Modeling of MIMO Cooperative Relaying System Based on Amplify-and-Forward Schemes Through Convex Optimization

Abstract: This paper presents performance analysis of cooperative multiple-input multiple-output (MIMO) relaying system with a single relay. The MIMO scheme is based on Alamouti space time block coding (STBC) over Rayleigh flat fading channels. The source node, equipped with two transmit antennas, simply broadcasts each STB code to the relay and the destination nodes. Then, the relay node, equipped with multiple antennas, amplifies-and-forwards (AF) the received STB codes. Finally, the destination node uses maximum ratio combining (MRC) and exploits the diversity gain obtained through the direct and the indirect links simultaneously. The moment generating functions (MGF) of the signal-to-noise ratio (SNR) for the direct and the indirect link are given in a closed form. These statistical results are then applied to derive a lower bound of the symbol error probability (SEP) for a particular signal of M-ary-quadrature-amplitude modulation (M-QAM) and to obtain the outage probability. Subsequently, simulation results of the SEP and the outage probabilities are presented to illustrate the performance improvement given by the MIMO cooperative diversity systems based on STBC schemes.

key words: MIMO relay channel, space-time block coding, MRC scheme, moment generating function (MGF), outage probability.

I. INTRODUCTION

Cooperative communication has recently attracted a lot of interest due to its ability to realize the performance gains and coverage extension [1] [3] . Typically, it concerns a system where users share and coordinate their resources to enhance the transmission quality and to optimize the power allocation. The combination of relaying system with MIMO processing is a natural extension of both concepts.

Moreover, this combination gives additional degrees of freedom to improve the capacity of the overall cooperative system [4] [5] .

Recently, it has been demonstrated that cooperation based on space time block codes (STBC) represents an effective way to introduce spatial diversity in wireless scenarios where we can not take the full benefit of the uncorrelated channels from the multi-antenna systems. Cooperative diversity gains can be achieved through creating distributed virtual antennas across

Different terminals in the network. Actually, there exist two ways to apply STBC technologies in cooperative system. In the first way, cooperation using distributed STBC is applied in order to create a virtual transmit array in a distributed multiple relay network [6] - [8] .

Fig. 1 : MIMO cooperation scheme with source node broadcasting Alamouti STBC matrix.

For example, in [8] , the Alamouti space time block code [9]

is employed with a distributed manner in a cooperative relay network over Rayleigh fading environment. In the second way, the STBC matrix is completely broadcasted to the relay and the destination. Several time slots are employed during the transmission of each STBC matrix. For Alamouti STBC, cooperative relay network with a single relay system and two receive antennas can be considered as a virtual MIMO system with four receive antennas, in consequence, the performance of the cooperative diversity system can be improved without increasing the number of receive antennas. In [10] , for a MIMO relay system, the authors neglect the source destination link and consider that one source transmits an STBC matrix via one relay node using AF protocol. Specifically, they derive the exact SEP for maximum likelihood (ML) decoding of orthogonal STBC in dual-hop relay channels.

Another work related to the use of STBC as in the second way is given in [11] . In their paper, the authors have investigated the performances of MIMO relaying systems with decode-and-forward (DF) protocol where the source, the relay and the destination are multiple-antenna nodes.

Specifically, the authors derive a closed-form expression for the outage probability..

In [13-15], the authors enlist some recent papers related to our contribution which give performance analysis of cooperative diversity system based on STBC scheme over some particular scenarios [13] [14] . In [13] , Safari and Uysal derive an upper-bound on the pairwise error probability (PEP) for cooperative diversity schemes over log-normal fading channels and the distributed SIMO, MISO and MIMO systems. In their contributions, the authors derive a Chernoff bound on PEP and a union bound on bit error rate (BER) performance where each node is equipped with single antenna. In [14] , Muhaidat and Uysal give a derivation of the PEP by including an extension to multiple antennas nodes. In their paper, the authors derive a closed form of the PEP for dual-hop relaying scheme and

Modeling of MIMO Cooperative Relaying System Based on Amplify-and-Forward Schemes Through Convex Optimization

Priyanka¹(M.Tech Scholar) , Garima mathur²

1, 2 JEC,Jaipur, India

[email protected], [email protected]

channel state information (CSI) assisted AF, blind AF and DF. In their work, the source node transmits a general STBC matrix and the destination applies ML decoding considering only the indirect link (the direct link is neglected). However, this derivation seems to be incomplete since in most realistic cases the diversity is increased by considering both the direct and indirect links. In [15] , Song and Hong, derive the SEP for MIMO cooperative AF relay system based on orthogonal STBC and M-ary phase-shift keying (M-PSK) modulation when ML detection is employed at the receiver.

They focus their analysis on SEP behavior in the asymptotic regimes of the number of relay antennas and SNR. In their derivation, the SEP is given in an integral form and it is limited to ML decoding which is known by its huge complexity for a big constellation and important number of antennas.

In this paper, we present performance analysis of a cooperative MIMO relay system based on Alamouti scheme (Fig. 1). The MGF of the signal-to-noise ratio for the direct and the indirect link are given in a closed forms. These MGF are then applied to derive a closed form of the lower bound of the SEP and to obtain the outage probability.

To the best of the authors’ knowledge, there is no published work that derives the SEP in a complete and exact form of MIMO relaying system using Alamouti STBC matrix codes and amplify-and-forward relaying. We complete the contributions of [14] as follows:

1) We derive the SEP for more general cooperative diversity by including the direct and indirect links.

2) We avoid the complexity of the ML decoder by using an MRC based decoder determining the SEP at the demapper front end for general M-QAM constellation.

The paper is organized as follows. In Section II, we introduce system model. Section III presents the SNR analysis at the output of the destination node considering only the direct link. In Section IV, we develop an upper bound of the SNR at the output of the destination nodefor the indirect link. In Section V, using the MGF of the SNR for the direct and indirect links, we give a lower bound on the SEP of the cooperative link taking into account the diversity given by these two links. Performances related to the outage probabilities and the diversity gains are detailed in section VI. We discuss and analyze numerical results in Section VII. Finally, conclusions are drawn in Section VIII.

II. S^YSTEM MODEL

In this section, we describe a MIMO cooperative system consisting of a source, relay and destination nodes equipped with multiple antennas. We consider the amplify-and- forward cooperative MIMO relay channel as shown in Fig.

1. The source, relay and destination nodes have

N

_t,

N

_r

and

N

_r antennas, respectively. In order to provide an efficient coding rate, we use

2  2

Alamouti matrix code, we consider

N

_t

= 2

transmits antennas. With a slight modification in the MIMO STBC model for each link, we have transformed the channel matrix

H

into a modified channel matrix

H [ X

₂

]

with orthogonal columns and

2

2 Nr 

entries. According to the same modification in

MIMO STBC channel model, the source-destination, the source-relay and the relay-destination channel matrices are

2 2 2

]

[ X  C

^Nr

H

,

D [ X

₂

]  C

^2Nr2 and

r

r N

N 2 2 2

]

[ X C

^

F

, respectively.

We assume a half duplex relaying protocol therefore, the transmission from the source to the destination is made in two separate time-slots as in time division multiplexing (TDM) systems [17] . In the first time-slot, the source sends its Alamouti encoded signal

x

to the relay and the destination, where

x = [ s

₁

, s

₂

]

^T is the vector symbols that composes the Alamouti matrix. The relay simply amplifies the received signal before forwarding it to the destination during the second time-slot. Finally, the destination combines the signals of two time-slots coming from the relay and the source nodes using MRC.

The vector signal transmitted by the source and received by the destination is given as follows:

0 2

0

= E . H [ ] x n

y

_s

X 



sd ⁽¹⁾

where

n

₀ is the modified

2 Nr  1

noise vector measured at the destination which is composed by zero mean complex Gaussian random variables with variance

N

₀.

E

_s is the average power of each symbol which is equal to

  ^x

ⁿ

^{P N}

^s

^/

^t

 

, where Ps is the total transmit power per symbol and



_sd is the pathloss factor applied at the direct link, respectively.

At the destination node, according to the MIMO STBC modelization given above, the received signal is a vector

y

₀

given by

 

 

 

 





 

 

 

 







 

 

 





 

 

 









 

 

 

 





 

 

 

 





* ,2 ,1

* 2,1 1,1

* ,1

* ,2

,2 ,1

* 1,1

* 1,2

1,2 1,1

* ,2 ,1

* 1,2 1,1

=

Nr Nr

Nr Nr

Nr Nr

sd

Nr Nr

n n n n

h h

h h

h h

h h

y y y y

x E

_s



⁽²⁾

If we consider the relayed link, the vector signals observed at the relay is

r sr

r

E D x n

y = 

_s

. [ X

₂

] 

⁽³⁾

where



_sr is the path loss factor related to the source relay link (first hop link), and

n

_r is the

2 Nr  1

modified vector noise applied at the relay, which is composed by

2 N

_r zero- mean complex random variables with variance

N

₀. At the relay, the system applies AF protocol with a matrix gain

G

defined as

Nr sr F

rd

2 0

2

.

= I

N D E G E

s r

  

(4) where

|| D ||

_F is the Frobenius norm ¹ of the matrix

D

and

1The Frobenius norm is defined as

|| D ||

_F

= Tr ( DD

^H

)

.

Nr

I

2 is the

2 N

_r

 2 N

_r identity matrix.. In order to simplify the analysis, we assume a fixed gain,

Nr Nr

sr

rd ₂

g

₂

0

= .

= I I

N E G E

s r

  

_.

We notice here, that fixed gain

G

used for AF relaying is also considered in many other papers in the literature, (see for example [18] ). At the destination, the relayed signal is given by

d r 2 2

s

FGD[ ] x FG[ ]n n

E

y

_d

⁼ 

_sr

X  X 

⁽⁵⁾

The equivalent virtual MIMO,

4 N

r

 2

, representation of the proposed MIMO relay system is given as follows



















 















 



 





d r s

s

n n n I FG 0

0 0

x I FGD E

H E y

y ⁰

2 2 2

2 0 2

] ] [

[ ]

= [

r r

N N

sr sd

d X X

X



(6) while the equivalent source-relay-destination channel matrix is given by

U [ X

₂

]  = FGD [ X

₂

]

^let

U = FD [ X

₂

]

.

.

= ] [

* ,1

* ,2

,2 ,1

* 1,1

* 1,2

1,2 1,1

2

 

 

 





 

 

 









Nr Nr

Nr Nr

u u

u u

u u

u u

U X

(7)

with the

FD

matrix product,

u

_i,j are defined as

2) (1 ), (1 , , 1

=

,^j

= 

_k^{N ik k j} _ⁱ_^Nr _^j_

i

f d

u

r

(8) The equivalent noise measured at the output of the relayed link is

d r

e

FG n n

n = [ X

₂

] 

₍₉₎

In order to evaluate the SEP, at the output of the MRC at the destination, we decompose the problem into two sub- problems. In the first one, we derive the SNR of the MRC output considering the direct link only. In the second sub- problem, we calculate the SNR of the MRC output for the relayed link and finally we use the MGF of each SNR to give the SEP and to evaluate the outage probability of the cooperative and non cooperative links.

III. SNR ^{OF THE} DIRECT LINK

The modified matrix

H [ X

₂

]

describing the equivalent (source-destination) channel has orthogonal columns [15, p.

285]. Applying the MRC to the received vector signal

y

₀, we have as output

0 2

,2 2 ,1 1

= 0

2 0

| ~

|

|

|

= . ] [

~ = H y E

_s

  x  n

  

_i^{N i}



ⁱ

sd

H

h h

y



r

X

(10)

where

~ n

₀ is the equivalent noise measured at the output of the MRC given as

 





 







 

^*

,2 ,1 ,1

* ,2

* ,2 ,2 ,1 ,1

1

= 0 2

0

= [ ]. =

~

i i i i

i i i i N

i H

n h n h

n h n h

r

n H

n X

The covariance matrix of the equivalent noise

n ~

₀is

  ^{~ n}

⁰

^{~ n}

^HF

^{= N}

⁰

^{|| H ||}

^2F

^I

²

E

  ^{~ n}

⁰

^{= 0}

^2,1

E

where

E {.}

is the expectation of

{.}

. The effective channel for the data symbols

s

_i

,

_{i{1, ,M}} is

} , {1, 0

2

~ ,

||

||

=

_{i i M}

i i F sd

i

s n s

z  E

_s

H 

_ ⁽¹³⁾

Hence, according to (12) and (13), the SNR of the signal transmitted through the source-destination link and measured at the output of the MRC is given by

0

||

2

||

= .

N

F sd

sd

H E

_s

 

⁽¹⁴⁾

Since the channel of the direct link with matrix

H

is flat Rayleigh fading then all the entries

h

_i,j of

H

are complex Gaussian random variables each with

N (0,1)

distribution.

Hence,



_sd is a random variable equal to the sum of

4 N

_r

Gaussian random variables each of which with

N (0,1)

distribution. Then,



_sd is a Chi-squared random variable with

4 N

_r degrees of freedom. Thus, the probabiliy density function (pdf) of



_sd ^is

2 ) ( ) exp

(2 2

= 1 )

(

₂

1 2 2

sd N

N

r

sd N r

r

r

N

f  



 









(15)

where = . (2 ²)

0

h sd

sd Nr

N 



 ^Es is the average SNR of the

source-destination link and

 (.)

is the Gamma function [18, eq. (8.310.1)] defined as

 ( n ) = ( n  1)!

where

n

is an integer,

n > 0

. The MGF,

( M ( s ))

sd , of



_sd is then given by

^M

_sd

^{( s ) =} 

^

^f

_sd

⁽  ^{) exp (}  ^s  ^{) d} 

(16) Since



_sd is a Chi-squared random variable with mean



sd, then using ^{( 1)}

0^

exp ( ) = ( ! )

^{ }

 ^

ⁿ

^{ d  n }

^{n , the}

MGF of



_sd can be easily found as

)

0

2 (1

= )

(

_sd ^m

sd

s s

M



 

^

(17) where

m

₀

= 2 N

_r

IV. SNR ^{OF THE} INDIRECT LINK

In this section, we evaluate the SNR of the indirect link (



srd) at the output of the MRC. The modified matrix

] [ X

₂

U

describing the equivalent relayed link

FD

has orthogonal columns. Applying the MRC to the vector signal

y

d, the output of the MRC is

d H

d

U y

y = [ ] .

~

X

2



= E

_s

|

_,1

|

²

|

_,2

|

²

x n ~

1

=

 

 

 

_i^{N i}



ⁱ

sr

u u



r ⁽¹⁸⁾

where

n ~

is the equivalent noise channel measured at the

output of the MRC

n

e

U

n = [ ].

~

X

2



H ₍₁₉₎

Substituting (9) in (19), the covariance matrix of the equivalent noise

_ n ~

is given by

 ^  ^ _ ^{ } _

²

^ _ ^

1

= 2 2

2 2 1 1

=

0

| | | | 1 | |

=

~ }

{ ~

_ik

Nr

k i

i N

i

H

N u u g f

r

n

 n

(20) Then, the effective channel for the data symbols

} ,

,

_i {1, _M

s

i _ is

} , {1,

2

~

||

||

.

=

_{sr F i i i M}

r

i

g s n

z  E

_s

U 

_{ (21)}

By letting

0

= N

sr sr

E

s

 

as the mean SNR of the source- relay link, the instantaneous SNR



_srd of the source-relay- destination link can be given by

 

_^





 

 





 



 







2 1

= 2 2

2 2 1 1

=

2 2 2 2 1 1

= 2

|

| 1

|

|

|

|

|

|

|

|

=

ik N

k i

i N

i

i i

N

i sr

srd

f g u

u

u u

g

r r

 r



(22)

Deriving a closed-form expression of the PDF of the instantaneous SNR



_srd is too hard to accomplish. Hence, we use an upper bound of the SNR and compare the analytical results with the exact simulation results. The upper bound of the SNR in (22) is obtained by neglecting

the term ²

1

= 2 ^Nr

|

_ik

|

k

f

g 

in (22) as follows





 



 



^sr

^.

²



_i^N₌₁

^|

ⁱ¹

^|

²

^|

ⁱ²

^|

²

srd

g u u



r



(23) Since

F

and

D

are random matrices with complex Gaussian entries,

f

_i,l and

d

_i,lcan be expressed as

il il il il

il

il

a jb d a j b

f =  and =   

(24) where

a

_il

, b

_il

, a

_il

 and b

_il



are N(0,1) random variables.

Substituting (24) in (8) and evaluating the expression in more compact form we have



^{ik kl ik kl}

 

^{ik kl ik kl}



Nr

k l i l i l

i

j a a b b j a b b a

u

^,

^{= }

^,

 ^

^,

⁼ 

₌₁

      

2) (1 ),

(1iNr l (25)

where



_i,l ^and



i,l are the real part the imaginary part of

l

u

i_, , respectively. Without loss of generality, omitting the indices

i, k

and

l

, the random variables

a , a  and , b b 

are

^N (0, 

²

)

. Then, according to the Gaussian random variable properties,

x = a a 

is a random variable equal to the product of two independent Gaussian random variables with zero mean and variance



₁ ^and



2 respectively.

According to [17] , this product is a zero mean random variable and its pdf is given by





2 1 0 2 1

|

|

= 1 )

(  

K x x

p_X

(26)

Where

K

₀

(.)

is the Bessel function of the second kind and order zero Fig. 2 illustrates the pdf of

X

obtained analytically and by Monte-Carlo simulation for verification.

Actually both



_i,l ^and



i,l are equal to the sum of

2 Nr

zero mean random variables with modified Bessel function

distribution of order 0. According to the central-limit theorem,



_i,l ^and



i,l can be approximated by Gaussian random variables with zero mean and variance

) (

2 Nr 

²_f



_d² . As shown in Fig. 3, the pdf of



_i,l ^{is very}

Fig. 2: Pdf of the the random variable

X = ab

, given by (26)

Fig. 3: pdf of



_i,l^.

Fig. 4: SEP versus SNR for the cooperative link for

2,3,

=

Nr

and

4

antennes, with 4-QAM modulation.

close to the Gaussian distribution. Hence,

| u

_il

|

²

= 

²_il

 

_il²

can be approximated by exponential random variables.

Thus,

|| U ||

²_F is equal to the sum of four exponential random variables with parameter

2 

², is a Chi-squared random variable with

2  2 N

_rdegrees of freedom. Thus, the pdf of



_srd is upper bounded by

 

_^ _^





srd Nr

srd Nr

srd Nr Nr

f

 



 

 exp 2

2 2

= 1 )

( _{(2 )}

1) (2 )

(2

(27) where

2 2 2

0

)(2 (

= ^sr _{f d}

srd g Nr

N  

  ^Es ).

Since the SNR at the output of the relayed link



_srd ^{is a}

Chi-squared random variable with degrees of freedom

4 N

_r

and using the same derivation as for equation (17), the MGF of



_srd is given by denoting

m

₁

= 2 N

_r

)

1

2 (1

= )

(

_srd ^m

srd

s s

M



 

^

(28) V.

SEP

OF THE COOPERATIVE SCHEME

The cooperation is based on the use of two independent branches: the direct and indirect links. The SEP must average the two branches conditional over the pdf of



_sd

and



_srd. For M-QAM constellation, the average SEP expression, obtained by the MGF method, can be written as the sum of two terms, denoted by

I

_{1 and}

I

2 _{[21] ,}

2

2 QAM 2

4 QAM 0 2

1

2 QAM 2

2 QAM 0

sin sin

4

sin sin

4 .

= ) ( SEP

I I

G G

G G

 





 



 













d M

q M

d M

q M

srd sd

srd sd























(29) where

G

_QAM

= 3/[2( M  1)]

_and

q = 1  4/ M

^{. For}

the first term in (29), if we substitute (17) and (28) in (29) and using the change of variable

^{t = cos}

²



, after some manipulations, we obtain ¹

^{= 2} q M

sd



^QAM

  M

srd ^QAM



₀¹

t

^21

⁽¹ t ⁾

^{21m0m1}

  G G

I

_{ }



dt t t

m

srd m

sd

1

QAM 0

QAM

2. . 1 1 1 2. .

1 1 1









 





 





 





 



 ^G

G

(30) For the second term, upon making the change of variable

 1 tan

= 

²

t

, we obtain 2 ^{0 1}

1 2

1 1

QAM 0 QAM

2

2

4 (2. ) (2. ) (1 )

=

^{m m}

srd

sd

M t t

q M

^{   }

  G G

I

_{ }



dt t t

t

m

srd srd m

sd sd

1 1

QAM QAM 0

QAM QAM

2 1 1 4. .

1 2.

1 1 4. .

1 2.

1 1

 



 

 

  

 



 

 





 

 



 

 





 









G G G

G

(31)

In order to continue the derivation of the SEP, we use the Lauricella multivariate hypergeometric function

F

_D⁽ⁿ⁾ [22]

as

) , ,

;

; , , , (

=

_{1 1}

) (

n n

n

D

a b b c x x

F

!

! )

(

) ( ) ( )

(

1 1 1

1 1 1 0 1

= ,

1, n

in n i

in i

in n i in i in

i

i

x i x c

b b

a



 







1

<

|}

|

| {|

max x

₁

x

_n

 ¹  ^.

) 1(1 ) .

( ) (

)

= (

1

= 1 1

0

t t x t dt

a c a

c

b_i

i L

i a

c 







 



  

^



0

>

) (

>

)

( c e a

e R

R

where

( a )

_n

=  ( a  n )/  ( a )

is the Pochhammer symbol, with

( a )

₀

= 1

. Therefore, with the help of (32), the average

SEP

of square M-QAM constellation can be deduced as

 ² ₁      ^{( 1} ₂ ^{, , ;1 ;}

2 1

=

_{QAM QAM} ² _{0 1 0 1}

1 0

1 0

m m m m F M

m M m

m m q

SEP

_D

srd

sd

 







 

 



  



G

G

_





) (2 ) (2 2

) 1 2

1 , 1 2

1 1

QAM QAM

1 0 2

QAM QAM

G G

G

G

_{sd srd}

M

^sd

M

^srd

m m q









 



 



  

 



2 ) , 1 4

1 2 , 1 4

1 2

; 1 2

,1; 3 , (1,

QAM QAM

QAM QAM 1

0 1

0 3

srd srd

sd sd

D

m m m m

F 







G G G

G







 



(33)

If we consider the indirect link only, the SEP is simply given by

   

2 ) 1

; 1

;1 2 , ( 1 1

2 2 1

=

QAM 1

1 1 QAM 1

1

srd srd D

m

m F M

m m q

SEP





^

G

G

 





 

 

  



4 1

2 , 1 2 ;

,1; 3 (1, ) (2 2

1

_QAM

QAM 1

1 2 QAM 1

2

srd srd

srd

F

D

m m

M m q







^

^G

G

G 

 

 

 

  



(34)

It is easy to generalize the derivation of the SEP for a MIMO cooperative diversity system with k relays based on Alamouti STBC sheme. The SNR of each indirect link will be upperbounded as in equation (23) and then the lowerbound of the SEP is as

 

0



^QAM

 

^QAM



0 0

1 2 2 1

= G G

skd d

s k k

M M

m m

m m

q

SEP

_{ }

    

 

 



   



2 1 , 1 2 ,

1

; 1 , , 2 , ( 1

QAM 0

QAM 0

0 1) (

d s k

k k

D

m m m m

F

^

   ^G   ^G 

) (2 )

(2 2

1

^{0 QAM QAM}

0 2

G G

skd d

s k

M M

m m

q



 





 



   



2 ) , 1 4

1 2 , 1 4 ,

1 2

; 1 2

,1; 3 , , (1,

QAM QAM

0 QAM

0 QAM 0

0 2) (

skd skd

d s

d s k

k k

D

m m m m

F 







G G G

G







 







(35)

where the index term

k

in

m

_k and in



_skd refers to the source destination link for

k = 0

and to the source-

k

th- relay-destination link for

k >= 1

.

Fig. 5: SEP versus SNR for the cooperative for 4-QAM, 16- QAM, and 32-QAM for

Nr = 2

.

Fig.e 6: Outage probability of the cooperative MIMO system (

Nr = 2

,

3

and

4

receive antennas)



_th, pathloss of the direct and indirect links are



sd

 0 . 5

^and

5 .

 0



srd ^.

Fig. 7: Outage probability of the cooperative MIMO STBC system (

2

=

Nr

,

3

and

4

receive antennas)



_th, pathloss of the direct and indirect links are



sd

 1

^and



srd

 ¹

^.

VI. O^UTAGE PROBABILITY PERFORMANCE In addition to the average SEP, outage probability, denoted by

P

_out

( 

_th

)

, is another standard performance criterion of cooperative diversity systems. It is defined as the probability that the the instantaneous combined SNR



_cop falls below a certain specified threshold (



_th^{), i.e.,}

cop cop cop

th th cop

out

P p d

P  



 



( )

= ] [0

= ^{ } 

0 ⁽³⁶⁾

where

(

_cop

)

p

cop



is the probability density function of



cop. Mathematically speaking, the outage probability coincides with the cumulative distribution function (CDF) of



cop evaluated at



_th, which is equal to the inverse Laplace transform

L

^1

(  )

of the ratio

s s

cop

(  )/

M

 evaluated at



th ^[23]

th cop

out

s

s M P





|

1

( )

=  





 







L

₍₃₇₎

According to the assumption of mutually independent channels, the MGF,

cop

M

of the combined SNR can be expressed as

).

( ) (

= )

( s s s

srd sd

cop  



M M

M

₍₃₈₎

where

(s )

sd

M

_and

(s )

srd

M

are the MGF of the direct and the indirect links SNRs given by the results in (17) and (28). We notice here, that the inverse Laplace transform can be derived analytically or using simple numerical techniques. Using the results in [24] , equation (37) can be developed as

 

 





k e K

P P

K

th k A K cop th

out

0

=

2

/2

= ) (

=



 