The 2010 International Conference on Advanced Technologies for Communications

Performance Analysis of Amplify-Forward Relay in Mixed Nakagami-m and Rician Fading Channels

Arun K. Gurung, Fawaz s. AI-Qahtani1, and Zahir M. Hussain, and Hussein Alnuweiri1 School of Electrical & Computer Engineering, RMIT University, Melbourne, Australia

{ arun.gurung@student.rmit.edu.au,zmhussain@ieee.org} 1 Electrical & Computer Engineering Program, Texas A&M University at Qatar, Doha, Qatar

{fawaz.al-qahtani,hussein.alnuweiri }@qatar.tamu.edu

Abstract-The performance of a dual-hop amplify-and forward relay system is analyzed in terms of outage probability and average symbol error rate. The source-relay and relay­destination channels experience mixed fading distributions namely, Rician and Nakagami-m. Analytical expressions for Cumulative Distribution Function (CDF) and Probability Density Function (PDF) of end-end signal-to-noise-ratio (SNR) are derived and confirmed with Monte-Carlo simulation. Approximate lower bound for outage probability is also derived, which becomes tight at high SNR values. The expressions are given in terms of infinite sum series of modified Bessel function, which converges after finite iterations. The derived expressions are valid for different fading scenarios (depending on m factor), and specialize to previously published results for m = 1.

-Keywords: Dual-Hop Relay, Asymmetric Fading, Performance Analysis

I. INTRODUCTION

Relays are shown to significantly improve the link quality, thereby increasing the coverage area of many infrastructure or ad-hoc networks alike. Relays receive signals from the source and relay them towards the destination based on some pro­tocols such as amplify-and-forward (AF), decode-and-forward (DF), etc. Some noteworthy papers on relay networks include [1]-[3] and the references therein.

Symmetric fading channels are intuitive choice for many of the researchers for the performance analysis of the relay networks (dual-hop, multi-hop, and multi-branch). Recently, there is an increased research attention on the performance of dual-hop relay networks where fading channels of two hops are asymmetric some recent works include [4]-[7]. Dissimilar or asymmetric Nakagami-m fading was assumed in [4] for decode-forward relay, and in [5] for amplify-forward case. Duong et. al. [6] presented the symbol error probability of dual-hop Rayleigh-Rican fading channels where the source employs orthogonal space-time block codes (OSTBCs) and the relay operates in AF mode. Himal et. at. [7] considered a dual-hop system with asymmetric fading channels which re­sembles practical scenarios of macro/micro cellular networks. The channel between the base station (source) and the relay has line-of-sight, whereas the channel between the relay to the mobile unit (destination) might not necessarily be same. The authors have derived exact expressions of the outage

probability and the average bit error probability (ABEP) in terms of infinite series.

In this paper, we extend [7] to relay channels subject to a line-of-sight (Ric ian) and a Nakagami-m fading. Inclusion of Nakagami-m distribution generalizes the analysis as the Rayleigh fading is its special case. Our contributions are summarized as follows: 1) Derivation of closed-form analytical expressions for SNR statistics such as cumulative distribu­tion function (CDF) and probability density function (PDF), and 2) System analysis in terms of closed-form expressions for outage probability and ASEP of dual-hop Nakagami-m­Rician fading channels. The final expressions specialize to the previously published work. Furthermore, approximation is presented which matches very well with exact result at high SNR region.

The paper is organized as follows: Next section briefly describes the basic relay model and associated channel dis­tributions. Then we derive SNR statistics. These statistics are used to compute the outage probability and the error rate in the following section. Few numerical results are shown to illustrate the system performance over asymmetric fading channels willi different severity values. The summary section concludes the paper.

Relay

Source Destination

Fig. 1. A dual-hop relay transmission system

II. SYSTEM DESCRIPTION

A dual-hop relay, as shown in Fig. 1, consists of the source (8) sending signals towards the destination (D) via the

978-1-4244-8876-6/101$26.00 ©2010 IEEE 321

assistance of an AF relay (R). The total transmission involves two time-slots. In AF relay systems, R amplifies the received signal before forwarding to the D node. Here, we assume that no direct link exists between S and D. The instantaneous end­to-end signal-to-noise-ratio (SNR) 'Yeq at D can be shown as [3]

'Ysr'Yrd 'Yeq = _--'...:.C'--'-'-=---_ 'Ysr + 'Yrd + C (1)

where 'Ysr = Ihsrl2 Ps/No and 'Yrd = Ihrdl2 Pd/No. Ps and Pr are the transmit powers at S and R, and No is the power of additive white Gaussian noise, assumed equal at both R and D. The hsr and hrd are the fading envelopes of the S - R and R - D links respectively. Finally c is a constant, equal to either 1 or 0 according to the gain G = 1/(h�r + No) or G = l/h�r.

We consider an asymmetric fading scenario where the S -R link experiences Nakagami-m fading, and the R - D link is subject to Rician fading (line-of-sight) fading. Due to the symmetry of link SNRs in (1), the assumption is valid other way around as well. If a link experiences Nakagami-m fading, the PDF and the CDF of 'Ysr are given by,

(2)

(3)

where m = 1,2 ... is fading severity parameter, 'Ysr =

E{lhsrI2}Ps/No is average SNR of S - R hop. Note that when m = 1, the distribution becomes Rayleigh. And the R - D SNR 'Yrd is a noncentral-x2 distributed, given by

( )_(K+l)e-K _

{K_+lh T (2 K(K+l) ) (4) P"(rd 'Y - e "Ird -,o 'Yrd 'Yrd

where K is the ratio of the powers of the line-of-sight com­ponent to the scattered components, 'Yrd = E{lhrdI2}Pr/No is the average SNR of R - D hop, and 10 (.) is the zeroth order modified Bessel function of the first kind [8, pp 900].

III. SNR STATISTICS

A. Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) statistics of received SNR at the destination can be obtained as,

where 0"(00"' ('Y) = 1 - F,,(",oo ('Y) is complementary CDF of 'Yxx. Inserting (4) and (3) into above equation yields

K+l -K- [ �+l + _rn l "( 1 - --e "Ird "Isr 'Yrd 100 - {K+l)w

_ "'{"I+ch ( X e '1rd '1srw 10 2 o

K(K + 1)) 'Yrd

x � i (�'Y) i (1 + C+ 'Y) idW

i=O �· 'Ysr W (5)

( Z )2k Using infinite-series representation of 10(z) = 2::� 0 (t!)2 [8, 8.44.1] and binomial expansion for the last term [Le. (x + y)n = 2::;=0 (;)xn-pyp], we get

where A = (�+1) + _m • With the help of [8, 3.471.9] to '"Yrd 'Ysr evaluate II, the CDF of end-end SNR for amplify-forward dual-hop relay under mixed Nakagami-m and Rician fading can be written as

F"(eq("() = 1- 2e-K-A"( f: �: t ({) � t (i) j=O J 1=0 i=O p=o \y

where (h = 2i+ l;-p+1 , (h =

2i- 1tP+1, and Kl(.) is 1-order modified Bessel function of second kind [8, 8.446]. For m=l, (7) specializes to [7, 8] as expected. Figs. 2 and 3 plots the CDFs for two different scenarios. In the first case, the average SNR in R - D link (Rician fading) is twice that of in S - R link. The Rician factor K = 5dB and the Nakagami fading parameter m = 3 are assumed. The analytical plots (continuous and dot curves) were drawn using (7) for C = ° and c = 1. Obviously, higher link average SNR improves the system performance. The difference between the curves for c = ° and c = 1 is small, and diminishes as threshold SNR 'Y increases. Therefore c = 0, which aids mathematical analysis, can be used for system performance analysis without sacrificing much of accuracy. Fig. 3 confirms the observation for different settings where Rayleigh-Rayleigh fading (K = - lOdB and m = 1) is compared with less severe fading (K = IOdB and m = 3). Both links have same average SNR in this case. Monte-carlo simulation results are also shown and validates the analytical results.

322

"dB

Fig. 2. Comparison of Cumulative Distribution Functions (CDFs) for c = 0 and c = 1, K = 5dB, m = 3, • indicates Monte-Carlo simulation points.

0.9

0.8 .

0.7 .

0.6 . CDP 0.5 .

0.4 .

0.3 .

0.2 .

0.1 .

2 4 6 8 10 "dB

Fig. 3. Comparison of Cumulative Distribution Functions (CDFs) for c = 0 and c = 1, )'sr = )'rd = lOdB, • indicates Monte-Carlo simulation points.

B. Probability Density Function (PDF)

The probability density function (PDF) can be obtained by taking derivative of (7) with respect to '"'I as shown below

P'Yeq ('"'f) = 2e-K -A'Y f �� t ({) E t (i) j=O J 1=0 i=O p=o P

X (� ) 81 (K

_ + 1) 82 '"'Ii��+1 [KI-P+1 (2'"'1() '"'Isr '"'Ird 2 .

where ( = r;(K;y+l). Note that we have employed sr rd zdK,;z(z ) + vKv(z) + zKv- 1(z) = 0 [9, 8.486-12] to arrive above expression. Two special cases can be deduced: Rayleigh-Rician and Rayleigh-Rayleigh fading scenarios.

Case I: When m = 1, we get Rayleigh-Rician fading scenario considered in [7] where only CDF was derived. In this case, i = P = 0 thereby yielding

00 . j 2;-1+1 P'Yeq('"'f) 2e-K- A'Y L �� L (j) (K

_+ 1) 2

j=O J. 1=0 l '"'Ird

x �; [KI+1 (2'"'1() { oX + l � j} + 2(KI(2'"'10] '"'Isr

(9)

Case II: When m = 1 and K = 0, fading scenario becomes Rayleigh-Rayleigh case considered in [3]. It can be easily seen that the infinite sum reduces to a single term (j = 0), thereby (9) specializes to [3, 19].

Fig. 4 shows the comparison of PDF plots for four dif­ferent settings. The first two cases include Rayleigh-Rayleigh (K = -lOdB, m = 1, 1sr = 1rd = 5dB) and Rayleigh­Rician (K = lOdB, m = 1, 1sr = 1rd = 5dB) fadings. Next two cases were considered to observe the impact of Nakagami fading parameter m and unbalanced link SNR values on the end-end statistics. Higher m reduces the variance, however doubling the R

-D SNR has a opposite effect. The analytical

results are exactly in accordance with the simulation confirm­ing accuracy of analysis.

0.9

0.8

1.5 2 2.5 "dB

• sil1lllatim -- tllll'Y

3 3.5 4

Fig. 4. Comparison of Probability Density Functions (PDFs), c = 0

C. Moment Generating Function (MGF)

The moment generating function (MGF) is an important statistics, which can be used to derive error probabilities

X { oX + l - P � i - j} + 2(KI-p(2'"'10] (8�23

efficiently for a varieties of modulation types. MGF is related to PDF by M'Yeq(s) = fo oo e- SYP'Yeq(y)dy. Here, the MGF can be easily derived in the form of hyper geometric functions with the aid of [9, 6.621-3].

D. When the Direct Link Exists

When 8 � D link is not in deep fade and active, we can easily extend above analysis to offer cooperative diversity at the destination. Two most popular combining schemes are Maximal Ratio Combining (MRC) and Selection Combining (SC). The MRC requires continuous channel estimation of all branches (8 � R � D and 8 � D), and provides opti­mum gain whereas SC is a sub-optimum scheme. Assuming independence between the relayed 8 � R � D and the direct link (8 � D), and when MRC is employed at the destination, the MGF of the total SNR becomes, M'Ytot ('Y) =

M'Yeq(,,().M'YSd("(). Once the MGF is known, ASEP can be easily computed. Taking inverse Laplace transform of the MGF results the PDF. If SC is used at the destination instead, we can write the CDF of the total SNR as F'Ytot("() = F'Yeq(,,().F'YsA'Y). Similarly the PDF can be calculated as f'Ytot ("() = F'Yeq (,,().f'YSd ("() + f'Yeq (,,().F'Ysd ("(). We have not included this scenario in the performance analysis next though. This is to keep the focus of the contribution to the mixed fading scenarios in a relayed system.

IV. PERFORMANCE ANALYSIS

In this section, we use the analysis presented in the previous section to compute two important measures - outage probabil­ity (OP) and average symbol error probability (ASEP).

A. Outage Probability (OP)

The outage probability quantifies the likelihood of the instantaneous end-to-end SNR falling below some predeter­mined threshold 'Yth, given by Pout = F'Yeq ('Y < 'Yth) .

A lower bound of outage probability can be obtained by simple approximation as shown below. This approximation takes account of the lower link SNR and independence of two inks, and has been used in several papers. It simplifies the analysis and is tight at medium to high SNR values.

Pout Pr [min("(STl 'Yrd) < 'Yl 1 -C'Ysr ("( )C'Yrd ("()

1- e 'YsrQl v 2K, - = ( r;:;u

'"'.! m'Y � ., -

m-l ( ) i

i=O �· 'Ysr

2(K + 1h) 'Yrd

(10)

where Ql(.,.) is the first-order Marcum Q-function [10,

4.33] and has following series representation Ql (a, (3) =

",2+/32 ( ) k e-----r- L�o � Ik(af3) where Ik(x) is the kth-order Bessel function of the first kind.

B. Average Symbol Error Probability (ASEP)

Average symbol error probability (ASEP) is an important system performance metric useful for system designers. The ASEP of communication link over fading channels is nor­mally computed by averaging the conditional symbol error probability (SEP) in A WGN channel over the range of fading distribution. Specifically,

Ps(e) = 100 Q (J7iY) P'Yeq(,,()d'Y (11)

where Q (x) = vb fxoo e- t2/2dt is the Gaussian Q-function

and essentially represents the conditional SEP over A WGN channel, and 13 is a constant specific to modulation type (BPSK, 13

= 2; QPSK, 13 = 1, etc). We further write (11)

into following form [7] which allows to directly use CDF (7)

to compute ASEP,

Ps(e) = � 100 F'Yeq (�) e-%dt 1 100 ( t ) t 1 -- F'Y - e-2C2dt 2J27T 0 eq 13

Inserting (7) into (12) yields

Ps(e) = 1- e-2: f �: t ({) Et (i) J27T j=O J 1=0 i=O p=o \y

X f3- ( i�Hl) (� ) 81 (K

_ + 1) 82

� 'Ysr 'Yrd x 100 tHH!e-C;p(3)t

�---, (2t m(K + 1) ) X Kp+l-H1 -13 - -'Ysr'Yrd

which using the following identity [8, 6.621.3],

(12)

(13)

r oo xi'-le-OIXK (f3'x)dx = v-;r(2f3't r(f.-t + v)r(f.-t - v)

10 v (a+ f3') i'+v r(f.-t+ !) 2Fl (f.-t+ v,v+ �;f.-t+ �; :��;) Re(a+ f3') > 0

becomes (c=0),

1 00 (K1'sr)j(K + l)j+1 j (i) m-l i Ps(e) = 2" -1P L ·!2 L l L L

j=O J 1=0 i=O p=o (i) 41-p(m1'rd)l+i-p r(03)r(04) P i! qq83r(05)

2Fl (03'P + l - i + �; 05, �) (15)

where1P = 2me-K1'sr1'rdV2f3'Ysr'Yrd, 03 = i+i+l-p+�, 04 = i+ i -l+p+ !, 05 = i+ i+ 2, v = (2A+ (3)1'sr1'rd/2+ 2i,(K + 1hsr'!rd, W = v - 4J.(K + 1hsr'Yrd, r(X) � fo tX-le- tdt IS the gamma functIOn, and 2Fl(a,b;c;x) IS the Gauss hypergeometric function defined in [8, 9.111]. Using approximated CDF (10), it is easy to derive an expression for approximated ASEP (not shown here).

(14)

324

The final ASEP expression (15) specializes to [7, 14] for

Rayleigh fading (m=I). Modified Bessel function in (7) and

Gauss hypergeometric function in (15) are available as built­

in functions in standard software packages such as MATLAB,

MATHEMATICA, and therefore facilitates efficient computa­

tion of the final expressions.

V. NUMERICAL RESULTS AND DISCUSSION

We present some results based on the analysis in previous

sections. We seek to understand the system behavior for

different settings of severity parameters m and K, and average

link SNRs. The figures were drawn using (7), (10) and (IS)

in Matlab. The infinite summation was computed for first 20

terms (j = 0, I, 2 ... ,19) which was found sufficient for the

effective SNR range considered here. Please note that the terms

1/ j!2 and l/i! in the final expressions ,which makes the contri­

butions of higher terms negligible. Mathematical justification

is possible however out of scope of this contribution.

The Fig. 1 shows the outage probability of dual-hop relay

system with mixed Nakagami-m and Rician fading channels.

Equal average link SNRs are assumed. We consider m = 1, 5

for K = 5dB and I'th = 10dB first. The gain with higher m (less severe fading) is obvious and remains same (shown by

equal slopes) irrespective of link SNRs at high SNR values.

With higher K = lOdB and lower threshold I'th = 5dB, the

gain becomes more prominent. The outage behavior remains

similar to previous case at very high SNR region (not shown

here). The approximate outage probability is also depicted and

matches well with the exact one at high SNR region (> 20dB).

Fig. 2 shows the ASEP for QPSK modulation (/3 = 1) for

similar settings. For m = 1 which is Rayleigh fading case,

K = 6dB offers constant gain of 2.5dB over K = OdB in high

SNR region. And with a fixed K = 6dB, higher m provides

only marginal (almost none for m > 3) gain as illustrated by

m = 2, 3. Similar observation was taken for K = OdB and

m = 2, 3,5 (not shown here to avoid ambiguity in the figure).

Another interesting observation is the diversity order of the

system. The diversity order of the system is independent of

m, and equal to one. This is similar to the observation in [7].

VI. CONCLUSION

The system performance of dual-hop relay system over

mixed fading scenarios has been considered. The analysis

is carried over Nakagami-m and Rician fading channels for

amplify-and-forward scheme. The SNR statistics have been

derived and validated through computer simulation. Further­

more, closed-form expressions for the OP and the ASEP are

derived, which are in terms of infinite series. The diversity

order of the system is one and does not depend on severity

parameter m. The derived expressions are in more general

forms than the previously published results.

ACKNOWLEDGMENT

Authors would like to thank T. Q. Duong (with the Blekinge

Institute of Technology, Sweden) for his help on manuscript

OP

............ 10-3 :::::: :::",'" ,: ,: 'i : .. ........................ .

--exact ,,,,,,,,appx

10-1 5 10 15 20

tth =5dB

25 30 'Ysr='Yrd,dB

35 40

Fig. 5. Comparison of Outage Probabilities, equal link average SNRs i.e., 1 sr = 1rd

100 ":":".""""."""" :"":".:""":. """:.""" .......................... ..................

ASEP

........ : .. · f(=6dB;m?=2;3. 10-6 "" " : , , "",,:0 "",,:; " ",,: ; "", , : ; :::::::;:::::: ..

o 5 10 15 20 25 30 35 40

Fig. 6. Comparison of ASEP, equal link average SNRs i.e., 1sr = 1rd

editing and comments which improved the presentation of the

paper. This work in part is supported by Qatar National Research

Fund (QNRF) grant through National Priority Research Pro­

gram (NPRP) No. 08-577-2-241. QNRF is an initiative of

Qatar Foundation and supported in part by a grant from Qtel.

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