Performance Analysis of Fixed Gain Relaying Systems in Nakagami-m Fading Channels

Rui Zhao and Luxi Yang School of Information Science and Engineering

Southeast University Nanjing, China

{zhaorui, lxyang}@seu.edu.cn

Abstract—The performance of dual-hop communication systems with fixed gain amplify-and-forward relay is analyzed over non-identical Nakagami-m fading channels. First, Closed-form expressions for outage probability and average symbol error rate (ASER) are derived based on probability density function (PDF) method for the system without cooperative diversity. Then, the expressions for outage probability and ASER and closed-form expression of generalized moments of received SNR are derived based on moment generating function (MGF) method for the system with cooperative diversity. Simulation results show that, the derived expressions match well with the numerical simulations, and cooperative diversity and larger m values are both of benefit to the improvement of system performance, and the channel quality of the first hop and that of the second hop have different effects on the system performance.

Keywords-amplify-and-forward relay; cooperative diversity; Nakagami-m fading channel; outage probability; average symbol error rate (ASER)

I. INTRODUCTION Cooperative communications with relaying techniques can

enhance communication reliability, expand the coverage, and increase the sum rate without high power usage at the transmitter, and has attracted much attention. Amplify-and-forward (AF) relay has been widely used due to low complexity of signal processing. AF relay can be classified into two categories, namely, 1) variable gain relays which use instantaneous channel state information (CSI) of the first hop to control the gain and 2) fixed gain (semi-blind) relays which control the gain by using statistical CSI about the first hop. Although systems with fixed gain relays can not perform as well as systems with variable gain relays, their low complexity and ease of deployment make them attractive from a practical standpoint [1]. In addition, by employing maximum ratio combining (MRC) technique to combine the signals from the relay and the source, cooperative diversity (CD) can be exploited to further improve the system performance.

For Rayleigh fading channel, the performances of outage probability and bit error rate of variable gain and fixed gain relay systems were analyzed in [2] and [1] respectively, and the effect of multiple antennas at the source on outage probability was investigated in [3]. For Nakagami-m fading channel, the performance of variable gain relay systems was analyzed in

[4], and the generalized moments of end-to-end SNR in fixed gain relay systems was derived in [5], and the asymptotic closed-form expression of ASER was given in multiple relays networks in [6].

In this paper, we analyze the performance of fixed gain relay system with and without CD in non-identical Nakagami-m fading channels. First, we derive the closed-form expressions for PDF and cumulative density function (CDF) of received SNR without CD, and present the closed-form expression of the ASER by using PDF-based performance analysis method. Then, we derive the closed-form expression of MGF of received SNR with CD, and analyze the outage probability and ASER of system by using MGF-based performance analysis method, and present the generalized moments of received SNR. For 1m = , the expressions derived in this paper reduce to that in Rayleigh channels. Our analytical results are confirmed through comparison with Monte Carlo simulations, and the effects of channel imbalance on system performance are also investigated.

II. SYSTEM MODEL

1h

0h

2h

Fig. 1. Dual-hop cooperative communication system

We consider a dual-hop relaying system as shown in Fig. 1. The source transmits a signal to the destination with the help of one AF relay. All nodes are equipped with one antenna. Let 0h ,

1h and 2h denote the complex channel coefficients with Nakagami parameters 0m , 1m and 2m respectively. Assume that three links instantaneous SNRs are denoted by 0γ , 1γ and

2γ . Assuming MRC at the destination, the equivalent end-to-end SNR with CD is shown to be

1 20 0

2eq r C

γ γγ γ γ γγ

= + = ++

(1)

This work was supported by the National Basic Research Program of China (2007CB310603); the National Natural Science Foundation of China (60672093, 60496310, 60702029, 60902012); the National High Technology Research and Development Program of China (2007AA01Z262) and the Natural Science Foundation of Jiangsu Province (BK2005061).

978-1-4244-5668-0/09/$25.00 © 2009 IEEE

where rγ denotes the received SNR without CD, and

( )1

201C G N= where G is the fixed gain of relay and

10N is the single-sided power spectral density of the additive white Gaussian noise (AWGN) at the first hop [5]. Because all three links are subject to Nakagami-m fading, iγ follows a gamma distribution so that the PDF and MGF of iγ are given by [7]

( ) ( )1

expi i

i i

m mi i i i

i mii i

m mp

mγγ γγ

γγ

− ⎛ ⎞= ⋅ −⎜ ⎟Γ ⎝ ⎠

(2)

( ) 1i

i

m

i

i

sM s

mγγ

−⎛ ⎞

= +⎜ ⎟⎝ ⎠

(3)

where ( )Γ ⋅ is the gamma function [8, eq. 8.310.1].

III. PERFORMANCE ANALYSIS In this section, we analyze the performance of dual-hop

fixed gain AF relay system. We first obtain the CDF of rγ , through which we get the PDF of rγ . Then we analyze the outage probability, ASER and generalized moments of received SNR of the system with and without CD by using MGF-based and PDF-based approaches respectively.

Theorem 1: Let 1m be natural number, the closed-form expression of the CDF of rγ is given by

( ) ( )

22 1

2

1 22 1 1 2

0 0 2 112 2

21

!r

m im km k

ith m k

k i

km m CmP C

i mkmγγγγγγ

−−

= =

⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟Γ ⎝ ⎠ ⎝ ⎠∑ ∑

2

2

1 1 22

1 1 2

exp 2m i

k th thth m i

m Cm mK

γ γγγ γ γ

−+

−

⎛ ⎞⎛ ⎞⋅ − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(4)

where thγ is SNR threshold and ( )vK ⋅ is the v order modified Bessel function of the second kind defined in [9, eq. 9.6.22] which can be found in most popular computing softwares such as Matlab.

Proof: See Appendix A.

Corollary 1: The closed-form expression of the PDF of rγ is given by

( ) ( )( ) ( )

1 2 1 1

1 2

11 2 1 1 1

01 2 1 2

2 expr

m m m m

m mk

m m x m x mp x

km mγγ

γ γ

−

=

− ⎛ ⎞= ⎜ ⎟Γ Γ ⎝ ⎠∑

2

2

21 2 1 2

1 2 1 2

2

m k

km k

m C x m m CxC K

mγ

γ γ γ

−

−

⎛ ⎞⎛ ⎞⋅ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(5)

Proof: See Appendix B.

A. Performance of Relaying Systems Without CD 1) Outage probability

Outage probability can be defined as the probability that the received signal falls below a predefined threshold thγ . The

closed-form expression for the outage probability is the CDF of rγ as (4), i.e., ( )1

rout thP Pγ γ= .

2) Average symbol error rate For various commonly used modulations, the ASER at

certain SNR has a unified expression

( ){ }1 22eP Qγ β β γ= E (6)

where ( )Q ⋅ is Gaussian-Q function with definition

( ) ( ) ( )21 2 exp 2Q x dxα

α π∞

= −∫ and 1β and 2β are

modulation specific constants. For example, BPSK ( 1 1β = ,

2 1β = ), BFSK with orthogonal signaling ( 1 1β = , 2 0.5β = )

and M-ary PAM ( ( )1 2 1M Mβ = − , ( )22 3 1Mβ = − ). We

can rewrite the ASER expression (6) directly in terms of the CDF of the received SNR as follows [10]

( )2

1 2

02

x

eeP P x dx

x

β

γ

β βπ

−∞= ∫ (7)

By substituting (4) into (7) and performing variable substitution y x= , the system ASER can be calculated as

( )2

1 21

02 r

x

eeP P x dx

x

β

γ

β βπ

−∞= ∫

( )

22 1

2

1 22 1 21 1 1 2

10 0 2 112 22 !

m im km k

ikm

k i

m km CmC I

i mkmβ ββ γ

γγπγ

−−

= =

⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟Γ ⎝ ⎠ ⎝ ⎠∑ ∑ (8)

where

2

2

2 21 1 21 20

1 1 2

2 exp 2k m im i

m m m CI y y K y dyβ

γ γ γ∞ + −

−

⎛ ⎞⎛ ⎞⎛ ⎞= − + ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠∫ (9)

The integral in (9) can be easily solved using [8, eq. 6.631.3], thus the closed-form expression of ASER of system without CD is given by

( )

22 1

2

1 22 1 21 1 1 1 2

0 0 2 112 22 !2

m im km k

ie km

k i

m km CmP C

i mkmβ ββ γ

γγπγ

−−

= =

⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟Γ ⎝ ⎠ ⎝ ⎠∑ ∑

( )21 122 2

1 1 22 2

1 1 2

1 12 2

k m im m m C

k m i kβγ γ γ

− + − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ + Γ + − + Γ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

( ) ( )2 2

1 2 1 21 12 ,

2 1 2 1 2 2 1 2 1 22 2

exp2 2 k m i m i

m m C m m CW

m mβ γ γ γ β γ γ γ− + − −

⎛ ⎞ ⎛ ⎞⋅ ⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

(10)

where ( ),kW λ ⋅ is Whittaker function [8, eq. 9.222].

B. Performance of Relaying Systems With CD MGF-based approach is an easy and efficient way to

evaluate the system performance in fading channels [7]. In the following, we analyze the outage probability and ASER

performance of the system with CD. We first give the MGF of eqγ as follows.

Theorem 2: The closed-form of the MGF of eqγ is given by

( ) ( )

21 2 1

1 2

12 2

11 2 1 2 1 2

01 2 1 21 2 2eq

m km m m

km m

k

mm m m m C m CM s C

k mmγγ

γ γ γγ γ

−−

=

⎛ ⎞ ⎛ ⎞⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎜ ⎟Γ ⎝ ⎠⎝ ⎠ ⎝ ⎠∑

( )( )1 2

1 2 12

1 2 11 2

2 1 1 2 1

exp2 2

m m km m C m

m m k sm sγ γ γ γ

− + − −⎛ ⎞⎛ ⎞

⋅Γ + − +⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠

( ) ( )

0

1 2 2

0 1 21 12 1 ,

0 2 1 1 22 2

1m

m m k m k

s m m CW

m m sγ

γ γ γ

−

− + − − −

⎛ ⎞ ⎛ ⎞⋅ +⎜ ⎟ ⎜ ⎟+⎝ ⎠⎝ ⎠

(11)

Proof: See Appendix C.

1) Outage probability The outage probability is the Laplace inverse transform of

( )eq

M s sγ at thγ :

( )

2 1 eq

th

out

M sP

sγ

γ

−⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

L (12)

where 1−L denotes the Laplace inverse transform and thγ is SNR threshold. It is difficult to get the closed-form expression of 2

outP . However, we can get the numerical results of Laplace inverse transform by using Euler summation technique [11].

2) Average symbol error rate We can obtain the ASER for a variety of commonly used

modulations using MGF-based approach. For example, for M-PAM modulation, the ASER is

( )2 2

20

2 1sineq

MPAMe

M gP M d

M

π

γ θπ θ− ⎛ ⎞= ⎜ ⎟

⎝ ⎠∫ (13)

where ( )23 1MPAMg M= − . Furthermore, the ASER for M-PAM can be upper bounded by a simple form:

( )2 11eqe MPAMP M g

M γ⎛ ⎞≤ −⎜ ⎟⎝ ⎠

.

3) Generalized moments of received SNR The first- and second-order moments of the received SNR

are important parameters which can be efficiently used to evaluate system performance measures, such as the average output SNR and the amount of fading [7, eq. 2.5]. The higher-order moments are also useful in signal processing algorithms for signal detection, classification and estimation.

Theorem 3: The generalized moments of eqγ can be expressed as

( )2

0 1 2

0 0 1 2

k n k mnneq

k

n CmE

k m mγ γγ

γ

−

=

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠∑

( ) ( )

( ) ( ) ( ) ( )20 1 1,2 2

2,120 1 2 2

1,1 mm k m n kG

n k mm m m n k Cmγ⎡ + ⎤Γ + Γ + −

⋅ ⎢ ⎥− +Γ Γ Γ Γ − ⎣ ⎦ (14)

where ( )mnpqG ⋅ is the Meijer’s G function [8, eq. 9.301], which

is the built-in function in most popular computing softwares, such as Maple and Mathematica.

Proof: See Appendix D.

IV. NUMERICAL RESULTS AND DISCUSSIONS In this section, we confirm our analytical results through

comparison with Monte Carlo simulations, and compare the performances of the outage probability, ASER and average received SNR of system with and without CD.

5 10 15 20 2510

-4

10-3

10-2

10-1

100

γ, Average SNR per hop (dB)

Out

age

prob

abili

ty

m1=1,m

2=1

m1=1,m

2=2

m1=2,m

2=1

m1=2,m

2=2

m1=2,m

2=3

m1=3,m

2=2

m1=3,m

2=3

Analytical

Fig. 2. Outage probability of system without CD for different values

of 1m and 2m ( 1 2γ γ γ= = , 0thγ = dB)

Fig. 2 shows the outage probability of system without CD for different values of 1m and 2m . It is shown that the analytical curves match well with the simulated results. Comparing the case of 1 21, 2m m= = and that of

1 22, 1m m= = , we can see that the first hop channel has greater effect on the improvement of outage probability performance than the second hop channel, which can also be shown by comparing the case of 1 22, 3m m= = and that of

1 23, 2m m= = . When the channel qualities of two hops both improve (i.e., 1 22, 2m m= = ), the outage probability improves remarkably.

0 5 10 15 20 2510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

γ, Average SNR per hop (dB)

Out

age

prob

abili

ty

m=1,2,3

m=1,2,3

Analytical,without CDAnalytical,with CDSimulation

Fig. 3. Comparison of outage probabilities with and without CD

( 0 1 2m m m m= = = ， 0 1 2γ γ γ γ= = = ， 0thγ = dB)

Fig. 3 compares the outage probabilities with and without CD. For 1m = , the curve of outage probability without CD coincide with the curve of Fig. 2 in [1] corresponding to “Fixed Gain Relays”. Compared with Fig. 1 in [4], the fixed gain relay is inferior to the variable gain relay. The outage probability of the system with CD outperforms the system without CD because of the cooperative diversity gain caused by the direct link.

5 10 15 20 25 3010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

γ0, Average SNR of direct link (dB)

Ave

rage

sym

bol e

rror

rat

e

m=1

m=2

m=5

γ1=5γ

0,γ

2=γ

0

γ1=γ

0,γ

2=5γ

0

γ1=γ

0,γ

2=γ

0

Fig. 4. Comparison of ASER of system with CD for different SNRs

and m (4-PAM modulation, 0 1 2m m m m= = = )

Fig. 4 compares the ASER performance of system with CD for three cases of SNRs of dual hops. Case 1: 1 05γ γ= ,

2 0γ γ= ; Case 2: 1 0γ γ= , 2 05γ γ= ; Case 3: 1 0γ γ= , 2 0γ γ= . For 1m = , the ASER performance of Case 1 outperforms that of Case 2 and Case 3. We can see that the transmission quality (average received SNR) of the first hop has greater effect on the ASER performance than that of the second hop. When m =2 and 5, the ASER of Case 1 is better than that of Case 2 and Case 3 at low 0γ . With the increase of 0γ , the ASER of Case 2 is better than that of the other two cases.

0 5 10 15 200

5

10

15

20

25

30

35

40

γ1, Average SNR of first hop (dB)

Ave

rage

end

-to-

end

SN

R

γ1=γ

2=γ

0,with CD

γ1=γ

2,without CD

Fig. 5. Comparison of average received SNR with and without CD

( 0 1 2 2m m m= = = )

Fig. 5 compares the average received SNRs of two cases: system with CD (eq. (14) for 1n = ) and system without CD [5]. From Fig. 5, we can see the average received SNR with CD outperforms that without CD evidently.

V. CONCLUSIONS In this paper, we have analyzed the performance of dual-

hop communication system with fixed gain amplify-and-forward relay over non-identical Nakagami-m fading channels. We have derived the closed-form expressions for outage probability and ASER of system without CD based on PDF-based approach. Furthermore, we also derived the closed-form expressions for outage probability, ASER and generalized moments of received SNR based on MGF-based approach. Monte Carlo simulations are provided to assess the accuracy of the closed-form expressions. The system with CD outperforms that without CD in the performance of outage probability, ASER and average received SNR. The channel qualities of the first hop and the second hop have different effects on the system performance.

APPENDIX A Since 1γ and 2γ are independent of each other, the joint

PDF of 1γ and 2γ is given by

( ) ( ) ( )1 2 1 2

1 2 1 2

1 11 2 1 2 1 1 2 2

, 1 21 21 2 1 2

, expm m m m

m m

m m m mp

m mγ γγ γ γ γγ γ

γ γγ γ

− − ⎛ ⎞= ⋅ − −⎜ ⎟Γ Γ ⎝ ⎠

(15)

Then the CDF of rγ can be derived as follows

( ) ( )

( )21 2

1 2

2

, 1 2 1 20 0

Pr Pr

,

r

thth

th r th th

C

PC

p d d

γ

γγγ

γ γ

γ γγ γ γ γγ

γ γ γ γ∞ +

⎛ ⎞= ≤ = ≤⎜ ⎟+⎝ ⎠

= ∫ ∫

( ) ( )1 2

2

1 2

11 2 2 220

21 2 1 2

expm m

mm m

m m mm m

γγγγ γ

∞ − ⎛ ⎞= ⋅ −⎜ ⎟Γ Γ ⎝ ⎠

∫

121 1 1

1 1 201

expth

thC

m md d

γγγ γγ γ γ

γ+ − ⎛ ⎞

⋅ −⎜ ⎟⎝ ⎠

∫ (16)

( )11 2

2

1 2

( )11 2 1 2 2

2 201 21 2 2

expmm ma

mm m

m m md

mmγ γγ γ

γγ γ∞ −

⎛⎛ ⎞ ⎛ ⎞⎜= ⋅ −⎜ ⎟ ⎜ ⎟⎜Γ ⎝ ⎠ ⎝ ⎠⎝

∫

11

1

11 1

0 01 1

exp!

m km ki kth

thm kk i

kmC

ik mγ γ γ

γ

−−

−= =

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

∑ ∑

2 1 1 2 22 20

1 2 2

expm i thCm md

γ γγ γγ γ γ

∞ − − ⎞⎛ ⎞⋅ − − ⎟⎜ ⎟ ⎟⎝ ⎠ ⎠∫ (17)

where (a) is solved using [8, eq. 3.351.1] and binomial theorem [8, eq. 1.111] with condition: 1m is natural number. With the help of [8, eq. 3.381.4] and [8, eq. 3.471.9], we yield the desired result in (4).

APPENDIX B

We can get the PDF of rγ by differentiating (4) w.r.t. thγ , but the differentiation process is complicated. A simple way is differentiating (16) w.r.t. thγ , and the derivation process is

( ) ( )Prr

r thth

th

dp

dγ

γ γγ

γ<

=

( ) ( )1 2

2

1 2

11 2 2 220

2 21 2 1 2

exp 1m m

mm m

m m m Cm m

γγγ γγ γ

∞ − ⎛ ⎞ ⎛ ⎞= − ⋅ +⎜ ⎟ ⎜ ⎟Γ Γ ⎝ ⎠ ⎝ ⎠

∫

1 1

12

2 1 2

expm

th thth th

C Cmd

γ γγ γ γγ γ γ

− ⎛ ⎞⎛ ⎞ ⎛ ⎞⋅ + − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

(18)

After some algebraic manipulations, (18) can be written as

( ) ( )( ) ( )

1 2 1

1 2

11 2 1 1

01 2 1 2

expr

m m mth th

th m m

m m mp I

m mγ

γ γ γγ

γ γ

− −=

Γ Γ (19)

where 1

21 1 12 20 2 20

0 2 1 2

= expm

m kk th

k

m m CmI C d

kγγγ γ

γ γ γ∞ − −

=

⎛ ⎞⎛ ⎞ − −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∫ .

With the help of [8, eq. 3.471.9] and substituting th xγ = , (10) can be obtained.

APPENDIX C The MGF of rγ can be expressed in terms of the PDF of

rγ as

( ) { } ( )0

r

r r r

s sxM s e p x e dxγγ γ γ

∞− −= = ∫E

( ) ( )

21 2 1

1 2

211 2 1 2

20 1 21 2 1 2

2m k

m m mk

m mk

mm m m CC I

k mm mγ

γγ γ

−

=

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟Γ Γ ⎝ ⎠ ⎝ ⎠∑ (20)

where 2

1

2

1 1 1 222 0

1 1 2

exp 2m k

m

m km m m Cx

I x s x K dxγ γ γ

−+ −∞

−

⎛ ⎞⎛ ⎞⎛ ⎞= − + ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠∫ (21)

The closed-form expression of ( )r

M sγ is obtained by

performing variable substitution y x= , using [8, eq. 6.631.3] and substituting (21) into (20). Since the MGF of equivalent SNR 0eq rγ γ γ= + is given by ( ) ( ) ( )

0eq rM s M s M sγ γ γ= ⋅ ,

(11) can be obtained from (3) and (20).

APPENDIX D The generalized moments of received SNR with CD is

given by

( ) 1 200 0 0

2

nneqE

Cγ γγ γ

γ∞ ∞ ∞ ⎛ ⎞

= +⎜ ⎟+⎝ ⎠∫ ∫ ∫

( ) ( ) ( )0 1 20 1 2 0 1 2p p p d d dγ γ γγ γ γ γ γ γ⋅ (22)

( )00 0 00

0

nk

k

np d

k γγ γ γ∞

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑ ∫

( ) ( )1 2

1 21 2 1 20 0

2

n k

p p d dC γ γγ γ γ γ γ γ

γ

−∞ ∞ ⎛ ⎞

⋅ ⎜ ⎟+⎝ ⎠∫ ∫ (23)

( )( ) ( )

( )00

0 0 0

knan kr

k

m knE

k m mγ γ −

=

Γ +⎛ ⎞⎛ ⎞= ⋅⎜ ⎟⎜ ⎟ Γ⎝ ⎠⎝ ⎠∑ (24)

where (a) is solved using [8, eq. 3.381.4]. Substituting the closed-form expression of ( )n k

rE γ − given in [5], we can get (14).

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