sep of rectangular qam in composite fading channels
TRANSCRIPT
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ARTICLE IN PRESSG ModelEUE-51287; No. of Pages 7
Int. J. Electron. Commun. (AEÜ) xxx (2014) xxx–xxx
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International Journal of Electronics andCommunications (AEÜ)
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EGULAR PAPER
EP of rectangular QAM in composite fading channels
etros S. Bithasa,∗, George P. Efthymogloub, Athanasios G. Kanatasb
Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens, GreeceDepartment of Digital Systems, University of Piraeus, Greece
r t i c l e i n f o
rticle history:eceived 30 November 2013ccepted 15 September 2014
a b s t r a c t
In this paper, we analytically evaluate the average symbol error probability (SEP) of rectangular quadra-ture amplitude modulation (QAM) signalling in composite fading channels modelled by the generalized-K(KG) distribution. The analysis is based on a fast converging infinite series representation of the average
eywords:omposite fading/shadowing channelsaussian Q-functioneneralized-K distributionuadrature amplitude modulation (QAM)
of the product of two Gaussian-Q functions over KG fading that has been extracted. Considering integerand a half values for the distribution’s shaping parameters, exact closed-form expressions have beenalso derived. Numerical evaluated results, complemented by equivalent computer simulated ones, arepresented to verify the accuracy of the proposed analysis.
© 2014 Elsevier GmbH. All rights reserved.
ymbol error probability
. Introduction
Contemporary cellular communication systems should supporthe continuously increasing data rate demands of a higher numberf subscribers. Towards this objective, a quite promising approachhat is found to considerable increase the spectral efficiency rep-esent the non-constant envelope modulation schemes. In thisontext, quadrature amplitude modulation (QAM) constellations a class of non-constant modulation scheme that can achieveigher data rates as compared to constant envelope schemes, e.g.,hase shift keying (PSK). Such a characteristic is very important forumerous of communication systems, where bandwidth efficiency
s more important than power efficiency. In addition to the band-idth efficiency offered, rectangular (or square) QAM can be easily
enerated as two independent pulse amplitude modulation (PAM)ignals, while its demodulation is also easy [1,2]. This is the maineason why QAM schemes have been employed in many practi-al communication systems, e.g., Digital Video Broadcasting (DVB)igital cable transmission [3], DVB-satellite to handheld [4] andVB-digital terrestrial television [5].
Among the various QAM constellations that have been pro-osed and studied in the past, rectangular QAM has gained an
ncreased interest as it is proved by the various contributions that
Please cite this article in press as: Bithas PS, et al. SEP of rectangular Q(2014), http://dx.doi.org/10.1016/j.aeue.2014.09.011
ave been reported on this topic. In particular, rectangular QAM is aeneric modulation technique which includes various modulationchemes as special cases, namely square QAM, binary PSK (BPSK),
∗ Corresponding author.E-mail addresses: [email protected], [email protected] (P.S. Bithas),
[email protected] (G.P. Efthymoglou), [email protected] (A.G. Kanatas).
ttp://dx.doi.org/10.1016/j.aeue.2014.09.011434-8411/© 2014 Elsevier GmbH. All rights reserved.
orthogonal binary frequency-shift keying, quadrature PSK and mul-tilevel amplitude shift-keying modulation techniques [6]. Thus,its performance has been evaluated in various communicationscenarios, including single channel reception, e.g., [1,7,8], mul-tichannel reception, e.g., [9–11], multiple-input-multiple-output(MIMO) systems, e.g., [12–14] and cooperative communications,e.g., [15–18]. More specifically, in [11], assuming a L-branch selec-tion combiner operating over Nakagami-m fading channels, a novelclosed-form expression was derived for the average symbol errorrate (ASER) of general order rectangular QAM. In [13], the ASER ofgeneral rectangular QAM in Rayleigh fading was studied for MIMOmaximal ratio combining (MRC) systems, with arbitrary number ofantennas on both sides. Finally in [18], a lower bound of the ASER foramplify and forward cooperative systems with best-relay selectionover Rayleigh fading channels is derived for general order rectan-gular QAM. A common observation in all these works is that theproposed analysis takes into consideration only the small scale fad-ing (multipath) that indeed represents a major impairment factorin mobile communication links.
However, in many practical communication scenarios largescale fading effects (shadowing) become dominant, originatingwhat is known as composite multipath/shadowing environment.This scenario arises in situations of slow moving or stationaryusers, for example congested downtown areas with slow movingpedestrians and vehicles, where the receiver is unable to aver-age over the effects of fading and a composite distribution isnecessary for evaluating link performance and other quantities
AM in composite fading channels. Int J Electron Commun (AEÜ)
[19,20]. Depending upon the statistical description of the shad-owing effects, several families of composite fading distributionshave been proposed, such as the lognormal-based ones, for exam-ple Rayleigh, Nakagami-lognormal [19], and gamma based ones,
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ARTICLEEUE-51287; No. of Pages 7
P.S. Bithas et al. / Int. J. Electron
or example K, generalized-K (KG) and Weibull-gamma [21,19,22].mploying gamma distribution as an alternative to the log-normalne leads to simpler composite distributions and mathematicalore tractable analytical approach. In the past, many researchers
ave used the K and KG distributions for investigating varioushenomena and parameters of the composite fading propagationnvironments, e.g., [21,23–27], and this is the case that is also goingo be investigated in the current paper.
In particular the scope of this paper is to analyze the perfor-ance of rectangular QAM in composite fading environments. To
his aim, novel exact expressions are derived for the average sym-ol error probability (ASEP) of rectangular QAM constellation inomposite fading channels, modelled by the KG distribution. Theroposed approach is based on fast converging infinite series repre-entation of the average of the product of two Gaussian-Q functionsver KG fading that has been derived. In addition our analysis alsoncludes simplified closed-form expressions for the case of specialalues of the distribution’s parameters.
The remainder of this paper is organized as follows. The systemnd channel model are described in Section 2. In Section 3, exactolutions for an integral representing the average of the product ofwo Gaussian Q-functions over KG fading are presented. These solu-ions are employed in Section 4 to analyze the ASEP for both generals well as special cases. In Section 5, the numerically evaluated per-ormance results are provided, while the concluding remarks areiven in Section 6.
. System and channel model
The KG distribution is general enough to accurately describearious fading and shadowing phenomena encountered in mobileommunication systems [23]. The main advantage of this compos-te fading distribution is that it makes mathematical performancenalysis much simpler to be handled, as compared to lognormal-ased models, such as Rayleigh or Nakagami-lognormal models21]. Let the fading envelope R be modelled as a KG random vari-ble. Then, the probability density function (PDF) of R is given by23]
R (x) = 4 m(m+k)/2xm+k−1
�(m)�(k)�(m+k)/2Kk−m
[2(
m
�
)1/2x
], x ≥ 0 (1)
here k, m are the distribution’s shaping parameters, Kk−m(·) ishe second kind modified Bessel function of order (k − m) [28, Eq.9.6.1)], �(·) is the gamma function [29, Eq. (8.310/1)] and � is the
ean power obtained as � = E〈R2〉/k. By using different values of mnd k, (1) can describe a great variety of short-term and long-termshadowing) fading conditions, respectively. For example, as k→ ∞,R(x) approximates the well known Nakagami-m fading channel
odel [30], for m = 1, it becomes the K-distribution and approachesayleigh-lognormal shadowing fading channel [30,31], while for→ ∞ and k → ∞, (1) approaches the additive white Gaussian noise
AWGN) (i.e., no fading) channel.Capitalizing on [32, Eq. (03.04.04.0003.01)], i.e., K−v(z) = Kv(z),
ssuming |k − m| = n + 1/2, where n ∈ N, and by employing [29, Eq.8.468)], the PDF expression in (1) can be re-written as
R (x) =√
�
�(m)�(k)
n∑i=0
(i + n)!i!(n − i)!
21−2i(
m
�
) k+m−i−1/22
× xk+m−i−3/2 exp
[−2(
m)1/2
x
]. (2)
Please cite this article in press as: Bithas PS, et al. SEP of rectangular Q(2014), http://dx.doi.org/10.1016/j.aeue.2014.09.011
�
In the next section, the average of the product of two Gaussian-functions over KG fading channels will be studied.
PRESSun. (AEÜ) xxx (2014) xxx–xxx
3. Average of the product of two Gaussian-Q functions overKG fading
For obtaining the ASEP of a communication system employingrectangular QAM and operating over KG composite fading channel,the average of the product of two Gaussian-Q functions over KG fad-ing needs to be evaluated. Mathematically speaking, the followingintegral needs to be solved
I =∫ ∞
0
fR (x)Q(A1x)Q(A2x)dx (3)
where Q( · ) denotes the Gaussian Q-function [19, Section 4.1.1]and A1, A2 are real value constants. By using the definition of theGaussian Q-function, that is
Q(x) =∫ ∞
x
exp(−x2/2)√2�
dx =∫ ∞
x
GQ (x)dx (4)
and interchanging the order of iterated integration [7], (3) can berewritten as
I =∫ ∞
0
GQ (x1)
︷ ︸︸ ︷∫ A2A1
x1
0
GQ (x2)
∫ x2A2
0
fR(�)d�︸ ︷︷ ︸I1a
dx2
I1b
dx1
︸ ︷︷ ︸I1
+∫ ∞
0
∫ A1A2
x2
0
∫ x1A1
0
GQ (x2)GQ (x1)fR(�)d�dx1dx2︸ ︷︷ ︸I2
. (5)
Next, depending upon the values that shadowing parameter kcan take, two distinct cases will be investigated, namely the generalcase with arbitrary values for the shaping parameters and a specialcase with integer and a half values for them.
3.1. Arbitrary values for the shaping parameters
Substituting (1) in (5) the resulting expression is very difficultto be mathematical manipulated. An alternative approach, whichis adopted here, is to employ an infinite series representation forthe modified Bessel function as follows [32, Eq. (03.04.06.0002.01)]
Kv(z) = �csc (�v)2
( ∞∑k=0
1�(k − v + 1)k!
(z
2
)2k−v
−∞∑
k=0
1�(k + v + 1)k!
(z
2
)2k+v)
. (6)
Using (6) in (5) a straight-forward solution for the first integralcan be obtained and thus (5) can be expressed as
I1 = m(m+k)/2
�(m)�(k)�(m+k)/2
∫ ∞
0
exp
(−x2
12
)∫ A2A1
x1
0
exp
(−x2
22
)
×{
csc[�(k − m)]∞∑
i=0
(m/�)[2i−(k−m)]/2
� [i − (k − m) + 1] i!
(x2/A2
)2m+2i
2m + 2i
AM in composite fading channels. Int J Electron Commun (AEÜ)
−csc[�(k−m)]∞∑
i=0
(m/�)[2i+(k−m)]/2
� [i + (k − m) + 1] i!
(x2/A2
)2k+2i
2k+2i
}dx2dx1.
(7)
ARTICLE ING ModelAEUE-51287; No. of Pages 7
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Table 1Minimum number of terms, hmin, for convergence of (9) with an accuracy betterthan ±10−6.
� (dB) m = 1 m = 3
k = 0.5 k = 2.5 k = 0.5 k = 2.5
0 8 10 18 205 6 6 10 11
10 3 4 6 715 3 3 4 5
(
I
)
w(sb
I
)
wEamaseai
feA
× �s + 1
2,
12
A2
A1x1 + 2
A2
(m
�
)1/2
− �
(s + 1
2,
12
(2A2
(m
�
)1/2)2)]
. (13)
Using the solutions provided in (12) and (13) in I1 of (5) and bymultiplying them with GQ (x1), the following three integrals needto be solved
20 2 2 3 3
The second integral can be solved in closed form using [29, Eq.3.381/1)], resulting
1 = m(m+k)/2
�(m)�(k)�(m+k)/2
∫ ∞
0
exp
(−x2
12
)�csc[�(k − m)]
4√
2
×{ ∞∑
i=0
(m/�)[2i−(k−m)]/2
� [i − (k − m) + 1] i!
(2/A2
2
)m+i
m + i�
(12
+ i + m,A2
2x21
2A21
)
−∞∑
i=0
(m/�)[2i+(k−m)]/2
� [i + (k − m) + 1] i!
(2/A2
2
)k+i
k + i�
(12
+i + k,A2
2x21
2A21
)}dx1
(8
here �(· , ·) is the lower incomplete gamma function [29, Eq.8.350/1)]. Finally, by employing [29, Eq. (6.455/2)] a closed-formolution can be also derived for the third integral and thus I1 cane finally expressed as
1 = csc[�(k − m)]�(m)�(k)
A1
4A2
×{ ∞∑
i=0
� (1 + i + m)� [i − (k − m) + 1] i!
[2m/(�A2
2)]m+i
(m + i)(1/2 + i + m)
×(
A22
A21 + A2
2
)1+i+m
2F1
(1, 1 + i + m;
32
+ i + m;A2
2
A21 + A2
2
)
−∞∑
i=0
� (1 + i + k)� (i + (k − m) + 1) i!
[2m/(�A2
2)]k+i
(k + i)(1/2 + i + k)
(A2
2
A21 + A2
2
)1+i+k
×2F1
(1, 1 + i + k;
32
+ i + k;A2
2
A21 + A2
2
)}(9
here 2F1(· , · ; · ; ·) denotes the Gauss hypergeometric function [29,q. (9.100)]. The rate of convergence of the infinite series of thebove expression is investigated in Table 1. In this table, we sum-arized the minimum number of terms, hmin, needed to achieve
n accuracy better than ±10−6, after the truncation of the infiniteeries, for several values of �, k, and m. From this table, it can beasily understood that only a few terms are needed in order tochieve the target accuracy, while the required terms increase byncreasing m, k and/or decreasing �.
Following a similar approach as the one for obtaining solutionor the integral I1 in (5), the solution to the integral I2 in the same
Please cite this article in press as: Bithas PS, et al. SEP of rectangular Q(2014), http://dx.doi.org/10.1016/j.aeue.2014.09.011
quation can be also provided by replacing A1 with A2 and A2 with1 in (9).
PRESSun. (AEÜ) xxx (2014) xxx–xxx 3
3.2. Integer and a half values for the shaping parameters
Assuming integer and a half values for the distribution’s shapingparameters, ignoring the constant terms in (2) and by substitutingit in (5), an integral of the following form needs to be solved1
I1a =∫ x2/A2
0
x2m+n−i−1 exp
[−2(
m
�
)1/2x
]dx. (10)
This integral, with the aid of [29, Eq. (3.351/1)], can be solved inclosed form as
I1a = (2m + n − i − 1)![2(m/�)1/2
]2m+n−i
︸ ︷︷ ︸C1b1
− exp
[−2(
m
�
)1/2 x2
A2
]︸ ︷︷ ︸
C1b2
×2m+n−i−1∑
q=0
(2m + n − i − 1)!q!
(x2/A2
)q[2(m/�)
]2m+n−i−q︸ ︷︷ ︸C1b2
. (11)
The second integral of I1 in (5), can be solved by substitutingC1b1
and C1b2in I1b
. Following such an approach and using [29, Eq.
(3.321/2)], the first part of I1bcan be easily derived as
I1b1= 2i−2m−n−1
(m/�)2m+n−i
2
(2m + n − i − 1)!erf
(A2√2A1
x1
)(12)
where erf(·) is the error function [29, Eq. (8.250/1)]. For the second
part, after substituting C1b2in I1b
, setting x = y − 2A2
(m�
)1/2and per-
forming several mathematical manipulations yields the followingconvenient closed-form expression
I1b2=
2m+n−i−1∑q=0
q∑s=0
(2m + n − i − 1)!√
�q!A2q−s2
×(
q
s
)22q+i−2m−n−s/2−1
(m/�)2m + n − i + s − 2q
2
exp
(2m
A22�
)(−1)q−s
[ ( ( )2)
AM in composite fading channels. Int J Electron Commun (AEÜ)
1 It is noted that without losing the generality and for the convenience of thepresentation, in the following analysis it is assumed k − m = n + 1/2. The same analysiscan be also applied in case where m − k = n + 1/2.
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ARTICLEEUE-51287; No. of Pages 7
P.S. Bithas et al. / Int. J. Electron
I1c1=∫ ∞
0
erf
(A2√2A1
x1
)exp
(−x2
12
)dx1
I1c2=∫ ∞
0
�
(s + 1
2,
12
(A2
A1x1 + 2
A2
(m
�
)1/2)2)
exp
(−x2
12
)dx1
I1c3=∫ ∞
0
exp
(−x2
12
)dx1.
(14)
Using the definition of the complementary error func-ion, i.e., erfc(x) = 1 − erf(x) [29, Eq. (8.250/4)], I1c1
in (14)an be reexpressed with integrals of the form I1c3
and
∞0
erfc(
A2√2A1
x1
)exp
(−x2
12
)dx1. These integrals can be solved
y employing [29, Eq. (3.310)] and [29, Eq. (6.285/1)], respectively.ased on these solutions, employing [29, Eq. (1.623/2)] and follow-
ng some mathematical manipulations, a closed-form expressionsor I1c1
can be derived as
1c1=√
2�
arctan(
A2
A1
). (15)
The integral I1c2can be simplified by using [29, Eq. (8.356/3)]
nd employing integration by parts as follows
1c2= �(H1)
∫ ∞
0
exp
(−x2
12
)dx1
−√
2�HH12
∫ ∞
0
(H3 + x)s exp[−H2(H3 + x)2] erf
(x√2
)dx1︸ ︷︷ ︸
I1c2a
(16)
here H1 = (s + 1)/2, H2 = A22/(2A2
1) and H3 =(
2A1/A22
)(m/�)1/2.
he first integral in (16) can be easily solved by using [29, Eq.3.326/1)]. The second one, can be further simplified by usinghe binomial identity and after some mathematical simplificationsields
1c2a=
√2�HH1
2
s∑k=0
Hk3
(s
k
)exp
(−H2H2
3
)
×∫ ∞
0
xs−k exp(−H2x2 − 2H3H2x)erf(
x√2
)dx1︸ ︷︷ ︸
I1c2b
. (17)
In (17), I1c2bcan be solved in closed form by using [33, Eq.
2.8.6/5)]. Thus, after some mathematical manipulations, it can bexpressed as
1c2b= H
s−k−32
2
[1√
2�H2
�(
s − k
2+ 1)
�1
(s − k
2+ 1,
12
;32
,12
;
− 12H2
, H23H2
)− 2H3√
2��(
s − k + 32
)(
s − k + 3 1 3 3 1 2)]
Please cite this article in press as: Bithas PS, et al. SEP of rectangular Q(2014), http://dx.doi.org/10.1016/j.aeue.2014.09.011
× �1 2,
2;
2,
2; −
2H2, H3H2 (18)
here �1(·) denotes the Humbert hypergeometric series of twoariables [33, pp. 749]. Therefore, combining the results presented
PRESSun. (AEÜ) xxx (2014) xxx–xxx
in Eqs. (10)–(18), the final closed-form expression for integral I1 isthe following
I1 =n∑
i=0
(i + n)!i!(n − i)!
(m/�)k − m − n − 1/2
2�(k)�(m)
(2m + n − i − 1)!2i+2m+n+1/2
×{√
�
2−√
2�
arctan(
A1
A2
)−
2m+n−i−1∑q=0
q∑s=0
(q
s
)
× 22q−
s
2 (m/�)q−
s
2
q!√
�A2q−s2
exp
(2m
A22�
)(−1)q−s
×
⎡⎣�(H1)
√�
2−√
�
2�
(H1,
2m
�A22
)−
s∑j=0
Hj/22 Hj
3
(s
j
)
× exp(−H2H2
3
)[�[(s − j + 2)/2
]√H2
× �1
(s − j + 2
2,
12
;32
,12
; − 12H2
, H2H23
)−2H3�
(s − j + 3
2
)
× �1
(s − j + 3
2,
12
;32
,32
; − 12H2
, H2H23
)]]}. (19)
The exact expressions for the average of the product of twoGaussian-Q functions over KG fading presented in this section, willbe used in the next section to evaluate the ASEP of general orderrectangular QAM in this channel.
4. ASEP of rectangular QAM over KG fading
QAM is formed using two independent quadrature M-ary PAMsignals, and thus its ASEP is given by [7]
Pse =∫ ∞
0
Pe(AIx, AQ x)fR(x)dx (20)
where
Pe(AIx, AQ x) = 2(
1 − 1MI
)Q (AIx) + 2
(1 − 1
MQ
)Q (AQ x)
− 4(
1 − 1MI
)(1 − 1
MQ
)Q (AIx)Q (AQ x). (21)
In this equation MI and MQ denotes the in-phase and quadra-ture PAM signals and AI = dI/�, AQ = dQ/�, with �2 being the noisepower and dI, dQ representing the in-phase and quadrature decisiondistances, respectively. Substituting (21) in (20) yields
Pse = 2(
1 − 1MI
)∫ ∞
0
Q (AIx)fR(x)dx︸ ︷︷ ︸NI
+ 2
(1 − 1
MQ
)
×∫ ∞
0
Q (AQ x)fR(x)dx︸ ︷︷ ︸NQ
− 4(
1 − 1MI
)(1 − 1
MQ
)
∫ ∞
AM in composite fading channels. Int J Electron Commun (AEÜ)
×0
Q (AIx)Q (AQ x)fR(x)dx︸ ︷︷ ︸I
. (22)
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Q
(B
K
t
N
f(
N
s
N
wreSAb
4
(t
P1
Q
)csc[�(k − m)]
�(m)�(k)
{ ∞∑i=0
� (1 + i + m)� [i − (k − m) + 1] i!
[m/(�A2)
]m+i
(m + i)(1/2 + i + m)
k, decreases for higher values of k. Also in this figure it is shownthat the worst performance is obtained for � = 21/5, whilst the cor-responding ASEPs for � = 1 and � =
√21/5 are quite similar, with
ARTICLEEUE-51287; No. of Pages 7
P.S. Bithas et al. / Int. J. Electron
The integral I appearing in (22) is solved using the frame-ork presented in Section 3. Moreover, the integrals of the formi =∫ ∞
0Q (Aix)fR(x)dx, with i ∈ {I, Q} appearing in the same equa-
ion can be solved in closed form by using the following approach.he Gaussian Q-function is expressed as complementary error func-ion, which is turns by using [32, Eq. (06.27.26.0006.01)] can bexpressed in terms of the Meijer G-function, that is
(Aix) = 12
erfc(
Aix
2
)= 1
2√
�G2,0
1,2
(A2
ix2
2
∣∣∣10, 12
)with Gm,n
p,q [ · | · ] denoting the Meijer’s G-function [29, Eq.9.301)]. Furthermore, employing [32, Eq. (03.04.26.0006.01)] theessel function is expressed as a Meijer G-function, that is
k−m
[2(
m�
)1/2x]
= 12 G2,0
0,2
(m�
x2∣∣∣−k−m
2 ,− k−m2
). Therefore, based on
hese representations, Ni can re-expressed as
i = mm+k
2
√��(m)�(k)�
m+k2
∫ ∞
0
xk+m−1G2,00,2
(m
�x2∣∣∣−k−m
2 ,− k−m2
)
× G2,01,2
(A2
ix2
2
∣∣∣10, 12
)dx. (23)
The integral appearing in (23) can be solved in closedorm by making a change of variables and employing [32, Eq.07.34.21.0011.01)], yielding
i = 12√
� �(m)�(k)G2,2
3,2
(A2
i�
2m
∣∣∣1−k,1−m,10, 1
2
). (24)
Furthermore, employing [32, Eq. (07.34.03.0897.01)] and afterome mathematics, (24) simplifies to
i =√
�csc [�(m − k)]2
×{
�(
k + 12
)�(m)
(A2
i�
2m
)−k
× P FQ
[k, k + 1
2; k − m + 1, k + 1;
(A2
i�
2m
)−1]
−�(
m + 12
)�(k)
(A2
i�
2m
)−m
× P FQ
[m, m + 1
2; m − k + 1, m + 1;
(A2
i�
2m
)−1]}
(25)
here P FQ ( · ) represents the regularized generalized hypergeomet-ic function [32, Eq. (07.32.02.0001.01)]. Therefore, based on (22),mploying (24) (or 25) and the analytical framework presented inection 3 for evaluating integrals I1 and I2, exact expressions for theSEP of general order rectangular QAM in KG fading channels haveeen finally extracted. Next two special cases will be presented.
.1. Equal in-phase and quadrature decision distances
Assuming that dI = dQ as well as AI = AQ = A and by substituting9) and (24) in (22), the following simplified exact expression forhe ASEP of rectangular MIxMQ QAM is obtained as
¯ se =
(2 − 1
MI− 1
MQ
)√
� �(m)�(k)G2,2
3,2
(A2�
2m
∣∣∣1−k,1−m,10, 1
2
)−(
1 − 1MI
)(1 −
M
Please cite this article in press as: Bithas PS, et al. SEP of rectangular Q(2014), http://dx.doi.org/10.1016/j.aeue.2014.09.011
×2F1
(1, 1 + i + m;
32
+ i + m;12
)−
∞∑i=0
� (1 + i + k)� (i + (k − m) + 1) i!
[m
(k +
PRESSun. (AEÜ) xxx (2014) xxx–xxx 5
It should be noted that (26) only employs the Meijer-G and 2F1(·)hypergeometric functions that are both being widely used in theperformance evaluation of digital modulations over fading chan-nels. Also, these functions are available in common mathematicalsoftware, such as Mathematica.
4.2. Square QAM
Assuming MI = MQ = √M, the ASEP of square QAM over KG fad-
ing further simplifies to
Pse =
(2 − 2√
M
)√
� �(m)�(k)G2,2
3,2
(A2�
2m
∣∣∣1−k,1−m,10, 1
2
)−(
1 − 1√M
)2
× csc[�(k − m)]�(m)�(k)
×{ ∞∑
i=0
� (1 + i + m)� [i − (k − m) + 1] i!
[m/(�A2)
]m+i
(m + i)(1/2 + i + m)
×2F1
(1, 1 + i + m;
32
+ i + m;12
)−
∞∑i=0
� (1 + i + k) /i!� (i + (k − m) + 1)
×[m/(�A2)
]k+i
(k + i)(1/2 + i + k) 2F1
(1, 1 + i + k;
32
+ i + k;12
)}.
(27)
5. Numerical results and discussion
In this section, using the previously derived analysis, variousnumerically evaluated performance results, considering differentfading and shadowing conditions, will be presented and discussed.Similar to other studies in the past, e.g., [7], we consider (8 × 4)QAM constellation. In Fig. 1, the ASEP is plotted as a function of the10 log 10(ET/�2), where ET is the average total energy per symbol, fordifferent values of the fading parameter m as well as the in-phaseto quadrature decision distance ratio � = AQ/AI and k = 1. For (8 × 4)
QAM, the signal to noise ratio (SNR) is defined as ET�2 = (21+5�2)�A2
I2 .
In this figure, it is depicted a considerable improvement on the ASEPperformance when m increases, i.e., the fading severity decreases.Furthermore, it is interesting to note that the worst performanceis obtained for � = 21/5, whilst the corresponding performances for� = 1 and � =
√21/5 are quite similar, with the first one having
always the lower ASEP.In Fig. 2, the ASEP of the same modulation scheme is also plot-
ted as a function of the 10 log 10(ET/�2), for different values ofthe shadowing parameter k as well as the in-phase to quadraturedecision distance ratio � = AQ/AI and m = 1. In this figure it is alsodepicted that the performance improves as the shadowing condi-tions become better, i.e., k increases. An interesting observation isthat the performance improvement induced due to the increase of
AM in composite fading channels. Int J Electron Commun (AEÜ)
/(�A2)]k+i
i)(1/2 + i + k)× 2F1
(1, 1 + i + k;
32
+ i + k;12
)}. (26)
ARTICLE ING ModelAEUE-51287; No. of Pages 7
6 P.S. Bithas et al. / Int. J. Electron. Comm
Fig. 1. ASEP of 8 × 4 QAM in composite fading channels for different values of m.
F
tfiaKshati
6
ab
[
[
[
[
[
[
[
[
[
[
[20] Stüber GL. Principles of mobile communication. 2nd ed. USA: Kluwer; 2001.
ig. 2. ASEP of 8 × 4 QAM in composite fading channels for different values of k.
he first one having also the best performance. In the same figure,or comparison purposes, based on [7, Eq. (12)], the correspond-ng ASEP performance in Rayleigh fading environment has beenlso evaluated. It is shown that as the shaping parameter of theG distribution, i.e., k, increases the ASEP approaches the corre-ponding performance of Rayleigh fading. Further research effortsave confirmed that for higher values of k, the ASEP of KG tightlypproximates the one of Rayleigh fading. Finally, computer simula-ion performance results are also included in the figures, verifyingn all cases the validity of the proposed theoretical approach.
. Conclusions
Please cite this article in press as: Bithas PS, et al. SEP of rectangular Q(2014), http://dx.doi.org/10.1016/j.aeue.2014.09.011
Exact expressions for ASEP are derived for a general order rect-ngular QAM signalling over composite fading channels modelledy the KG distribution. These expressions are based on the exact
[
[
PRESSun. (AEÜ) xxx (2014) xxx–xxx
solution derived for the first time for the average of the productof two Gaussian-Q functions over KG fading. The final expres-sions are obtained in terms of well known mathematical functions.In addition, for the special case of integer and a half values ofthe distribution shaping parameters, simplified expressions arealso provided. Numerical evaluated results depict the performancedegradation induced to the system under investigation due to thefading/shadowing phenomena.
Acknowledgements
This work has been co-financed by the European Union (Euro-pean Social Fund-ESF) and Greek national funds through theOperational Program “Education and Lifelong Learning” of theNational Strategic Reference Framework (NSRF)-Research FundingProgram THALES INTENTION (MIS: 379489). Investing in knowl-edge society through the European Social Fund.
References
[1] Karagiannidis GK. On the symbol error probability of general order rectangularQAM in Nakagami-m fading. IEEE Commun Lett 2006;10:745–7.
[2] Yoon D, Cho K. General bit error probability of rectangular quadrature ampli-tude modulation. Electron Lett 2002;38:131–3.
[3] ETSI-TS-102-991, Digital Video Broadcasting (DVB); Implementation guide-lines for a second generation digital cable transmission system, TechnicalReport, European Telecommunications Standards Institute, v1.2.1; 2011.
[4] ETSI-TS-102-584, Digital Video Broadcasting (DVB); DVB-SH Implementationguidelines, Technical Report, European Telecommunications Standards Insti-tute, v1.3.1; 2011.
[5] ETSI-TS-102-831, Digital Video Broadcasting (DVB); Implementation guide-lines for a second generation digital terrestrial television broadcasting system(DVB-T2), Technical Report, European Telecommunications Standards Insti-tute, v1.2.1; 2012.
[6] Proakis JG. Digital communications. 3rd ed. McGraw-Hill; 1995.[7] Beaulieu NC. A useful integral for wireless communication theory and its appli-
cation to rectangular signalling constellation error rates. IEEE Trans Commun2006;54:802–5.
[8] Suraweera HA, Armstrong J. A simple and accurate approximation to the SEP ofrectangular QAM in arbitrary Nakagami-m fading channels. IEEE Commun Lett2007;11:426–8.
[9] Lei X, Fan P, Hao L. Performance analysis of general rectangular QAM with MRCdiversity over Nakagami-m fading channels with arbitrary fading parameters.In: International Conference on Wireless Communications, Networking andMobile Computing. 2007. p. 641–4.
10] Maaref A, Aissa S. Exact error probability analysis of rectangular QAM for single-and multichannel reception in Nakagami-m fading channels. IEEE Trans Com-mun 2009;57:214–21.
11] Dixit D, Sahu P. Performance analysis of rectangular QAM with SC receiver overNakagami-m fading channels. IEEE Commun Lett 2014;18:1262–5.
12] Romero-Jerez J, Pea-Martin J. Closed-form ASER results of rectangular QAM inMIMO MRC with arbitrary number of antennas. In: IEEE International Confer-ence on Communications (ICC). 2010. p. 1–5.
13] Romero-Jerez J, Pena-Martin J. ASER of rectangular MQAM in noise-limitedand interference-limited MIMO MRC systems. IEEE Wirel Commun Lett2012;1:18–21.
14] Jang W. Diversity order and coding gain of general-order rectangular QAMin MIMO relay with TAS/MRC in Nakagami-m fading. IEEE Trans Veh Technol2014;63:3157–66.
15] Al-Qahtani FS, Duong TQ, Gurung AK, Bao VNQ. Selection decode-and-forwardrelay networks with rectangular QAM in Nakagami-m fading channels. In:Wireless Communications and Networking Conference. 2010. p. 1–4.
16] Asghari V, Maaref A, Aissa S. Symbol error probability analysis for multihoprelaying over Nakagami fading channels. In: Wireless Communications andNetworking Conference. 2010. p. 1–6.
17] Trigui I, Affes S, Stephenne A. On the performance of dual-hop fixed gainrelaying systems over composite multipath/shadowing channels. In: VehicularTechnology Conference. 2010. p. 1–5.
18] Dixit D, Sahu PR. Symbol error rate of rectangular QAM with best-relay selec-tion in cooperative systems over Rayleigh fading channels. IEEE Commun Lett2012;16:466–9.
19] Simon MK, Alouini M-S. Digital communication over fading channels. 2nd ed.New York: Wiley; 2005.
AM in composite fading channels. Int J Electron Commun (AEÜ)
21] Abdi A, Kaveh M. Comparison of DPSK and MSK bit error rates for K andRayleigh-lognormal fading distributions. IEEE Commun Lett 2000;4:122–4.
22] Bithas PS. Weibull-gamma composite distribution: alternative multi-path/shadowing fading model. Electron Lett 2009;45:749–51.
ING ModelA
. Comm
[
[
[
[
[
[
[
[
[lognormal distribution in fading-shadowing wireless channels. Electron Lett
ARTICLEEUE-51287; No. of Pages 7
P.S. Bithas et al. / Int. J. Electron
23] Bithas PS, Sagias NC, Mathiopoulos PT, Karagiannidis GK, Rontogiannis AA. Onthe performance analysis of digital communications over Generalized-K fadingchannels. IEEE Commun Lett 2006;5:353–5.
24] Al-Ahmadi S, Yanikomeroglu H. On the approximation of the generalized-Kdistribution by a gamma distribution for modeling composite fading channels.IEEE Wirel Commun 2010;9:706–13.
25] Kostic I. Analytical approach to performance analysis for channel subject toshadowing and fading. IEEE Proc – Commun 2005;152:821–7.
26] Zhu C, Mietzner J, Schober R. On the performance of non-coherent transmission
Please cite this article in press as: Bithas PS, et al. SEP of rectangular Q(2014), http://dx.doi.org/10.1016/j.aeue.2014.09.011
schemes with equal-gain combining in generalized K-fading. IEEE Trans WirelCommun 2010;9:1337–49.
27] Bissias N, Efthymoglou GP, Aalo VA. Performance analysis of dual-hop relaysystems with single relay selection in composite fading channels. AEU – Int JElectron Commun 2012;66:39–44.
[[
PRESSun. (AEÜ) xxx (2014) xxx–xxx 7
28] Abramowitz M, Stegun IA. Handbook of mathematical functions, withformulas, graphs, and mathematical tables. 9th ed. New York: Dover;1972.
29] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. 6th ed. NewYork: Academic Press; 2000.
30] Shankar PM. Error rates in generalized shadowed fading channels. Wirel PersCommun 2004;28:233–8.
31] Abdi A, Kaveh M. K distribution: an appropriate substitute for Rayleigh-
AM in composite fading channels. Int J Electron Commun (AEÜ)
1998;34:851–2.32] The Wolfram Functions Site; 2014 http://functions.wolfram.com33] Prudnikov AP, Brychkov YA, Marichev OI. Integrals and series. 2nd ed. Gordon
and Breach Science Publishers; 1986.