Performance Analysis of Rectangular QAM With SC Receiver Over Nakagami- $m$ Fading Channels

Abstract

Error performance of a $L$-branch selection combining (SC) receiver is analyzed over independent and non-identically distributed Nakagami- $m$ fading channels. A novel exact closed-form expression for average symbol error rate (ASER) for general order rectangular quadrature amplitude modulation (RQAM) scheme is derived using cumulative distribution function based approach. Further, exact closed-form ASER expressions for differentially encoded quadri-phase shift-keying (DE-QPSK) and $pi/4$-QPSK modulation schemes have been presented. Numerically evaluated results are plotted and compared with Monte Carlo simulation results to verify the correctness of the derivations.

Full text

1262 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 7, JULY 2014
Performance Analysis of Rectangular QAM With SC Receiver Over
Nakagami-
m Fading Channels
Dharmendra Dixit, Member, IEEE, and P. R. Sahu, Member, IEEE
Abstract—Error performance of a L-branch selection combin-
ing ( SC) receiver is analyzed over independent and non-identically
distributed Nakagami-m fading channels. A novel exact closed-
form expression for average symbol error rate (ASER) for general
order rectangular quadrature amplitude modulation (RQAM)
scheme is derived using cumulative distribution function based
approach. Further, exact closed-form ASER expressions for dif-
ferentially encoded quadri-phase shift-keying (DE-QPSK) and
π/4-QPSK modulation schemes have been presented. Numeri-
cally evaluated results are plotted and compared with Monte Carlo
simulation results to verify the correctness of the derivations.
Index Terms—Average symbol error rate (ASER),
Nakagami-m fading, rectangular quadrature amplitude
modulation (RQAM) scheme, selection combining (SC) receiver.
I. INTRODUCTION
P
ERFORMANCE analysis of selection combining (SC)
receiver over fading channels has drawn lot of research
interest in recent years [1], [2]. Considerable effort has been
and is being devoted to obtain analytical expressions for av-
erage symbol error rate (ASER) of SC receiver over fading
channels which can be used to address wireless system design
issues without going for extensive Monte Carlo simulations.
Nakagami-m distribution is used to model practical fading
channels which includes Rayleigh (m =1) and one-sided
Gaussian (m =0.5) fading channels as special cases. Further, it
can closely approximate the Rician and Hoyt distributions and
is also suitable to model fading channels worse than Rayleigh
model (0.5 ≤ m<1) [3].
Among various modulation schemes employed to trans-
mit digital data, rectangular quadrature amplitude modulation
(RQAM) has drawn considerable attention due to its high band-
width efficiency characteristics [4], [5]. RQAM is a generic
modulation s cheme as it includes square QAM (SQAM), bi-
nary phase shift keying (BPSK), orthogonal binary frequency
shift keying, quadri phase shift keying (QPSK) and multilevel
amplitude shift keying as special cases. It finds applications
in microwave communications, high speed mobile communi-
cation systems, asymmetric subscriber loop and telephone-line
modems etc. [1], [2].
Manuscript received October 9, 2013; revised March 16, 2014; accepted
March 19, 2014. Date of publication April 4, 2014; date of current version
July 8, 2014. The associate editor coordinating the review of this paper and
approving it for publication was Z. Hadzi-Velkov.
D. Dixit is with the LNM Institute of Information Technology (LNMIIT),
Rajasthan 302031, India (e-mail: ddixit@lnmiit.ac.in).
P. R. Sahu is with the School of Electrical Sciences, Indian Institute of Tech-
nology Bhubaneswar, Bhubaneswar 751 013, India (e-mail: prs@iitbbs.ac.in).
Digital Object Identifier 10.1109/LCOMM.2014.2315616
In literature, research works on the error performance analy-
sis of diversity combining receivers over Nakagami-m fading
channels for a variety of modulation schemes are available
in [3]–[11]. These analyses use well known methods such as
either moment generating function (MGF) based approach [3]–
[8] or cumulative distribution function (CDF) based approach
[9] or probability density function (PDF) based approach [1],
[10]. In particular, closed-form ASER expressions for maximal-
ratio combining (MRC), equal-gain combining (EGC) and SC
receivers using different modulations are presented in [3].
In [4], ASER expression for the general order RQAM is
derived for MRC receiver over L independent and identi-
cally distributed Nakagami-m fading channels for arbitrary m.
In [5], exact closed-form ASER expressions have been pre-
sented for arbitrary RQAM for single- and multichannel di-
versity reception over independent and non-identically (i.n.d)
distributed Nakagami-m fading channels. In [6], closed-form
ASER expressions for binary and M-ary signals are given
for arbitrary m. In [7], Feminas presents exact closed-form
average bit error rate (ABER) expressions for a SC receiver
with added switching constraints feature. In [8], Aalo presents
exact closed-form ABER expressions for MRC, EGC and
SC receivers with binary modulation signals. In [9], using
CDF based approach exact closed-form ASER expressions are
presented for SQAM and binary modulation schemes with
transmit antenna selection in Nakagami-m fading channels with
arbitrary parameters. In [10], PDF based performance analyses
of general order selection combining over i.n.d. Weibull and
Nakagami-m fading channels are presented. In [11], perfor-
mance analysis of maximal-ratio transmission/receive antenna
selection in Nakagami-m fading channels with channel esti-
mation errors and feedback delay are given. To the best of
our knowledge, ASER performance analysis of the SC receiver
for RQAM, differentially encoded QPSK (DE-QPSK) and
π/4-QPSK modulation schemes over i.n.d Nakagami-m fading
channels for arbitrary m is not addressed in literature.
In this letter, we present novel exact ASER expressions for
L branch SC receiver for general order RQAM, DE-QPSK
and π/4-QPSK modulation schemes over i.n.d Nakagami-m
fading channels. We apply the CDF based approach to obtain
closed-form expressions which contain Lauricella hypergeo-
metric function (LHF) of n variables, F
(n)
A
(·) and is valid for
noninteger m. LHFs can efficiently evaluated using either its
converging infinite series representation or by alternate efficient
methods proposed in [9], [12].
The rest of the letter is organized as follows. In Section II,
system and channel model are discussed and in Section I II the
error performance analysis of the SC receiver is presented. In
Section IV, numerical results and discussion are given. The
letter is concluded in Section V.
1089-7798 © 2014
IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

DIXIT AND SAHU: PERFORMANCE ANALYSIS OF RECTANGULAR QAM WITH SC RECEIVER OVER NAKAGAMI-m FADING CHANNELS 1263
II. SYSTEM AND CHANNEL MODEL
We consider a L-branch SC receiver in i.n.d Nakagami-m
fading channels. In the SC receiver, the signal to noise ra-
tio (SNR) of received s ignals from all diversity branches are
continuously monitored and the received signal having highest
SNR is selected for detection [1]. Let γ
o
denote the output SNR
of L-branch SC receiver. Mathematically, γ
o
can be stated as
γ
o
=
max
1 ≤ i ≤ L{γ
i
}, (1)
where γ
i
= α
2
i
(E
s
/N
0
) is the instantaneous SNR of i-th
branch, E
s
is the energy per symbol, N
0
represents one
sided power spectral density of additive white Gaussian noise
(AWGN) and α
i
is the fading envelope of the i-th branch which
follows a Nakagami-m distribution with fading parameter m.
An expression for t he PDF of γ
i
is given as [1]
f
γ
i
(γ
i
)=
(m/¯γ
i
)
m
Γ(m
i
)
γ
m−1
i
exp (−(m
i
/¯γ
i
) γ
i
) , (2)
where m ≥ 0.5, Γ(·) is the gamma function and ¯γ
i
is the
average SNR of i-th branch. The corresponding expressoion
for the CDF can be obtained, by applying the formula
[13, (8.351.2)], as
F
γ
i
(γ
i
)
=
((m
i
/¯γ
i
) γ
i
)
m
i
exp (−(m
i
/¯γ
i
) γ
i
)
m
i
Γ(m
i
)
1
F
1

1; m
i
+1;
m
i
¯γ
i
γ
i

,
(3)
where
1
F
1
(·; ·; ·) is the confluent hypergeometric function
[13, (9.21.1)]. Assuming received fading signals at the input
receiving antennas of SC receiver are independent, the CDF of
γ
o
can be obtained, by the product of L CDFs in (3), as
F
γ
o
(γ
o
)=
L

i=1
F
γ
i
(γ
o
)
= R
m,L
γ

L
i=1
m
i
o
exp (−(m
L
)γ
o
)
×
L

i=1
1
F
1

1; m
i
+1;
m
i
¯γ
i
γ
o

, (4)
where
m
L
=

L
i=1
(m
i
/ ¯γ
i
) and R
m,L
=

L
i=1
((m
i
/¯γ
i
)
m
i
/
m
i
Γ(m
i
)).
III. E
RROR PERFORMANCE ANALYSIS
Mathematically, ASER for a digital modulation scheme can
be computed as
P
s
(e)=−
∞

0
P

s
(e|γ)F
γ
(γ)dγ, (5)
where P

s
(e|γ) is the first derivative of the conditional symbol
error rate (SER),
1
P
s
(e|γ) of a modulation scheme in AWGN.
This approach of the ASER evaluation is known as the CDF
based approach.
1
w.r.t the received instantaneous SNR γ.
A. RQAM Scheme
The conditional SER expression of M
I
× M
Q
-ary RQAM
scheme in AWGN channels is given as [14]
P
RQAM
s
(e|γ)
=2[pQ(a
√
γ)+qQ(b
√
γ) − 2pqQ(a
√
γ)Q(b
√
γ)] , (6)
where M
I
and M
Q
are the number of in-phase and quadrature-
phase constellation points, respectively, p =1− (1/M
I
), q =
1 − (1/M
Q
), a =

6/((M
2
I
− 1) + (M
2
Q
− 1)β
2
), b = βa
and β = d
Q
/d
I
is the quadrature-to-in-phase decision distance
ratio with d
I
and d
Q
being the in-phase and quadrature decision
distance, respectively, and Q(x) is the Gaussian Q-function
given by Q(x)=(1/
√
2π)

∞
x
exp(−(t
2
/2))dt.Tobeableto
use the ASER formula in (5) we need to obtain the first deriva-
tive of P
RQAM
s
(e|γ) in (6) w. r. t γ. Applying the identities in
[13, (6.285.1)] and [15, (7.1.21)] followed by some algebraic
manipulations, we can obtain P

RQAM
s
(e|γ) as
P

RQAM
s
(e|γ)=

ap(q − 1)/(
√
2π)

γ
−1/2
exp(−a
2
γ/2)
+

b(p − 1)q/(
√
2π)

γ
−1/2
exp(−b
2
γ/2)
− [abpq/π]exp

−(a
2
+ b
2
)γ/2

×
1
F
1
(1; 1.5; a
2
γ/2)
− [abpq/π]exp

−(a
2
+ b
2
)γ/2

×
1
F
1
(1; 1.5; b
2
γ/2). (7)
Substituting (4) and (7) i nto (5) and integrating the resulting
expression, using formulas [13, (7.622.3,9.220.2)], a closed-
form ASER expression for RQAM scheme at the output of the
SC receiver (i.e. γ = γ
o
) can be obtained as
P
RQAM
s
(e)
= R
m,L

ap(1 − q)Γ (Δ
L
(0.5)) /(
√
2π)

× [
m
L
+0.5a
2
]
−Δ
L
(0.5)
F
(L)
A

Δ
L
(0.5); {1}|
1:L
; {m
j
+1}|
L
j=1
;
{Λ
j
(a, 0)}|
L
j=1

+

b(1 − p)qΓ(Δ
L
(0.5)) /(
√
2π)

× [
m
L
+0.5b
2
]
−Δ
L
(0.5)
F
(L)
A

Δ
L
(0.5); {1}|
1:L
; {m
j
+1}|
L
j=1
;
{Λ
j
(0,b)}|
L
j=1

+[abpqΓ(Δ
L
(1)) /π]
×

m
L
+0.5(a
2
+ b
2
)

−Δ
L
(1)
F
(L+1)
A

Δ
L
(1); {1}|
1:L
;1.5, {m
j
+1}|
L
j=1
;
λ
L
(a, b), {Λ
j
(a, b)}|
L
j=1

.
+[abpqΓ(Δ
L
(1)) /π]
×

m
L
+0.5(a
2
+ b
2
)

−Δ
L
(1)
F
(L+1)
A

Δ
L
(1); {1}|
1:L+1
;1.5, {m
j
+1}|
L
j=1
;
λ
L
(b, a), {Λ
j
(a, b)}|
L
j=1

(8)

1264 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 7, JULY 2014
where {1}|
1:L
=1, 1,...,1

 
L numbers
, {x
j
+1}|
L
j=1
=x
1
+1,x
2
+1,...,
x
L
+1, {x
j
}|
L
j=1
= x
1
,x
2
,...,x
L
, Δ
L
(n)=

L
i=1
m
i
+ n,
Λ
k
(a, b)=(m
k
/¯γ
1
)/(m
L
+0.5(a
2
+ b
2
)), {Λ
j
(a, b)}|
L
j=1
=
Λ
1
(a, b), Λ
2
(a, b),...Λ
L
(a, b), λ
L
(a, b)=0.5a
2
/(m
L
+
0.5(a
2
+ b
2
)) and F
(n)
A
(β; {x
j
}|
n
j=1
; {y
j
}|
n
j=1
; {z
j
}|
n
j=1
) is
the LHF of n variables. The LHF can be evaluated by using
either the converging infinite series representation given in
[9, (11)] or by alternate efficient methods proposed in [9],
[12]. The expression (8) is valid for any arbitrary value of
m whereas the ASER expression in [10, (37)] for SQAM is
valid for i nteger m only. For the special case of M -ary SQAM
scheme, i.e., when M
I
=M
Q
=
√
M and β =1, it can be shown
that the expression in (8) reduces to the previously reported
ASER expression [9, (34)]. Further, for BPSK modulation, i.e.,
M
I
=2, M
Q
=1, p =0.5, q =0, a =
√
2 and β =0,(8)
reduces to the expression [9, (26)].
Both coherent, DE-QPSK and π/4-QPSK modulation
schemes have the same expression for the conditional SER and
can be given as [1, (8.39)]
P
QP SK
s
(e|γ)=4Q(
√
γ)−8Q
2
(
√
γ)+8Q
3
(
√
γ)−4Q
4
(
√
γ).
(9)
Following the steps used for RQAM scheme, an ASER expres-
sion for DE-QPSK and π/4-QPSK modulation schemes can be
obtained as
P
QP SK
s
(e)
= R
m,L

[Γ (Δ
L
(1)) /π][m
L
+1]
−Δ
L
(1)
F
(L+1)
A

Δ
L
(1); {1}|
1:(L+1)
;1.5, {m
j
+1}|
L
j=1
;
λ
L
(1, 1), {Λ
j
(1, 1)}|
L
j=1

+

2Γ (Δ
L
(2)) /π
2

[
m
L
+2]
−Δ
L
(2)
F
(L+3)
A

Δ
L
(2); {1}|
1:(L+3)
;1.5, 1.5, 1.5,
{m
j
+1}|
L
j=1
;0.5λ
L
(
√
2,
√
2),
0.5λ
L
(
√
2,
√
2), 0.5λ
L
(
√
2,
√
2),

Λ
j
(
√
2,
√
2)




L
j=1

. (10)
IV. N
UMERICAL AND SIMULATION RESULTS
Derived expressions are numerically evaluated and plotted to
illustrate and study the impact of L and m
i
on ASER. In the
figures, ASER vs. ¯γ
i
curves are shown to illustrate error per-
formance. Monte Carlo simulation results are plotted with the
numerical results to verify the correctness of the mathematical
derivations.
In the numerical evaluation, we evaluate the F
(n)
A
(·) function
by approximating the involved infinite series by a finite number
of terms [9, (11)]. Each infinite series is truncated to N terms
such that an accuracy at the 7th place of decimal digit is
achieved in the evaluation of ASER.
Figs. 1 and 2 illustrate the ASER performance of 4×2-QAM
scheme over i.i.d and i.n.d Nakagami-m fading channels, re-
Fig. 1. ASER for 4 × 2 RQAM scheme over i.i.d fading channels.
Fig. 2. ASER for 4 × 2 RQAM scheme over i.n.d fading channels.
spectively. As an important observation, it can be noticed from
the figures that ASER improves with increase in m.Infact,a
save in SNR can be noted with the increase in m for a desired
ASER. For example, from Fig. 1 with L =2, for an ASER of
10
−3
, the SNR that we save by moving from a location with
m =1 (Rayleigh fading) to a location with m =2 is 5 dB
(approx.). This observation may be useful for a limited power
scenario, as in s ensor network transceivers, where it is required
the ASER of a receiver not to fall below a certain minimum
value. Assuming the transceiver is at a place having m =2,
it can inform the transmitter to reduce the transmitted power
suitably and hence save battery power. This technique, if used
adaptively is likely to improve average life of sensors. Increase
in L also improves the error performance, as expected. For
example, from Fig. 2 for an ASER of 10
−3
, the SNR gain that
can be achieved by increasing L from 2 to 3 ranges between
1 dB to 4 dB (depending on m). Thus, a suitable combination
of L and m can be appropriate for a desired performance.
In Table I, we have tabulated the minimum values of N
to achieve an accuracy at 7th place of decimal digit in the
evaluation of ASER expression in (8). It can be observed that a
large N is required to obtain the desired accuracy at poor SNR.
Thus, the computation time is high at low SNR.
In Fig. 3, ASER curves are shown for DE-QPSK and
π/4-QPSK modulations by numerically evaluating (10). It can
be observed that the effect of m and L on ASER follows similar
trend as the ASER in Fig. 1. In this case the SNR gain that can
be achieved by increasing L from 3 to 5 to maintain an ASER
at 10
−3
,is4dBform =1and reduces with increase in m i.e.,
for m =1.5, the SNR gain falls to ∼3dB.

DIXIT AND SAHU: PERFORMANCE ANALYSIS OF RECTANGULAR QAM WITH SC RECEIVER OVER NAKAGAMI-m FADING CHANNELS 1265
TAB LE I
R
EQUIRED NUMBER OF TERMS N , IN THE NUMERICAL EVA L UAT I O N OF
(8) TO ACHIEVE AN ACCURACY AT THE 7TH P LACE OF DECIMAL DIGIT
Fig. 3. ASER for DE-QPSK and π/4-QPSK modulation schemes over i.i.d
fading channels.
A. More Observations on the Effect of L and m on ASER
ASER improves with increase in L and/or m but the percent-
age improvement falls with every increase in L or m.Fromthe
ASER curves, it can be seen that for a given L the percentage
increase in ASER gradually decreases with increase in m.For
example, in Fig. 1 for L =2and ¯γ
i
=15dB, the percentage
improvement in ASER when m increases from 1 to 2 is 65.39
whereas the percentage improvement when m increases from 2
to 3 is 41.58. In a similar manner, for m =2and ¯γ
i
=15dB,
the percentage increase in ASER for L increasing from 2 to 3
is 67.31 and decreases to 49.66 with increase in L from 3 to 4.
Another important observation, supported by simulation re-
sults, is that for L ≥3 significant ASER improvement is not no-
ticed at SNR below 10 dB even in better fading conditions. This
can be interpreted as follows: With increase in L the receiver
gets sufficient opportunity to switch to higher SNRs at a mod-
erate fading conditions and hence the ASER remains constant
for most of the time. This also indicates that it is not a good idea
to increase L beyond 3 to achieve better ASER at low SNRs.
V. C
ONCLUSION
We present novel closed-form expressions of exact ASER for
RQAM, DE-QPSK and π/4-QPSK digital modulation schemes
with SC receiver over i.n.d Nakagami-m fading channels.
Obtained ASER expressions are valid for all values of fading
parameter m. Obtained ASER expressions are compared with
the available expressions in literature and numerically evaluated
results are verified against Monte Carlo simulation results.
R
EFERENCES
[1] M. K. Simon and M.-S. Alouini, Digital Communication Over Fading
Channels, 2nd ed. New York, NY, USA: Wiley, 2005.
[2] T. S. Rappaport, Wireless Communication, 2nd ed. Chennai, India:
Pearson, 2011.
[3] A. Annamalai and C. Telumbura, “Error rates for Nakagami-m fad-
ing multichannel reception of binary and M-ary signals,” IEEE Trans.
Commun., vol. 49, no. 1, pp. 58–68, Jan. 2001.
[4] X. Lei, P. Fan, and L. Hao, “Exact symbol error probability of general
order rectangular QAM with MRC diversity reception over Nakagami-m
fading channels,” IEEE Commun. Lett., vol. 11, no. 12, pp. 958–960,
Dec. 2007.
[5] A. Maaref and S. Aissa, “Exact error probability analysis of rectangu-
lar QAM for single- and multichannel reception in Nakagami-m fad-
ing channels,” IEEE Trans. Commun., vol. 57, no. 1, pp. 214–221,
Jan. 2009.
[6] H. Shin and J. H. Lee, “On the error probability of binary and M -ary
signals in Nakagami-m fading channels,” IEEE Trans. Commun., vol. 52,
no. 4, pp. 536–539, Apr. 2004.
[7] G. Femenias, “MGF-based performance analysis of selection diversity
with switching constraints in nakagami fading,” IEEE Trans. Wireless
Commun., vol. 5, no. 9, pp. 2328–2333, Sep. 2006.
[8] V. A. Aalo and G. P. Efthymoglou, “On the MGF and BER of linear di-
versity schemes in Nakagami fading channels with arbitrary parameters,”
in Proc. IEEE VTC Spring, Apr. 26–29, 2009, pp. 1–5.
[9] J. M. Romero-Jerez and A. J. Goldsmith, “Performance of multichan-
nel reception with transmit antenna selection in arbitrarily distributed
Nagakami fading channels,” IEEE Trans. Wireless Commun., vol. 8, no. 4,
pp. 2006–2013, Apr. 2009.
[10] R. Kwan and C. Leung, “General order selection combining for Nakagami
and Weibull fading channels,” IEEE Trans. Wireless. Commun.,vol.6,
no. 6, pp. 2027–2033, Jun. 2007.
[11] A. F. Coskun and O. Kucur, “Performance analysis of maximal-ratio
transmission/receive antenna selection in Nakagami-m fading channels
with channel estimation errors and feedback delay,” IEEE Trans. Veh.
Technol., vol. 61, no. 3, pp. 1099–1108, Mar. 2012.
[12] Q. Shi and Y. Karasawa, “Some applications of lauricella hypergeometric
function F
A
in performance analysis o f wireless communications,” IEEE
Commun. Lett., vol. 16, no. 5, pp. 581–584, May 2012.
[13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, Products,
6th ed. New York, NY, USA: Academic, 2000.
[14] N. C. Beaulieu, “A useful integral for wireless communication theory and
its application to rectangular signaling constellation error rates,” IEEE
Trans. Commun., vol. 54, no. 5, pp. 802–805, May 2006.
[15] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions
With Formulas, Graphs, Mathematical Tables, 9th ed. New York, NY,
USA: Dover, 1970.