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This is the published version of a paper published in Wireless Communications & Mobile Computing.

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

Access to the published version may require subscription.

N.B. When citing this work, cite the original published paper.

Permanent link to this version:

http://urn.kb.se/resolve?urn=urn:nbn:se:bth-6740

(2)

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

(3)

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

F

presents 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

0

is 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

2

F

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

M

g be the input information sequence of the encoder at SU

TX

including 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

1

antennas, these symbols are encoded into an N

1

 L

1

OSTBC matrix S where L

1

is 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

(4)

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

R

and SU

RX

have per- fect channel state information to decouple and decode their received signals.

In the first hop, the SU

TX

transmits an OSTBC matrix S over L

1

time slots with the code rate of the OSTBC encoder R

C1

D M =L

1

. As a result, the N

2

 L

1

received signal matrix Y

1

at SU

R

and the N

3

 L

1

received signal matrix Y

3

at SU

RX

are, respectively, given by

Y

1

D H

1

S C Z

1

(1)

Y

3

D H

3

S C Z

3

(2)

where H

1

is the N

2

 N

1

channel coefficient matrix from SU

TX

to SU

R

and H

3

is the N

3

 N

1

channel coefficient matrix from SU

TX

to SU

RX

. Furthermore, Z

1

and Z

3

are, respectively, the N

2

 L

1

and N

3

 L

1

AWGN matrices at SU

R

and SU

RX

whose elements are complex Gaussian RVs with zero mean and variance N

0

. Let P

1

and P

T1

be the average transmit power per symbol and the total trans- mit power through N

1

antennas, respectively; the transmit power of the OSTBC matrix S must satisfy E ˚

kSk

2F

 D N

1

MP

1

D L

1

P

T1

or P

1

D P

T1

=.N

1

R

C1

/. Denoting Q as the interference power threshold at PU, the total inter- ference that PU can tolerate in L

1

time slots is QL

1

. Therefore, E ˚

kH

4

Sk

2F



D N

1

MP

1

kH

4

k

2F

D QL

1

where H

4

is the N

4

 N

1

channel coefficient matrix from SU

TX

to PU. Hence, the transmit power of each symbol at SU

TX

must satisfy the following constraint:

P

1

D Q

N

1

R

C1

kH

4

k

2F

(3) In the second hop, the SU

R

decomposes the received sig- nals by using the squaring approach [25]. Consequently, the i -th decoupled signal Q x

i

corresponding to the i -th symbol x

i

of the source can be expressed as

Q

x

i

D kH

1

k

2F

x

i

C n

1

(4) where n

1

is the noise component of the decoupled signal.

As in [25, Equation (19)], n

1

is an AWGN with zero mean and variance N

0

kH

1

k

2F

, denoted as CN 

0; N

0

kH

1

k

2F

 . Considering M decoupled signals f Q x

i

g

Mi D1

as the input, the OSTBC encoder of SU

R

generates an N

2

 L

2

OSTBC matrix QS with code rate R

C2

D M =L

2

. Then, SU

R

mul- tiplies this OSTBC with a scalar gain G and forwards the signal to the SU

RX

. Let P

2

and P

T2

, respectively, denote the average transmit power per symbol and the total transmit power through N

2

antennas at the SU

R

, that is, L

2

P

T2

D N

2

MP

2

or P

2

D P

T2

=.N

2

R

C2

/. The gain factor G is determined as E ˚

jjG Q x

i

jj

2



D P

2

. From (4), we have

G

2

 P

2

P

1

kH

1

k

4F

(5)

As a result, the N

3

 L

2

received signal matrix Y

2

at the SU

RX

is given by

Y

2

D H

2

G QS C Z

2

(6) where H

2

is an N

3

 N

2

channel coefficient matrix from SU

R

to SU

RX

and Z

2

stands for the N

3

 L

2

AWGN matrix at SU

RX

whose elements are complex Gaussian RVs with zero mean and variance N

0

. Let H

5

be the N

4

 N

2

channel coefficient matrix from SU

R

to PU. To prevent the interference caused by the SU

R

transmission to the PU from being beyond the predefined threshold Q, the transmit power of each symbol at SU

R

must satisfy E n

kH

5

QSk

2F

o

D N

2

MP

2

kH

5

k

2F

D QL

2

or

P

2

D Q

N

2

R

C2

kH

5

k

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;i

D GkH

1

k

2F

kH

2

k

2F

x

i

C GkH

2

k

2F

n

1

C n

2

(8) Q

x

D;i

D kH

3

k

2F

x

i

C n

3

(9)

where n

2

and n

3

are, respectively, the noise compo-

nents of the decoupled signals Q x

R;i

and Q x

D;i

, that is,

(5)

CN 

0; N

0

kH

2

k

2F



and CN 

0; N

0

kH

3

k

2F



. Therefore, the instantaneous signal-to-noise ratio (SNR) per symbol at the SU

RX

from the relaying link is given by



D1

D E n

jG  H

1

 

2F

  H

2

 

2

F

x

i

j

2

o E n

jG kH

2

k

2F

n

1

C n

2

j

2

o (10)

Substituting (3), (5), and (7) into (10), after some manip- ulations, we finally obtain the instantaneous SNR at the SU

RX

from the relaying link as



D1

D Q N

0

 H

1

 

2F

  H

2

 

2

F

N

1

R

C1

 H

2

 

2F

  H

4

 

2F

C N

2

R

C2

 H

1

 

2F

  H

5

 

2F

(11) Similarly, the instantaneous SNR at the SU

RX

from the direct link is expressed as



D2

D E ˚

jGkH

3

k

2F

x

i

j

2

 Efjn

3

j

2

g D Q

N

0

kH

3

k

2F

N

1

R

C1

kH

4

k

2F

(12)

For the sake of brevity, we denote kH

l

k

2F

D X

l

; l 2 f1; 2; 3; 4; 5g,  D

NQ

0

, a D N

1

R

C1

, and b D N

2

R

C2

. Then, we rewrite the instantaneous SNRs at the SU

RX

for the relay link and the direct link as, respectively,



D1

D  X

1

X

2

aX

2

X

4

C bX

1

X

5

D 

C

(13)



D2

D  X

3

aX

4

(14)

where 

C

D X

1

X

2

=.aX

2

X

4

C bX

1

X

5

/. We assume that all channel coefficients are independent and identical dis- tributed within each hop with fading severity parameter m

l

and channel mean power 

l

. Clearly, X

l

is the sum of M

l

gamma RVs with parameter set 

m

l

; ˛

l1



where ˛

l

D

ml

and M

1

D N

1

N

2

; M

2

D N

2

N

3

; M

3

D N

1

N

3

; M

4

D

l

N

1

N

4

, and M

5

D N

2

N

4

. Consequently, the PDF and CDF of X

l

are written as

f

Xl

.x/ D ˛

lMlml

 .M

l

m

l

/ x

Mlml1

exp .˛

l

x/ (15)

F

Xl

.x

l

/ D 1 

Mlml1

X

qD0

˛

ql

x

lq

qŠ exp .˛

l

x

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

RX

to select the best signal through the relaying and direct transmissions, the instantaneous SNR at the SU

RX

after SC is determined as



D

D max 



D1

; 

D2



(17)

As observed from (13) and (14), X

4

appears in both terms 

D1

and 

D2

, which leads to statistical dependence among them. The reason for mutually statistical depen- dence of 

D1

and 

D2

in 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 

D

cannot be obtained as for conventional relay networks, meaning that F

D

. / ¤ F

1

D

. /F

2 D

. /.

However, thanks to the statistical independence among X

4

and the other terms X

1

, X

2

, X

3

, and X

5

, we first calculate F

1

D

. j

X4

/ and F

2

D

. j

X4

/ and then attain F

D

. j

X4

/ as

F

D

. j

X4

/ D F

1

D

. j

X4

/F

2

D

. j

X4

/ (18) where F

1

D

  j

X4



is obtained from (13) as F

1

D

. j

X4

/ D F

C







j

X4



and F

2

D

. j

X4

/ is obtained from (14) as F

2

D

. j

X4

/ D F

X3



 ax

4



. As a result, the unconditional CDF of 

D

is given by

F

D

. / D Z

1

0

F

1

D

. j

X4

/F

2

D

. 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

1

 J

2

C J

3

(20) where

J

1

D

N1N3m31

X

i D0

a

i

˛

3i

˛

4N1N4m4



N1N4m4

i Š

  .N

1

N

4

m

4

C i /

 .N

1

N

4

m

4

/



i

4

 C ˛

3

a /

N1N4m4Ci

(21)

(6)

J

2

D

N1N

X

2m11 pD0

1 pŠ

X

p qD0

C

qp

N2N

X

3m21 rD0

C

rN2N3m21

a

pCrqC1

b

N2N3m2

˛

1pCrqC1

˛

N22N3m2

˛

4N1N4m4

˛

5N2N4m5

 

N2N4m5CN1N4m4

.N

1

N

4

m

4

C p/.N

2

N

3

m

2

C N

2

N

4

m

5

 r C q  1/.N

2

N

3

m

2

C N

2

N

4

m

5

/

.N

2

N

4

m

5

/.N

2

N

3

m

2

/.N

1

N

4

m

4

C N

2

N

3

m

2

C N

2

N

4

m

5

C p/

 .N

1

N

4

m

4

C p C r  q C 1/

.N

1

N

4

m

4

/



N2N3m2CpCrqC1

5

 C ˛

2

b /

N1N4m4N2N3m2N2N4m5CpCrqC1

4

˛

5



2

C ˛

1

˛

5

a C ˛

2

˛

4

b C ˛

1

˛

2

ab

2

/

.N1N4m4CpCrqC1/



2

F

1



N

1

N

4

m

4

C p C r  q C 1; N

2

N

3

m

2

C N

2

N

4

m

5

; N

1

N

4

m

4

C N

2

N

3

m

2

(22)

C N

2

N

4

m

5

C pI ˛

4

˛

5



2

C ˛

1

˛

5

a C ˛

2

˛

4

b

˛

4

˛

5



2

C ˛

1

˛

5

a C ˛

2

˛

4

b C ˛

1

˛

2

ab

2

J

3

D

N1N2m11

X

pD0

1 pŠ

X

p qD0

C

qp

N2N3m21

X

rD0

C

rN2N3m21

N1N3m31

X

i D0

a

pCrCi qC1

b

N2N3m2

˛

pCrqC11

˛

2N2N3m2

˛

3i

i Š

 ˛

4N1N4m4

˛

5N2N4m5



N2N4m5CN1N4m4

.N

2

N

3

m

2

C N

2

N

4

m

5

/ .N

1

N

4

m

4

C i C p/

.N

2

N

3

m

2

/.N

1

N

4

m

4

C N

2

N

3

m

2

C N

2

N

4

m

5

C p C i /

 .N

1

N

4

m

4

C i C p C r  q C 1/ .N

2

N

3

m

2

C N

2

N

4

m

5

 r C q  1/ 

N2N3m2Ci CpCrqC1

.N

1

N

4

m

4

/.N

2

N

4

m

5

/

 .˛

5

 C ˛

2

b /

N1N4m4N2N3m2N2N4m5Ci CpCrqC1

 ˛

4

˛

5



2

C ˛

5

˛

3

a C ˛

1

˛

5

a C ˛

2

˛

4

b C ˛

2

˛

3

ab

2

C ˛

1

˛

2

ab

2



N1N4m4Ci CpCrqC1



2

F

1

N

1

N

4

m

4

C i C p C r  q C 1; N

2

N

3

m

2

C N

2

N

4

m

5

; N

1

N

4

m

4

C N

2

N

3

m

2

C N

2

N

4

m

5

Cp C i I ˛

4

˛

5



2

C ˛

5

˛

3

a C ˛

1

˛

5

a C ˛

2

˛

4

b C ˛

2

˛

3

ab

2

˛

4

˛

5



2

C ˛

5

˛

3

a C ˛

1

˛

5

a C ˛

2

˛

4

b C ˛

2

˛

3

ab

2

C ˛

1

˛

2

ab

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 

th

as argument of the CDF given in (20) as P

out

D 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 

D

as follows:

P

e

D a

1

p b

1

2 p

 Z

1

0

F

D

. /

12

exp.b

1

 /d  (24)

where a

1

and b

1

are parameters depending on the modula- tion scheme, that is, for M -PSK a

1

D 2; b

1

D sin

2





M

 . 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

e

D a

1

2  Q

1

 Q

2

C Q

3

(25) where

Q

1

D a

1

p b

1

2 p



N1N

X

3m31 i D0

1 i Š

˛

1 2 4



12

a

12

˛

312

.N

1

N

4

m

4

C i /  i C

12



.N

1

N

4

m

4

/

 U

 i C 1

2 ; N

1

N

4

m

4

C 3 2 ; b

1

˛

4



˛

3

a

(26)

(7)

Q

2

D a

1

p b

1

2 p



N2N

X

3m21 rD0

1 rŠ

N1N

X

2m11 pD0

1 pŠ

˛

4N1N4m4

˛

N52N4m5



N1N4m4CN2N4m5

a

N1N4m4

b

N2N4m5

˛

N11N4m4

˛

2N2N4m5

.N

2

N

4

m

5

C r/

.N

2

N

4

m

5

/



.N

1

N

4

m

4

C p/ 

r C p C

12



.N

1

N

4

m

4

/

2 4

N2N

X

4m5Cr kD1



rk

 ˛

5



˛

2

b

rCpC12k

U



r C p C 1

2 ; r C p C 3

2  k; b

1

˛

5



˛

2

b

C

N1N

X

4m4Cp lD1



pl

 ˛

4



˛

1

a

rCpC12l

U



r C p C 1

2 ; r C p C 3

2  l; b

1

˛

4



˛

1

a 3

5 (27)

Q

3

D a

1

p b

1

2 p



N2N3m21

X

rD0

1 rŠ

N1N2m11

X

pD0

1 pŠ

N1N3m31

X

i D0

1 i Š

˛

1p

˛

i3

˛

4N1N4m4

˛

5N2N4m5



N1N4m4CN2N4m5

a

N1N4m4

b

N2N4m5

˛

2N2N4m5

1

C ˛

3

/

N1N4m4Ci Cp



.N

2

N

4

m

5

C r/ .N

1

N

4

m

4

C p C i /  

r C i C p C

12



.N

2

N

4

m

5

/.N

1

N

4

m

4

/

2 4

N2N

X

3m3Cr j D1

rj

 ˛

5



˛

2

b

rCi Cp C12j

 U



r C i C p C 1

2 ; r C i C p C 3

2  j ; b

1

˛

5



˛

2

b

C

N1N3

X

m5Ci Cp t D1

ı

pt

 ˛

4

 a ˛

1

C a ˛

3

rCi CpC12t

 U



r C i C p C 1

2 ; r C i C p C 3

2  t ; b

1

˛

4

 a ˛

1

C a ˛

3

3

5 (28)

and 

rk

; 

pl

;

rj

, and ı

pt

are 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

out1

D .G

c

/

Gd

C o 



Gd



(29)

where  D Q=N

0

is 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

d

is 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

d

is given by

G

d

D lim



!1



 log P

out

log 

(30)

Furthermore, G

c

is 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 

D

as 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

(8)

expression for outage probability of the considered system, P

out1

, can be easily obtained as P

out1

D 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

. /

 !0

D 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/



N1N2m1CN1N3m3

N1N2m1CN1N3m3

I if N

1

N

2

m

1

< N

2

N

3

m

2

˛2N2N3m2˛N1N3m33 aN1N3m3bN2N3m2

˛4N1N3m3˛5N2N3m2

.N2N3m2CN2N4m5/

.N2N4m5/ .N1N3m3C1/ .N2N3m2C1/



.N1N3m3CN1N4m4/

N1N3m3CN2N3m2

N1N3m3CN2N3m2

N1N3m3CN2N3m2

I if N

1

N

2

m

1

> N

2

N

3

m

2

˛1N1N2m1˛N1N3m33 aN1N2m1CN1N3m3

˛N1N2m1CN1N3m3 4

.N1N2m1CN1N3m3CN1N4m4/

.N1N2m1C1/ .N1N3m3C1/ .N1N4m4/



N1N2m1CN1N3m3

N1N2m1CN1N3m3

C

˛

N2N3m2

2 ˛3N1N3m3aN1N3m3bN2N3m2

˛N1N3m34 ˛5N2N3m2.N2N4m5/

.N2N3m2CN2N4m5/

.N1N3m3C1/



.N.N1N3m3CN1N4m4/

2N3m2C1/.N1N4m4/

N1N3m3CN2N3m2

N1N3m3CN2N3m2

I if N

1

N

2

m

1

D N

2

N

3

m

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

1E

D 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

1

N

2

m

1

< N

2

N

3

m

2 a1

2p

 bN1N3m3CN2N3m2 1

˛2N2N3m2˛N1N3m33 aN1N3m3bN2N3m2

˛N1N3m34 ˛N2N3m25

.N2N3m2CN2N4m5/

.N2N4m5/ .N1N3m3C1/



.N1N3m3CN1N4m4/ 



N1N3m3CN2N3m2C12

.N2N3m2C1/ .N1N4m4/ N1N3m3CN2N3m2

I if N

1

N

2

m

1

> N

2

N

3

m

2 a1

2p

 bN1N3m3CN1N2m1 1

˛1N1N2m1˛N1N3m33 aN1N2m1CN1N3m3

˛N1N2m1CN1N3m3 4

.N1N2m1CN1N3m3CN1N4m4/

.N1N2m1C1/







N1N3m3CN1N2m1C12

.N1N3m3C1/.N1N4m4/N1N2m1CN1N3m3

C

a1

2p

bN1N3m3CN2N3m2 1

˛

2N2N3m2

˛

3N1N3m3



aN1N3m3bN2N3m2

˛N1N3m34 ˛N2N3m25

.N2N3m2CN2N4m5/

.N2N4m5/ .N1N3m3C1/

.N1N3m3CN1N4m4/ 

N1N3m3CN2N3m2C12

.N2N3m2C1/ .N1N4m4/ N1N3m3CN2N3m2

I if N

1

N

2

m

1

D N

2

N

3

m

2

(32)

As we can see from (31) the diversity gain of the con- sidered system is equal to min.N

1

N

3

m

3

C N

1

N

2

m

1

; N

1

N

3

m

3

C N

2

N

3

m

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

TX

to PU and SU

R

to 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.

(9)

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

4

and 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

C1

D 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



D

D ˇ X

1

X

2

X

2

C X

1

(33)

ˇ D P

s

=.R

C1

N

1

N

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

out

D 1 

N2N

X

3m31 qD0

˛

2q

 

t h

ˇ

q

exp



 ˛

2



t h

ˇ

C

N2N

X

3m21 i D0

C

iN2N3m21

˛

2N2N3m2i 1

.i C 1/

.N

2

N

3

m

2

/



 

t h

ˇ

N2N3m21i

exp



 ˛

2



t h

ˇ

 2

N1N2m11

X

pD0

N2N3m2Cp1

X

j D0

C

iN2N3m2Cp1

˛

N2N3m2C2pi 2 1

.N

2

N

3

m

2

/

 ˛

N2N3m2Ci 2 2

 

t h

ˇ

N2N3m2CpCj i

exp



 .˛

1

C ˛

2

/ 

t h

ˇ

K

N2N3m2i

 2 p

˛

1

˛

2



t h

ˇ

(34)

P

e

D a

1

2 

N2N3m31

X

qD0

˛

2q



12

  q C

12

 qŠ.b

1

 C ˛

2

/

qC12

C

N2N3m21

X

i D0

C

iN2N3m21

˛

2N2N3m2i 1

 

N

2

N

3

m

2

 i 

12



.N

2

N

3

m

2

/

 .i C 1/

12

.b

1

 C ˛

2

/

N2N3m2i 12

 2 p



N1N

X

2m11 pD0

N2N3

X

m2Cp1 j D0

4

N2N3m2i

C

iN2N3m2Cp1



12

 

p C j C

12



.N

2

N

3

m

2

/



˛

1N2N3m2Cpi

˛

N22N3m2Ci

 

2N

2

N

3

m

2

C p C j  2i C

12



 b

1

 C ˛

1

C ˛

2

C 2 p

˛

1

˛

2



2N2N3m2CpCj 2i C12

.N

2

N

3

m

2

C p C j  i C 1/



2

F

1



2N

2

N

3

m

2

CpCj 2i C 1

2 ; N

2

N

3

m

2

 i C 1

2 ; N

2

N

3

m

2

C p C j  i C 1; b

1

 C .˛

1

C ˛

2

/  2 p

˛

1

˛

2

b

1

 C .˛

1

C ˛

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

1

D N

2

D N

3

D N

4

D 1.

Furthermore, the code rates R

C1

at the source and R

C2

at

the relay will be moved out of these equations as OSTBC

will no longer be deployed for this system, that is, a D

N

1

R

C1

D 1 and b D N

2

R

C2

D 1.

(10)

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 h

D 5 dB. Furthermore, we consider quadrature phase shift keying (QPSK) modulation, for which a

1

D 2 and b

1

D sin

2





4

 , 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

l

D 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

0

for different antenna configurations.

Figure 3. Symbol error rate of QPSK modulation versus

interference power-to-noise ratio Q=N

0

for 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

1

N

2

m

1

D N

2

N

3

m

2

, Cases 3 and 5 fulfil the relationship N

1

N

2

m

1

<

N

2

N

3

m

2

, and Case 4 complies with the relationship N

1

N

2

m

1

> N

2

N

3

m

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

TX

and 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

; 

5

I 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

; 

5

I 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

; 

5

I 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

l

D 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

RX

are 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

l

D 0:5 given in Case 7 to m

l

D 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

(11)

Figure 4. Outage probability versus interference power-to-noise

ratio Q=N

0

for different fading channel parameters.

Figure 5. Symbol error rate of QPSK modulation versus inter-

ference power-to-noise ratio Q=N

0

for 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

1

D 0:5, d

2

D 0:5, d

3

D 0:7, d

4

D 1:0, and

Figure 6. Comparing outage probability versus interference

power-to-noise ratio Q=N

0

for 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

0

for the relaying link,

the direct link, and combining both links by SC.

d

5

D 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

l

D 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.

(12)

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

RX

to 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 

C

given in (13) conditioned on X

4

as

F

C

. j

X4

/ D Z

1

0

Z

1

0

Pr

X

1

x

2

ax

2

x

4

C bX

1

x

5

 

f

X2

.x

2

/f

X5

.x

5

/dx

2

dx

5

(A.1) To further calculate F

C

. j

X4

/, we separate the integra- tion domain of x

2

in (A.1) into two subsets as .x

2

< b x

5

/ and .x

2

 b x

5

/. After some algebraic manipulations, we rewrite F

C

. j

X4

/ as

F

C

. j

X4

/ D Z

1

0

F

X2

.b x

5

/f

X5

.x

5

/dx

5

„ ƒ‚ …

K

C Z

1

0

I

1

f

X5

.x

5

/dx

5

„ ƒ‚ …

I

(A.2)

where

I

1

D Z

1

0

F

X1

a x

4

C ab

2

x

4

x

5

x

2

!

f

X2

.x

2

C b x

5

/dx

2

(A.3)

By substituting (15) and (16) into (A.3), we have

I

1

D 1  F

X2

. b x

5

/ 

N1N

X

2m11 pD0

1 pŠ

X

p qD0

C

qp



N2N3m21

X

rD0

C

rN2N3m21

 a

p

b

N2N3m2Cqr1

.N

2

N

3

m

2

/

 ˛

p1

˛

N22N3m2



N2N3m2CpCqr1

 x

4p

x

5N2N3m2Cqr1

 exp.˛

1

a  x

4

/ exp.˛

2

 bx

5

/

 Z

1

0

x

rq2

exp.˛

2

x

2

/

 exp  ˛

1



2

a b x

4

x

5

x

2

!

dx

2

(A.4)

where C

ab

denotes the binomial coefficient, that is, C

ab

D

bŠ.ab/Š

. Applying [22, Equation (3.471.9)] to solve the remaining integral of (A.4), after rearranging terms, we obtain I

1

as

I

1

D 1  F

X2

. bx

5

/  2

N1N

X

2m11 pD0

1 pŠ

X

p qD0

C

qp



N2N3m21

X

rD0

C

rN2N3m21

 ˛

2pCrqC1 2

1

˛

2N2N3m2Cqr1 2 2

.N

2

N

3

m

2

/

 a

2pCrqC12

b

2N2N3m2Cqr1 2

 

N2N3m2Cp

x

2pCrqC1

4 2

x

2N2N3m2Cqr1 5 2

 exp .˛

1

a x

4

/ exp .˛

2

 bx

5

/

 K

rqC1

 2

q

˛

1

˛

2



2

abx

4

x

5

(A.5)

Now, we are ready to solve the outer integral I . Substitut-

ing (A.5) and (15) into (A.2) yields

References

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