SPECTRAL EFFICIENCY EVALUATION FOR SELECTION COMBINING DIVERSITY SCHEMES UNDER WORST CASE OF FADING SCENARIO

Abstract

Abstract: The results of spectral efficiencies for optimum rate adaptation with constant transmit power (ORA) and channel inversion with fixed rate (CIFR) schemes over uncorrelated diversity branch with Selection Combining (SC) available so far in literature are not applicable for Nakagami-0.5 fading channels. This paper derived closed-form expressions for the spectral efficiency of dual-branch SC over uncorrelated Nakagami-0.5 fading channels. This spectral efficiency is evaluated under ORA and CIFR schemes. Since, the spectral efficiency expression under ORA scheme contains an infinite series, hence bounds on the errors resulting from truncating the infinite series have been derived 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 spectral efficiency results which have been previously published for ORA and CIFR schemes. It has been observed that by employing SC, spectral efficiency improves under ORA, but does not improve under CIFR.

Specification

Description:FORM 2
THE PATENTS ACT, 1970
(39 OF 1970)
&
THE PATENTS RULES, 2003
COMPLETE SPECIFICATION
(See section 10; rule 13)
Title: Spectral Efficiency Evaluation for Selection Combining Diversity Schemes under
Worst Case of Fading Scenario
APPLICANT DETAILS:
(a) NAME: GRAPHIC ERA DEEMED TO BE UNIVERSITY
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(c) ADDRESS: 566/6, Bell Road, Society Area, Clement Town, Dehradun - 248002,
Uttarakhand, India
PREAMBLE TO THE DESCRIPTION:
The following specification particularly describes the nature of this invention and the manner
in which it is to be performed.
123
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
Spectral Efficiency Evaluation for Selection
Combining Diversity Schemes under Worst Case of
Fading Scenario
Mohammad Irfanul Hasan1, 2 and Sanjay Kumar1
1Department of Electronics and Communication Engineering, Birla Institute of Technology, Mesra, Ranchi, India
2
Department of Electronics and Communication Engineering, Graphic Era University, Dehradun, India
irfanhasan25@rediffmail.com, skumar@bitmesra.ac.in
Abstract: The results of spectral efficiencies for optimum rate
adaptation with constant transmit power (ORA) and channel
inversion with fixed rate (CIFR) schemes over uncorrelated
diversity branch with Selection Combining (SC) available so far in
literature are not applicable for Nakagami-0.5 fading channels. This
paper derived closed-form expressions for the spectral efficiency of
dual-branch SC over uncorrelated Nakagami-0.5 fading channels.
This spectral efficiency is evaluated under ORA and CIFR schemes.
Since, the spectral efficiency expression under ORA scheme
contains an infinite series, hence bounds on the errors resulting
from truncating the infinite series have been derived 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 spectral efficiency results which have been previously
published for ORA and CIFR schemes. It has been observed that by
employing SC, spectral efficiency improves under ORA, but does
not improve under CIFR.
Keywords: Dual-branch, Channel inversion with fixed rate,
Nakagami-0.5, Optimum rate adaptation with constant transmit
power, Selection combining, Spectral efficiency.
1. Introduction
Wireless communication services, such as wireless personal
area networks, satellite-terrestrial services, wireless mobile
communication services, wireless local-area networks, and
internet access have been growing at a rapid pace in recent
years. These services require high data rate. Thus, channel
capacity is of fundamental importance in the design of
wireless communication systems as it determines the
maximum achievable data rate of the system. Since wireless
mobile channels are subjected to fading, which degrades the
data rate performance. The channel capacity in fading
environment can be improved by employing diversity
combining and / or adaptive transmission schemes [1]-[5].
Diversity combining is known to be a powerful technique
that can be used to combat fading in wireless mobile
environment. Maximal ratio combining, equal gain
combining and SC are most prevalent diversity combing
techniques [3]-[4].
Adaptive transmission is another effective scheme that can
be used to overcome fading. Adaptive transmission requires
accurate channel estimation at the receiver and a reliable
feedback path between the estimator and the transmitter [6].
There are four adaptation transmission schemes such as
ORA, CIFR, optimum power and rate adaptation (OPRA)
and truncated channel inversion with fixed Rate (TIFR) [6]-
[8]. Numerous researchers have worked on the study of
channel capacity over different fading channels. We discuss
here some representative examples. Specifically, [3]-[4]
discuss the channel capacity over correlated Nakagamim ( m = &1 m < )1 fading channels under ORA and CIFR
schemes with different diversity combining techniques. In
[7], the channel capacity over uncorrelated Nakagamim ( m = )1 fading channels with MRC and without diversity
under different adaptive transmissions schemes was
analyzed. Expressions for the capacity over uncorrelated
Rayleigh fading channels with MRC and SC under different
adaptive transmission schemes were obtained in [8]. An
analytical performance study of the channel capacity for
correlated generalized gamma fading channels with dualbranch SC under the different power and rate adaptation
schemes was introduced in [9]. The channel capacity of
Nakagami- m ( m = )1 fading channel without diversity was
derived in [10] for different adaptive transmission schemes.
In [11], channel capacity of dual-branch SC and MRC
systems over correlated Hoyt fading channels using
different adaptive transmission schemes was presented. In
[12], expression for the ergodic capacity of MRC over
arbitrarily correlated Rician fading channels was derived. In
[13], 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. The analytical study of the capacity under
k - µ fading and Weibull fading channels with OPRA,
ORA, CIFR and TIFR adaptation transmission schemes
using different diversity systems was presented in [14]. In
[15], an analytical performance study of the channel
capacity for uncorrelated Nakagami-0.5 with dual-branch
MRC using OPRA and TIFR was obtained. In [16], the
channel capacity over correlated Nakagami-0.5 fading
channels under OPRA and TIFR schemes with MRC was
discussed. An analytical performance study of the channel
capacity for uncorrelated Nakagami-0.5 fading channels
with dual-branch SC under OPRA, and TIFR was
introduced in [17]. The Nakagami-0.5 model has been
widely used in general to study wireless mobile
communication system performance. Results obtained for
Nakagami-0.5 will have great practical usefulness, they will
be of theoretical interest as a worst fading case. This paper
fills this gap by presenting an analytical performance study
124
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
of the channel capacity of dual-branch SC over uncorrelated
Nakagami-0.5 fading channels using ORA, and CIFR
schemes.
In this paper, SC has been considered which is one of the
least complex diversity combining techniques [18].
The remainder of this paper is organized as follows: In
Section 2, the channel model is defined. In Section 3, spectral
efficiency of no diversity and dual-branch SC over
Nakagami-0.5 fading channels are derived for ORA and
CIFR schemes. In Section 4, several numerical results are
presented and analyzed, whereas in Section 5, concluding
remarks are given.
2. Channel Model
We assume slowly-varying Nakagami- m flat fading channel.
The probability distribution function (pdf) of instantaneous
received SNR ? )( of this fading channel is gamma distributed
given by [7]
exp , &0 5.0
)(
)(
1
= = ?
?
?
?
?
?
?
? -
?
?
?
?
?
?
?
?
G
=
-
m
m m
m
p
m m
?
?
?
?
?
? ?
(1)
where m is the Nakagami fading parameter, which measures
the amount of fading, ? is the average received SNR, and
G(.) is the gamma function. For different values of m , this
expression simplifies to several important distributions
describing fading models. Like m = 5.0 corresponds to the
highest amount of fading, m = 1 corresponds to Rayleigh
distribution, m = 1 corresponds to Rician distribution, and as
m ? 8 , the distribution converges to a nonfading AWGN
from [19].
In case of no diversity the pdf under worst case of fading
using (1) is
, 0
2
5.0
exp
)( =
?
?
?
?
?
?
?
? -
= ?
??p
?
?
? p?
(2)
Assuming independent branch signals and equal average
received SNR, the pdf of the received SNR at the output of
dual-branch SC under Nakagami- m fading channels is given
by [19]-[20] is
, 0
2
exp 1 ,0
)(
2
)(
1
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- ?
?
?
?
?
?
?
?
- ?
?
?
?
?
?
?
?
G
=
-
?
?
?
?
?
?
?
? ?
m
Q
m m
m
p m
m m
(3)
where ? is the average received SNR, m (m = )5.0 is the
fading parameter, and Q m (.) is the MarcumQ -function,
which can be represented, when m is not an integer, as
given in [19]
)(
,
2
,0
m
m
m
m
Qm
G
?
?
?
?
?
?
?
?
G
=
?
?
?
?
?
?
?
? ?
?
?
?
where G .],[. is the complementary incomplete gamma
function.
As we consider worst case of fading, then by [21]
?
?
?
?
?
?
?
?
=
G
?
?
?
?
?
?
?
?
G
=
?
?
?
?
?
?
?
? ×
?
? ?
?
?
? 5.0
)5.0(
5.0
,5.0
5.02
,0 5.0 Q erfc
where erfc(.) is called complementary error function. So
that
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
-
?
?
?
5.0 ? 5.0
1 erfc erf
Hence, the pdf of dual-branch SC under worst case of
fading using above mathematical transformation as given in
[17]
, 0
5.0 5.0
exp
2
)( =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - ?
?
?
?
?
??p
? ? p erf (4)
3. Spectral Efficiency
In this section, we present closed-form expressions for the
spectral efficiency of uncorrelated Nakagami-0.5 fading
channels with dual-branch SC and no diversity under ORA,
and CIFR schemes. It is assumed that, for the above
considered adaptation scheme, there exist perfect channel
estimation and an error-free delayless feedback path, similar
to the assumption made in [8].
3.1 ORA
The average channel capacity of fading channel with
received SNR distribution ? )( p?
under ORA scheme
(CORA [bit/sec]) is defined in [6] as
?
8
= +
0
2
log 1( ? ) ? )( ? ? CORA B p d (5)
where B [Hz] is the channel bandwidth.
In fact, (5) represents the capacity of the fading channel
without transmitter feedback (i.e., with the channel fade
level known at the receiver only).
3.1.1 Spectral efficiency in case of no diversity
Substituting (2) into (5), the average channel capacity
becomes
?
8 ?
?
?
?
?
?
?
? -
= +
0
2
2
5.0
exp
log 1( ) ?
??p
?
?
CORA B ? d
?
8 ?
?
?
?
?
?
?
? -
= +
0
5.0
exp
log 1( )
2
443.1
?
?
?
?
?
?p
d
B
CORA
The integral can be solved using partial integration as follows
( ) ( ) ? ?
8
8? ?
8
= - -
0
0
0
dvu lim vu lim vu duv
? ?
Let u = log 1( +? )
then
?
?
+
=
1
d
du
125
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
Now, let ?
?
?
?
dv d
?
?
?
?
?
?
?
? -
=
5.0
exp
After performing integral using [21], we obtain
?
?
?
?
?
?
?
?
=
?
?
?p
5.0
v 2 erf
Evaluating the above integral by using partial integration and
after some mathematical transformation using [21]-[22], we
obtain
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- +
?
?
?
?
?
?
?
?
+ × ?
?
?
?
?
?
?
?
-
=
2
log
2
1
2
1
;2,
2
3
;,1,1 2 2
443.1
?
? ?
?
p?
?
?
e
F ierf
B
CORA
where (.,.;.,.;.) 2 F2
is the generalized hypergeometric
function and erf i(.) is the imaginary error function.
Using that result, we obtain average channel capacity per unit
bandwidth i.e. B
CORA [bit/sec/Hz] said to be spectral
efficiency as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- +
?
?
?
?
?
?
?
?
+ × ?
?
?
?
?
?
?
?
-
= =
2
log
2
1
2
1
;2,
2
3
2 2 ;,1,1
443.1
?
? ?
?
p?
?
?
?
ierf e
F
B
CORA ORA

(6)
where e is the Euler-Mascheroni constant having the value
approximately equal to 0.577215665 given in [19].
3.1.2 Spectral efficiency in case of dual-branch SC
Substituting (4) into (5), the average channel capacity of
dual-branch SC over uncorrelated Nakagami-0.5 fading
channels is
?
8
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= + -
0
5.0 5.0
exp
2
1( ) 2
log ?
?
?
?
?
??p
CORA B ? erf d
(7)
As we know that error function can be represented by [21] as
?
8
=
+
+
-
=
0
2 1
2(! )1
2 )1(
)(
n
n n
n n
z
erf z
p
(8)
Substituting (8) in (7), CORA after some mathematical
transformation
?
?
8
?
?
?
?
?
?
+ -
×
8
=
+
+
-
=
0
5.0
1(log ) exp
0
1
2(! )2()1
)1(
837.1
?
?
?
??
?
d
n
n
n
n n
n
CORA B

Using that result we obtain spectral efficiency i.e.
B
CORA [bit/sec/Hz] as
?
?
8
?
?
?
?
?
?
+ -
×
8
=
+
+
-
=
0
5.0
1(log ) exp
0
1
2(! )2()1
)1(
837.1
?
?
?
??
?
?
d
n
n
n
n n
n
ORA B
(9)
The integral can be solved using partial integration as follows
( ) ( ) ? ?
8
8? ?
8
= - -
0
0
0
dvu lim vu lim vu duv
? ?
Let u = log 1( +? )
then
?
?
+
=
1
d
du
Now let ? ?
?
?
dv d
n
?
?
?
?
?
?
?
?
= -
5.0
exp
After performing integral using [21], we obtain
k kn
n
k
n k
n
v
+ -
=
? -
?
?
?
?
?
?
?
?
-= - ? ?
?
? 1
0
)2(
( )!
5.0 !
exp
Evaluating integral by using partial integral and some
mathematical transformation using [21]-[22], we obtain
? ?
8
= =
+
?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
=
0 0
2( )1
5.0
,
5.0 5.0
)1( exp
837.1
n
n
z
z
n
ORA
n
z
? ? ?
? (10)
The computation of the spectral efficiency according to (10)
requires the computation of an infinite series. To efficiently
compute the series, we truncate the series, and present
bounds for the spectral efficiency.
The spectral efficiency in (10) can be written
as?ORA =?ORA, N +? ORA, E
, where ?ORA, N is the
expression in (10) with the infinite series truncated at the
N th term as
? ?
= =
+
?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
=
N
n
n
z
n
z
z
n
ORA N
0 0
2( )1
5.0
,
5.0 5.0
)1( exp
837.1
,
? ? ?
?
and ? ORA, E
is the truncation error resulting from truncating
the infinite series in (10) at n = N .
The lower bound for ?ORA is derived as
?ORA >?ORA, N +?ORA,E-low
where ?ORA,E-low, which is the lower bound of ? ORA, E
The lower bound for the spectral efficiency can be derived by
using the relationship between the area of the pdf and the
expression of the spectral efficiency as discuss in [13].
As we know that area of pdf )(? ? p is equal to unity.
?
8
= =
0
P p?
? )( d? 1 (11)
Substituting (4) into (11), we get
1
5.0 5.0
exp
2
0
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - ?
8
?
?
?
?
?
??p
P erf d
126
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
After integrating and using manipulation we get
1
2( )1
4 )1(
0
=
+
-
= ?
8
n=
n
n
P
p
Let ?
-
=
-
+
-
=
1
0
1
2( )1
4 )1(
N
n
n
N
n
P
p
(12)
And let
2( )1
4 )1(
1
+
-
? - = ×
N
P
N
N
p
(13)
Then
-1 -1
= ?+
N N
PN P P
Similarly, from (10) let
? ?
-
= =
+
?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- =
1
0 0
2( )1
5.0
,
5.0 5.0
exp)1(
837.1
, 1

N
n
n
z
n
z
z
n
ORA N
? ? ?
? (14)
And
?
=
- ?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
-
? =
N
z
z
N
ORA N z
N
0
, 1
5.0
,
5.0
2( )1
5.0
)1( exp
837.1
? ?
?
? (15)
Dividing (15) by (13), yields
?
=
-
?
?
?
?
?
?
?
?
G - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
-
N
z
z
ORA N
z
P
N 0
, 1 5.0
,
5.0 5.0
443.1 exp

1
? ? ?
?
(16)
Observing that ?
?
?
?
?
?
?
?
?
?
-
-
1
, 1

N
P
?ORA N
monotonically increases with
increasing N , i.e.
1
, , 1
-
-
?
?
>
?
?
N
ORA N
i
ORA i
P P
? ?
for i = N
( ) N
N
ORA N
Ni Ni
i
N
ORA N
ORA i P
P
P
P
-
?
?
? =
?
?
? >
-
-
8
=
8
=
-
- ? ? 1
1
, 1
1
, 1
,
? ?
?
(17)
Hence, the spectral efficiency in (10) can be lower bounded
?ORA,E-low by using (16) and (17) as
?
?
?
?
?
?
?
?
+
-
-
?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
> +
?
?
=
=
N
n
n
N
z
z
ORA ORA N
n
z
0
0
,
2( )1
4 )1(
1
5.0
,
5.0 5.0
443.1 exp
p
? ? ?
? ?
(18)
The upper bound for ? ORA is derived as
?ORA <?ORA, N +?ORA, -upE
where ?ORA, -upE , which is the upper bound of ? ORA, E
The expression in (9) can be written as
?
?
8
8
+=
?
?
?
?
?
?
?
?
-
?
?
?
?
?
?
+ -
×
+
=
0
1
,
!
5.0
5.0
1(log exp)
2( )1
1
2
837.1
?
?
?
?
?
?
?
?
d
n
n
n
Nn
ORA E
(19)
Let
2 1
1
+
=
n
an
,
Then 1
2 3
1 2 1
<
+
+
=
+
n
n
a
a
n
n
i.e. n
a monotonically decreases with increase of n ,
therefore, ?ORA, E
can be upper bounded as

? ?
8 8
+=
?
?
?
?
?
?
?
?
-
?
?
?
?
?
?
?
?
+ -
×
+
<
0 1
,
!
5.0
5.0
1(log exp)
2(2 )3
837.1
Nn
n
ORA E
d
n
N
?
?
?
?
?
?
?
?
? ? ?
8 8
= =
?
?
?
?
??
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
-
-
?
?
?
?
?
?
?
?
-
?
?
?
?
?
?
?
?
+ -
×
+
<
0 0 0
,
!
5.0
!
5.0
5.0
log(1 exp)
2(2 )3
837.1
?
?
?
?
?
?
?
?
?
?
d
n n
N
n
N
n
n n
ORA E
(20)
After evaluating the integral (20) and some mathematical
manipulations using [20]-[21], we obtain the upper bound
?ORA, -upE for, ?ORA, E
as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-G?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
<
??
= =
N
n
n
z
z
n
ORA E
z
E
N
0 0
1
,
5.0
,
5.0
exp
5.0
)1(
1 1
exp5.0
2 3
837.1
? ? ?
? ?
? (21)
Therefore, the spectral efficiency in (10) can be upper
bounded as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-G?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
×
+
= + < +
? ?
= =
N
n
n
z
z
n
ORA ORA N ORA E ORA N
z
E
N
0 0
1
, , ,
5.0
,
5.0
exp
5.0
)1(
1 1
exp5.0
2 3
837.1
? ? ?
? ?
? ? ? ?
(22)
where (.) E1
is the exponential integral of first order.
Hence, the spectral efficiency is bounded using (18) and (22)
as
127
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
?
?
?
?
?
?
?
?
+
-
-
×?
?
?
?
?
?
?
?
G - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
> >
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
G - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
?
??
=
=
= =
N
n
n
N
z
z
ORA N
ORA
N
n
n
z
z
n
ORA N
n
z
z
E
N
0
0
,
0 0
1
,
2( )1
4 )1(
1
5.0
,
5.0 5.0
443.1 exp
5.0
,
5.0
exp
5.0
)1(
1 1
exp5.0
2 3
837.1
p
? ? ?
?
?
? ? ?
? ?
?

(23)
3.2 CIFR
The average channel capacity of fading channel with
received SNR distribution ? )( p?
under CIFR scheme
(CCIFR [bit/sec]) is defined in [6] as
, 0
)(
1
log 1
0
2 =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= +
?
8
?
?
?
? ?
d
p
CCIFR B (24)
Channel inversion with fixed rate is the least complex
technique to implement, assuming good channel estimates are
available at the transmitter and receiver.
3.2.1 Spectral Efficiency in case of no Diversity
The pdf for Nakagami-0.5 fading channel is given in (2) as
, 0
2
5.0
exp
)( =
?
?
?
?
?
?
?
? -
= ?
??p
?
?
? ? p
Hence,
, 0
1
2
5.0
exp )(
× =
?
?
?
?
?
?
?
? -
= ?
??p ?
?
?
?
? ? p
(25)
Integrating (25) over an interval as shown below
? ?
8 8
×
?
?
?
?
?
?
?
? -
=
0 0
1
2
5.0
exp )(
?
??p ?
?
?
?
?
? ?
d d
p
Evaluating integral by some manipulation using [21], we
obtain
8
8
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? -
-
?
?
?
?
?
?
?
?
= -
?
0
0
5.0
exp2
2 5.0
2
)( 1
?
?
?
?
?
?
p
?p
?
?
? ?
d erf
p
As we know that
erf (0) = 0 , erf (8) =1,
And
0
5.0
exp
lim =
?
?
?
?
?
?
?
? -
8? ?
?
?
?
, as ? > 0
Hence
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - - - 8- ?
8
0( )1 0(2 )
2
2
)( 1
0
?
p
?p
?
?
? ?
d
p
8= ?
8
0
)(
?
?
? ?
d
p
Putting this value of integral in (24), we get
log )1( 0 CCIFR = B 2 = (26)
So, the spectral efficiency with no diversity under Nakagami0.5 fading channel is zero.
3.2.2 Spectral Efficiency in case of dual-branch SC
The pdf of dual-branch SC over uncorrelated Nakagami-0.5
fading channel is given in (4) is
, 0
5.0 5.0
exp
2
)( =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - ?
?
?
?
?
??p
? ? p erf
Hence
, 0
5.0 5.0 1
exp
)( 2
× =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - ?
? ?
?
?
?
? ??p
? ?
erf
p
(27)
Integrating the (27) over an interval as shown below
? ?
8 8
×
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= -
0 0
5.0 5.0 1
exp
)( 2
?
? ?
?
?
?
??p
?
?
? ?
d erf d
p
After evaluating the integral using [21]-[22], we obtain
8 8
=
+
8
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- G
+
-
?
=?
0 0
1
0
5.0
)2( ,
)2( 2(! )1
)( )1(4
?
?
?
p ?
?
?
? ?
n
n n
d
p
n
n
n
n
As we know that
G n )0,( = G n)(
And
lim G ),( = 0
8?
xn
x
G× 8=
+
-
?
=?
8
=
8
)(
2(!)2( )1
)( )1(4
0 0
n
n n
d
p
n
n
p ?
?
?
? ?
Putting this value of integral in (24), we get
log )1( 0 CCIFR = B 2 = (28)
4. Numerical Results and Analysis
In this section, various performance evaluation results for the
spectral efficiency have been obtained using dual-branch SC
and no diversity under worst fading condition. These results
also focus on spectral efficiency comparisons between the
different adaptive transmission schemes.
In Fig. 1, the spectral efficiency under ORA scheme is
plotted as a function of the average received SNR per
128
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
branch ? . As expected, by increasing ? and/or employing
diversity, spectral efficiency improves.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ bit / sec / Hz ]
Dual - branch SC
No diversity
It is seen in Table. 1 that as the truncation error bounds
becomes tighter as the truncation level, N , increases. It
means that as the truncation level increases the difference
between upper and lower bounds for each average received
SNR per branch ? decreases and hence calculated spectral
efficiency becomes more appropriate. That's why the infinite
series in ?ORA has been truncated at the 15th term to
calculate the spectral efficiency for the Fig. 1.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ b i t / sec / H z ]
Dual - branch SC for m = 1
No diversity for m = 1
Dual - branch SC for m = 0 . 5
No diversity for m = 0 . 5
In Fig. 2, the spectral efficiency under ORA scheme is
plotted as a function of the average received SNR per
branch ? . For comparison, the spectral efficiency of
uncorrelated Rayleigh fading channels with dual-branch SC
and without diversity, which was obtained in [8, Eq. (44)]
and [8, Eq. (34)] respectively, is also presented in Fig. 2. As
expected, as the channel fading conditions improves, i.e, m
and / or ? increases, spectral efficiency improves. It is very
interesting to observe that the spectral efficiency without
diversity for ? -= 5.7 dB remains same as we move from
worst fading condition to Rayleigh. Similarly, dual-branch
SC for ? -= 5.2 dB , gives almost identical performance even
we move from worst fading condition to Rayleigh.
Fig. 3 states that spectral efficiency versus average received
SNR per branch ? over Nakagami-0.5 fading channel
remains zero as we go from no diversity to dual-branch SC
under CIFR scheme.
-10 -5 0 5 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ bit / sec / Hz ]
No diversity or Dual - branch SC under CIFR
Figure 1. Spectral Efficiency for a worst case of fading
versus average received SNR? under ORA.
Figure 2. Spectral Efficiency versus average received
SNR? under ORA.
Figure 3. Spectral Efficiency for a worst-case of
fading versus average received SNR? .
Table 1. Comparison of?ORA,N , ?ORA, -upE ,and
?ORA, E-low at two different values of N for worst
case of fading.
N = 5
? [dB] ?ORA, N ? ORA, -upE ?ORA, E-low
-10 0.143708 0.061883 0.0581541295
-5 0.411463 0.120879 0.1153123659
0 0.977032 0.196274 0.1896116138
5 1.89534 0.27954 0.2723395591
10 3.10794 0.36565 0.3582797952
N =15
? [dB] ?ORA, N ?ORA, -upE ?ORA, E-low
-10 0.164232 0.041359 0.04064235544
-5 0.463140 0.069202 0.06833679633
0 1.07295 0.10041 0.09942977640
5 2.04205 0.13283 0.13180969046
10 3.30793 0.16564 0.164623141056
129
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ b i t / sec / H z ]
Dual - branch SC for m = 1
No diversity for m = 0.5 , 1 or Dual - branch SC for m = 0.5
In Fig. 4, the spectral efficiency under CIFR is plotted as a
function of the average received SNR per branch ? . For
comparison, the spectral efficiency of uncorrelated Rayleigh
fading channels with dual-branch SC, which was obtained in
[8, Eq. (52)], is also presented in Fig. 4. It is observed that,
as the channel fading conditions improve, i.e, m and / or ?
increases, spectral efficiency with no diversity remains zero.
However, employing a dual-branch SC system improves the
channel capacity as we go from worst case of fading
conditions to Rayleigh fading conditions.
In Fig. 5, the spectral efficiency of uncorrelated Nakagami0.5 fading channels with and without diversity is plotted as a
function of ? , considering ORA, OPRA, and TIFR
adaptation schemes with the aid of (6), (23), [17, Eq. (27)],
[17, Eq. (35)], [15, Eq. (8)], and [15, Eq. (22)]. It shows that,
the spectral efficiency with no diversity under ORA scheme
improves over TIFR for ? =5dB . It is also observed that the
spectral efficiency with dual-branch SC under ORA scheme
improves over TIFR for ? =0dB and OPRA scheme provides
better efficiency under worst case of fading. It is also
interesting to observe that for ? -= 5.7 dB ORA scheme with
dual-branch SC gives inferior performance with respect to
TIFR scheme without diversity.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ b i t / sec / H z ]
Dual - branch SC under OPRA
No diversity under OPRA
Dual - branch SC under ORA
No diversity under ORA
Dual - branch SC under TIFR
No diversity under TIFR
5. Conclusions
In this paper, closed-form expressions for the spectral
efficiency of dual-branch SC and no diversity under ORA
and CIFR schemes have been obtained and analyzed. Error
bounds have been derived for truncated infinite series under
ORA. Numerical results illustrate that the bounds can be used
effectively to determine the number of terms needed to
achieve desirable level of accuracy. Results have been
plotted, which show that by increasing ? and/or employing
diversity, spectral efficiency improves under ORA scheme. It
is also observed that the spectral efficiency under CIFR is
zero under worst case of fading even when a dual-branch SC
is utilized. It is important to note that the spectral efficiency
using ORA scheme remains almost same even when fading
conditions improve from Nakagami-0.5 to Rayleigh either
under dual-branch SC for ? -= 5.2 dB or under no diversity
for ? -= 5.7 dB . This paper finally concludes that under
worst case of fading TIFR scheme is a better choice for low
average received SNR and ORA scheme is for high average
received SNR even employing diversity. Therefore it is
recommended that under worst fading condition, ORA
scheme is not always a better choice over TIFR.
Figure 4. Spectral Efficiency versus average received
SNR? under CIFR.
Figure 5. Spectral Efficiency versus average received
SNR? under different adaptive transmission schemes.
130
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
References
[1] A. Iqbal, A.M. Kazi, "Integrated Satellite-Terrestrial System
Capacity Over Mix Shadowed Rician and Nakagami
Channels," International Journal of Communication Networks
and Information Security (IJCNIS), Vol. 5, No. 2, pp. 104-
109, 2013.
[2] R.Saadane, M. Wahbi, "UWB Indoor Radio Propagation
Modelling in Presence of Human Body Shadowing Using Ray
Tracing Technique,'' International Journal of Communication
Networks and Information Security (IJCNIS) Vol. 4, No. 2,
2012.
[3] S. Khatalin, J.P.Fonseka, "Channel capacity of dual-branch
diversity systems over correlated Nakagami-m fading with
channel inversion and fixed rate transmission scheme," IET
Communications, Vol. 1, No.6, pp.1161-1169, 2007.
[4] S.Khatalin, J.P.Fonseka, "Capacity of Correlated Nakagami-m
Fading Channels With Diversity Combining Techniques,"
IEEE Transactions on Vehicular Technology.," Vol. 55,
No.1, pp.142-150, 2006.
[5] V. Hentinen, "Error performance for adaptive transmission on
fading channels," IEEE Transactions on Communications,
Vol. 22, No. 9, pp. 1331-1337, 1974.
[6] A. J. Goldsmith, P. P. Varaiya, "Capacity of Fading Channels
with Channel Side Information," IEEE Transactions on
Information Theory," Vol. 43, No. 6, pp. 1986-1992, 1997.
[7] M. S. Alouini, A. Goldsmith, "Capacity of Nakagami
multipath fading channels," Proceedings of the IEEE
Vehicular Technology Conference., Phoenix, AZ, pp. 358-
362, 1997.
[8] M. S. Alouini, A.J. Goldsmith, "Capacity of Rayleigh Fading
Channels Under Different Adaptive Transmission and
Diversity-Combining Techniques," IEEE Transactions on
Vehicular Technology, Vol. 48, No. 4, pp.1165-1181, 1999.
[9] P.S.Bithas, P.T.Mathiopoulos, "Capacity of Correlated
Generalized Gamma Fading With Dual-Branch Selection
Diversity," IEEE Transactions on Vehicular Technology, Vol.
58, No. 9, pp.5258-5263, 2009.
[10] M .S. Alouini, A. Goldsmith, "Adaptive Modulation over
Nakagami Fading channels," Wireless Personal
Communication, Vol. 13, pp. 119-143, 2000.
[11] R. Subadar, P. R. Sahu, "Capacity Analysis of Dual-SC andMRC Systems over Correlated Hoyt Fading Channels," IEEE
Conference, TENCON, pp.1-5, 2009.
[12] K. A. Hamdi, "Capacity of MRC on Correlated Rician Fading
Channels," IEEE Transaction. on Communication, Vol. 56,
No. 5, pp. 708-711, 2008.
[13] S. Khatalin, J.P.Fonseka, "On the Channel Capacity in Rician
and Hoyt Fading Environments With MRC Diversity," IEEE
Transactions on Vehicular Technology, Vol.55, No.1,
pp.137-141, 2006.
[14] M. Stefanovic, J. Anastasov, S. Panic,.P. Spalevic, C.
Dolicanin, Chapter 10 "Channel Capacity Analysis Under
Various Adaptation Policies and Diversity Techniques over
Fading Channels," in the book Wireless Communications and
Networks- Recent Advances, In Tech, 2012.
[15] M.I.Hasan, S.Kumar, "Channel Capacity of Dual-Branch
Maximal Ratio Combining under worst case of fading
scenario," WSEAS Transactions on Communications, Vol.
13, pp. 162-170, 2014.
[16] M.I.Hasan, S.Kumar, "Average Channel Capacity of
Correlated Dual-Branch Maximal Ratio Combining Under
Worst Case of Fading Scenario," Wireless Personal
Communication, Vol. 83, No. 4, pp. 2623-2646, 2015.
[17] M.I.Hasan, S.Kumar, "Average Channel Capacity Evaluation
for Selection Combining Diversity Schemes over Nakagami0.5 Fading Channels," International Journal of
Communication Networks and Information Security
(IJCNIS), Vol. 7, No. 2, pp. 69-79, 2015.
[18] D.Brennan, "Linear Diversity Combining techniques,"
Proceedings of IEEE, Vol.91, No.2, pp.331-354. 2003.
[19] M. K. Simon, M. S. Alouini, Digital Communication over
Fading Channels, 2nd ed., New York: Wiley, 2005.
[20] G. Fodele, I.Izzo, M. Tanda, "Dual diversity reception of Mary DPSK signals over Nakagami fading channels," In
Proceedings of the IEEE International Symposium Personal,
Indoor, and Mobile Radio Communications, Toronto, Ont.,
Canada, pp.1195-1201, 1995.
[21] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, series, and
products, 6th ed., New York: Academic Press, 2000.
[22] Wolfram, The Wolfram functions site (2015), Internet (online),
http://functions.wolfram.com. , Claims:
Wireless communication services, such as wireless personal
area networks, satellite-terrestrial services, wireless mobile
communication services, wireless local-area networks, and
internet access have been growing at a rapid pace in recent
years. These services require high data rate. Thus, channel
capacity is of fundamental importance in the design of
wireless communication systems as it determines the
maximum achievable data rate of the system. Since wireless
mobile channels are subjected to fading, which degrades the
data rate performance. The channel capacity in fading
environment can be improved by employing diversity
combining and / or adaptive transmission schemes [1]-[5].
Diversity combining is known to be a powerful technique
that can be used to combat fading in wireless mobile
environment. Maximal ratio combining, equal gain
combining and SC are most prevalent diversity combing
techniques [3]-[4].
Adaptive transmission is another effective scheme that can
be used to overcome fading. Adaptive transmission requires
accurate channel estimation at the receiver and a reliable
feedback path between the estimator and the transmitter [6].
There are four adaptation transmission schemes such as
ORA, CIFR, optimum power and rate adaptation (OPRA)
and truncated channel inversion with fixed Rate (TIFR) [6]-
[8]. Numerous researchers have worked on the study of
channel capacity over different fading channels. We discuss
here some representative examples. Specifically, [3]-[4]
discuss the channel capacity over correlated Nakagamim ( m = &1 m < )1 fading channels under ORA and CIFR
schemes with different diversity combining techniques. In
[7], the channel capacity over uncorrelated Nakagamim ( m = )1 fading channels with MRC and without diversity
under different adaptive transmissions schemes was
analyzed. Expressions for the capacity over uncorrelated
Rayleigh fading channels with MRC and SC under different
adaptive transmission schemes were obtained in [8]. An
analytical performance study of the channel capacity for
correlated generalized gamma fading channels with dualbranch SC under the different power and rate adaptation
schemes was introduced in [9]. The channel capacity of
Nakagami- m ( m = )1 fading channel without diversity was
derived in [10] for different adaptive transmission schemes.
In [11], channel capacity of dual-branch SC and MRC
systems over correlated Hoyt fading channels using
different adaptive transmission schemes was presented. In
[12], expression for the ergodic capacity of MRC over
arbitrarily correlated Rician fading channels was derived. In
[13], 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. The analytical study of the capacity under
k - µ fading and Weibull fading channels with OPRA,
ORA, CIFR and TIFR adaptation transmission schemes
using different diversity systems was presented in [14]. In
[15], an analytical performance study of the channel
capacity for uncorrelated Nakagami-0.5 with dual-branch
MRC using OPRA and TIFR was obtained. In [16], the
channel capacity over correlated Nakagami-0.5 fading
channels under OPRA and TIFR schemes with MRC was
discussed. An analytical performance study of the channel
capacity for uncorrelated Nakagami-0.5 fading channels
with dual-branch SC under OPRA, and TIFR was
introduced in [17]. The Nakagami-0.5 model has been
widely used in general to study wireless mobile
communication system performance. Results obtained for
Nakagami-0.5 will have great practical usefulness, they will
be of theoretical interest as a worst fading case. This paper
fills this gap by presenting an analytical performance study
124
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
of the channel capacity of dual-branch SC over uncorrelated
Nakagami-0.5 fading channels using ORA, and CIFR
schemes.
In this paper, SC has been considered which is one of the
least complex diversity combining techniques [18].
The remainder of this paper is organized as follows: In
Section 2, the channel model is defined. In Section 3, spectral
efficiency of no diversity and dual-branch SC over
Nakagami-0.5 fading channels are derived for ORA and
CIFR schemes. In Section 4, several numerical results are
presented and analyzed, whereas in Section 5, concluding
remarks are given.
2. Channel Model
We assume slowly-varying Nakagami- m flat fading channel.
The probability distribution function (pdf) of instantaneous
received SNR ? )( of this fading channel is gamma distributed
given by [7]
exp , &0 5.0
)(
)(
1
= = ?
?
?
?
?
?
?
? -
?
?
?
?
?
?
?
?
G
=
-
m
m m
m
p
m m
?
?
?
?
?
? ?
(1)
where m is the Nakagami fading parameter, which measures
the amount of fading, ? is the average received SNR, and
G(.) is the gamma function. For different values of m , this
expression simplifies to several important distributions
describing fading models. Like m = 5.0 corresponds to the
highest amount of fading, m = 1 corresponds to Rayleigh
distribution, m = 1 corresponds to Rician distribution, and as
m ? 8 , the distribution converges to a nonfading AWGN
from [19].
In case of no diversity the pdf under worst case of fading
using (1) is
, 0
2
5.0
exp
)( =
?
?
?
?
?
?
?
? -
= ?
??p
?
?
? p?
(2)
Assuming independent branch signals and equal average
received SNR, the pdf of the received SNR at the output of
dual-branch SC under Nakagami- m fading channels is given
by [19]-[20] is
, 0
2
exp 1 ,0
)(
2
)(
1
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- ?
?
?
?
?
?
?
?
- ?
?
?
?
?
?
?
?
G
=
-
?
?
?
?
?
?
?
? ?
m
Q
m m
m
p m
m m
(3)
where ? is the average received SNR, m (m = )5.0 is the
fading parameter, and Q m (.) is the MarcumQ -function,
which can be represented, when m is not an integer, as
given in [19]
)(
,
2
,0
m
m
m
m
Qm
G
?
?
?
?
?
?
?
?
G
=
?
?
?
?
?
?
?
? ?
?
?
?
where G .],[. is the complementary incomplete gamma
function.
As we consider worst case of fading, then by [21]
?
?
?
?
?
?
?
?
=
G
?
?
?
?
?
?
?
?
G
=
?
?
?
?
?
?
?
? ×
?
? ?
?
?
? 5.0
)5.0(
5.0
,5.0
5.02
,0 5.0 Q erfc
where erfc(.) is called complementary error function. So
that
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
-
?
?
?
5.0 ? 5.0
1 erfc erf
Hence, the pdf of dual-branch SC under worst case of
fading using above mathematical transformation as given in
[17]
, 0
5.0 5.0
exp
2
)( =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - ?
?
?
?
?
??p
? ? p erf (4)
3. Spectral Efficiency
In this section, we present closed-form expressions for the
spectral efficiency of uncorrelated Nakagami-0.5 fading
channels with dual-branch SC and no diversity under ORA,
and CIFR schemes. It is assumed that, for the above
considered adaptation scheme, there exist perfect channel
estimation and an error-free delayless feedback path, similar
to the assumption made in [8].
3.1 ORA
The average channel capacity of fading channel with
received SNR distribution ? )( p?
under ORA scheme
(CORA [bit/sec]) is defined in [6] as
?
8
= +
0
2
log 1( ? ) ? )( ? ? CORA B p d (5)
where B [Hz] is the channel bandwidth.
In fact, (5) represents the capacity of the fading channel
without transmitter feedback (i.e., with the channel fade
level known at the receiver only).
3.1.1 Spectral efficiency in case of no diversity
Substituting (2) into (5), the average channel capacity
becomes
?
8 ?
?
?
?
?
?
?
? -
= +
0
2
2
5.0
exp
log 1( ) ?
??p
?
?
CORA B ? d
?
8 ?
?
?
?
?
?
?
? -
= +
0
5.0
exp
log 1( )
2
443.1
?
?
?
?
?
?p
d
B
CORA
The integral can be solved using partial integration as follows
( ) ( ) ? ?
8
8? ?
8
= - -
0
0
0
dvu lim vu lim vu duv
? ?
Let u = log 1( +? )
then
?
?
+
=
1
d
du
125
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
Now, let ?
?
?
?
dv d
?
?
?
?
?
?
?
? -
=
5.0
exp
After performing integral using [21], we obtain
?
?
?
?
?
?
?
?
=
?
?
?p
5.0
v 2 erf
Evaluating the above integral by using partial integration and
after some mathematical transformation using [21]-[22], we
obtain
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- +
?
?
?
?
?
?
?
?
+ × ?
?
?
?
?
?
?
?
-
=
2
log
2
1
2
1
;2,
2
3
;,1,1 2 2
443.1
?
? ?
?
p?
?
?
e
F ierf
B
CORA
where (.,.;.,.;.) 2 F2
is the generalized hypergeometric
function and erf i(.) is the imaginary error function.
Using that result, we obtain average channel capacity per unit
bandwidth i.e. B
CORA [bit/sec/Hz] said to be spectral
efficiency as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- +
?
?
?
?
?
?
?
?
+ × ?
?
?
?
?
?
?
?
-
= =
2
log
2
1
2
1
;2,
2
3
2 2 ;,1,1
443.1
?
? ?
?
p?
?
?
?
ierf e
F
B
CORA ORA

(6)
where e is the Euler-Mascheroni constant having the value
approximately equal to 0.577215665 given in [19].
3.1.2 Spectral efficiency in case of dual-branch SC
Substituting (4) into (5), the average channel capacity of
dual-branch SC over uncorrelated Nakagami-0.5 fading
channels is
?
8
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= + -
0
5.0 5.0
exp
2
1( ) 2
log ?
?
?
?
?
??p
CORA B ? erf d
(7)
As we know that error function can be represented by [21] as
?
8
=
+
+
-
=
0
2 1
2(! )1
2 )1(
)(
n
n n
n n
z
erf z
p
(8)
Substituting (8) in (7), CORA after some mathematical
transformation
?
?
8
?
?
?
?
?
?
+ -
×
8
=
+
+
-
=
0
5.0
1(log ) exp
0
1
2(! )2()1
)1(
837.1
?
?
?
??
?
d
n
n
n
n n
n
CORA B

Using that result we obtain spectral efficiency i.e.
B
CORA [bit/sec/Hz] as
?
?
8
?
?
?
?
?
?
+ -
×
8
=
+
+
-
=
0
5.0
1(log ) exp
0
1
2(! )2()1
)1(
837.1
?
?
?
??
?
?
d
n
n
n
n n
n
ORA B
(9)
The integral can be solved using partial integration as follows
( ) ( ) ? ?
8
8? ?
8
= - -
0
0
0
dvu lim vu lim vu duv
? ?
Let u = log 1( +? )
then
?
?
+
=
1
d
du
Now let ? ?
?
?
dv d
n
?
?
?
?
?
?
?
?
= -
5.0
exp
After performing integral using [21], we obtain
k kn
n
k
n k
n
v
+ -
=
? -
?
?
?
?
?
?
?
?
-= - ? ?
?
? 1
0
)2(
( )!
5.0 !
exp
Evaluating integral by using partial integral and some
mathematical transformation using [21]-[22], we obtain
? ?
8
= =
+
?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
=
0 0
2( )1
5.0
,
5.0 5.0
)1( exp
837.1
n
n
z
z
n
ORA
n
z
? ? ?
? (10)
The computation of the spectral efficiency according to (10)
requires the computation of an infinite series. To efficiently
compute the series, we truncate the series, and present
bounds for the spectral efficiency.
The spectral efficiency in (10) can be written
as?ORA =?ORA, N +? ORA, E
, where ?ORA, N is the
expression in (10) with the infinite series truncated at the
N th term as
? ?
= =
+
?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
=
N
n
n
z
n
z
z
n
ORA N
0 0
2( )1
5.0
,
5.0 5.0
)1( exp
837.1
,
? ? ?
?
and ? ORA, E
is the truncation error resulting from truncating
the infinite series in (10) at n = N .
The lower bound for ?ORA is derived as
?ORA >?ORA, N +?ORA,E-low
where ?ORA,E-low, which is the lower bound of ? ORA, E
The lower bound for the spectral efficiency can be derived by
using the relationship between the area of the pdf and the
expression of the spectral efficiency as discuss in [13].
As we know that area of pdf )(? ? p is equal to unity.
?
8
= =
0
P p?
? )( d? 1 (11)
Substituting (4) into (11), we get
1
5.0 5.0
exp
2
0
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - ?
8
?
?
?
?
?
??p
P erf d
126
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
After integrating and using manipulation we get
1
2( )1
4 )1(
0
=
+
-
= ?
8
n=
n
n
P
p
Let ?
-
=
-
+
-
=
1
0
1
2( )1
4 )1(
N
n
n
N
n
P
p
(12)
And let
2( )1
4 )1(
1
+
-
? - = ×
N
P
N
N
p
(13)
Then
-1 -1
= ?+
N N
PN P P
Similarly, from (10) let
? ?
-
= =
+
?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- =
1
0 0
2( )1
5.0
,
5.0 5.0
exp)1(
837.1
, 1

N
n
n
z
n
z
z
n
ORA N
? ? ?
? (14)
And
?
=
- ?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
-
? =
N
z
z
N
ORA N z
N
0
, 1
5.0
,
5.0
2( )1
5.0
)1( exp
837.1
? ?
?
? (15)
Dividing (15) by (13), yields
?
=
-
?
?
?
?
?
?
?
?
G - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
-
N
z
z
ORA N
z
P
N 0
, 1 5.0
,
5.0 5.0
443.1 exp

1
? ? ?
?
(16)
Observing that ?
?
?
?
?
?
?
?
?
?
-
-
1
, 1

N
P
?ORA N
monotonically increases with
increasing N , i.e.
1
, , 1
-
-
?
?
>
?
?
N
ORA N
i
ORA i
P P
? ?
for i = N
( ) N
N
ORA N
Ni Ni
i
N
ORA N
ORA i P
P
P
P
-
?
?
? =
?
?
? >
-
-
8
=
8
=
-
- ? ? 1
1
, 1
1
, 1
,
? ?
?
(17)
Hence, the spectral efficiency in (10) can be lower bounded
?ORA,E-low by using (16) and (17) as
?
?
?
?
?
?
?
?
+
-
-
?
?
?
?
?
?
?
?
-G ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
> +
?
?
=
=
N
n
n
N
z
z
ORA ORA N
n
z
0
0
,
2( )1
4 )1(
1
5.0
,
5.0 5.0
443.1 exp
p
? ? ?
? ?
(18)
The upper bound for ? ORA is derived as
?ORA <?ORA, N +?ORA, -upE
where ?ORA, -upE , which is the upper bound of ? ORA, E
The expression in (9) can be written as
?
?
8
8
+=
?
?
?
?
?
?
?
?
-
?
?
?
?
?
?
+ -
×
+
=
0
1
,
!
5.0
5.0
1(log exp)
2( )1
1
2
837.1
?
?
?
?
?
?
?
?
d
n
n
n
Nn
ORA E
(19)
Let
2 1
1
+
=
n
an
,
Then 1
2 3
1 2 1
<
+
+
=
+
n
n
a
a
n
n
i.e. n
a monotonically decreases with increase of n ,
therefore, ?ORA, E
can be upper bounded as

? ?
8 8
+=
?
?
?
?
?
?
?
?
-
?
?
?
?
?
?
?
?
+ -
×
+
<
0 1
,
!
5.0
5.0
1(log exp)
2(2 )3
837.1
Nn
n
ORA E
d
n
N
?
?
?
?
?
?
?
?
? ? ?
8 8
= =
?
?
?
?
??
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
-
-
?
?
?
?
?
?
?
?
-
?
?
?
?
?
?
?
?
+ -
×
+
<
0 0 0
,
!
5.0
!
5.0
5.0
log(1 exp)
2(2 )3
837.1
?
?
?
?
?
?
?
?
?
?
d
n n
N
n
N
n
n n
ORA E
(20)
After evaluating the integral (20) and some mathematical
manipulations using [20]-[21], we obtain the upper bound
?ORA, -upE for, ?ORA, E
as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-G?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
<
??
= =
N
n
n
z
z
n
ORA E
z
E
N
0 0
1
,
5.0
,
5.0
exp
5.0
)1(
1 1
exp5.0
2 3
837.1
? ? ?
? ?
? (21)
Therefore, the spectral efficiency in (10) can be upper
bounded as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-G?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
×
+
= + < +
? ?
= =
N
n
n
z
z
n
ORA ORA N ORA E ORA N
z
E
N
0 0
1
, , ,
5.0
,
5.0
exp
5.0
)1(
1 1
exp5.0
2 3
837.1
? ? ?
? ?
? ? ? ?
(22)
where (.) E1
is the exponential integral of first order.
Hence, the spectral efficiency is bounded using (18) and (22)
as
127
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
?
?
?
?
?
?
?
?
+
-
-
×?
?
?
?
?
?
?
?
G - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
> >
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
G - ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
?
??
=
=
= =
N
n
n
N
z
z
ORA N
ORA
N
n
n
z
z
n
ORA N
n
z
z
E
N
0
0
,
0 0
1
,
2( )1
4 )1(
1
5.0
,
5.0 5.0
443.1 exp
5.0
,
5.0
exp
5.0
)1(
1 1
exp5.0
2 3
837.1
p
? ? ?
?
?
? ? ?
? ?
?

(23)
3.2 CIFR
The average channel capacity of fading channel with
received SNR distribution ? )( p?
under CIFR scheme
(CCIFR [bit/sec]) is defined in [6] as
, 0
)(
1
log 1
0
2 =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= +
?
8
?
?
?
? ?
d
p
CCIFR B (24)
Channel inversion with fixed rate is the least complex
technique to implement, assuming good channel estimates are
available at the transmitter and receiver.
3.2.1 Spectral Efficiency in case of no Diversity
The pdf for Nakagami-0.5 fading channel is given in (2) as
, 0
2
5.0
exp
)( =
?
?
?
?
?
?
?
? -
= ?
??p
?
?
? ? p
Hence,
, 0
1
2
5.0
exp )(
× =
?
?
?
?
?
?
?
? -
= ?
??p ?
?
?
?
? ? p
(25)
Integrating (25) over an interval as shown below
? ?
8 8
×
?
?
?
?
?
?
?
? -
=
0 0
1
2
5.0
exp )(
?
??p ?
?
?
?
?
? ?
d d
p
Evaluating integral by some manipulation using [21], we
obtain
8
8
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? -
-
?
?
?
?
?
?
?
?
= -
?
0
0
5.0
exp2
2 5.0
2
)( 1
?
?
?
?
?
?
p
?p
?
?
? ?
d erf
p
As we know that
erf (0) = 0 , erf (8) =1,
And
0
5.0
exp
lim =
?
?
?
?
?
?
?
? -
8? ?
?
?
?
, as ? > 0
Hence
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - - - 8- ?
8
0( )1 0(2 )
2
2
)( 1
0
?
p
?p
?
?
? ?
d
p
8= ?
8
0
)(
?
?
? ?
d
p
Putting this value of integral in (24), we get
log )1( 0 CCIFR = B 2 = (26)
So, the spectral efficiency with no diversity under Nakagami0.5 fading channel is zero.
3.2.2 Spectral Efficiency in case of dual-branch SC
The pdf of dual-branch SC over uncorrelated Nakagami-0.5
fading channel is given in (4) is
, 0
5.0 5.0
exp
2
)( =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - ?
?
?
?
?
??p
? ? p erf
Hence
, 0
5.0 5.0 1
exp
)( 2
× =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= - ?
? ?
?
?
?
? ??p
? ?
erf
p
(27)
Integrating the (27) over an interval as shown below
? ?
8 8
×
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
= -
0 0
5.0 5.0 1
exp
)( 2
?
? ?
?
?
?
??p
?
?
? ?
d erf d
p
After evaluating the integral using [21]-[22], we obtain
8 8
=
+
8
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- G
+
-
?
=?
0 0
1
0
5.0
)2( ,
)2( 2(! )1
)( )1(4
?
?
?
p ?
?
?
? ?
n
n n
d
p
n
n
n
n
As we know that
G n )0,( = G n)(
And
lim G ),( = 0
8?
xn
x
G× 8=
+
-
?
=?
8
=
8
)(
2(!)2( )1
)( )1(4
0 0
n
n n
d
p
n
n
p ?
?
?
? ?
Putting this value of integral in (24), we get
log )1( 0 CCIFR = B 2 = (28)
4. Numerical Results and Analysis
In this section, various performance evaluation results for the
spectral efficiency have been obtained using dual-branch SC
and no diversity under worst fading condition. These results
also focus on spectral efficiency comparisons between the
different adaptive transmission schemes.
In Fig. 1, the spectral efficiency under ORA scheme is
plotted as a function of the average received SNR per
128
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
branch ? . As expected, by increasing ? and/or employing
diversity, spectral efficiency improves.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ bit / sec / Hz ]
Dual - branch SC
No diversity
It is seen in Table. 1 that as the truncation error bounds
becomes tighter as the truncation level, N , increases. It
means that as the truncation level increases the difference
between upper and lower bounds for each average received
SNR per branch ? decreases and hence calculated spectral
efficiency becomes more appropriate. That's why the infinite
series in ?ORA has been truncated at the 15th term to
calculate the spectral efficiency for the Fig. 1.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ b i t / sec / H z ]
Dual - branch SC for m = 1
No diversity for m = 1
Dual - branch SC for m = 0 . 5
No diversity for m = 0 . 5
In Fig. 2, the spectral efficiency under ORA scheme is
plotted as a function of the average received SNR per
branch ? . For comparison, the spectral efficiency of
uncorrelated Rayleigh fading channels with dual-branch SC
and without diversity, which was obtained in [8, Eq. (44)]
and [8, Eq. (34)] respectively, is also presented in Fig. 2. As
expected, as the channel fading conditions improves, i.e, m
and / or ? increases, spectral efficiency improves. It is very
interesting to observe that the spectral efficiency without
diversity for ? -= 5.7 dB remains same as we move from
worst fading condition to Rayleigh. Similarly, dual-branch
SC for ? -= 5.2 dB , gives almost identical performance even
we move from worst fading condition to Rayleigh.
Fig. 3 states that spectral efficiency versus average received
SNR per branch ? over Nakagami-0.5 fading channel
remains zero as we go from no diversity to dual-branch SC
under CIFR scheme.
-10 -5 0 5 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ bit / sec / Hz ]
No diversity or Dual - branch SC under CIFR
Figure 1. Spectral Efficiency for a worst case of fading
versus average received SNR? under ORA.
Figure 2. Spectral Efficiency versus average received
SNR? under ORA.
Figure 3. Spectral Efficiency for a worst-case of
fading versus average received SNR? .
Table 1. Comparison of?ORA,N , ?ORA, -upE ,and
?ORA, E-low at two different values of N for worst
case of fading.
N = 5
? [dB] ?ORA, N ? ORA, -upE ?ORA, E-low
-10 0.143708 0.061883 0.0581541295
-5 0.411463 0.120879 0.1153123659
0 0.977032 0.196274 0.1896116138
5 1.89534 0.27954 0.2723395591
10 3.10794 0.36565 0.3582797952
N =15
? [dB] ?ORA, N ?ORA, -upE ?ORA, E-low
-10 0.164232 0.041359 0.04064235544
-5 0.463140 0.069202 0.06833679633
0 1.07295 0.10041 0.09942977640
5 2.04205 0.13283 0.13180969046
10 3.30793 0.16564 0.164623141056
129
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ b i t / sec / H z ]
Dual - branch SC for m = 1
No diversity for m = 0.5 , 1 or Dual - branch SC for m = 0.5
In Fig. 4, the spectral efficiency under CIFR is plotted as a
function of the average received SNR per branch ? . For
comparison, the spectral efficiency of uncorrelated Rayleigh
fading channels with dual-branch SC, which was obtained in
[8, Eq. (52)], is also presented in Fig. 4. It is observed that,
as the channel fading conditions improve, i.e, m and / or ?
increases, spectral efficiency with no diversity remains zero.
However, employing a dual-branch SC system improves the
channel capacity as we go from worst case of fading
conditions to Rayleigh fading conditions.
In Fig. 5, the spectral efficiency of uncorrelated Nakagami0.5 fading channels with and without diversity is plotted as a
function of ? , considering ORA, OPRA, and TIFR
adaptation schemes with the aid of (6), (23), [17, Eq. (27)],
[17, Eq. (35)], [15, Eq. (8)], and [15, Eq. (22)]. It shows that,
the spectral efficiency with no diversity under ORA scheme
improves over TIFR for ? =5dB . It is also observed that the
spectral efficiency with dual-branch SC under ORA scheme
improves over TIFR for ? =0dB and OPRA scheme provides
better efficiency under worst case of fading. It is also
interesting to observe that for ? -= 5.7 dB ORA scheme with
dual-branch SC gives inferior performance with respect to
TIFR scheme without diversity.
-10 -5 0 5 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Received SNR [ dB ] per Branch
Spectral Efficiency [ b i t / sec / H z ]
Dual - branch SC under OPRA
No diversity under OPRA
Dual - branch SC under ORA
No diversity under ORA
Dual - branch SC under TIFR
No diversity under TIFR
5. Conclusions
In this paper, closed-form expressions for the spectral
efficiency of dual-branch SC and no diversity under ORA
and CIFR schemes have been obtained and analyzed. Error
bounds have been derived for truncated infinite series under
ORA. Numerical results illustrate that the bounds can be used
effectively to determine the number of terms needed to
achieve desirable level of accuracy. Results have been
plotted, which show that by increasing ? and/or employing
diversity, spectral efficiency improves under ORA scheme. It
is also observed that the spectral efficiency under CIFR is
zero under worst case of fading even when a dual-branch SC
is utilized. It is important to note that the spectral efficiency
using ORA scheme remains almost same even when fading
conditions improve from Nakagami-0.5 to Rayleigh either
under dual-branch SC for ? -= 5.2 dB or under no diversity
for ? -= 5.7 dB . This paper finally concludes that under
worst case of fading TIFR scheme is a better choice for low
average received SNR and ORA scheme is for high average
received SNR even employing diversity. Therefore it is
recommended that under worst fading condition, ORA
scheme is not always a better choice over TIFR.
Figure 4. Spectral Efficiency versus average received
SNR? under CIFR.
Figure 5. Spectral Efficiency versus average received
SNR? under different adaptive transmission schemes.
130
International Journal of Communication Networks and Information Security (IJCNIS) Vol. 7, No. 3, December 2015
References
[1] A. Iqbal, A.M. Kazi, "Integrated Satellite-Terrestrial System
Capacity Over Mix Shadowed Rician and Nakagami
Channels," International Journal of Communication Networks
and Information Security (IJCNIS), Vol. 5, No. 2, pp. 104-
109, 2013.
[2] R.Saadane, M. Wahbi, "UWB Indoor Radio Propagation
Modelling in Presence of Human Body Shadowing Using Ray
Tracing Technique,'' International Journal of Communication
Networks and Information Security (IJCNIS) Vol. 4, No. 2,
2012.
[3] S. Khatalin, J.P.Fonseka, "Channel capacity of dual-branch
diversity systems over correlated Nakagami-m fading with
channel inversion and fixed rate transmission scheme," IET
Communications, Vol. 1, No.6, pp.1161-1169, 2007.
[4] S.Khatalin, J.P.Fonseka, "Capacity of Correlated Nakagami-m
Fading Channels With Diversity Combining Techniques,"
IEEE Transactions on Vehicular Technology.," Vol. 55,
No.1, pp.142-150, 2006.
[5] V. Hentinen, "Error performance for adaptive transmission on
fading channels," IEEE Transactions on Communications,
Vol. 22, No. 9, pp. 1331-1337, 1974.
[6] A. J. Goldsmith, P. P. Varaiya, "Capacity of Fading Channels
with Channel Side Information," IEEE Transactions on
Information Theory," Vol. 43, No. 6, pp. 1986-1992, 1997.
[7] M. S. Alouini, A. Goldsmith, "Capacity of Nakagami
multipath fading channels," Proceedings of the IEEE
Vehicular Technology Conference., Phoenix, AZ, pp. 358-
362, 1997.
[8] M. S. Alouini, A.J. Goldsmith, "Capacity of Rayleigh Fading
Channels Under Different Adaptive Transmission and
Diversity-Combining Techniques," IEEE Transactions on
Vehicular Technology, Vol. 48, No. 4, pp.1165-1181, 1999.
[9] P.S.Bithas, P.T.Mathiopoulos, "Capacity of Correlated
Generalized Gamma Fading With Dual-Branch Selection
Diversity," IEEE Transactions on Vehicular Technology, Vol.
58, No. 9, pp.5258-5263, 2009.
[10] M .S. Alouini, A. Goldsmith, "Adaptive Modulation over
Nakagami Fading channels," Wireless Personal
Communication, Vol. 13, pp. 119-143, 2000.
[11] R. Subadar, P. R. Sahu, "Capacity Analysis of Dual-SC andMRC Systems over Correlated Hoyt Fading Channels," IEEE
Conference, TENCON, pp.1-5, 2009.
[12] K. A. Hamdi, "Capacity of MRC on Correlated Rician Fading
Channels," IEEE Transaction. on Communication, Vol. 56,
No. 5, pp. 708-711, 2008.
[13] S. Khatalin, J.P.Fonseka, "On the Channel Capacity in Rician
and Hoyt Fading Environments With MRC Diversity," IEEE
Transactions on Vehicular Technology, Vol.55, No.1,
pp.137-141, 2006.
[14] M. Stefanovic, J. Anastasov, S. Panic,.P. Spalevic, C.
Dolicanin, Chapter 10 "Channel Capacity Analysis Under
Various Adaptation Policies and Diversity Techniques over
Fading Channels," in the book Wireless Communications and
Networks- Recent Advances, In Tech, 2012.
[15] M.I.Hasan, S.Kumar, "Channel Capacity of Dual-Branch
Maximal Ratio Combining under worst case of fading
scenario," WSEAS Transactions on Communications, Vol.
13, pp. 162-170, 2014.
[16] M.I.Hasan, S.Kumar, "Average Channel Capacity of
Correlated Dual-Branch Maximal Ratio Combining Under
Worst Case of Fading Scenario," Wireless Personal
Communication, Vol. 83, No. 4, pp. 2623-2646, 2015.
[17] M.I.Hasan, S.Kumar, "Average Channel Capacity Evaluation
for Selection Combining Diversity Schemes over Nakagami0.5 Fading Channels," International Journal of
Communication Networks and Information Securit

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