nakagami-0.5 fading channels under
TIFR scheme as a function of the equal average received SNR
per branch ( ¯ ?1 = ¯?2 = ¯? ) for ? = 0.2 and ? = 0.6. For comparison,
the spectral efficiency of uncorrelatedNakagami-0.5
fading channels with dual-branch SC and without diversity,
which was obtained in [24, Eq. (35)] and [22, Eq. (22)]
respectively, is also presented in Fig. 8. As expected, by
increasing ¯ ? and/or employing diversity, spectral efficiency
improves. It is also observed in Table 3, that the spectral
efficiency with dual-branch SC is largest when ? = 0 and
123
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Spectral efficiency of dual diversity selection...
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
Cutoff SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
First branch average received SNR = - 10 dB
First branch average received SNR = - 5 dB
First branch average received SNR = 0 dB
First branch average received SNR = 5 dB
First branch average received SNR = 10 dB
Fig. 7 Spectral efficiency of an unequal SNR ( ¯ ?1 = 2 ¯ ?2) dual-branch
SC versus the cutoff SNR over Nakagami-0.5 fading channels using
TIFR for ? = 0.6
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Average Received SNR [ dB] per Branch
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 8 Spectral efficiency over correlated Nakagami-0.5 fading channels
as a function of equal average received SNR ( ¯ ?1 = ¯?2 = ¯? ) using
TIFR
decreases as correlation coefficient ? increases except for
¯ ? = -7.5 dB, gives almost identical performance even when
correlation coefficient ? increased to ? = 0.2, and same is
shown in Fig. 8.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
First Branch Average Received SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 9 Spectral efficiency over correlated Nakagami-0.5 fading channels
as a function of first branch average received SNR ( ¯ ?1) using TIFR
Figure 9 depicts the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under TIFR scheme with unequal average received SNR
( ¯ ?1 = 2 ¯ ?2) for ? = 0, ? = 0.2 and ? = 0.6. For comparison,
the spectral efficiency of Nakagami-0.5 fading channels
without diversity, which was obtained in [22, Eq. (22)], is
also presented in Fig. 9. Similarly, the spectral efficiency
with dual-branch SC as a function of first branch average
received ( ¯ ?1) is largest when ? = 0. It is again very interesting
to observe in Table 4, that the spectral efficiency without
diversity under TIFR scheme gives almost identical performance
for ¯ ? = 2.5 dB even when employing diversity as
a function of first branch average received SNR ( ¯ ?1) with
correlation coefficient decrease to zero (? = 0), and same is
shown in Fig. 9.
Figure 10 shows the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under ORA scheme as a function of the equal average
received SNR per branch ( ¯ ?1 = ¯?2 = ¯? ) for ? = 0.2 and
? = 0.6. For comparison, the spectral efficiency without
diversity and uncorrelated dual-branch SC with equal average
received SNRper branch ( ¯ ?1 = ¯?2 = ¯? ) for ? = 0 under
Nakagami-0.5 fading channel, which was obtained in [25,
Table 3 Comparison of ?T IFR
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with equal SNR ( ¯ ?1 = ¯?2 = ¯? )
¯ ? (dB) ?T IFR for ? = 0 ?T IFR for ? = 0.2 ?T IFR for ? = 0.6 ?T IFR for no diversity
-10 0.313607599 0.307224921 0.294457017 0.2491
-5 0.641415299 0.628441118 0.59348451 0.4945
0 1.200999905 1.168007708 1.10200389 0.9039
5 2.038088171 1.972422736 1.871163546 1.5144
10 3.136902858 3.058132377 2.901971304 2.3323
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M. I. Hasan, S. Kumar
Table 4 Comparison of ?T IFR
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with unequal SNR ( ¯ ?1 = 2 ¯ ?2)
¯ ?1 (dB) ?T IFR for ? = 0 ?T IFR for ? = 0.2 ?T IFR for ? = 0.6 ?T IFR for no diversity
-10 0.2491 0.2491 0.2491 0.2491
-5 0.494502 0.494501 0.4945008 0.4945
0 0.90395 0.90393 0.90392 0.9039
5 1.60619677 1.56838859 1.552782 1.5144
10 2.77435225 2.64525 2.58713565 2.3323
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Average Received SNR [ dB] per Branch
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 10 Spectral efficiency over correlatedNakagami-0.5 fading channels
as a function of equal average received SNR ( ¯ ?1 = ¯?2 = ¯? ) using
ORA
Eqs. 6 and 23], is also presented in Fig. 10. As expected, by
increasing ¯ ? and/or employing diversity, spectral efficiency
improves. It is also observed in Table 5 that the spectral
efficiency with dual-branch SC is largest when ? = 0 and
decreases as ? increases, and same is shown in Fig. 10.
Figure 11 shows the spectral efficiency of a dual-branch
SC system over correlated Nakagami-0.5 fading channels
under ORA scheme as a function of the unequal average
received SNR ( ¯ ?1 = 2 ¯ ?2) for ? = 0,? = 0.2 and ? = 0.6.
For comparison, the spectral efficiency without diversity
under Nakagami-0.5 fading channel using [25] is also presented
in Fig. 11. Similarly, the spectral efficiency with
dual-branch SC as a function of first branch average received
SNR ( ¯ ?1) is largest when ? = 0. But the spectral effi-
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
First Branch Average Received SNR [ dB]
Spectral Efficiency [ bit/sec/Hz]
Dual-branch SC with = 0
Dual-branch SC with = 0.2
Dual-branch SC with = 0.6
No Diversity
Fig. 11 Spectral efficiency over correlatedNakagami-0.5 fading channels
as a function of first branch average received SNR ( ¯ ?1) using ORA
ciency without diversity gives almost identical performance
for ¯ ? = 1.2 dB even when employing diversity as a function
of first branch average received SNR( ¯ ?1) with correlation
coefficient decrease to zero (? = 0) as observe in Table 6
and shown in Fig. 11.
5 Conclusions
This research paper derives and analyzes the spectral effi-
ciency expressions over correlated Nakagami-0.5 fading
channels under different adaptive transmission schemes with
unequal average received SNRof dual-branch SC.By numerical
evaluations it has been found that the spectral efficiency
improves by increasing ¯ ? , decreasing correlation coefficient
Table 5 Comparison of ?ORA
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with equal SNR ( ¯ ?1 = ¯?2 = ¯? )
¯ ? (dB) ?ORA for ? = 0 ?ORA for ? = 0.2 ?ORA for ? = 0.6 ?ORA for no diversity
-10 0.205587 0.197396 0.177011 0.127571
-5 0.532337 0.512258 0.461776 0.337317
0 1.1733 1.13373 1.036258 0.769773
5 2.17481 2.11313 1.936045 1.49328
10 3.47351 3.39399 3.177 2.50593
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Spectral efficiency of dual diversity selection...
Table 6 Comparison of ?ORA
for different values of ? under
worst case fading scenario using
dual-branch SC and no diversity
with unequal SNR ( ¯ ?1 = 2 ¯ ?2)
¯ ?1 (dB) ?ORA for ? = 0 ?ORA for ? = 0.2 ?ORA for ? = 0.6 ?ORA for no diversity
-10 0.127571 0.127571 0.127571 0.127571
-5 0.337318 0.33731707 0.33731703 0.337317
0 0.769798 0.769783 0.769777 0.769773
5 1.558 1.5222 1.50 1.49328
10 2.678 2.567 2.532 2.50593
?, and/ or employing diversity with unequal to equal average
received SNRin all the cases of considered adaptive transmission
schemes, OPRA, tifr and ORA. However the magnitude
of improvement is slightly higher in case of OPRA. It has
been observed that for OPRA scheme, spectral efficiency
without diversity is almost same for ¯ ?1 = 0 dB, even though
employing uncorrelated dual-branch diversity with unequal
average received SNR ( ¯ ?1 = 2 ¯ ?2). It has also been observed
that for TIFR scheme, spectral efficiency without diversity
is almost same for ¯ ?1 = 2.5 dB, even though employing
uncorrelated dual-branch diversity with unequal average
received SNR ( ¯ ?1 = 2 ¯ ?2). It has also been observed that
for ORA scheme, spectral efficiency without diversity is
almost same for ¯ ?1 = 1.2 dB, even though correlation coef-
ficient decreases and employing dual-branch diversity with
unequal average received SNR ( ¯ ?1 = 2 ¯ ?2). Therefore it is
very important to recommend that under worst fading condition,
the proper antenna spacing at the receiver end, required
for uncorrelated diversity path for obtaining the optimum
spectral efficiency in each of the scheme, OPRA, TIFR, or
ORA scheme is not an important issue for low value of average
received SNR, particularly, ¯ ?1 = 0 dB in case of OPRA,
¯ ?1 = 2.5 dB in case of TIFR, or ¯ ?1 = 1.2 dB in case of
ORA This paper also states that under Nakagami-0.5 fading
channels with equal average received SNR, TIFR scheme
is a better choice over ORA for low average received SNR
and ORA scheme is for high average received SNR even
employing diversity. It is very interesting to finally conclude
that TIFR scheme with unequal average received SNR under
worst condition of fading is always a better option overORA.
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MohammadIrfanul Hasan was
born in Samastipur, India on
March 01, 1977. He received
his B.Tech. degree in Electronics
& Communications Engineering
from Magadh University, Bihar,
and M.Tech. degree from Institute
of Advanced Studies in
Education University, Rajasthan.
Since February 2012, he has
been working towards his Ph.D.
degree in Electronics and Communication
Engineering at Birla
Institute of Technology, Mesra,
Ranchi, India. He is currently
working as an Assistant Professor in the Department of Electronics and
Communications Engineering atGraphic Era University,Dehradun. His
research interests are in the field of Wireless Communication Technology
primarily in the area of diversity techniques.
Sanjay Kumar was born in
Ranchi, India on January 18,
1967. In 1994 he received
MBA degree from Pune University,
and M.Tech. in Electronics
and Communication Engineering
from Guru Nanak Dev Engineering
College, Ludhiana, India
in the year 2000. Subsequently
he obtained his Ph.D. degree in
Wireless Communication from
Aalborg University, Denmark in
the year 2009. He served the
Indian Air Force from the 1985 to
2000, where he was involved in
the technical supervision and maintenance activities of telecommunications
and radar systems. Hewas a guest researcher at AalborgUniversity
during 2006 to 2009, where he worked in close cooperation with Nokia
Siemens Networks and Centre for TeleInFrastruktur. He also worked
as a guest lecturer in the department of Electronics Systems at Aalborg
University during the years 2007 to 2009. Presently he is working as an
Associate Professor in the Department of Electronics and Communications
Engineering at Birla Institute of Technology, Mesra, Ranchi. His
current interest lies in the field ofWireless Communication Technology.
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