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Citation for the original published paper (version of record):
Chu, T., Phan, H., Zepernick, H. (2015)
Performance analysis of MIMO cognitive amplify-and-forward relay networks with orthogonal space–time block codes.
Wireless Communications & Mobile Computing, 15: 1659-1679 http://dx.doi.org/10.1002/wcm.2449
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Published online 28 January 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.2449
RESEARCH ARTICLE
Performance analysis of MIMO cognitive
amplify-and-forward relay networks with orthogonal space–time block codes
Thi My Chinh Chu
*, Hoc Phan and Hans-Jürgen Zepernick
Blekinge Institute of Technology, Karlskrona, Sweden
ABSTRACT
In this paper, we study the performance of multiple-input multiple-output cognitive amplify-and-forward relay networks using orthogonal space–time block coding over independent Nakagami-m fading. It is assumed that both the direct trans- mission and the relaying transmission from the secondary transmitter to the secondary receiver are applicable. In order to process the received signals from these links, selection combining is adopted at the secondary receiver. To evaluate the system performance, an expression for the outage probability valid for an arbitrary number of transceiver antennas is presented. We also derive a tight approximation for the symbol error rate to quantify the error probability. In addition, the asymptotic performance in the high signal-to-noise ratio regime is investigated to render insights into the diversity behavior of the considered networks. To reveal the effect of network parameters on the system performance in terms of outage probability and symbol error rate, selected numerical results are presented. In particular, these results show that the performance of the system is enhanced when increasing the number of antennas at the transceivers of the secondary network. However, increasing the number of antennas at the primary receiver leads to a degradation in the secondary system performance. Copyright © 2014 John Wiley & Sons, Ltd.
KEYWORDS
multiple-input multiple-output (MIMO); amplify-and-forward (AF); relay networks; orthogonal space–time block codes (OSTBCs);
cognitive radio networks; outage probability; symbol error rate (SER)
*Correspondence
Thi My Chinh Chu, Blekinge Institute of Technology, Karlskrona, Sweden.
E-mail: thi.my.chinh.chu@bth.se
1. INTRODUCTION
With the growing demand for multimedia services in wireless communication, nowadays, frequency resources become more and more exhausted. In addition, the transmission of high data rate services faces fundamental limitations because of impairments inflicted by multipath fading channels. Hence, achieving high data rates with reliable transmission is a major challenge for a wireless communication system design. Recently, the studies of cognitive radio (CR) technology and cooperative tech- niques have emerged as promising approaches to improve the efficiency of frequency spectrum utilization, to enhance the transmission reliability, and to extend the coverage [1–11]. Specifically, the literature [1–3] provided several important aspects of a cognitive relay network (CRN) such as interference temperature estimation, spectrum hold detection, and dynamic spectrum management. In [4,5], power allocation strategies for CRNs with multiple relays
have been proposed to optimize throughput of CRNs. Fur- thermore, the works of Han et al. [6], Si et al. [7], and Lee et al. [8] investigated system performance in terms of outage probability and symbol error rate (SER) for cognitive decode-and-forward relay networks under inter- ference constraints. In [9], the performance of cognitive amplify-and-forward (AF) relay networks with selection combining (SC) was investigated. In addition, the study of Bohara et al. [10] proposed a spectrum sharing scheme for an overlay cognitive network wherein secondary transmission only opportunistically accesses the licensed spectrum. On the contrary, Louni and Khalaj [11] have focused on deriving a power control algorithm for another cognitive approach, namely the underlay scheme, in which secondary communication is allowed to coexist with the primary transmission.
Besides cooperative transmission, diversity techniques
such as maximum ratio transmission and orthogonal
space–time block codes (OSTBCs) in multiple-input
multiple-output (MIMO) systems are shown as powerful approaches to combat the adverse effect of fading, increase capacity and extend radio coverage [12,13]. Implementing OSTBCs can obtain full spatial diversity as well as low implementation complexity because of its linear decoding [13,14]. Specifically, Lee and Kim [15] have shown that maximum diversity gain can be achieved for decouple- and-forward relay networks with OSTBC transmission.
In [16,17], outage probability and SER of MIMO dual- hop AF relay networks with OSTBC transmission have been investigated. Furthermore, the works of Chalise and Vandendorpe [18] and Yang et al. [19] have analyzed the outage probability and diversity gain of decode-and- forward multiple relay networks using OSTBC transmis- sion. However, all of the aforementioned works studied the utilization of OSTBC transmission for cooperative relay networks only. With characteristics of achieving full diver- sity and low complexity, integrating OSTBC transmission for a CRN, which suffers from many restrictions imposed by the primary network, can be a promising approach to improve system performance. Recently, the studies of Axell and Larsson [20] and Wei et al. [21] have con- sidered the incorporation of OSTBC transmission into cognitive networks. Specifically, Axell and Larsson [20]
proposed a number of detectors to sense signals encoded with OSTBC, and Wei et al. [21] investigated the through- put for overlay cognitive networks. Both authors [20,21]
investigated OSTBC transmission in one hop overlay cog- nitive networks. The disadvantage of the overlay scheme is that the secondary transmission is allowed to access spectrum only if the primary transmission is idle. This policy reduces the opportunity for overlay cognitive net- works to occupy the licensed spectrum. On the con- trary, the underlay scheme allows the secondary user to occupy the licensed spectrum at any time provided that its transmit power is regulated to meet the interference power constraint given by the primary user [8]. There- fore, underlay cognitive systems can enhance the effective- ness of spectrum utilization as compared with the overlay scheme. Nonetheless, restricting the transmitter power in an underlay cognitive network will somewhat degrade its performance. This disadvantage can be alleviated by incor- porating OSTBC transmission into MIMO CRNs. To the best of our knowledge, there exist no works focusing on applying OSTBC transmission for MIMO cognitive AF relay networks.
In this paper, we therefore study the use of OSTBC transmission in a MIMO cognitive AF relay network wherein the direct transmission of the secondary relay sub- system has been taken into consideration. It is assumed that SC is utilized to process the received signals at the secondary receiver. Further, we focus on Nakagami-m fad- ing, which captures a wide variety of fading models. In particular, we derive an expression for the outage prob- ability and a tight bound for the SER of the examined networks. In order to reveal the contribution of each link on the overall system performance, we also make perfor- mance comparisons among the cases of employing only the
direct transmission (no relay), only the relay transmission (no direct links), and both relaying and direct transmis- sions with SC at the destination (the considered network).
To gain insights into the spatial diversity behavior of the considered networks, an asymptotic analysis has been con- ducted. Particularly, asymptotic expressions for the out- age probability and SER are presented, which enables us to evaluate the diversity gain and coding gain of the considered network.
Our paper is organized as follows. The system and channel model are presented in Section 2, in which the related concepts of the considered MIMO cognitive AF relay network with OSTBC transmission are introduced. In Section 3, we present the performance analysis for the examined network. An asymptotic analysis for the outage probability and SER is presented in Section 4. Section 5 presents special cases of the system. Numerical results and discussions are provided in Section 6.
Notation: Throughout this paper, the following notations are used. Bold lower and upper case letters denote a vec- tor and a matrix, respectively. The superscript H stands for the transpose conjugate of a vector or matrix. In addition, f
X./ and F
X./ indicate the probability density function (PDF) and the cumulative distribution function (CDF) of a random variable (RV) X , respectively. The operator k k
Fpresents the Frobenius norm of a vector or matrix, and Efg is the expectation operator. Then, an additive white Gaussian noise (AWGN) RV with zero mean and vari- ance N
0is denoted as CN .0; N
0/. We use .n/ as the gamma function [22, Equation (8.310.1)] and .n; x/ as the incomplete gamma function [22, Equation (8.350.2)].
Further, K
n./ and W
;.x/ represent the n-th order modified Bessel function of the second kind [22, Equation (8.432.1)] and the Whittaker function [22, Equation (9.222)], respectively. Finally, U .a; bI x/ stands for the confluent hypergeometric function [22, Equation (9.211.4)], and
2F
1.a; bI cI x/ denotes the Gauss hypergeometric function defined in [23, Equation (2.12.1)].
2. SYSTEM AND CHANNEL MODEL
We consider a MIMO cognitive AF relay network using OSTBC transmission consisting of an N
1-antenna sec- ondary transmitter, SU
TX, an N
2-antenna secondary relay, SU
R, an N
3-antenna secondary receiver, SU
RX, and an N
4- antenna primary receiver, PU, as shown in Figure 1. Let X D fx
1; x
2; : : : ; x
Mg be the input information sequence of the encoder at SU
TXincluding M symbols selected from any modulation scheme such as pulse amplitude mod- ulation (PAM), phase shift keying (PSK), or quadrature amplitude modulation (QAM). To be transmitted through N
1antennas, these symbols are encoded into an N
1L
1OSTBC matrix S where L
1is the block length of the
codeword, that is, the number of time slots used to trans-
mit the codeword. We utilize the generalized orthogonal
designs [13,24] to generate the OSTBC matrices, which
are designed for both real and complex constellations. In
Figure 1. System model of the considered MIMO cognitive AF
relay network with OSTBC transmission.
the case of real constellations such as PAM, the struc- ture of OSTBC matrices is given in Section 4 of [13] and Section 4.5 of [24]. In the case of complex constellations such as PSK and QAM, the structure of OSTBC matri- ces is given in Section 5 of [13] and Section 4.7 of [24].
Assume that all terminals operate in half-duplex mode and all channels are subject to quasi-static Nakagami-m fading.
Furthermore, we also assume that SU
Rand SU
RXhave per- fect channel state information to decouple and decode their received signals.
In the first hop, the SU
TXtransmits an OSTBC matrix S over L
1time slots with the code rate of the OSTBC encoder R
C1D M =L
1. As a result, the N
2L
1received signal matrix Y
1at SU
Rand the N
3L
1received signal matrix Y
3at SU
RXare, respectively, given by
Y
1D H
1S C Z
1(1)
Y
3D H
3S C Z
3(2)
where H
1is the N
2N
1channel coefficient matrix from SU
TXto SU
Rand H
3is the N
3N
1channel coefficient matrix from SU
TXto SU
RX. Furthermore, Z
1and Z
3are, respectively, the N
2L
1and N
3L
1AWGN matrices at SU
Rand SU
RXwhose elements are complex Gaussian RVs with zero mean and variance N
0. Let P
1and P
T1be the average transmit power per symbol and the total trans- mit power through N
1antennas, respectively; the transmit power of the OSTBC matrix S must satisfy E ˚
kSk
2FD N
1MP
1D L
1P
T1or P
1D P
T1=.N
1R
C1/. Denoting Q as the interference power threshold at PU, the total inter- ference that PU can tolerate in L
1time slots is QL
1. Therefore, E ˚
kH
4Sk
2FD N
1MP
1kH
4k
2FD QL
1where H
4is the N
4N
1channel coefficient matrix from SU
TXto PU. Hence, the transmit power of each symbol at SU
TXmust satisfy the following constraint:
P
1D Q
N
1R
C1kH
4k
2F(3) In the second hop, the SU
Rdecomposes the received sig- nals by using the squaring approach [25]. Consequently, the i -th decoupled signal Q x
icorresponding to the i -th symbol x
iof the source can be expressed as
Q
x
iD kH
1k
2Fx
iC n
1(4) where n
1is the noise component of the decoupled signal.
As in [25, Equation (19)], n
1is an AWGN with zero mean and variance N
0kH
1k
2F, denoted as CN
0; N
0kH
1k
2F. Considering M decoupled signals f Q x
ig
Mi D1as the input, the OSTBC encoder of SU
Rgenerates an N
2L
2OSTBC matrix QS with code rate R
C2D M =L
2. Then, SU
Rmul- tiplies this OSTBC with a scalar gain G and forwards the signal to the SU
RX. Let P
2and P
T2, respectively, denote the average transmit power per symbol and the total transmit power through N
2antennas at the SU
R, that is, L
2P
T2D N
2MP
2or P
2D P
T2=.N
2R
C2/. The gain factor G is determined as E ˚
jjG Q x
ijj
2D P
2. From (4), we have
G
2P
2P
1kH
1k
4F(5)
As a result, the N
3L
2received signal matrix Y
2at the SU
RXis given by
Y
2D H
2G QS C Z
2(6) where H
2is an N
3N
2channel coefficient matrix from SU
Rto SU
RXand Z
2stands for the N
3L
2AWGN matrix at SU
RXwhose elements are complex Gaussian RVs with zero mean and variance N
0. Let H
5be the N
4N
2channel coefficient matrix from SU
Rto PU. To prevent the interference caused by the SU
Rtransmission to the PU from being beyond the predefined threshold Q, the transmit power of each symbol at SU
Rmust satisfy E n
kH
5QSk
2Fo
D N
2MP
2kH
5k
2FD QL
2or
P
2D Q
N
2R
C2kH
5k
2F(7) At the SU
RX, using the squaring approach [25] to decom- pose the received signals, the i -th decoupled symbol from the relaying link SU
TX! SU
R! SU
RX, Q x
R;i, and the i -th decoupled symbol from the direct link SU
TX! SU
RX, Q x
D;i, are, respectively, given by
Q
x
R;iD GkH
1k
2FkH
2k
2Fx
iC GkH
2k
2Fn
1C n
2(8) Q
x
D;iD kH
3k
2Fx
iC n
3(9)
where n
2and n
3are, respectively, the noise compo-
nents of the decoupled signals Q x
R;iand Q x
D;i, that is,
CN
0; N
0kH
2k
2Fand CN
0; N
0kH
3k
2F. Therefore, the instantaneous signal-to-noise ratio (SNR) per symbol at the SU
RXfrom the relaying link is given by
D1D E n
jG H
12F
H
22
F
x
ij
2o E n
jG kH
2k
2Fn
1C n
2j
2o (10)
Substituting (3), (5), and (7) into (10), after some manip- ulations, we finally obtain the instantaneous SNR at the SU
RXfrom the relaying link as
D1D Q N
0H
12F
H
22
F
N
1R
C1H
22F
H
42F
C N
2R
C2H
12F
H
52F
(11) Similarly, the instantaneous SNR at the SU
RXfrom the direct link is expressed as
D2D E ˚
jGkH
3k
2Fx
ij
2Efjn
3j
2g D Q
N
0kH
3k
2FN
1R
C1kH
4k
2F(12)
For the sake of brevity, we denote kH
lk
2FD X
l; l 2 f1; 2; 3; 4; 5g, D
NQ0
, a D N
1R
C1, and b D N
2R
C2. Then, we rewrite the instantaneous SNRs at the SU
RXfor the relay link and the direct link as, respectively,
D1D X
1X
2aX
2X
4C bX
1X
5D
C(13)
D2D X
3aX
4(14)
where
CD X
1X
2=.aX
2X
4C bX
1X
5/. We assume that all channel coefficients are independent and identical dis- tributed within each hop with fading severity parameter m
land channel mean power
l. Clearly, X
lis the sum of M
lgamma RVs with parameter set
m
l; ˛
l1where ˛
lD
mland M
1D N
1N
2; M
2D N
2N
3; M
3D N
1N
3; M
4D
lN
1N
4, and M
5D N
2N
4. Consequently, the PDF and CDF of X
lare written as
f
Xl.x/ D ˛
lMlml.M
lm
l/ x
Mlml1exp .˛
lx/ (15)
F
Xl.x
l/ D 1
Mlml1
X
qD0
˛
qlx
lqqŠ exp .˛
lx
l/ (16)
3. END-TO-END PERFORMANCE ANALYSIS
In this section, we first derive an expression for the CDF of the instantaneous SNR
D, which enables us to obtain anexpression for the outage probability. Then, we derive a
tight bound for the CDF of the instantaneous SNR to attain an approximation for the SER.
3.1. Outage probability
Because SC is utilized at the SU
RXto select the best signal through the relaying and direct transmissions, the instantaneous SNR at the SU
RXafter SC is determined as
DD max
D1;
D2(17)
As observed from (13) and (14), X
4appears in both terms
D1and
D2, which leads to statistical dependence among them. The reason for mutually statistical depen- dence of
D1and
D2in a cognitive cooperative system is deduced from the fact that both the relaying trans- mission and the direct transmission is constrained by the interference power threshold of the PU. Consequently, the CDF of
Dcannot be obtained as for conventional relay networks, meaning that F
D. / ¤ F
1D
. /F
2 D. /.
However, thanks to the statistical independence among X
4and the other terms X
1, X
2, X
3, and X
5, we first calculate F
1D
. j
X4/ and F
2D
. j
X4/ and then attain F
D. j
X4/ as
F
D. j
X4/ D F
1D
. j
X4/F
2D
. j
X4/ (18) where F
1D
j
X4is obtained from (13) as F
1D
. j
X4/ D F
Cj
X4and F
2D
. j
X4/ is obtained from (14) as F
2D
. j
X4/ D F
X3 ax4
. As a result, the unconditional CDF of
Dis given by
F
D. / D Z
10
F
1D
. j
X4/F
2D
. j
X4/f
X4.x
4/dx
4(19) Then, the following theorem can be utilized to derive an expression for F
D. /.
Theorem 1. The CDF of the instantaneous SNR of the considered network can be given by
F
D. / D 1 J
1J
2C J
3(20) where
J
1D
N1N3m31
X
i D0
a
i˛
3i˛
4N1N4m4 N1N4m4i Š
.N
1N
4m
4C i /
.N
1N
4m
4/
i.˛
4C ˛
3a /
N1N4m4Ci(21)
J
2D
N1N
X
2m11 pD01 pŠ
X
p qD0C
qpN2N
X
3m21 rD0C
rN2N3m21a
pCrqC1b
N2N3m2˛
1pCrqC1˛
N22N3m2˛
4N1N4m4˛
5N2N4m5N2N4m5CN1N4m4
.N
1N
4m
4C p/.N
2N
3m
2C N
2N
4m
5r C q 1/.N
2N
3m
2C N
2N
4m
5/
.N
2N
4m
5/.N
2N
3m
2/.N
1N
4m
4C N
2N
3m
2C N
2N
4m
5C p/
.N
1N
4m
4C p C r q C 1/
.N
1N
4m
4/
N2N3m2CpCrqC1.˛
5C ˛
2b /
N1N4m4N2N3m2N2N4m5CpCrqC1.˛
4˛
52C ˛
1˛
5a C ˛
2˛
4b C ˛
1˛
2ab
2/
.N1N4m4CpCrqC1/ 2F
1N
1N
4m
4C p C r q C 1; N
2N
3m
2C N
2N
4m
5; N
1N
4m
4C N
2N
3m
2(22)
C N
2N
4m
5C pI ˛
4˛
52C ˛
1˛
5a C ˛
2˛
4b
˛
4˛
52C ˛
1˛
5a C ˛
2˛
4b C ˛
1˛
2ab
2J
3D
N1N2m11
X
pD0
1 pŠ
X
p qD0C
qpN2N3m21
X
rD0
C
rN2N3m21N1N3m31
X
i D0
a
pCrCi qC1b
N2N3m2˛
pCrqC11˛
2N2N3m2˛
3ii Š
˛
4N1N4m4˛
5N2N4m5N2N4m5CN1N4m4.N
2N
3m
2C N
2N
4m
5/ .N
1N
4m
4C i C p/
.N
2N
3m
2/.N
1N
4m
4C N
2N
3m
2C N
2N
4m
5C p C i /
.N
1N
4m
4C i C p C r q C 1/ .N
2N
3m
2C N
2N
4m
5r C q 1/
N2N3m2Ci CpCrqC1.N
1N
4m
4/.N
2N
4m
5/
.˛
5C ˛
2b /
N1N4m4N2N3m2N2N4m5Ci CpCrqC1˛
4˛
52C ˛
5˛
3a C ˛
1˛
5a C ˛
2˛
4b C ˛
2˛
3ab
2C ˛
1˛
2ab
2N1N4m4Ci CpCrqC1 2F
1N
1N
4m
4C i C p C r q C 1; N
2N
3m
2C N
2N
4m
5; N
1N
4m
4C N
2N
3m
2C N
2N
4m
5Cp C i I ˛
4˛
52C ˛
5˛
3a C ˛
1˛
5a C ˛
2˛
4b C ˛
2˛
3ab
2˛
4˛
52C ˛
5˛
3a C ˛
1˛
5a C ˛
2˛
4b C ˛
2˛
3ab
2C ˛
1˛
2ab
2!
(23)
Proof . See Appendix A.
The outage probability of the considered network, that is, the probability that the instantaneous SNR falls below a specified threshold
th, is found by using
thas argument of the CDF given in (20) as P
outD F
D.
th/.
3.2. Symbol error rate
As reported in [26], SER can be directly expressed in terms of the CDF of the instantaneous SNR
Das follows:
P
eD a
1p b
12 p
Z
10
F
D. /
12exp.b
1/d (24)
where a
1and b
1are parameters depending on the modula- tion scheme, that is, for M -PSK a
1D 2; b
1D sin
2M
. Clearly, deriving an expression for the SER from the exact
closed-from expression of F
D. / in (20) is very challeng- ing. Thus, we adopt a tight approximation for the instanta- neous SNR of the considered network, denoted as
DU, to analyze the SER performance. The SER expression can be obtained as given in the following theorem.
Theorem 2. A tight approximation for the SER can be found as
P
eD a
12 Q
1Q
2C Q
3(25) where
Q
1D a
1p b
12 p
N1N
X
3m31 i D01 i Š
˛
1 2 4
12a
12˛
312.N
1N
4m
4C i / i C
12.N
1N
4m
4/
U
i C 1
2 ; N
1N
4m
4C 3 2 ; b
1˛
4˛
3a
(26)
Q
2D a
1p b
12 p
N2N
X
3m21 rD01 rŠ
N1N
X
2m11 pD01 pŠ
˛
4N1N4m4˛
N52N4m5N1N4m4CN2N4m5a
N1N4m4b
N2N4m5˛
N11N4m4˛
2N2N4m5.N
2N
4m
5C r/
.N
2N
4m
5/
.N
1N
4m
4C p/
r C p C
12.N
1N
4m
4/
2 4
N2N
X
4m5Cr kD1 rk˛
5˛
2b
rCpC12k
U
r C p C 1
2 ; r C p C 3
2 k; b
1˛
5˛
2b
C
N1N
X
4m4Cp lD1 pl˛
4˛
1a
rCpC12l
U
r C p C 1
2 ; r C p C 3
2 l; b
1˛
4˛
1a 3
5 (27)
Q
3D a
1p b
12 p
N2N3m21
X
rD0
1 rŠ
N1N2m11
X
pD0
1 pŠ
N1N3m31
X
i D0
1 i Š
˛
1p˛
i3˛
4N1N4m4˛
5N2N4m5N1N4m4CN2N4m5a
N1N4m4b
N2N4m5˛
2N2N4m5.˛
1C ˛
3/
N1N4m4Ci Cp.N
2N
4m
5C r/ .N
1N
4m
4C p C i /
r C i C p C
12.N
2N
4m
5/.N
1N
4m
4/
2 4
N2N
X
3m3Cr j D1rj
˛
5˛
2b
rCi Cp C12j
U
r C i C p C 1
2 ; r C i C p C 3
2 j ; b
1˛
5˛
2b
C
N1N3
X
m5Ci Cp t D1ı
pt˛
4a ˛
1C a ˛
3rCi CpC12t
U
r C i C p C 1
2 ; r C i C p C 3
2 t ; b
1˛
4a ˛
1C a ˛
33
5 (28)
and
rk;
pl;
rj, and ı
ptare partial fraction coefficients defined in Appendix B as (B.21), (B.22), and (B.23), and (B.24), respectively.
Proof . See Appendix B.
4. ASYMPTOTIC PERFORMANCE ANALYSIS
It can be concluded that the expressions for the outage probability in (20) and SER in (25) are too complicated to render insights into the system performance. Therefore, in this section, asymptotic approximations for the outage probability and SER are provided to examine the diversity gain and coding gain of the considered system.
4.1. Asymptotic outage probability
As pointed out in [27], in order to obtain the diversity gain of the considered system, we need to present the outage probability in the high SNR regime as
P
out1D .G
c/
GdC o
Gd(29)
where D Q=N
0is the average interference power-to- noise ratio and o.
Gd/ denotes the higher order terms of P
out1, that is, lim
!1o.Gd/Pout1
D 0. Next, G
dis the diversity gain of the network. It is a crucial performance metric in MIMO systems, which determines the slope of the curve of the outage probability or SER versus average SNR in a log scale. From (29), G
dis given by
G
dD lim
!1
log P
outlog
(30)
Furthermore, G
cis the coding gain of the network, which determines the shift of the asymptotic outage probability curve as compared with the benchmark curve
Gd.
Usually, the asymptotic performance can be investigated
by deploying the MacLaurin series of the CDF of the
instantaneous SNR F
D. / around zero value. Neverthe-
less, because of the mathematical complexity of (20), it
is difficult to deploy the MacLaurin expansion directly
to F
D. /. Instead, we will start our asymptotic analy-
sis by deriving a bound of the CDF of
Das F
DU. /
given in (B.9). Then, we adopt the MacLaurin expansion to
F
DU. /. With this approach, the asymptotic expression for
F
DU. / is given in Theorem 3. As a result, the asymptotic
expression for outage probability of the considered system, P
out1, can be easily obtained as P
out1D F
DU.
th/.
Theorem 3. Assuming that the average interference power-to-noise ratio approaches infinity, ! 1, an asymptotic expression for F
DU. / can be given by
F
DU. /
!0D 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ <
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :
˛1N1N2m1˛N1N3m33 aN1N2m1CN1N3m3
˛N1N2m1CN1N3m3 4
.N1N2m1CN1N3m3CN1N4m4/
.N1N2m1C1/ .N1N3m3C1/ .N1N4m4/
N1N2m1CN1N3m3N1N2m1CN1N3m3
I if N
1N
2m
1< N
2N
3m
2˛2N2N3m2˛N1N3m33 aN1N3m3bN2N3m2
˛4N1N3m3˛5N2N3m2
.N2N3m2CN2N4m5/
.N2N4m5/ .N1N3m3C1/ .N2N3m2C1/
.N1N3m3CN1N4m4/N1N3m3CN2N3m2
N1N3m3CN2N3m2
N1N3m3CN2N3m2
I if N
1N
2m
1> N
2N
3m
2˛1N1N2m1˛N1N3m33 aN1N2m1CN1N3m3
˛N1N2m1CN1N3m3 4
.N1N2m1CN1N3m3CN1N4m4/
.N1N2m1C1/ .N1N3m3C1/ .N1N4m4/
N1N2m1CN1N3m3N1N2m1CN1N3m3
C
˛N2N3m2
2 ˛3N1N3m3aN1N3m3bN2N3m2
˛N1N3m34 ˛5N2N3m2.N2N4m5/
.N2N3m2CN2N4m5/
.N1N3m3C1/
.N.N1N3m3CN1N4m4/2N3m2C1/.N1N4m4/
N1N3m3CN2N3m2
N1N3m3CN2N3m2
I if N
1N
2m
1D N
2N
3m
2(31)
Proof . See Appendix C.
4.2. Asymptotic symbol error rate
Substituting (31) in (24) and utilizing [22, [Equation (3.381.4)]] yield the asymptotic expression for the SER as
P
1ED 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
<
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :
a1 2p
bN1N3m3CN1N2m1 1
˛1N1N2m1˛N1N3m33 aN1N2m1CN1N3m3
˛N1N2m1CN1N3m3 4
.N1N2m1CN1N3m3CN1N4m4/
.N1N2m1C1/
N1N3m3CN1N2m1C12
.N1N3m3C1/.N1N4m4/N1N2m1CN1N3m3
I if N
1N
2m
1< N
2N
3m
2 a12p
bN1N3m3CN2N3m2 1
˛2N2N3m2˛N1N3m33 aN1N3m3bN2N3m2
˛N1N3m34 ˛N2N3m25
.N2N3m2CN2N4m5/
.N2N4m5/ .N1N3m3C1/
.N1N3m3CN1N4m4/
N1N3m3CN2N3m2C12
.N2N3m2C1/ .N1N4m4/ N1N3m3CN2N3m2
I if N
1N
2m
1> N
2N
3m
2 a12p
bN1N3m3CN1N2m1 1
˛1N1N2m1˛N1N3m33 aN1N2m1CN1N3m3
˛N1N2m1CN1N3m3 4
.N1N2m1CN1N3m3CN1N4m4/
.N1N2m1C1/
N1N3m3CN1N2m1C12
.N1N3m3C1/.N1N4m4/N1N2m1CN1N3m3
C
a12p
bN1N3m3CN2N3m2 1
˛
2N2N3m2˛
3N1N3m3 aN1N3m3bN2N3m2˛N1N3m34 ˛N2N3m25
.N2N3m2CN2N4m5/
.N2N4m5/ .N1N3m3C1/
.N1N3m3CN1N4m4/
N1N3m3CN2N3m2C12
.N2N3m2C1/ .N1N4m4/ N1N3m3CN2N3m2
I if N
1N
2m
1D N
2N
3m
2(32)
As we can see from (31) the diversity gain of the con- sidered system is equal to min.N
1N
3m
3C N
1N
2m
1; N
1N
3m
3C N
2N
3m
2/. Thus, the diversity gain only depends on the number of antennas at the terminals and
the fading severity parameter of each hop in the secondary relay network. On the contrary, the coding gain of the con- sidered system does not only depend on the number of antennas at the terminals and the fading severity parame- ters of the hops in the secondary relay network but also depends on the number of antennas at PU and fading severity parameters of the links from SU
TXto PU and SU
Rto PU.
5. SPECIAL CASES OF THE CONSIDERED SYSTEM
As mentioned before, in this paper, we deploy OSTBC for MIMO cognitive cooperative networks by considering both direct and relaying transmissions. In the sequel, we
also provide derivations for special cases of our consid-
ered system, i.e., the combination of MIMO and cogni-
tive radio, MIMO and cooperative communications, and
cognitive radio and cooperative communications.
5.1. Combination of multiple-input multiple-output and cognitive
Considering the combination of MIMO and CRs, deploy- ing only direct communication, expressions for the out- age probability and the SER of the system are derived as given in (D.4) and (D.10) of Appendix D, respectively. Fur- thermore, it can be seen from the numerical results that the outage probability and SER of the considered MIMO cognitive cooperative network substantially outperform that of the MIMO CR networks.
5.2. Combination of multiple-input multiple-output and cooperation
In the case of deploying OSTBC for MIMO coopera- tive communication systems, without considering CR, the transmit powers at the source and the relay are not con- strained by the interference power threshold; that is, the source can transmit the signal with average power P
s. Furthermore, the terms X
4and X
5, which respectively represent the channel power gains from the source and the relay to the primary receiver will no longer exist in (13). If the source and the relay utilize the same code rate R
C1D R
C2, the system model here is consistent with the system model of [17]. Also, if we consider only the relaying transmission as in [17], the received SNR at the destination will be given by
DD ˇ X
1X
2X
2C X
1(33)
ˇ D P
s=.R
C1N
1N
0/. Based on (33), the analysis of the outage probability and SER will be much more sim- ple. Following the approach used in this paper, we can derive expressions for the outage probability and SER as the following:
P
outD 1
N2N
X
3m31 qD0˛
2qqŠ
t h
ˇ
qexp
˛
2t hˇ
C
N2N
X
3m21 i D0C
iN2N3m21˛
2N2N3m2i 1.i C 1/
.N
2N
3m
2/
t h
ˇ
N2N3m21i
exp
˛
2t hˇ
2
N1N2m11
X
pD0
N2N3m2Cp1
X
j D0
C
iN2N3m2Cp1pŠ
˛
N2N3m2C2pi 2 1
.N
2N
3m
2/
˛
N2N3m2Ci 2 2
t h
ˇ
N2N3m2CpCj i
exp
.˛
1C ˛
2/
t hˇ
K
N2N3m2i2 p
˛
1˛
2t hˇ
(34)
P
eD a
12
N2N3m31
X
qD0
˛
2q12q C
12qŠ.b
1C ˛
2/
qC12C
N2N3m21
X
i D0
C
iN2N3m21˛
2N2N3m2i 1N
2N
3m
2i
12.N
2N
3m
2/
.i C 1/
12.b
1C ˛
2/
N2N3m2i 122 p
N1N
X
2m11 pD0N2N3
X
m2Cp1 j D04
N2N3m2iC
iN2N3m2Cp112pŠ
p C j C
12.N
2N
3m
2/
˛
1N2N3m2Cpi˛
N22N3m2Ci2N
2N
3m
2C p C j 2i C
12b
1C ˛
1C ˛
2C 2 p
˛
1˛
22N2N3m2CpCj 2i C12.N
2N
3m
2C p C j i C 1/
2F
12N
2N
3m
2CpCj 2i C 1
2 ; N
2N
3m
2i C 1
2 ; N
2N
3m
2C p C j i C 1; b
1C .˛
1C ˛
2/ 2 p
˛
1˛
2b
1C .˛
1C ˛
2/ C 2 p
˛
1˛
2(35) It can be seen that the expressions of outage probability and SER are the same as those derived in [17].
5.3. Combination of cognitive radio and cooperation
For the case of a cognitive cooperative system without deploying MIMO techniques, the expressions derived for outage probability given in (20), SER given in (25), asymp- totic expressions on outage probability given in (31), and SER given in (32) are still applicable. However, in this case, the number of antennas at all terminals is now reduced to one, that is, N
1D N
2D N
3D N
4D 1.
Furthermore, the code rates R
C1at the source and R
C2at
the relay will be moved out of these equations as OSTBC
will no longer be deployed for this system, that is, a D
N
1R
C1D 1 and b D N
2R
C2D 1.
6. NUMERICAL RESULTS
In this section, we provide the numerical results, includ- ing analysis results and Monte Carlo simulations, for the performance metrics derived in the previous sections. For all considered scenarios, the outage threshold is set as
t hD 5 dB. Furthermore, we consider quadrature phase shift keying (QPSK) modulation, for which a
1D 2 and b
1D sin
24
, in all examples related to the SER.
In order to illustrate the effect of the number of antennas at the terminals on the system performance of the consid- ered networks, we depict the outage probability and SER for different antenna configurations as in Figures 2 and 3.
In these examples, the fading severity parameters and chan- nel mean powers are set as m
lD 0:5; l 2 f1; : : : ; 5g and .
1;
2;
3;
4;
5/ D .1:0; 1:0; 0:625; 0:5; 0:417/, respectively. Furthermore, the examined cases of the num- ber of antennas at the terminals are selected as follows:
Figure 2. Outage probability versus interference power-to-noise
ratio Q=N
0for different antenna configurations.
Figure 3. Symbol error rate of QPSK modulation versus
interference power-to-noise ratio Q=N
0for different antenna
configurations.
Case 1: .N
1; N
2; N
3; N
4/ D .2; 2; 2; 4/
Case 2: .N
1; N
2; N
3; N
4/ D .2; 2; 2; 2/
Case 3: .N
1; N
2; N
3; N
4/ D .2; 2; 4; 4/
Case 4: .N
1; N
2; N
3; N
4/ D .4; 4; 2; 2/
Case 5: .N
1; N
2; N
3; N
4/ D .2; 2; 8; 2/
Referring to (31) and (32), the antenna configurations in Cases 1 and 2 satisfy the relationship N
1N
2m
1D N
2N
3m
2, Cases 3 and 5 fulfil the relationship N
1N
2m
1<
N
2N
3m
2, and Case 4 complies with the relationship N
1N
2m
1> N
2N
3m
2. By comparing Case 2 with Case 3, Case 2 with Case 4, and Case 2 with Case 5 in Fig- ures 2 and 3, we notice that when the number of antennas at the terminals in the secondary network increases, the out- age probability and SER are decreased. This performance improvement in the outage probability and SER can be understood by the fact that increase in the number of anten- nas at the transceivers of the secondary system leads to increase in the coding gain and diversity gain. As expected, from Figures 2 and 3, we can see that the best performance is obtained with Case 5 giving the highest diversity gain. It can be also seen that Case 3 provides better performance than Cases 1 and 2 because Case 3 has higher diversity gain. With the antenna configurations in Cases 1 and 2, the respective networks have the same diversity gain. How- ever, the performance of Case 1 is a bit worse than that of Case 2. This performance reduction can be interpreted from the fact that Case 1 with a larger number of antennas at the PU than Case 2 leads to a stricter interference con- straint being imposed on the secondary users. In particular, the secondary users, that is, SU
TXand SU
R, must reduce their transmit powers to prevent the interference power at the PU receiver from being beyond the interference power threshold.
To investigate the effect of fading parameters on the sys- tem performance, we show the outage probability and SER as in Figures 4 and 5 for the three following cases:
Case 6: .
1;
2;
3;
4;
5I m
1; m
2; m
3; m
4; m
5/ D .0:7; 0:7; 0:5; 0:5; 0:5I 0:5; 0:5; 0:5; 0:5; 0:5/
Case 7: .
1;
2;
3;
4;
5I m
1; m
2; m
3; m
4; m
5/ D .1:0; 1:0; 0:7; 0:5; 0:5I 0:5; 0:5; 0:5; 0:5; 0:5/
Case 8: .
1;
2;
3;
4;
5I m
1; m
2; m
3; m
4; m
5/ D .1:0; 1:0; 0:7; 0:5; 0:5I 1:0; 1:0; 1:0; 1:0; 1:0/
wherein the antenna configurations at the terminals are fixed as .N
1; N
2; N
3; N
4/ D .2; 2; 2; 2/. In Figures 4 and 5, comparing Case 6 with Case 7, which have the same fading severity parameters m
lD 0:5; l 2 f1; : : : ; 5g, we can see that the higher channel mean powers of the links SU
TX! SU
R, SU
R! SU
RX, and SU
TX! SU
RXare the lower outage probability and SER are obtained. In addi- tion, fixing the values of the channel mean powers of all the links, when the fading severity parameters increase from m
lD 0:5 given in Case 7 to m
lD 1:0 given in Case 8, the outage probability and SER are improved significantly.
To reveal the performance advantage of the considered
network using both the direct and relaying transmissions
Figure 4. Outage probability versus interference power-to-noise
ratio Q=N
0for different fading channel parameters.
Figure 5. Symbol error rate of QPSK modulation versus inter-
ference power-to-noise ratio Q=N
0for different fading channel
parameters.
over the network using only the relaying transmission and over the network using only the direct transmission, we make performance comparisons in terms of the outage probability and SER among these three networks as in Figures 6 and 7. Assume that the SC scheme is utilized to combine the received signals from the relaying and direct transmissions of the considered network. We denote the normalized distances of the links SU
TX! SU
R, SU
R! SU
RX, SU
TX! SU
RX, SU
TX! PU, and SU
R! PU as d
1; d
2; d
3; d
4, and d
5, respectively.
Suppose that the channel mean powers for all the chan- nels attenuate according to the exponential decaying model with path-loss exponent of 4, d
l4; l 2 f1; : : : ; 5g, rep- resenting a suburban environment. For these illustrations, we set d
1D 0:5, d
2D 0:5, d
3D 0:7, d
4D 1:0, and
Figure 6. Comparing outage probability versus interference
power-to-noise ratio Q=N
0for the relaying link, the direct link,
and combining both links by SC.
Figure 7. Comparing symbol error rate of QPSK modulation ver-
sus interference power-to-noise ratio Q=N
0for the relaying link,
the direct link, and combining both links by SC.
d
5D 1:0. Moreover, the number of antennas at all ter-
minals is fixed as .N
1; N
2; N
3; N
4/ D .2; 2; 2; 2/, and
the fading severity parameters for all the channels are the
same m
lD 2:0; l 2 f1; : : : ; 5g. As can be seen from
Figures 6 and 7, respectively, the outage probability and
SER of the considered network substantially outperform
that of the network using only the relaying transmission
and that of the network using only the direct transmis-
sion. This performance benefit can be explained by the
fact that the relaying or direct transmission can contribute
its respective diversity gain to the system performance
of the considered network. Therefore, its diversity gain
will be greater than the networks using either relaying
transmission or direct transmission.
7. CONCLUSIONS
In this paper, we have investigated a MIMO cognitive dual- hop AF relay network with OSTBC transmission where selection combing is deployed at the SU
RXto select the signal with the maximum SNR among the relaying and the direct transmissions. Specifically, we have derived an expression for the outage probability and a tight approx- imation for the SER of the considered network over Nakagami-m fading. Moreover, an asymptotic analysis has been conducted for the outage probability and the SER to reveal the diversity gain of the network. Numerical results are also provided to illustrate the effect of fading param- eters and the number of antennas at the terminals on the system performance. In these selected examples, the per- formance of the examined network is improved relative to the increase in the number of antennas at the terminals and is degraded relative to the decrease in the fading severity parameter of the secondary channels. In contrast, increas- ing the number of antennas at PU results in a reduction of system performance. Finally, the considered network pro- vides significant performance improvement as compared with a network either using solely relaying transmission or using only direct transmission.
APPENDIX A: PROOF OF THEOREM 1
In this appendix, we derive the CDF of the instantaneous SNR
D. To this end, we write the CDF of
Cgiven in (13) conditioned on X
4as
F
C. j
X4/ D Z
10
Z
10
Pr
X
1x
2ax
2x
4C bX
1x
5f
X2.x
2/f
X5.x
5/dx
2dx
5(A.1) To further calculate F
C. j
X4/, we separate the integra- tion domain of x
2in (A.1) into two subsets as .x
2< b x
5/ and .x
2b x
5/. After some algebraic manipulations, we rewrite F
C. j
X4/ as
F
C. j
X4/ D Z
10
F
X2.b x
5/f
X5.x
5/dx
5„ ƒ‚ …
K
C Z
10
I
1f
X5.x
5/dx
5„ ƒ‚ …
I
(A.2)
where
I
1D Z
10
F
X1a x
4C ab
2x
4x
5x
2!
f
X2.x
2C b x
5/dx
2(A.3)
By substituting (15) and (16) into (A.3), we have
I
1D 1 F
X2. b x
5/
N1N
X
2m11 pD01 pŠ
X
p qD0C
qpN2N3m21
X
rD0
C
rN2N3m21a
pb
N2N3m2Cqr1.N
2N
3m
2/
˛
p1˛
N22N3m2N2N3m2CpCqr1x
4px
5N2N3m2Cqr1exp.˛
1a x
4/ exp.˛
2bx
5/
Z
10
x
rq2exp.˛
2x
2/
exp ˛
12a b x
4x
5x
2!
dx
2(A.4)
where C
abdenotes the binomial coefficient, that is, C
abD
aŠ
bŠ.ab/Š
. Applying [22, Equation (3.471.9)] to solve the remaining integral of (A.4), after rearranging terms, we obtain I
1as
I
1D 1 F
X2. bx
5/ 2
N1N
X
2m11 pD01 pŠ
X
p qD0C
qpN2N3m21
X
rD0
C
rN2N3m21˛
2pCrqC1 2
1
˛
2N2N3m2Cqr1 2 2
.N
2N
3m
2/
a
2pCrqC12b
2N2N3m2Cqr1 2N2N3m2Cp
x
2pCrqC1
4 2
x
2N2N3m2Cqr1 5 2