AVERAGE CHANNEL CAPACITY OF CORRELATED DUAL-BRANCH MAXIMAL RATIO COMBINING UNDER WORST CASE OF FADING SCENARIO

Abstract

The channel capacity and probability of outage results for optimum power and rate adaptation (OPRA) and truncated channel inversion with fixed rate (TIFR) schemes over correlated diversity branch obtained so far in literature are applicable only for m C 1. This paper derived closed-form expressions for the average channel capacity and probability of outage of dual-branch maximal ratio combining (MRC) over correlated Nakagami- 0.5 (m\1) fading channels. This channel capacity and probability of outage are evaluated under OPRA and TIFR schemes. Since, the capacity and probability of outage expressions of dual-branch MRC under OPRA and TIFR schemes contain an infinite series; bounds on the errors resulting from truncating the infinite series have been derived for both average channel capacity and probability of outage. The corresponding expressions for Nakagami-0.5 fading are called expressions under worst fading condition with severe fading. Finally, numerical results are presented, which are then compared to the capacity and probability of outage results that previously published for OPRA and TIFR schemes. It has been observed that OPRA provides improved average channel capacity and probability of outage, as compared to TIFR under worst case of fading. It is also observed that probability of outage under TIFR scheme is not improved adequately than the probability of outage under OPRA even as employing diversity.

Specification

Description:FORM 2
THE PATENTS ACT, 1970
(39 OF 1970)
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THE PATENTS RULES, 2003 5
COMPLETE SPECIFICATION
(See section 10; rule 13)
10
Title: Average Channel Capacity of Correlated Dual-Branch Maximal Ratio Combining
Under Worst Case of Fading Scenario
15
APPLICANT DETAILS:
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Uttarakhand, India
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in which it is to be performed.
Average Channel Capacity of Correlated Dual-Branch
Maximal Ratio Combining Under Worst Case of Fading
Scenario
Mohammad Irfanul Hasan1,2
• Sanjay Kumar1
 Springer Science+Business Media New York 2015
Abstract The channel capacity and probability of outage results for optimum power and
rate adaptation (OPRA) and truncated channel inversion with fixed rate (TIFR) schemes
over correlated diversity branch obtained so far in literature are applicable only for m C 1.
This paper derived closed-form expressions for the average channel capacity and probability
of outage of dual-branch maximal ratio combining (MRC) over correlated Nakagami-
0.5 (m\1) fading channels. This channel capacity and probability of outage are
evaluated under OPRA and TIFR schemes. Since, the capacity and probability of outage
expressions of dual-branch MRC under OPRA and TIFR schemes contain an infinite
series; bounds on the errors resulting from truncating the infinite series have been derived
for both average channel capacity and probability of outage. The corresponding expressions
for Nakagami-0.5 fading are called expressions under worst fading condition with
severe fading. Finally, numerical results are presented, which are then compared to the
capacity and probability of outage results that previously published for OPRA and TIFR
schemes. It has been observed that OPRA provides improved average channel capacity and
probability of outage, as compared to TIFR under worst case of fading. It is also observed
that probability of outage under TIFR scheme is not improved adequately than the probability
of outage under OPRA even as employing diversity.
Keywords Average channel capacity  Dual-branch  Maximal ratio combining  Nakagami-0.5 fading channels  Optimum power with rate adaptation  Truncated channel
inversion with fixed rate
& 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, Jharkhand 835215, India
2 Department of Electronics and Communication Engineering, Graphic Era University,
Dehradun 248002, Uttarakhand, India
123
Wireless Pers Commun
DOI 10.1007/s11277-015-2559-z
Author's personal copy
1 Introduction
The channel capacity implies maximum achievable data rate of a communication system.
Channel capacity is of fundamental importance in the design of wireless mobile communication
systems as the demand for wireless mobile communication services is growing
rapidly [1]. The wireless mobile channels are subjected to fading, which is undesirable.
The channel capacity in fading environment can be improved by employing diversity
combining and/or adaptive transmission schemes [1, 2]. Diversity combining, which is
known to be a powerful technique that can be used to mitigate fading in wireless mobile
environment. MRC, equal gain combining (EGC), selection combining (SC) are the most
fundamental diversity combining techniques. Adaptive transmission is another effective
scheme that can be used to mitigate fading. Adaptive transmission, which requires an
accurate channel estimation at the receiver and a reliable feedback path between the
estimator and the transmitter, provides great improvement to the average channel capacity
[1, 2]. The capacity of flat fading channels was derived in [3] for four different adaptive
transmission schemes such as optimum rate adaptation with constant transmit power
(ORA), OPRA, channel inversion with fixed rate transmission (CIFR) and TIFR. In case of
ORA scheme, the transmitter adapts the data rate according to the channel fading conditions
while the transmit power is constant [3]. In case of OPRA scheme, transmitter can
realize optimal capacity by transmitting appropriate power and data rate in accordance
with the channel variations [3]. In CIFR scheme, transmitter adapts its power to maintain
constant signal to noise ratio (SNR) at the receiver by inverting the channel gain, which
makes the channel to appear as a time invariant additive white Gaussian noise (AWGN)
channel [3]. Channel inversion with fixed rate suffers a large capacity penalty relative to
the other techniques, since a large amount of power is required to compensate for the deep
channel fades. A better approach is to use a modified inversion policy that inverts the
channel fading only above a cutoff fading level, which is called TIFR scheme [3].
Numerous researchers have worked on the study of channel capacity over different
fading channels. Specifically, [1, 2] discuss the channel capacity over correlated Nakagami-
m (m C 1 & m\1) fading channels under ORA and CIFR schemes with different
diversity combining techniques. In [4], the channel capacity over uncorrelated Nakagami-
m (m C 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 [5]. An analytical performance study of the channel
capacity for correlated generalized gamma fading channels with dual-branch SC under
the different power and rate adaptation schemes was introduced in [6]. The channel
capacity of Nakagami-m (m C 1) fading channel without diversity was derived in [7] for
different adaptive transmission schemes. In [8], channel capacity of dual-branch SC and
MRC systems over correlated Hoyt fading channels using ORA, OPRA, CIFR and TIFR
was presented. In [9], expression for the ergodic capacity of MRC over arbitrarily
correlated Rician fading channels was derived. In [10], 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 [11], an analytical performance
study of the channel capacity for uncorrelated Nakagami-0.5 with dual-branch
MRC using OPRA and TIFR was obtained. However, an analytical study of average
channel capacity over correlated Nakagami-0.5 fading channels under OPRA and TIFR
adaptation schemes with MRC has not been considered so far. The Nakagami-m model
M. I. Hasan, S. Kumar
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has been widely used in general to study wireless mobile communication system performance,
less attention 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
great practical usefulness, they will be of theoretical interest as a worst fading case. All
previously published literature related to the channel capacity over correlated Nakagamim
fading channels using OPRA and TIFR schemes are not applicable for m = 0.5. This
paper fills this gap by presenting an analytical performance study of the channel capacity
and probability of outage of dual-branch MRC over correlated Nakagami-0.5 fading
channels using OPRA, and TIFR schemes.
In this paper, we consider MRC, which is optimal combining scheme that provides
maximum performance improvement relative to all other combining techniques [12]. The
dual-branch diversity has been considered since it offers the least complexity and physical
space requirements [12]. Hence this paper gives the analysis of channel capacity for
optimum combining technique (MRC) using optimum adaptation scheme, OPRA and a
modified inversion scheme, TIFR under most practical challenging fading scenario means
Nakagami-0.5 fading channels said to be worst fading conditions.
The remainder of this paper is organized as follows: In Sect. 2, the channel model is
defined. In Sect. 3, average channel capacity and probability of outage of dual-branch
MRC over correlated Nakagami-0.5 fading channels are derived for OPRA and TIFR
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 received SNR, c, at the output of an L-branch
MRC combiner, pcðcÞ, for the correlated Nakagami-m fading channels is obtained in [13]
pcðcÞ ¼
mc
c  Lm1
exp mc
c 1 ffiffi q p ð Þ
 1F 1 m; Lm; Lm ffiffi q p c
c 1 ffiffi q p ð Þ1 ffiffi q p þL ffiffi q p ð Þ
 
c
m   1  ffiffiffi q p  Lmm
1  ffiffiffi q p þ L ffiffiffi q p  mC Lm ð Þ
; c0 ð1Þ
where c is the average received SNR, m (m C 0.5) is the fading parameter, Cð:Þ is the
gamma function, 1F 1½ : ; : ; :  is the confluent hypergeometric function of the first kind, and
q is the envelope or power correlation coefficient between any pair of signals.
For different values of m, this expression simplifies to several important distributions
used in fading channel modeling. Like m = 0.5 corresponds to the highest amount of
fading, m = 1 corresponds to Rayleigh distribution, m C 1 corresponds to Rician distribution,
and as m!1, the distribution converges to a nonfading AWGN [13].
Considering dual-branch MRC (i.e. L = 2) and replacing the 1F 1½ : ; : ; :  function with
its series representation as given in [14], and then the pdf under worst case of fading using
(1) becomes
pcðcÞ ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n !
exp 
0:5c
c 1  ffiffiffi q p  
!; c0 ð2Þ
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3 Average Channel Capacity
In this section, we present closed-form expressions for the average channel capacity of correlated
Nakagami-0.5 fading channels with dual-branch MRC under OPRA, and TIFR
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 [5].
3.1 Opra
The average channel capacity of fading channel with received SNR distribution pcðcÞ
under OPRA scheme (COPRA[bit/sec]) is defined in [3, 4] as
COPRA ¼ B Z
1
c0
log2
c
c0

pcðcÞ dc ð3Þ
where B [Hz] is the channel bandwidth and c0 is the optimum cutoff SNR level below
which data transmission is suspended. To obtain the optimal cutoff SNR, c0 must satisfy
the equation given by [3, 4] as
Z
1
c0
1
c0 
1
c

pc ðcÞ dc ð4Þ
To achieve the channel capacity (3), the channel fade level must be tracked at both the
receiver and transmitter, and the transmitter has to adapt its posswer and rate accordingly,
allocating high power levels and rate for good channel conditions (c large), and lower
power levels and rates for unfavorable channel conditions (c small). So, OPRA gives the
analysis of maximum channel capacity under Nakagami-m fading conditions.
When c\c0, no data is transmitted, the optimal scheme suffers a probability of outage
Pout, equal to the probability of no transmission, given by [3, 4] is
Pout ¼ Z
c0
0
pcðcÞ dc ¼ 1  Z
1
c0
pcðcÞ dc ð5Þ
Substituting (2) in (4) for optimal cutoff SNR c0, then
Z
1
c0
1
c0 
1
c

0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n !
exp 
0:5c
c 1  ffiffiffi q p  
!dc ¼ 1
Evaluating the above integral using some mathematical transformation by [14], we obtain
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
n !
2c 1  ffiffiffi q p    nþ1
c0
C n þ 1;
0:5c0
c 1  ffiffiffi q p  
! "
 2c 1  ffiffiffi q p    nC n ;
0:5c0
c 1  ffiffiffi q p  
" ##
¼ 1
ð6Þ
The numerical evaluation techniques have confirmed that, by solving (6), there is a unique
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positive value of c0 for each c satisfying (6) that takes values from c0 2 ½0; 1. Result
shows that c0 increases as c increases. The value of cutoff SNR c0 that satisfies (6) for each
c is used for finding the average channel capacity per unit bandwidth and probability of
outage.
Substituting (2) in (3), the average channel capacity of dual-branch MRC under Nakagami-
0.5 fading channel is
COPRA¼BZ
1
c0
log2
c
c0

0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1qÞ p cX
1
n¼0
Cðnþ0:5Þ
Cðnþ1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1 ffiffiffi q p  
!dc ð7Þ
COPRA¼B
1:4430:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1qÞ p cX
1
n¼0
Cðnþ0:5Þ
Cðnþ1Þ
ffiffi q p
cð1qÞ  n
n!
 Z
1
c0
logðcÞcn exp 
0:5c
c 1 ffiffiffi q p  
!dcZ
1
c0
logðc0Þcn exp 
0:5c
c 1 ffiffiffi q p  
!dc
264
375
ð8Þ
Following can be taken from the first part of above integral is
Z
1
c 0
log ðcÞ cn exp 
0:5c
c 1  ffiffiffi q p  
!dc
This can be solved using partial integration as follows
Z
1
c0
udv ¼ lim
c!1
u v ð Þlim
c!c0
u v ð Þ Z
1
c0
v du
Let u ¼ log c
then du ¼ dc
c
Now let dv ¼ cn exp  0:5c
c 1 ffiffi q p ð Þ

dc
Integrating this expression using [14], we obtain
v ¼  2c 1  ffiffiffi q p    nþ1C n þ 1;
0:5c
c 1  ffiffiffi q p  
" #
Evaluating first part of above integral (8) using this partial integration and some mathematical
transformation [14, 15], we obtain
Z
1
c0
log ðcÞ cn exp 
0:5c
c 1  ffiffiffi q p  
!dc ¼ log ðc0Þ  2c 1  ffiffiffi q p    nþ1C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" #
þX
n
k¼0
n !
n  k !  2c 1  ffiffiffi q p    nþ1C n  k;
0:5c0
c 1  ffiffiffi q p  
" #
ð9Þ
Second part of above integral (8) of can be solved using [14], we obtain
Average Channel Capacity of Correlated Dual-Branch Maximal…
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Z
1
c0
log ðc0Þ cn exp 
0:5c
c 1  ffiffiffi q p  
!dc
¼ log ðc0Þ  2c 1  ffiffiffi q p    nþ1C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" #
ð10Þ
Substituting (9) and (10) in (8), the average channel capacity of dual-branch MRC under
Nakagami-0.5 fading channels is
COPRA ¼
1:443  0:5  B
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X 1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
Using that result we obtain average channel capacity per unit bandwidth i.e. COPRA
B [bit/sec/
Hz] as
gOPRA ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X 1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k! ð11Þ
The computation of the average channel capacity per unit bandwidth according to (11)
requires the computation of an infinite series. To efficiently compute the series, we truncate
the series, and present bounds for the average channel capacity per unit bandwidth.
The average channel capacity per unit bandwidth in (11) can be written
asgOPRA ¼ gOPRA; N þ gOPRA; E, where gOPRA; N is the expression in (11) with the infinite
series truncated at the Nth term as
gOPRA; N ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
and gOPRA; E is the truncation error resulting from truncating the infinite series in (11) at
n = N.The lower bound for gOPRA is derived as
gOPRA[gOPRA ; N þ gOPRA ; Elow
where gOPRA; Elow, which is the lower bound of gOPRA; E
The lower bound for the capacity can be derived by using the relationship between the
area of the pdf and the expression of the average channel capacity per unit bandwidth as
discuss in [10].
As we know that area of pdf pcðcÞ is equal to unity.
P ¼ Z
1
0
pcðcÞ dc ¼ 1 ð12Þ
Substituting (2) into (12), we get
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Z
1
0
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n !
exp 
0:5c
c 1  ffiffiffi q p  
!dc ¼ 1
After performing integration using [14], we obtain
P ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X 1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1¼ 1
Let
PN1 ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1 ð13Þ
And let
DPN1 ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
ffiffiffi q p
1  q

NCðN þ 0:5Þ
CðN þ 1Þ
2 1 ffiffiffi p p    Nþ1 ð14Þ
then
PN ¼ PN1 þ DPN1
Similarly, from (11) let
gOPRA; N1 ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
ð15Þ
And
DgOPRA; N1 ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
ffiffiffi q p
1  q

NCðN þ 0:5Þ
CðN þ 1Þ
2 1 ffiffiffi p p    Nþ1X
N
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
ð16Þ
Dividing (16) by (14), yields
DgOPRA; N1
DPN1 ¼ 1:443X
N
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k! ð17Þ
Observing that
DgOPRA; N1
DPN1   monotonically increases with increasing N, that
is,
DgOPRA; i
DPi [ DgOPRA; N1
DPN1
for iN
X 1
i¼N
DgOPRA; i[
DgOPRA; N1
DPN1 X
1
i¼N
DPi ¼
DgOPRA; N1
DPN1
1  PN ð Þ ð18Þ
Hence, the average channel capacity per unit bandwidth in (11) can be lower bounded by
using (17) and (18) as
Average Channel Capacity of Correlated Dual-Branch Maximal…
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gOPRA[gOPRA; N þ 1:443X
N
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
 2 1 ffiffiffi p p    nþ1 !
ð19Þ
The upper bound for gOPRA is derived as
gOPRA\gOPRA; N þ gOPRA; Eup
where gOPRA; Eup, which is the upper bound of gOPRA; E
The expression in (7) can be written as
gOPRA ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
Z
1
c0
log
c
c0


X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!
þ
X 1
n¼Nþ1
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!
26666664
37777775
dc
Therefore, gOPRA; N can be obtained as
gOPRA; N ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
Z
1
c0
log
c
c0

X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Further, gOPRA; E can be presented as
gOPRA; E ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c X
1
n¼Nþ1
Cðn þ 0:5Þ
Cðn þ 1Þ
Z
1
c 0
log
c
c0
 
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Let an ¼ C½nþ0:5 C½nþ1
, then anþ1
an ¼ C½nþ1:5 C½nþ2  C½nþ1 C½nþ0:5
\1
i.e. an monotonically decreases with increase of n, therefore, gOPRA; E can be upper
bounded as
gOPRA; E1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c 
CðN þ 1:5Þ
CðN þ 2Þ
Z
1
c0
log
c
c0
 
 exp 
0:5c
c 1  ffiffiffi q p  
!X
1
n¼0
c ffiffi q p
cð1qÞ  n
n! X
N
n¼0
c ffiffi q p
cð1qÞ  n
n!
8<:
9=;
dc
M. I. Hasan, S. Kumar
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gOPRA; E\gOPRA; Eup ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
CðN þ 1:5Þ
CðN þ 2Þ
2 ffiffiffi q p þ 1  exp 0:5c0
ffiffiffi q p þ 1  c
!exp
c0
2 ffiffiffi q p þ c  
!E1
0:5c0
ffiffiffi q p þ 1  c
" #
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
266666664
377777775
where E1ð:Þ is the exponential integral of first order.
Therefore, the average channel capacity per unit bandwidth in (11) can be upper
bounded as
gOPRA ¼ gOPRA N þ gOPRA ;E\gOPRA ;N
þ
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
CðN þ 1:5Þ
CðN þ 2Þ
2 ffiffiffi q p þ 1  exp 0:5c 0
ffiffiffi q p þ 1  c
!exp
c 0
2 ffiffiffi q p þ c  
!E1
0:5c 0
ffiffiffi q p þ 1  c
" #
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
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ð20Þ
Hence, the average channel capacity per unit bandwidth is bounded using (19) and (20) as
gOPRA; N þ
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
CðN þ 1:5Þ
CðN þ 2Þ
2 ffiffiffi q p þ 1  exp 0:5c0
ffiffiffi q p þ 1  c
!exp
c0
2 ffiffiffi q p þ c  
!E1
0:5c0
ffiffiffi q p þ 1  c
" #
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
266666664
377777775
[gOPRA[gOPRA; Nþ
1:443X
N
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1 !
ð21Þ
When q = 0, (7) reduces to the corresponding expressions for uncorrelated dual-branch
MRC under worst case of fading channels. Note that for q = 0, the integration of (7) is
equivalent to [11, Eq. (11)]. Finally we compare (21) with [11, Eq. (11)] and channel
capacity without diversity in [11, Eq. (8)].
Substituting (2) in (5) for probability of outage, then
POPRA ¼ Z
c0
0
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc ð22Þ
After evaluating the above integral by using mathematical transformation using [14, 15],
we obtain
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POPRA ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X 1
n¼0
C½n þ 0:5
C½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n!
 C ½n þ 1  C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" # ( ) ð23Þ
The computation of probability of outage according to (23) requires the computation of an
infinite series. To efficiently compute the series, we truncate the series and derive bounds
for the probability of outage.
The probability of outage in (23) can be written as POPRA ¼ POPRA; N þ POPRA; E, where
POPRA; N is the expression in (23) with the infinite series truncated at the Nth term as
POPRA; N ¼
0:5
ffiffiffi p p ð1  qÞX
N
n¼0
C ½n þ 0:5
C ½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n!
C ½n þ 1  C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" # ( )
and POPRA; E is the truncation error resulting from truncating the infinite series in (23).The
lower bound for POPRA is derived as
POPRA[POPRA; N þ POPRA; Elow
where POPRA; Elow, which is the lower bound of POPRA; E
Similarly, from (23) let
POPRA; N1 ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X
N1
n¼0
C ½n þ 0:5
C ½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n !
 C ½n þ 1  C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" # ( )
ð24Þ
And
DPOPRA; N1 ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
ffiffiffi q p
1  q

NCðN þ 0:5Þ
CðN þ 1Þ
2 1 ffiffiffi p p    Nþ1

C ½N þ 1  C N þ 1; 0:5c0
c 1 ffiffi q p ð Þ
 

N!
ð25Þ
Dividing (25) by (14), yields
DPOPRA; N1
DPN1 ¼
C ½N þ 1  C N þ 1; 0:5c0
c 1 ffiffi q p ð Þ
 
N! ð26Þ
Observing that DPOPRA; N1
DPN1   monotonically increases with increasing N,
i.e. DPOPRA; i
DPi [ DPOPRA; N1
DPN1
for i C N
M. I. Hasan, S. Kumar
123
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X 1
i¼N
DPOPRA; i[
DPOPRA;N1
DPN1 X
1
i¼N
DPi ¼
DPOPRA;N1
DPN1
1  PN ð Þ ð27Þ
Hence, the probability of outage in (23) can be lower bounded by using (26) and (27) as
POPRA[POPRA;N þ
C½N þ 1  C N þ 1; 0:5c0
c 1 ffiffi q p ð Þ
 
N!
 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1 !
ð28Þ
The upper bound for POPRA is derived as
POPRA\POPRA ; N þ POPRA ; Eup
where POPRA; Eup, which is the upper bound of POPRA; E
The expression in (22) can be written as
POPRA ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
Z
c0
0
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
n!
cn exp 
0:5c
c 1  ffiffiffi q p  
!
þ X
1
n¼Nþ1
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!
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37777775
dc
Therefore, POPRA;N can be obtained as
POPRA;N ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
Z
c0
0
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Further, POPRA; E can be presented as
POPRA; E ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c X
1
n¼Nþ1
Cðn þ 0:5Þ
Cðn þ 1Þ
Z
c0
0
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Let an ¼ C½nþ0:5 C½nþ1
, then anþ1
an ¼ C½nþ1:5 C½nþ2  C½nþ1 C½nþ0:5
\1
i.e. an monotonically decreases with increase of n, therefore, POPRA; E can be upper
bounded as
POPRA; E0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c

CðN þ 1:5Þ
CðN þ 2Þ
Z
c0
0
exp 
0:5c
c 1  ffiffiffi q p  
!X
1
n¼0
c ffiffi q p
cð1qÞ  n
n! X
N
n¼0
c ffiffi q p
cð1qÞ  n
n!
8<:
9=;
dc
After evaluating the above integral by using mathematical transformation using [14], we
obtain
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123
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POPRA; E0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p 
CðN þ 1:5Þ
CðN þ 2Þ

exp c0 ffiffi q p
1q ð Þ c ð Þ  0:5c0
1 ffiffi q p ð Þc
 1
ffiffi q p
1q ð Þ 0:5
1 ffiffi q p ð Þ

X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
26666666664
37777777775
Therefore, the probability of outage in (23) can be upper bounded as
POPRA ¼ POPRA; N þ POPRA; E\POPRA; N þ
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
CðN þ 1:5Þ
CðN þ 2Þ

exp c0 ffiffi q p
1q ð Þ c ð Þ  0:5c0
1 ffiffi q p ð Þc
 1
ffiffi q p
1q ð Þ 0:5
1 ffiffi q p ð Þ
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
26666666664
37777777775
ð29Þ
Hence, the probability of outage is bounded using (28) and (29) as
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [dB] per Branch Average Channel Capacity per Unit Bandwidth [
bit/sec/Hz]
No diversity
Dual-branch MRC with = 0
Dual-branch MRC with = 0.2
Dual-branch MRC with = 0.6
Fig. 1 Average channel capacity per unit bandwidth over correlated Nakagami-0.5 fading channels as a
function of average received SNR using OPRA
M. I. Hasan, S. Kumar
123
Author's personal copy
POPRA; N þ
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
CðN þ 1:5Þ
CðN þ 2Þ
exp c0 ffiffi q p
1q ð Þ c ð Þ  0:5c0
1 ffiffi q p ð Þc
 1
ffiffi q p
1q ð Þ 0:5
1 ffiffi q p ð Þ
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
26666666664
37777777775
[POPRA[
POPRA ; N þ
C½N þ 1  C N þ 1; 0:5c0
c 1 ffiffi q p ð Þ
 
N!
1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1 !
ð30Þ
when q = 0, (22) reduces to the corresponding expressions for uncorrelated dual-branch
MRC under worst case of fading channels. Note that for q = 0, the integration of (22) is
equivalent to [11, Eq. (12)]. Finally we compare (30) with [11, Eq. (12)] and probability of
outage without diversity [11, Eq. (9)].
3.2 Tifr
The average channel capacity of fading channel with received SNR distribution pcðcÞ
under TIFR scheme (CTIFR[bit/sec]) is defined in [3, 4] as
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Average Received SNR [dB] per Branch
Average Channel Capacity per Unit Bandwidth [bit/sec/Hz]
No diversity under Rayleigh fading
Dual-branch MRC under Rayleigh fading with = 0
No diversity under Nakagami-0.5 fading
Dual-branch MRC under Nakagami-0.5 fading with = 0
Dual-branch MRC under Nakagami-0.5 fading with = 0.2
Dual-branch MRC under Nakagami-0.5 fading with = 0.6
Fig. 2 Average channel capacity per unit bandwidth versus average received SNR using OPRA
Average Channel Capacity of Correlated Dual-Branch Maximal…
123
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CTIFR ¼ B log2 1 þ
1
R
1
c0
pcðcÞ
c  dc
0BBB@
1CCCA
ð1  PoutÞ; c0 ð31Þ
where Pout is given in (5). The cutoff level c0, can be chosen to achieve a specified probability
of outage, Pout, or, alternatively, to maximize the average channel capacity (31).
Using the pdf of dual-branch MRC under correlated Nakagami-0.5 fading channels in
(2), we obtain
pcðcÞ
c ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
cn1
n !
exp 
0:5c
c 1  ffiffiffi q p  
!
ð32Þ
Evaluating the above integral using mathematical transformation by [14], we obtain
Z
1
c0
pcðcÞ
c
dc ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X 1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n 2 1 ffiffiffi p p    n
n!
C n;
0:5c0
c 1  ffiffiffi q p  
!
ð33Þ
Now, we evaluate the probability of outage using (2) is
Pout ¼ Z
c0
0
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
cn
n !
exp 
0:5c
c 1  ffiffiffi q p  
!dc
1  Pout ¼ 1  Z
c0
0
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
cn
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Evaluating the above integral by some mathematical transformation using [14, 15], we
obtain
1  Pout ¼ P cc0 ½ ¼ 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X 1
n¼0
C½n þ 0:5
C½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n!
C½n þ 1  C½n þ 1;
0:5c0
c 1  ffiffiffi q p  
( ) ð34Þ
Putting the value of (33) and (34) in (31), we get
CTIFR ¼ 1:443B log 1 þ
1
0:5
ffiffiffiffiffiffiffiffiffiffiffi pð1qÞ p P
1
n¼0
Cðnþ0:5Þ Cðnþ1Þ ffiffi q p
1q n on 2 1 ffiffip p ð Þ f gn
n ! C n; 0:5c0
c 1 ffiffi q p ð Þ


2664
3775
 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X 1
n¼0
C½n þ 0:5
C½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n !
(
 C½n þ 1  C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" # ( ))
M. I. Hasan, S. Kumar
123
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Using that result we obtain average channel capacity per unit bandwidth i.e. CTIFR
B [bit/sec/
Hz] as
gTIFR ¼ 1:443 log 1 þ
1
0:5
ffiffiffiffiffiffiffiffiffiffiffi pð1qÞ p P
1
n¼0
Cðnþ0:5Þ Cðnþ1Þ ffiffi q p
1q n on 2 1 ffiffip p ð Þ f gn
n ! C n; 0:5c0
c 1 ffiffi q p ð Þ


2664
3775
 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X 1
n¼0
C ½n þ 0:5
C ½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n !
(
 C½n þ 1  C½n þ 1;
0:5c0
c 1  ffiffiffi q p  
( ))
ð35Þ
The computation of average channel capacity per unit bandwidth according to (35) requires
the evaluation of two infinite series. It is difficult but not impossible to compute the
channel capacity under TIFR. To efficiently compute the series, we truncate the series
using numerical evaluation techniques.The expression of probability of outage in case of
TIFR is same as (30), except the cutoff level. In this case the cutoff level c0, can be chosen
to maximize the average channel capacity per unit bandwidth in (35).
4 Numerical Results and Analysis
In this section, various performance evaluation results for the average channel capacity per
unit bandwidth and probability of outage using dual-branch MRC operating over correlated
Nakagami-0.5 fading channels has been presented and analyzed. These results compare the
-10 -5 0 5 10
10 -2
10 -1
10 0
Average Received SNR [dB] per Branch Probability of Outage
No diversity
Dual-branch MRC with = 0
Dual-branch MRC with = 0.2
Dual-branch MRC with = 0.6
Fig. 3 Probability of outage of correlated Nakagami-0.5 fading channels versus average received SNR
using OPRA
Average Channel Capacity of Correlated Dual-Branch Maximal…
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-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Cutoff SNR [dB]
Average Channel Capacity per Unit Bandwidth [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 Average channel capacity per unit bandwidth of a dual-branch MRC system versus the cutoff SNR
over correlated Nakagami-0.5 fading channels using TIFR for q = 0.2
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Cutoff SNR [dB]
Average Channel Capacity per Unit Bandwidth [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. 5 Average channel capacity per unit bandwidth of a dual-branch MRC system versus the cutoff SNR
over correlated Nakagami-0.5 fading channels using TIFR for q = 0.6
M. I. Hasan, S. Kumar
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-10 -5 0 5 10
10 -2
10 -1
10 0
Average Received SNR [dB] per Branch
Probability of Outage
No diversity
Dual-branch MRC with = 0
Dual-branch MRC with = 0.2
Dual-branch MRC with = 0.6
Fig. 6 Probability of outage of correlated Nakagami-0.5 fading channels versus average received SNR
using TIFR
-10 -5 0 5 10
10-2
10-1
100
Average Received SNR [dB] per Branch
Probability of Outage
No diversity for TIFR
Dual-branch MRC for TIFR with = 0
Dual-branch MRC for TIFR with = 0.2
Dual-branch MRC for TIFR with = 0.6
No diversity for OPRA
Dual-branch MRC for OPRA with = 0
Dual-branch MRC for OPRA with = 0.2
Dual-branch MRC for OPRA with = 0.6
Fig. 7 Probability of outage of correlated Nakagami-0.5 fading channels versus average received SNR
Average Channel Capacity of Correlated Dual-Branch Maximal…
123
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different adaptive transmission schemes under worst fading channel condition and also
with previously published results for the same adaptisve transmission schemes.
Figure 1 show the average channel capacity per unit bandwidth of a dual-branch MRC
system over correlated Nakagami-0.5 fading channels under OPRA scheme as a function
of the average received SNR per branch c for q = 0.2 and q = 0.6. For comparison, the
average channel capacity per unit bandwidth of uncorrelated Nakagami-0.5 fading channels
with dual-branch MRC and without diversity, which was obtained in [11, Eq. (11)]
and [11, Eq. (8)] respectively, is also presented in Fig. 1. As expected, by increasing c and/
or employing diversity, average channel capacity per unit bandwidth improves. It is also
observed that the average channel capacity per unit bandwidth with dual-branch MRC is
largest when q = 0 but almost identical performance for c B -5 dB even when correlation
coefficient exists.
In Fig. 2, the average channel capacity per unit bandwidth under OPRA scheme improves
by increasing m and/or employing diversity using [5], [11]. But the channel capacity
without diversity is almost same for c B -1.25 dB and the channel capacity with
dual-branch MRC is almost same for c B -5 dB, even when fading parameter increases
from m = 0.5 to m = 1 and/or decreasing correlation coefficient q.
In Fig. 3, the probability of outage under OPRA scheme is plotted as a function of the
average received SNR per branch c for q = 0.2 and q = 0.6. For comparison, the probability
of outage of uncorrelated Nakagami-0.5 fading channels with dual-branch MRC and
without diversity, which was obtained in [11, Eq. (12)] and [11, Eq. (9)] respectively, is
also presented in Fig. 3. As expected, by increasing c and/or employing diversity, probability
of outage improves. It is also observed in Fig. 3 that the probability of outage with
diversity is largest when q = 0.6 and decreases as q decreases.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Average Received SNR [dB] per Branch Average Channel Capacity per Unit Bandwidth [
bit/sec/Hz]
No diversity
Dual-branch MRC with = 0
Dual-branch MRC with = 0.2
Dual-branch MRC with = 0.6
Fig. 8 Average channel capacity per unit bandwidth of correlated Nakagami-0.5 fading channels versus
average received SNR using TIFR
M. I. Hasan, S. Kumar
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In Fig. 4, the average channel capacity per unit bandwidth of dual-branch MRC for
q = 0.2 under TIFR scheme is plotted as a function of the cutoff SNR c0 for several values
of the average received SNR per branch c. As expected, by increasing c average channel
capacity per unit bandwidth improves.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [dB] per Branch
Average Channel Capacity per Unit Bandwidth [bit/sec/Hz]
No diversity under Rayleigh fading
Dual-branch MRC under Rayleigh fading with = 0
No diversity under Nakagami-0.5 fading
Dual-branch MRC under Nakagami-0.5 fading with = 0
Dual-branch MRC under Nakagami-0.5 fading with = 0.2
Dual-branch MRC under Nakagami-0.5 fading with = 0.6
Fig. 9 Average channel capacity per unit bandwidth versus average received SNR using TIFR
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [dB] per Branch
Average Channel Capacity per Unit Bandwidth [bit/sec/Hz]
No diversity for TIFR
Dual-branch MRC for TIFR with = 0
Dual-branch MRC for TIFR with = 0.2
Dual-branch MRC for TIFR with = 0.6
No diversity for OPRA
Dual-branch MRC for OPRA with = 0
Dual-branch MRC for OPRA with = 0.2
Dual-branch MRC for OPRA with = 0.6
Fig. 10 Average channel capacity per unit bandwidth over correlated Nakagami-0.5 fading channels versus
average received SNR using different adaptation schemes
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In Fig. 5, the average channel capacity per unit bandwidth of dual-branch MRC for
q = 0.6 under TIFR scheme is plotted as a function of the cutoff SNR c0 for several values
of the average received SNR per branch c. As expected, by increasing c average channel
capacity per unit bandwidth improves. It is also observed in Figs. 4 and 5 that the average
channel capacity per unit bandwidth is decreases as q increases for each average received
SNR.
In Fig. 6, the probability of outage under TIFR scheme is plotted as a function of the
average received SNR per branch c for q = 0.2 and q = 0.6. For comparison, the probability
of outage of uncorrelated Nakagami-0.5 fading channels with dual-branch MRC and
without diversity, which was obtained in [11], is also presented in Fig. 6. As expected, by
increasing c and/or employing diversity, probability of outage improves. It is also observed
in Fig. 6 that the probability of outage with diversity is largest when q = 0.6 and decreases
as q decreases.
In Fig. 7, it is depicted that under worst fading conditions, OPRA achieves improved
probability of outage compared to TIFR. It is also observed that probability of outage
under TIFR scheme is not improved adequately than the probability of outage under OPRA
even as employing diversity and/or decrease in correlation coefficient. Thus, the probability
of outage with dual-branch MRC using TIFR is higher than the probability of
outage with no diversity using OPRA even correlation coefficient becomes zero means
q = 0.
Figure 8 show the average channel capacity per unit bandwidth of a dual-branch MRC
system over correlated Nakagami-0.5 fading channels under TIFR scheme as a function of
Table 1 Comparison of gOPRA; N, gOPRA; Eup, and gOPRA; ELow at two different values of N for worst case of
fading and q = 0.2
c dB ½  N = 5 N = 10
gOPRA; N gOPRA; Eup gOPRA; ELow gOPRA; N gOPRA; Eup gOPRA; ELow
-10 0.348728402 0.0473088 0.03336408 0.38723345 0.01003795 0.00352367
-5 0.732760690 0.0637937 0.04832618 0.78520980 0.0052650 0.00456839
0 1.387196937 0.0840047 0.06669941 1.45674036 0.0066245 0.00584891
5 2.357131365 0.1083293 0.08816630 2.44725073 0.0082590 0.00739022
10 3.609588758 0.1364326 0.11436815 3.72347901 0.0101484 0.00917087
Table 2 Comparison of gOPRA; N, gOPRA; Eup, lxand gOPRA; ELow at two different values of N for worst case
of fading and q = 0.6
c dB ½  N = 10 N = 20
gOPRA; N gOPRA; Eup gOPRA; ELow gOPRA; N gOPRA; Eup gOPRA; ELow
-10 0.22074068 0.23016136 0.12729104 0.3544244 0.0564820 0.0425469
-5 0.53281732 0.32097199 0.20217760 0.72635405 0.0734707 0.05761250
0 1.08192771 0.42833878 0.29075644 1.34622714 0.0935546 0.07542510
5 1.91621947 0.553960689 0.39439374 2.2633188 0.1170535 0.09626582
10 3.02212399 0.69764360 0.51293025 3.463919845 0.1439312 0.12010265
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the average received SNR per branch c for q = 0.2 and q = 0.6. For comparison, the
average channel capacity per unit bandwidth of uncorrelated Nakagami-0.5 fading channels
with dual-branch MRC and without diversity, which was obtained in [11, Eq. (22)]
and [11, Eq. (18)] respectively, is presented in Fig. 6. As expected, by increasing c and/or
employing diversity, average channel capacity per unit bandwidth improves. It is also
observed in Fig. 8 that the average channel capacity per unit bandwidth with dual-branch
MRC is largest when q = 0 but almost identical performance for c 5dB even when
correlation exists.
In Fig. 9, the average channel capacity per unit bandwidth under TIFR scheme improves
by increasing m and/or employing diversity using [5], [11]. But the channel capacity
without diversity is almost same for c B -1.25 dB and the channel capacity with
dual-branch MRC is almost same for c B -5 dB, even when fading parameter increases
from m = 0.5 to m = 1 and/or decreasing correlation coefficient q.
In Fig. 10, the average channel capacity per unit bandwidth over correlated Nakagami-
0.5 fading channels is plotted as a function of average received SNR c, considering OPRA,
and TIFR adaptation schemes with the aid of (21), and (35). It shows that, for Nakagami-
0.5 fading channel conditions OPRA achieves the highest capacity, whereas TIFR achieves
the lowest capacity. It is also observed that the channel capacity with uncorrelated dualbranch
MRC under TIFR scheme is lower than the channel capacity with correlated dualbranch
MRC under OPRA. Since probability of outage improvement is not significant in
case of TIFR scheme.
Tables 1, 2, 3, 4, 5, 6 and 7 present gOPRA;N, gOPRA; Eup, and gOPRA; ELow, POPRA;N,
POPRA; Eup, and POPRA; ELow, PTIFR; N, PTIFR; Eup, and PTIFR; ELow, and gTIFR;N, respectively,
at two different levels of truncation for q = 0.2 and q = 0.6. It is seen in Tables 1,
2, 3, 4, 5 and 6 that the upper and lower bounds of the truncation errors become tighter as
Table 3 Comparison of POPRA; N, POPRA; Eup, and POPRA; ELow at two different values of N for worst case
of fading and q = 0.2
c dB ½  N = 5 N = 10
POPRA; N POPRA; Eup POPRA; ELow POPRA; N POPRA; Eup POPRA; ELow
-10 0.602933530 0.000105154 1.138573 9 10-5 0.602944361 3.3866 9 10-7 1.16747 9 10-9
-5 0.416665974 6.37603 9 10-6 3.49944 9 10-7 0.416666336 1.005074 9 10-8 4.48315 9 10-12
0 0.240476836 1.49666 9 10-7 1.113 9 10-7 0.240486840 1.35765 9 10-8 3.46691 9 10-15
5 0.114140294 1.37455 9 10-9 1.28790 9 10-9 0.114141294 1.0060 9 10-9 4.80847 9 10-19
10 0.045654787 5.2144 9 10-10 5.1689 9 10-10 0.045654799 3.88773 9 10-10 0
Table 4 Comparison of POPRA; N, POPRA; Eup, and POPRA; ELow at two different values of N for worst case
of fading and q = 0.6
c dB ½  N = 5 N = 10
POPRA; N POPRA; Eup POPRA; ELow POPRA; N POPRA; Eup POPRA; ELow
-10 0.629739794 0.0398077 5.274878 9 10-3 0.634813393 0.000178324 1.164845 9 10-5
-5 0.463075246 0.0044325 2.751503 9 10-4 0.463345326 1.20098 9 10-6 2.6553 9 10-8
0 0.286033430 0.000158553 4.427154 9 10-6 0.286037798 1.608864 9 10-8 1.39024 9 10-9
5 0.144896014 1.87721 9 10-6 6.50053 9 10-8 0.14489603 3.22836 9 10-8 2.80945 9 10-13
10 0.060937223 8.46147 9 10-9 2.49499 9 10-9 0.060937223 1.888088 9 10-9 1.12122 9 10-17
Average Channel Capacity of Correlated Dual-Branch Maximal…
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the truncation level, N, increases. Note that the truncation levels that were used to calculate
the average channel capacity and probability of outage for Fig. 1, 2, 3, 4, 5, 6 and 7 were
high enough to ensure the upper and lower bound of the truncation errors must give
appropriate accuracy. It is also seen in Tables 5 and 6 that upper and lower bounds are very
close to zero for truncation level N = 20. At the same time Table-7 gives very small
truncation error for N = 20 using numerical evaluation techniques.
5 Conclusions
This research paper derives and analyzes the average channel capacity and probability of
outage expressions over slowly varying correlated Nakagami-0.5 fading channels under
OPRA and TIFR schemes with dual-branch MRC. By numerical evaluations it has been
Table 5 Comparison of PTIFR; N, PTIFR; Eup, and PTIFR; ELow at two different values of N for worst case of
fading and q = 0.2
c dB ½  N = 10 N = 20
PTIFR; N PTIFR; Eup PTIFR; ELow PTIFR; N PTIFR; Eup PTIFR; ELow
-10 0.750785405 6.32610 9 10-4 6.46788 9 10-8 0.750885411 2.62898 9 10-9 2.08184 9 10-18
-5 0.6092705498 2.17342 9 10-5 1.38921 9 10-9 0.6093715499 1.681713 9 10-12 0
0 0.4589879651 2.33410 9 10-7 1.76379 9 10-11 0.4589880951 1.645351 9 10-16 0
5 0.3220259305 1.30533 9 10-9 1.40579 9 10-13 0.3220259826 0 0
10 0.2186994674 1.1519 9 10-11 1.07783 9 10-15 0.2186995794 0 0
Table 6 Comparison of PTIFR; N, PTIFR; Eup, and PTIFR; ELow at two different values of N for worst case of
fading and q = 0.6
c dB ½  N = 10 N = 20
PTIFR;N PTIFR; Eup PTIFR; ELow PTIFR; N PTIFR; Eup PTIFR; ELow
-10 0.769956582 0.00472594 6.32610 9 10-4 0.7715217718 4.82716 9 10-8 2.62898 9 10-9
-5 0.654779990 0.000299897 2.17342 9 10-5 0.6552003551 5.24137 9 10-11 1.68171 9 10-12
0 0.514421388 5.95149 9 10-6 2.33410 9 10-7 0.5144316838 8.95571 9 10-15 1.64535 9 10-16
5 0.3767986795 5.93354 9 10-8 1.30533 9 10-9 0.3767997998 0 0
10 0.2752653201 8.3927 9 10-10 1.1519 9 10-11 0.2752654310 0 0
Table 7 Comparison of gTIFR; N at two different values of N for worst case of fading
c dB ½  q = 0.2 q = 0.6
N = 10, gTIFR; N N = 20, gTIFR; N N = 10, gTIFR; N N = 20, gTIFR; N
-10 0.374338282 0.368470529 0.368475158 0.368470529
-5 0.715053606 0.715053606 0.715053606 0.715053606
0 1.30927068 1.30927068 1.26567848 1.26567848
5 2.179238947 2.179238947 2.068103813 2.068103813
10 3.302457758 3.302457758 3.116722868 3.116722868
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found that average channel capacity improves in both the cases of OPRA and TIFR by
increasing c, decreasing correlation coefficient q and/or employing diversity. However the
magnitude of improvement is slightly higher in case of OPRA. It has also been observed
that for either scheme of OPRA or TIFR the average channel capacity with dual-branch
MRC is almost the same for c B -5 dB, even though correlation coefficient decreases.
Therefore it is recommended that under worst fading condition, the proper antenna spacing
at the receiver end, required for uncorrelated diversity path is not an important issue for
low value of average received SNR, particularly for c B -5 dB. It has also been observed
that probability of outage improves for both the schemes OPRA and TIFR by increasing c,
decreasing correlation coefficient q and/or employing diversity. However this improvement
is not significant in case of TIFR scheme. Thus, the probability of outage with dualbranch
MRC under TIFR is higher than the probability of outage with no diversity using
OPRA, even when correlation coefficient becomes zero (q = 0). This paper also concludes
that Nakagami-0.5 fading channels using TIFR scheme remains in outage for longer duration
than using OPRA, even employing diversity and/or decreasing correlation coeffi-
cient. So, it is finally concluded that the channel capacity for correlated (q = 0) dualbranch
MRC with OPRA scheme is higher than the capacity for uncorrelated (q = 0) dualbranch
MRC with TIFR scheme under worst fading conditions.
References
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Nakagami-m fading with channel inversion and fixed rate transmission scheme. IET Communications,
1(6), 1161-1169.
2. Khatalin, S., & Fonseka, J. P. (2006). Capacity of correlated Nakagami-m fading channels with diversity
combining techniques. IEEE Transactions on Vehicular Technology, 55(1), 142-150.
3. Goldsmith, A. J., & Varaiya, P. P. (1997). Capacity of fading channels with channel side information.
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of the IEEE Vehicular Technology Conference, Phoenix, AZ, pp. 358-362.
5. Alouini, M. S., & Goldsmith, A. J. (1999). Capacity of Rayleigh fading channels under different
adaptive transmission and diversity-combining techniques. IEEE Transactions on Vehicular Technology,
48(4), 1165-1181.
6. Bithas, P. S., & Mathiopoulos, P. T. (2009). Capacity of correlated generalized gamma fading with
dual-branch selection diversity. IEEE Transactions on Vehicular Technology, 58(9), 5258-5263.
7. Alouini, M. S., & Goldsmith, A. (2000). Adaptive modulation over Nakagami fading channels. Wireless
Presonal Communication, 13, 119-143.
8. Subadar, R., & Sahu, P.R. (2009). Capacity analysis of dual-SC and-MRC systems over correlated Hoyt
fading channels, IEEE Conference, TENCON-2009, pp. 1-5.
9. Hamdi, K. A. (2008). Capacity of MRC on correlated Rician fading channels. IEEE Transactions on
Communications, 56(5), 708-711.
10. Khatalin, S., & Fonseka, J. P. (2006). On the channel capacity in Rician and Hoyt fading environment
with MRC diversity. IEEE Transactions on Vehicular Technology, 55(1), 137-141.
11. Hasan, M. I., & Kumar, S. (2014). Channel capacity of dual-branch maximal ratio combining under
worst case of fading scenario. WSEAS Transactions on Communications, 13, 162-170.
12. Brennan, D. (2003). Linear diversity combining techniques. Proceedings of the IEEE, 91(2), 331-354.
13. Simon, M. K., & Alouini, M. S. (2005). Digital communication over fading channels. New York:
Wiley.
14. Gradshteyn, I. S., & Ryzhik, I. M. (2000). Table of integrals, series, and products. New York: Academic
Press.
15. Wolfram, The Wolfram functions site (2014), Internet (online). http://functions.wolfram.com. , Claims:FORM 2
THE PATENTS ACT, 1970
(39 OF 1970)
&
THE PATENTS RULES, 2003 5
COMPLETE SPECIFICATION
(See section 10; rule 13)
10
Title: Average Channel Capacity of Correlated Dual-Branch Maximal Ratio Combining
Under Worst Case of Fading Scenario
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.
Average Channel Capacity of Correlated Dual-Branch
Maximal Ratio Combining Under Worst Case of Fading
Scenario
Mohammad Irfanul Hasan1,2
• Sanjay Kumar1
 Springer Science+Business Media New York 2015
Abstract The channel capacity and probability of outage results for optimum power and
rate adaptation (OPRA) and truncated channel inversion with fixed rate (TIFR) schemes
over correlated diversity branch obtained so far in literature are applicable only for m C 1.
This paper derived closed-form expressions for the average channel capacity and probability
of outage of dual-branch maximal ratio combining (MRC) over correlated Nakagami-
0.5 (m\1) fading channels. This channel capacity and probability of outage are
evaluated under OPRA and TIFR schemes. Since, the capacity and probability of outage
expressions of dual-branch MRC under OPRA and TIFR schemes contain an infinite
series; bounds on the errors resulting from truncating the infinite series have been derived
for both average channel capacity and probability of outage. The corresponding expressions
for Nakagami-0.5 fading are called expressions under worst fading condition with
severe fading. Finally, numerical results are presented, which are then compared to the
capacity and probability of outage results that previously published for OPRA and TIFR
schemes. It has been observed that OPRA provides improved average channel capacity and
probability of outage, as compared to TIFR under worst case of fading. It is also observed
that probability of outage under TIFR scheme is not improved adequately than the probability
of outage under OPRA even as employing diversity.
Keywords Average channel capacity  Dual-branch  Maximal ratio combining  Nakagami-0.5 fading channels  Optimum power with rate adaptation  Truncated channel
inversion with fixed rate
& 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, Jharkhand 835215, India
2 Department of Electronics and Communication Engineering, Graphic Era University,
Dehradun 248002, Uttarakhand, India
123
Wireless Pers Commun
DOI 10.1007/s11277-015-2559-z
Author's personal copy
1 Introduction
The channel capacity implies maximum achievable data rate of a communication system.
Channel capacity is of fundamental importance in the design of wireless mobile communication
systems as the demand for wireless mobile communication services is growing
rapidly [1]. The wireless mobile channels are subjected to fading, which is undesirable.
The channel capacity in fading environment can be improved by employing diversity
combining and/or adaptive transmission schemes [1, 2]. Diversity combining, which is
known to be a powerful technique that can be used to mitigate fading in wireless mobile
environment. MRC, equal gain combining (EGC), selection combining (SC) are the most
fundamental diversity combining techniques. Adaptive transmission is another effective
scheme that can be used to mitigate fading. Adaptive transmission, which requires an
accurate channel estimation at the receiver and a reliable feedback path between the
estimator and the transmitter, provides great improvement to the average channel capacity
[1, 2]. The capacity of flat fading channels was derived in [3] for four different adaptive
transmission schemes such as optimum rate adaptation with constant transmit power
(ORA), OPRA, channel inversion with fixed rate transmission (CIFR) and TIFR. In case of
ORA scheme, the transmitter adapts the data rate according to the channel fading conditions
while the transmit power is constant [3]. In case of OPRA scheme, transmitter can
realize optimal capacity by transmitting appropriate power and data rate in accordance
with the channel variations [3]. In CIFR scheme, transmitter adapts its power to maintain
constant signal to noise ratio (SNR) at the receiver by inverting the channel gain, which
makes the channel to appear as a time invariant additive white Gaussian noise (AWGN)
channel [3]. Channel inversion with fixed rate suffers a large capacity penalty relative to
the other techniques, since a large amount of power is required to compensate for the deep
channel fades. A better approach is to use a modified inversion policy that inverts the
channel fading only above a cutoff fading level, which is called TIFR scheme [3].
Numerous researchers have worked on the study of channel capacity over different
fading channels. Specifically, [1, 2] discuss the channel capacity over correlated Nakagami-
m (m C 1 & m\1) fading channels under ORA and CIFR schemes with different
diversity combining techniques. In [4], the channel capacity over uncorrelated Nakagami-
m (m C 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 [5]. An analytical performance study of the channel
capacity for correlated generalized gamma fading channels with dual-branch SC under
the different power and rate adaptation schemes was introduced in [6]. The channel
capacity of Nakagami-m (m C 1) fading channel without diversity was derived in [7] for
different adaptive transmission schemes. In [8], channel capacity of dual-branch SC and
MRC systems over correlated Hoyt fading channels using ORA, OPRA, CIFR and TIFR
was presented. In [9], expression for the ergodic capacity of MRC over arbitrarily
correlated Rician fading channels was derived. In [10], 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 [11], an analytical performance
study of the channel capacity for uncorrelated Nakagami-0.5 with dual-branch
MRC using OPRA and TIFR was obtained. However, an analytical study of average
channel capacity over correlated Nakagami-0.5 fading channels under OPRA and TIFR
adaptation schemes with MRC has not been considered so far. The Nakagami-m model
M. I. Hasan, S. Kumar
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has been widely used in general to study wireless mobile communication system performance,
less attention 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
great practical usefulness, they will be of theoretical interest as a worst fading case. All
previously published literature related to the channel capacity over correlated Nakagamim
fading channels using OPRA and TIFR schemes are not applicable for m = 0.5. This
paper fills this gap by presenting an analytical performance study of the channel capacity
and probability of outage of dual-branch MRC over correlated Nakagami-0.5 fading
channels using OPRA, and TIFR schemes.
In this paper, we consider MRC, which is optimal combining scheme that provides
maximum performance improvement relative to all other combining techniques [12]. The
dual-branch diversity has been considered since it offers the least complexity and physical
space requirements [12]. Hence this paper gives the analysis of channel capacity for
optimum combining technique (MRC) using optimum adaptation scheme, OPRA and a
modified inversion scheme, TIFR under most practical challenging fading scenario means
Nakagami-0.5 fading channels said to be worst fading conditions.
The remainder of this paper is organized as follows: In Sect. 2, the channel model is
defined. In Sect. 3, average channel capacity and probability of outage of dual-branch
MRC over correlated Nakagami-0.5 fading channels are derived for OPRA and TIFR
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 received SNR, c, at the output of an L-branch
MRC combiner, pcðcÞ, for the correlated Nakagami-m fading channels is obtained in [13]
pcðcÞ ¼
mc
c  Lm1
exp mc
c 1 ffiffi q p ð Þ
 1F 1 m; Lm; Lm ffiffi q p c
c 1 ffiffi q p ð Þ1 ffiffi q p þL ffiffi q p ð Þ
 
c
m   1  ffiffiffi q p  Lmm
1  ffiffiffi q p þ L ffiffiffi q p  mC Lm ð Þ
; c0 ð1Þ
where c is the average received SNR, m (m C 0.5) is the fading parameter, Cð:Þ is the
gamma function, 1F 1½ : ; : ; :  is the confluent hypergeometric function of the first kind, and
q is the envelope or power correlation coefficient between any pair of signals.
For different values of m, this expression simplifies to several important distributions
used in fading channel modeling. Like m = 0.5 corresponds to the highest amount of
fading, m = 1 corresponds to Rayleigh distribution, m C 1 corresponds to Rician distribution,
and as m!1, the distribution converges to a nonfading AWGN [13].
Considering dual-branch MRC (i.e. L = 2) and replacing the 1F 1½ : ; : ; :  function with
its series representation as given in [14], and then the pdf under worst case of fading using
(1) becomes
pcðcÞ ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n !
exp 
0:5c
c 1  ffiffiffi q p  
!; c0 ð2Þ
Average Channel Capacity of Correlated Dual-Branch Maximal…
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3 Average Channel Capacity
In this section, we present closed-form expressions for the average channel capacity of correlated
Nakagami-0.5 fading channels with dual-branch MRC under OPRA, and TIFR
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 [5].
3.1 Opra
The average channel capacity of fading channel with received SNR distribution pcðcÞ
under OPRA scheme (COPRA[bit/sec]) is defined in [3, 4] as
COPRA ¼ B Z
1
c0
log2
c
c0

pcðcÞ dc ð3Þ
where B [Hz] is the channel bandwidth and c0 is the optimum cutoff SNR level below
which data transmission is suspended. To obtain the optimal cutoff SNR, c0 must satisfy
the equation given by [3, 4] as
Z
1
c0
1
c0 
1
c

pc ðcÞ dc ð4Þ
To achieve the channel capacity (3), the channel fade level must be tracked at both the
receiver and transmitter, and the transmitter has to adapt its posswer and rate accordingly,
allocating high power levels and rate for good channel conditions (c large), and lower
power levels and rates for unfavorable channel conditions (c small). So, OPRA gives the
analysis of maximum channel capacity under Nakagami-m fading conditions.
When c\c0, no data is transmitted, the optimal scheme suffers a probability of outage
Pout, equal to the probability of no transmission, given by [3, 4] is
Pout ¼ Z
c0
0
pcðcÞ dc ¼ 1  Z
1
c0
pcðcÞ dc ð5Þ
Substituting (2) in (4) for optimal cutoff SNR c0, then
Z
1
c0
1
c0 
1
c

0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n !
exp 
0:5c
c 1  ffiffiffi q p  
!dc ¼ 1
Evaluating the above integral using some mathematical transformation by [14], we obtain
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
n !
2c 1  ffiffiffi q p    nþ1
c0
C n þ 1;
0:5c0
c 1  ffiffiffi q p  
! "
 2c 1  ffiffiffi q p    nC n ;
0:5c0
c 1  ffiffiffi q p  
" ##
¼ 1
ð6Þ
The numerical evaluation techniques have confirmed that, by solving (6), there is a unique
M. I. Hasan, S. Kumar
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positive value of c0 for each c satisfying (6) that takes values from c0 2 ½0; 1. Result
shows that c0 increases as c increases. The value of cutoff SNR c0 that satisfies (6) for each
c is used for finding the average channel capacity per unit bandwidth and probability of
outage.
Substituting (2) in (3), the average channel capacity of dual-branch MRC under Nakagami-
0.5 fading channel is
COPRA¼BZ
1
c0
log2
c
c0

0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1qÞ p cX
1
n¼0
Cðnþ0:5Þ
Cðnþ1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1 ffiffiffi q p  
!dc ð7Þ
COPRA¼B
1:4430:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1qÞ p cX
1
n¼0
Cðnþ0:5Þ
Cðnþ1Þ
ffiffi q p
cð1qÞ  n
n!
 Z
1
c0
logðcÞcn exp 
0:5c
c 1 ffiffiffi q p  
!dcZ
1
c0
logðc0Þcn exp 
0:5c
c 1 ffiffiffi q p  
!dc
264
375
ð8Þ
Following can be taken from the first part of above integral is
Z
1
c 0
log ðcÞ cn exp 
0:5c
c 1  ffiffiffi q p  
!dc
This can be solved using partial integration as follows
Z
1
c0
udv ¼ lim
c!1
u v ð Þlim
c!c0
u v ð Þ Z
1
c0
v du
Let u ¼ log c
then du ¼ dc
c
Now let dv ¼ cn exp  0:5c
c 1 ffiffi q p ð Þ

dc
Integrating this expression using [14], we obtain
v ¼  2c 1  ffiffiffi q p    nþ1C n þ 1;
0:5c
c 1  ffiffiffi q p  
" #
Evaluating first part of above integral (8) using this partial integration and some mathematical
transformation [14, 15], we obtain
Z
1
c0
log ðcÞ cn exp 
0:5c
c 1  ffiffiffi q p  
!dc ¼ log ðc0Þ  2c 1  ffiffiffi q p    nþ1C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" #
þX
n
k¼0
n !
n  k !  2c 1  ffiffiffi q p    nþ1C n  k;
0:5c0
c 1  ffiffiffi q p  
" #
ð9Þ
Second part of above integral (8) of can be solved using [14], we obtain
Average Channel Capacity of Correlated Dual-Branch Maximal…
123
Author's personal copy
Z
1
c0
log ðc0Þ cn exp 
0:5c
c 1  ffiffiffi q p  
!dc
¼ log ðc0Þ  2c 1  ffiffiffi q p    nþ1C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" #
ð10Þ
Substituting (9) and (10) in (8), the average channel capacity of dual-branch MRC under
Nakagami-0.5 fading channels is
COPRA ¼
1:443  0:5  B
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X 1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
Using that result we obtain average channel capacity per unit bandwidth i.e. COPRA
B [bit/sec/
Hz] as
gOPRA ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X 1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k! ð11Þ
The computation of the average channel capacity per unit bandwidth according to (11)
requires the computation of an infinite series. To efficiently compute the series, we truncate
the series, and present bounds for the average channel capacity per unit bandwidth.
The average channel capacity per unit bandwidth in (11) can be written
asgOPRA ¼ gOPRA; N þ gOPRA; E, where gOPRA; N is the expression in (11) with the infinite
series truncated at the Nth term as
gOPRA; N ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
and gOPRA; E is the truncation error resulting from truncating the infinite series in (11) at
n = N.The lower bound for gOPRA is derived as
gOPRA[gOPRA ; N þ gOPRA ; Elow
where gOPRA; Elow, which is the lower bound of gOPRA; E
The lower bound for the capacity can be derived by using the relationship between the
area of the pdf and the expression of the average channel capacity per unit bandwidth as
discuss in [10].
As we know that area of pdf pcðcÞ is equal to unity.
P ¼ Z
1
0
pcðcÞ dc ¼ 1 ð12Þ
Substituting (2) into (12), we get
M. I. Hasan, S. Kumar
123
Author's personal copy
Z
1
0
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n !
exp 
0:5c
c 1  ffiffiffi q p  
!dc ¼ 1
After performing integration using [14], we obtain
P ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X 1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1¼ 1
Let
PN1 ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1 ð13Þ
And let
DPN1 ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
ffiffiffi q p
1  q

NCðN þ 0:5Þ
CðN þ 1Þ
2 1 ffiffiffi p p    Nþ1 ð14Þ
then
PN ¼ PN1 þ DPN1
Similarly, from (11) let
gOPRA; N1 ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
ð15Þ
And
DgOPRA; N1 ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
ffiffiffi q p
1  q

NCðN þ 0:5Þ
CðN þ 1Þ
2 1 ffiffiffi p p    Nþ1X
N
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
ð16Þ
Dividing (16) by (14), yields
DgOPRA; N1
DPN1 ¼ 1:443X
N
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k! ð17Þ
Observing that
DgOPRA; N1
DPN1   monotonically increases with increasing N, that
is,
DgOPRA; i
DPi [ DgOPRA; N1
DPN1
for iN
X 1
i¼N
DgOPRA; i[
DgOPRA; N1
DPN1 X
1
i¼N
DPi ¼
DgOPRA; N1
DPN1
1  PN ð Þ ð18Þ
Hence, the average channel capacity per unit bandwidth in (11) can be lower bounded by
using (17) and (18) as
Average Channel Capacity of Correlated Dual-Branch Maximal…
123
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gOPRA[gOPRA; N þ 1:443X
N
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
 2 1 ffiffiffi p p    nþ1 !
ð19Þ
The upper bound for gOPRA is derived as
gOPRA\gOPRA; N þ gOPRA; Eup
where gOPRA; Eup, which is the upper bound of gOPRA; E
The expression in (7) can be written as
gOPRA ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
Z
1
c0
log
c
c0


X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!
þ
X 1
n¼Nþ1
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!
26666664
37777775
dc
Therefore, gOPRA; N can be obtained as
gOPRA; N ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
Z
1
c0
log
c
c0

X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Further, gOPRA; E can be presented as
gOPRA; E ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c X
1
n¼Nþ1
Cðn þ 0:5Þ
Cðn þ 1Þ
Z
1
c 0
log
c
c0
 
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Let an ¼ C½nþ0:5 C½nþ1
, then anþ1
an ¼ C½nþ1:5 C½nþ2  C½nþ1 C½nþ0:5
\1
i.e. an monotonically decreases with increase of n, therefore, gOPRA; E can be upper
bounded as
gOPRA; E1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c 
CðN þ 1:5Þ
CðN þ 2Þ
Z
1
c0
log
c
c0
 
 exp 
0:5c
c 1  ffiffiffi q p  
!X
1
n¼0
c ffiffi q p
cð1qÞ  n
n! X
N
n¼0
c ffiffi q p
cð1qÞ  n
n!
8<:
9=;
dc
M. I. Hasan, S. Kumar
123
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gOPRA; E\gOPRA; Eup ¼
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
CðN þ 1:5Þ
CðN þ 2Þ
2 ffiffiffi q p þ 1  exp 0:5c0
ffiffiffi q p þ 1  c
!exp
c0
2 ffiffiffi q p þ c  
!E1
0:5c0
ffiffiffi q p þ 1  c
" #
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
266666664
377777775
where E1ð:Þ is the exponential integral of first order.
Therefore, the average channel capacity per unit bandwidth in (11) can be upper
bounded as
gOPRA ¼ gOPRA N þ gOPRA ;E\gOPRA ;N
þ
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
CðN þ 1:5Þ
CðN þ 2Þ
2 ffiffiffi q p þ 1  exp 0:5c 0
ffiffiffi q p þ 1  c
!exp
c 0
2 ffiffiffi q p þ c  
!E1
0:5c 0
ffiffiffi q p þ 1  c
" #
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
266666664
377777775
ð20Þ
Hence, the average channel capacity per unit bandwidth is bounded using (19) and (20) as
gOPRA; N þ
1:443  0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
CðN þ 1:5Þ
CðN þ 2Þ
2 ffiffiffi q p þ 1  exp 0:5c0
ffiffiffi q p þ 1  c
!exp
c0
2 ffiffiffi q p þ c  
!E1
0:5c0
ffiffiffi q p þ 1  c
" #
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
266666664
377777775
[gOPRA[gOPRA; Nþ
1:443X
N
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1 !
ð21Þ
When q = 0, (7) reduces to the corresponding expressions for uncorrelated dual-branch
MRC under worst case of fading channels. Note that for q = 0, the integration of (7) is
equivalent to [11, Eq. (11)]. Finally we compare (21) with [11, Eq. (11)] and channel
capacity without diversity in [11, Eq. (8)].
Substituting (2) in (5) for probability of outage, then
POPRA ¼ Z
c0
0
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc ð22Þ
After evaluating the above integral by using mathematical transformation using [14, 15],
we obtain
Average Channel Capacity of Correlated Dual-Branch Maximal…
123
Author's personal copy
POPRA ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X 1
n¼0
C½n þ 0:5
C½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n!
 C ½n þ 1  C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" # ( ) ð23Þ
The computation of probability of outage according to (23) requires the computation of an
infinite series. To efficiently compute the series, we truncate the series and derive bounds
for the probability of outage.
The probability of outage in (23) can be written as POPRA ¼ POPRA; N þ POPRA; E, where
POPRA; N is the expression in (23) with the infinite series truncated at the Nth term as
POPRA; N ¼
0:5
ffiffiffi p p ð1  qÞX
N
n¼0
C ½n þ 0:5
C ½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n!
C ½n þ 1  C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" # ( )
and POPRA; E is the truncation error resulting from truncating the infinite series in (23).The
lower bound for POPRA is derived as
POPRA[POPRA; N þ POPRA; Elow
where POPRA; Elow, which is the lower bound of POPRA; E
Similarly, from (23) let
POPRA; N1 ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X
N1
n¼0
C ½n þ 0:5
C ½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n !
 C ½n þ 1  C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" # ( )
ð24Þ
And
DPOPRA; N1 ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
ffiffiffi q p
1  q

NCðN þ 0:5Þ
CðN þ 1Þ
2 1 ffiffiffi p p    Nþ1

C ½N þ 1  C N þ 1; 0:5c0
c 1 ffiffi q p ð Þ
 

N!
ð25Þ
Dividing (25) by (14), yields
DPOPRA; N1
DPN1 ¼
C ½N þ 1  C N þ 1; 0:5c0
c 1 ffiffi q p ð Þ
 
N! ð26Þ
Observing that DPOPRA; N1
DPN1   monotonically increases with increasing N,
i.e. DPOPRA; i
DPi [ DPOPRA; N1
DPN1
for i C N
M. I. Hasan, S. Kumar
123
Author's personal copy
X 1
i¼N
DPOPRA; i[
DPOPRA;N1
DPN1 X
1
i¼N
DPi ¼
DPOPRA;N1
DPN1
1  PN ð Þ ð27Þ
Hence, the probability of outage in (23) can be lower bounded by using (26) and (27) as
POPRA[POPRA;N þ
C½N þ 1  C N þ 1; 0:5c0
c 1 ffiffi q p ð Þ
 
N!
 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1 !
ð28Þ
The upper bound for POPRA is derived as
POPRA\POPRA ; N þ POPRA ; Eup
where POPRA; Eup, which is the upper bound of POPRA; E
The expression in (22) can be written as
POPRA ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
Z
c0
0
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
n!
cn exp 
0:5c
c 1  ffiffiffi q p  
!
þ X
1
n¼Nþ1
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!
26666664
37777775
dc
Therefore, POPRA;N can be obtained as
POPRA;N ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
Z
c0
0
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Further, POPRA; E can be presented as
POPRA; E ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c X
1
n¼Nþ1
Cðn þ 0:5Þ
Cðn þ 1Þ
Z
c0
0
c ffiffi q p
cð1qÞ  n
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Let an ¼ C½nþ0:5 C½nþ1
, then anþ1
an ¼ C½nþ1:5 C½nþ2  C½nþ1 C½nþ0:5
\1
i.e. an monotonically decreases with increase of n, therefore, POPRA; E can be upper
bounded as
POPRA; E0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c

CðN þ 1:5Þ
CðN þ 2Þ
Z
c0
0
exp 
0:5c
c 1  ffiffiffi q p  
!X
1
n¼0
c ffiffi q p
cð1qÞ  n
n! X
N
n¼0
c ffiffi q p
cð1qÞ  n
n!
8<:
9=;
dc
After evaluating the above integral by using mathematical transformation using [14], we
obtain
Average Channel Capacity of Correlated Dual-Branch Maximal…
123
Author's personal copy
POPRA; E0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p 
CðN þ 1:5Þ
CðN þ 2Þ

exp c0 ffiffi q p
1q ð Þ c ð Þ  0:5c0
1 ffiffi q p ð Þc
 1
ffiffi q p
1q ð Þ 0:5
1 ffiffi q p ð Þ

X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
26666666664
37777777775
Therefore, the probability of outage in (23) can be upper bounded as
POPRA ¼ POPRA; N þ POPRA; E\POPRA; N þ
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p c
CðN þ 1:5Þ
CðN þ 2Þ

exp c0 ffiffi q p
1q ð Þ c ð Þ  0:5c0
1 ffiffi q p ð Þc
 1
ffiffi q p
1q ð Þ 0:5
1 ffiffi q p ð Þ
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
26666666664
37777777775
ð29Þ
Hence, the probability of outage is bounded using (28) and (29) as
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [dB] per Branch Average Channel Capacity per Unit Bandwidth [
bit/sec/Hz]
No diversity
Dual-branch MRC with = 0
Dual-branch MRC with = 0.2
Dual-branch MRC with = 0.6
Fig. 1 Average channel capacity per unit bandwidth over correlated Nakagami-0.5 fading channels as a
function of average received SNR using OPRA
M. I. Hasan, S. Kumar
123
Author's personal copy
POPRA; N þ
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
CðN þ 1:5Þ
CðN þ 2Þ
exp c0 ffiffi q p
1q ð Þ c ð Þ  0:5c0
1 ffiffi q p ð Þc
 1
ffiffi q p
1q ð Þ 0:5
1 ffiffi q p ð Þ
X
N
n¼0
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1X
n
k¼0
C k; 0:5c0
c 1 ffiffi q p ð Þ


k!
26666666664
37777777775
[POPRA[
POPRA ; N þ
C½N þ 1  C N þ 1; 0:5c0
c 1 ffiffi q p ð Þ
 
N!
1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X
N
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n
2 1 ffiffiffi p p    nþ1 !
ð30Þ
when q = 0, (22) reduces to the corresponding expressions for uncorrelated dual-branch
MRC under worst case of fading channels. Note that for q = 0, the integration of (22) is
equivalent to [11, Eq. (12)]. Finally we compare (30) with [11, Eq. (12)] and probability of
outage without diversity [11, Eq. (9)].
3.2 Tifr
The average channel capacity of fading channel with received SNR distribution pcðcÞ
under TIFR scheme (CTIFR[bit/sec]) is defined in [3, 4] as
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Average Received SNR [dB] per Branch
Average Channel Capacity per Unit Bandwidth [bit/sec/Hz]
No diversity under Rayleigh fading
Dual-branch MRC under Rayleigh fading with = 0
No diversity under Nakagami-0.5 fading
Dual-branch MRC under Nakagami-0.5 fading with = 0
Dual-branch MRC under Nakagami-0.5 fading with = 0.2
Dual-branch MRC under Nakagami-0.5 fading with = 0.6
Fig. 2 Average channel capacity per unit bandwidth versus average received SNR using OPRA
Average Channel Capacity of Correlated Dual-Branch Maximal…
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CTIFR ¼ B log2 1 þ
1
R
1
c0
pcðcÞ
c  dc
0BBB@
1CCCA
ð1  PoutÞ; c0 ð31Þ
where Pout is given in (5). The cutoff level c0, can be chosen to achieve a specified probability
of outage, Pout, or, alternatively, to maximize the average channel capacity (31).
Using the pdf of dual-branch MRC under correlated Nakagami-0.5 fading channels in
(2), we obtain
pcðcÞ
c ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
cn1
n !
exp 
0:5c
c 1  ffiffiffi q p  
!
ð32Þ
Evaluating the above integral using mathematical transformation by [14], we obtain
Z
1
c0
pcðcÞ
c
dc ¼
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p
X 1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffiffi q p
1  q

n 2 1 ffiffiffi p p    n
n!
C n;
0:5c0
c 1  ffiffiffi q p  
!
ð33Þ
Now, we evaluate the probability of outage using (2) is
Pout ¼ Z
c0
0
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
cn
n !
exp 
0:5c
c 1  ffiffiffi q p  
!dc
1  Pout ¼ 1  Z
c0
0
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  qÞ p cX
1
n¼0
Cðn þ 0:5Þ
Cðn þ 1Þ
ffiffi q p
cð1qÞ  n
cn
n!
exp 
0:5c
c 1  ffiffiffi q p  
!dc
Evaluating the above integral by some mathematical transformation using [14, 15], we
obtain
1  Pout ¼ P cc0 ½ ¼ 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X 1
n¼0
C½n þ 0:5
C½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n!
C½n þ 1  C½n þ 1;
0:5c0
c 1  ffiffiffi q p  
( ) ð34Þ
Putting the value of (33) and (34) in (31), we get
CTIFR ¼ 1:443B log 1 þ
1
0:5
ffiffiffiffiffiffiffiffiffiffiffi pð1qÞ p P
1
n¼0
Cðnþ0:5Þ Cðnþ1Þ ffiffi q p
1q n on 2 1 ffiffip p ð Þ f gn
n ! C n; 0:5c0
c 1 ffiffi q p ð Þ


2664
3775
 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X 1
n¼0
C½n þ 0:5
C½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n !
(
 C½n þ 1  C n þ 1;
0:5c0
c 1  ffiffiffi q p  
" # ( ))
M. I. Hasan, S. Kumar
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Using that result we obtain average channel capacity per unit bandwidth i.e. CTIFR
B [bit/sec/
Hz] as
gTIFR ¼ 1:443 log 1 þ
1
0:5
ffiffiffiffiffiffiffiffiffiffiffi pð1qÞ p P
1
n¼0
Cðnþ0:5Þ Cðnþ1Þ ffiffi q p
1q n on 2 1 ffiffip p ð Þ f gn
n ! C n; 0:5c0
c 1 ffiffi q p ð Þ


2664
3775
 1 
0:5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð1  qÞ p
X 1
n¼0
C ½n þ 0:5
C ½n þ 1
ffiffiffi q p
1  q

n 2 1 ffiffiffi q p    nþ1
n !
(
 C½n þ 1  C½n þ 1;
0:5c0
c 1  ffiffiffi q p  
( ))
ð35Þ
The computation of average channel capacity per unit bandwidth according to (35) requires
the evaluation of two infinite series. It is difficult but not impossible to compute the
channel capacity under TIFR. To efficiently compute the series, we truncate the series
using numerical evaluation techniques.The expression of probability of outage in case of
TIFR is same as (30), except the cutoff level. In this case the cutoff level c0, can be chosen
to maximize the average channel capacity per unit bandwidth in (35).
4 Numerical Results and Analysis
In this section, various performance evaluation results for the average channel capacity per
unit bandwidth and probability of outage using dual-branch MRC operating over correlated
Nakagami-0.5 fading channels has been presented and analyzed. These results compare the
-10 -5 0 5 10
10 -2
10 -1
10 0
Average Received SNR [dB] per Branch Probability of Outage
No diversity
Dual-branch MRC with = 0
Dual-branch MRC with = 0.2
Dual-branch MRC with = 0.6
Fig. 3 Probability of outage of correlated Nakagami-0.5 fading channels versus average received SNR
using OPRA
Average Channel Capacity of Correlated Dual-Branch Maximal…
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-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Cutoff SNR [dB]
Average Channel Capacity per Unit Bandwidth [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 Average channel capacity per unit bandwidth of a dual-branch MRC system versus the cutoff SNR
over correlated Nakagami-0.5 fading channels using TIFR for q = 0.2
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Cutoff SNR [dB]
Average Channel Capacity per Unit Bandwidth [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. 5 Average channel capacity per unit bandwidth of a dual-branch MRC system versus the cutoff SNR
over correlated Nakagami-0.5 fading channels using TIFR for q = 0.6
M. I. Hasan, S. Kumar
123
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-10 -5 0 5 10
10 -2
10 -1
10 0
Average Received SNR [dB] per Branch
Probability of Outage
No diversity
Dual-branch MRC with = 0
Dual-branch MRC with = 0.2
Dual-branch MRC with = 0.6
Fig. 6 Probability of outage of correlated Nakagami-0.5 fading channels versus average received SNR
using TIFR
-10 -5 0 5 10
10-2
10-1
100
Average Received SNR [dB] per Branch
Probability of Outage
No diversity for TIFR
Dual-branch MRC for TIFR with = 0
Dual-branch MRC for TIFR with = 0.2
Dual-branch MRC for TIFR with = 0.6
No diversity for OPRA
Dual-branch MRC for OPRA with = 0
Dual-branch MRC for OPRA with = 0.2
Dual-branch MRC for OPRA with = 0.6
Fig. 7 Probability of outage of correlated Nakagami-0.5 fading channels versus average received SNR
Average Channel Capacity of Correlated Dual-Branch Maximal…
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different adaptive transmission schemes under worst fading channel condition and also
with previously published results for the same adaptisve transmission schemes.
Figure 1 show the average channel capacity per unit bandwidth of a dual-branch MRC
system over correlated Nakagami-0.5 fading channels under OPRA scheme as a function
of the average received SNR per branch c for q = 0.2 and q = 0.6. For comparison, the
average channel capacity per unit bandwidth of uncorrelated Nakagami-0.5 fading channels
with dual-branch MRC and without diversity, which was obtained in [11, Eq. (11)]
and [11, Eq. (8)] respectively, is also presented in Fig. 1. As expected, by increasing c and/
or employing diversity, average channel capacity per unit bandwidth improves. It is also
observed that the average channel capacity per unit bandwidth with dual-branch MRC is
largest when q = 0 but almost identical performance for c B -5 dB even when correlation
coefficient exists.
In Fig. 2, the average channel capacity per unit bandwidth under OPRA scheme improves
by increasing m and/or employing diversity using [5], [11]. But the channel capacity
without diversity is almost same for c B -1.25 dB and the channel capacity with
dual-branch MRC is almost same for c B -5 dB, even when fading parameter increases
from m = 0.5 to m = 1 and/or decreasing correlation coefficient q.
In Fig. 3, the probability of outage under OPRA scheme is plotted as a function of the
average received SNR per branch c for q = 0.2 and q = 0.6. For comparison, the probability
of outage of uncorrelated Nakagami-0.5 fading channels with dual-branch MRC and
without diversity, which was obtained in [11, Eq. (12)] and [11, Eq. (9)] respectively, is
also presented in Fig. 3. As expected, by increasing c and/or employing diversity, probability
of outage improves. It is also observed in Fig. 3 that the probability of outage with
diversity is largest when q = 0.6 and decreases as q decreases.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Average Received SNR [dB] per Branch Average Channel Capacity per Unit Bandwidth [
bit/sec/Hz]
No diversity
Dual-branch MRC with = 0
Dual-branch MRC with = 0.2
Dual-branch MRC with = 0.6
Fig. 8 Average channel capacity per unit bandwidth of correlated Nakagami-0.5 fading channels versus
average received SNR using TIFR
M. I. Hasan, S. Kumar
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In Fig. 4, the average channel capacity per unit bandwidth of dual-branch MRC for
q = 0.2 under TIFR scheme is plotted as a function of the cutoff SNR c0 for several values
of the average received SNR per branch c. As expected, by increasing c average channel
capacity per unit bandwidth improves.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [dB] per Branch
Average Channel Capacity per Unit Bandwidth [bit/sec/Hz]
No diversity under Rayleigh fading
Dual-branch MRC under Rayleigh fading with = 0
No diversity under Nakagami-0.5 fading
Dual-branch MRC under Nakagami-0.5 fading with = 0
Dual-branch MRC under Nakagami-0.5 fading with = 0.2
Dual-branch MRC under Nakagami-0.5 fading with = 0.6
Fig. 9 Average channel capacity per unit bandwidth versus average received SNR using TIFR
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [dB] per Branch
Average Channel Capacity per Unit Bandwidth [bit/sec/Hz]
No diversity for TIFR
Dual-branch MRC for TIFR with = 0
Dual-branch MRC for TIFR with = 0.2
Dual-branch MRC for TIFR with = 0.6
No diversity for OPRA
Dual-branch MRC for OPRA with = 0
Dual-branch MRC for OPRA with = 0.2
Dual-branch MRC for OPRA with = 0.6
Fig. 10 Average channel capacity per unit bandwidth over correlated Nakagami-0.5 fading channels versus
average received SNR using different adaptation schemes
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In Fig. 5, the average channel capacity per unit bandwidth of dual-branch MRC for
q = 0.6 under TIFR scheme is plotted as a function of the cutoff SNR c0 for several values
of the average received SNR per branch c. As expected, by increasing c average channel
capacity per unit bandwidth improves. It is also observed in Figs. 4 and 5 that the average
channel capacity per unit bandwidth is decreases as q increases for each average received
SNR.
In Fig. 6, the probability of outage under TIFR scheme is plotted as a function of the
average received SNR per branch c for q = 0.2 and q = 0.6. For comparison, the probability
of outage of uncorrelated Nakagami-0.5 fading channels with dual-branch MRC and
without diversity, which was obtained in [11], is also presented in Fig. 6. As expected, by
increasing c and/or employing diversity, probability of outage improves. It is also observed
in Fig. 6 that the probability of outage with diversity is largest when q = 0.6 and decreases
as q decreases.
In Fig. 7, it is depicted that under worst fading conditions, OPRA achieves improved
probability of outage compared to TIFR. It is also observed that probability of outage
under TIFR scheme is not improved adequately than the probability of outage under OPRA
even as employing diversity and/or decrease in correlation coefficient. Thus, the probability
of outage with dual-branch MRC using TIFR is higher than the probability of
outage with no diversity using OPRA even correlation coefficient becomes zero means
q = 0.
Figure 8 show the average channel capacity per unit bandwidth of a dual-branch MRC
system over correlated Nakagami-0.5 fading channels under TIFR scheme as a function of
Table 1 Comparison of gOPRA; N, gOPRA; Eup, and gOPRA; ELow at two different values of N for worst case of
fading and q = 0.2
c dB ½  N = 5 N = 10
gOPRA; N gOPRA; Eup gOPRA; ELow gOPRA; N gOPRA; Eup gOPRA; ELow
-10 0.348728402 0.0473088 0.03336408 0.38723345 0.01003795 0.00352367
-5 0.732760690 0.0637937 0.04832618 0.78520980 0.0052650 0.00456839
0 1.387196937 0.0840047 0.06669941 1.45674036 0.0066245 0.00584891
5 2.357131365 0.1083293 0.08816630 2.44725073 0.0082590 0.00739022
10 3.609588758 0.1364326 0.11436815 3.72347901 0.0101484 0.00917087
Table 2 Comparison of gOPRA; N, gOPRA; Eup, lxand gOPRA; ELow at two different values of N for worst case
of fading and q = 0.6
c dB ½  N = 10 N = 20
gOPRA; N gOPRA; Eup gOPRA; ELow gOPRA; N gOPRA; Eup gOPRA; ELow
-10 0.22074068 0.23016136 0.12729104 0.3544244 0.0564820 0.0425469
-5 0.53281732 0.32097199 0.20217760 0.72635405 0.0734707 0.05761250
0 1.08192771 0.42833878 0.29075644 1.34622714 0.0935546 0.07542510
5 1.91621947 0.553960689 0.39439374 2.2633188 0.1170535 0.09626582
10 3.02212399 0.69764360 0.51293025 3.463919845 0.1439312 0.12010265
M. I. Hasan, S. Kumar
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the average received SNR per branch c for q = 0.2 and q = 0.6. For comparison, the
average channel capacity per unit bandwidth of uncorrelated Nakagami-0.5 fading channels
with dual-branch MRC and without diversity, which was obtained in [11, Eq. (22)]
and [11, Eq. (18)] respectively, is presented in Fig. 6. As expected, by increasing c and/or
employing diversity, average channel capacity per unit bandwidth improves. It is also
observed in Fig. 8 that the average channel capacity per unit bandwidth with dual-branch
MRC is largest when q = 0 but almost identical performance for c 5dB even when
correlation exists.
In Fig. 9, the average channel capacity per unit bandwidth under TIFR scheme improves
by increasing m and/or employing diversity using [5], [11]. But the channel capacity
without diversity is almost same for c B -1.25 dB and the channel capacity with
dual-branch MRC is almost same for c B -5 dB, even when fading parameter increases
from m = 0.5 to m = 1 and/or decreasing correlation coefficient q.
In Fig. 10, the average channel capacity per unit bandwidth over correlated Nakagami-
0.5 fading channels is plotted as a function of average received SNR c, considering OPRA,
and TIFR adaptation schemes with the aid of (21), and (35). It shows that, for Nakagami-
0.5 fading channel conditions OPRA achieves the highest capacity, whereas TIFR achieves
the lowest capacity. It is also observed that the channel capacity with uncorrelated dualbranch
MRC under TIFR scheme is lower than the channel capacity with correlated dualbranch
MRC under OPRA. Since probability of outage improvement is not significant in
case of TIFR scheme.
Tables 1, 2, 3, 4, 5, 6 and 7 present gOPRA;N, gOPRA; Eup, and gOPRA; ELow, POPRA;N,
POPRA; Eup, and POPRA; ELow, PTIFR; N, PTIFR; Eup, and PTIFR; ELow, and gTIFR;N, respectively,
at two different levels of truncation for q = 0.2 and q = 0.6. It is seen in Tables 1,
2, 3, 4, 5 and 6 that the upper and lower bounds of the truncation errors become tighter as
Table 3 Comparison of POPRA; N, POPRA; Eup, and POPRA; ELow at two different values of N for worst case
of fading and q = 0.2
c dB ½  N = 5 N = 10
POPRA; N POPRA; Eup POPRA; ELow POPRA; N POPRA; Eup POPRA; ELow
-10 0.602933530 0.000105154 1.138573 9 10-5 0.602944361 3.3866 9 10-7 1.16747 9 10-9
-5 0.416665974 6.37603 9 10-6 3.49944 9 10-7 0.416666336 1.005074 9 10-8 4.48315 9 10-12
0 0.240476836 1.49666 9 10-7 1.113 9 10-7 0.240486840 1.35765 9 10-8 3.46691 9 10-15
5 0.114140294 1.37455 9 10-9 1.28790 9 10-9 0.114141294 1.0060 9 10-9 4.80847 9 10-19
10 0.045654787 5.2144 9 10-10 5.1689 9 10-10 0.045654799 3.88773 9 10-10 0
Table 4 Comparison of POPRA; N, POPRA; Eup, and POPRA; ELow at two different values of N for worst case
of fading and q = 0.6
c dB ½  N = 5 N = 10
POPRA; N POPRA; Eup POPRA; ELow POPRA; N POPRA; Eup POPRA; ELow
-10 0.629739794 0.0398077 5.274878 9 10-3 0.634813393 0.000178324 1.164845 9 10-5
-5 0.463075246 0.0044325 2.751503 9 10-4 0.463345326 1.20098 9 10-6 2.6553 9 10-8
0 0.286033430 0.000158553 4.427154 9 10-6 0.286037798 1.608864 9 10-8 1.39024 9 10-9
5 0.144896014 1.87721 9 10-6 6.50053 9 10-8 0.14489603 3.22836 9 10-8 2.80945 9 10-13
10 0.060937223 8.46147 9 10-9 2.49499 9 10-9 0.060937223 1.888088 9 10-9 1.12122 9 10-17
Average Channel Capacity of Correlated Dual-Branch Maximal…
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the truncation level, N, increases. Note that the truncation levels that were used to calculate
the average channel capacity and probability of outage for Fig. 1, 2, 3, 4, 5, 6 and 7 were
high enough to ensure the upper and lower bound of the truncation errors must give
appropriate accuracy. It is also seen in Tables 5 and 6 that upper and lower bounds are very
close to zero for truncation level N = 20. At the same time Table-7 gives very small
truncation error for N = 20 using numerical evaluation techniques.
5 Conclusions
This research paper derives and analyzes the average channel capacity and probability of
outage expressions over slowly varying correlated Nakagami-0.5 fading channels under
OPRA and TIFR schemes with dual-branch MRC. By numerical evaluations it has been
Table 5 Comparison of PTIFR; N, PTIFR; Eup, and PTIFR; ELow at two different values of N for worst case of
fading and q = 0.2
c dB ½  N = 10 N = 20
PTIFR; N PTIFR; Eup PTIFR; ELow PTIFR; N PTIFR; Eup PTIFR; ELow
-10 0.750785405 6.32610 9 10-4 6.46788 9 10-8 0.750885411 2.62898 9 10-9 2.08184 9 10-18
-5 0.6092705498 2.17342 9 10-5 1.38921 9 10-9 0.6093715499 1.681713 9 10-12 0
0 0.4589879651 2.33410 9 10-7 1.76379 9 10-11 0.4589880951 1.645351 9 10-16 0
5 0.3220259305 1.30533 9 10-9 1.40579 9 10-13 0.3220259826 0 0
10 0.2186994674 1.1519 9 10-11 1.07783 9 10-15 0.2186995794 0 0
Table 6 Comparison of PTIFR; N, PTIFR; Eup, and PTIFR; ELow at two different values of N for worst case of
fading and q = 0.6
c dB ½  N = 10 N = 20
PTIFR;N PTIFR; Eup PTIFR; ELow PTIFR; N PTIFR; Eup PTIFR; ELow
-10 0.769956582 0.00472594 6.32610 9 10-4 0.7715217718 4.82716 9 10-8 2.62898 9 10-9
-5 0.654779990 0.000299897 2.17342 9 10-5 0.6552003551 5.24137 9 10-11 1.68171 9 10-12
0 0.514421388 5.95149 9 10-6 2.33410 9 10-7 0.5144316838 8.95571 9 10-15 1.64535 9 10-16
5 0.3767986795 5.93354 9 10-8 1.30533 9 10-9 0.3767997998 0 0
10 0.2752653201 8.3927 9 10-10 1.1519 9 10-11 0.2752654310 0 0
Table 7 Comparison of gTIFR; N at two different values of N for worst case of fading
c dB ½  q = 0.2 q = 0.6
N = 10, gTIFR; N N = 20, gTIFR; N N = 10, gTIFR; N N = 20, gTIFR; N
-10 0.374338282 0.368470529 0.368475158 0.368470529
-5 0.715053606 0.715053606 0.715053606 0.715053606
0 1.30927068 1.30927068 1.26567848 1.26567848
5 2.179238947 2.179238947 2.068103813 2.068103813
10 3.302457758 3.302457758 3.116722868 3.116722868
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found that average channel capacity improves in both the cases of OPRA and TIFR by
increasing c, decreasing correlation coefficient q and/or employing diversity. However the
magnitude of improvement is slightly higher in case of OPRA. It has also been observed
that for either scheme of OPRA or TIFR the average channel capacity with dual-branch
MRC is almost the same for c B -5 dB, even though correlation coefficient decreases.
Therefore it is recommended that under worst fading condition, the proper antenna spacing
at the receiver end, required for uncorrelated diversity path is not an important issue for
low value of average received SNR, particularly for c B -5 dB. It has also been observed
that probability of outage improves for both the schemes OPRA and TIFR by increasing c,
decreasing correlation coefficient q and/or employing diversity. However this improvement
is not significant in case of TIFR scheme. Thus, the probability of outage with dualbranch
MRC under TIFR is higher than the probability of outage with no diversity using
OPRA, even when correlation coefficient becomes zero (q = 0). This paper also concludes
that Nakagami-0.5 fading channels using TIFR scheme remains in outage for longer duration
than using OPRA, even employing diversity and/or decreasing correlation coeffi-
cient. So, it is finally concluded that the channel capacity for correlated (q = 0) dualbranch
MRC with OPRA scheme is higher than the capacity for uncorrelated (q = 0) dualbranch
MRC with TIFR scheme under worst fading conditions.
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