This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2010 proceedings.

Selection Decode-and-Forward Relay Networks with Rectangular QAM in Nakagami-m Fading Channels Fawaz S. Al-Qahtani§ , Trung Q. Duong‡ , Arun K. Gurung§ and Vo Nguyen Quoc Bao†  King

Abdulaziz City for Science and Technology (KACST), National Satellite Technology Program, Kingdom of Saudi Arabia, E-mail:[email protected] ‡ Radio Communication Group, Blekinge Institute of Technology, Sweden E-mail: [email protected]  School of Electrical and Computer Engineering, RMIT University, Melbourne, Australia E-mail: [email protected] † Department of Electrical Engineering, University of Ulsan, South Korea E-mail: [email protected]

Abstract—In this paper, we investigate the average symbol error probability (SEP) of fixed decode-and-forward relay networks over independent but not identically distributed Nakagami-m fading channels. We have derived closed form expression of SEP for general rectangular quadrature amplitude modulation (QAM) under relay selection scheme where only the best relay forwards message from the source to the destination. The expressions are in terms of hypergeometric series which can be efficiently numerically evaluated. The numerical results are validated with Monte-Carlo simulations.

I. I NTRODUCTION In recent years, wireless relaying techniques have been shown to provide significant performance gain for being able to exploit spatial diversity to combat the channel fading. The spatial diversity is obtained by multiple communication nodes collaborating together to form a virtual antenna array. In cooperative communication systems, the source relies on a relay to convey the message to the destination [1], [2]. The performance depends on the number of relaying nodes and the processing operation at both relays and destination. One popular relaying scheme, decode-and-forward (DF), where relays decode the received signals from the source and retransmit them toward the destination, was analyzed in [1]–[4] and references therein. In fixed DF case, the relays always respond to all received signals by decoding, re-encoding, and transmitting them to the destination. The relays forward only the correct decoded messages by applying cyclic redundancy check (CRC) at the relay nodes in adaptive DF relay networks [2]. In [4], the performance of fixed DF cooperative networks with relay selection over independent but not necessarily identically distributed (i.n.i.d) Nakagami-m is presented. The authors derived the closed-form expression of SEP for M-ary phase-shift keying (M -PSK). On the other side, rectangular quadrature amplitude modulation (R-QAM) is gaining more attention [5], [6] as it generalizes the error analysis. Beaulieu has recently shown an exact closed form expression for the SEP of arbitrary QAM for a direct link [5].

In this contribution, we extend the previous work [4] on fixed selection DF relay networks to derive closed form expression of average SEP for rectangular QAM signals. Unlike [4] where the final SEP expression is in the integral form, we provide SEP in terms of finite sums which is simpler and more tractable. The final expression is in general form, expressions for Rayleigh fading and Square QAM cases can be derived as special cases. The paper is organized as follows: Next section briefly introduces the system under consideration. The SEP analysis is given in Section III. Numerical results are discussed next. The final section summarizes the main contributions of the paper. II. S YSTEM M ODEL Consider a wireless system where a source node S transmits to a destination D with assistance of K relay nodes {R1 , R2 , . . . Rk }. The source S transmits the signal (broadcasting phase) to K relays (S − Rk ). Then, only the best relay is selected to forward the source’s signal to the destination (Rk − D). Let us denote hSRk and hRk D as the channel coefficients of S − Rk and Rk − D links, respectively. In this paper, we assume that channels for all links are i.n.i.d Nakagami-m fading. Moreover, we denote the instantaneous signal-to-noise ratio (SNR) for S − Rk and Rk − D links as γSRk = γ|hSRk |2 and γRk D = γ|hRk D |2 , respectively, where γ is the average SNR. The effective channel powers |hSRk |2 and |hRk D |2 follow the gamma distribution with different fading parameters 1/ΩSRk , 1/ΩRk D and fading severity parameters m1k , m2k , respectively. In this paper, we consider the fixed DF relay protocol, where the selected relay terminal is always active to assist the direct communication. With the fixed DF relaying operation, the dual-hop S-Rk -D channel can be tightly approximated in the high SNR regime as follows [3], [4]: γeqk = min{γSRk , γRk D }

(1)

For the relay SC scheme, the relay with largest equivalent received SNR is selected, and thus, the instantaneous SNR at

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the destination of the SC system with fixed DF relays can be expressed as follows: γsc = max γeqk

(2)

k=1,...K

  The MGF of γsc is defined as Mγsc (s) = E esγsc with E[.] is the expectation operator, and can be written as [4, eq. (16)] Mγsc (s) =

K 

Ak nk1 !

nk 1 +1

Bk nk2 !

nk 2 +1

Ps

m1k ΩSRk γ ,

Ak =

×

u=0

m1n1 −1 m1nl −1





i1 =0

j1 =0

Bk =

×

m2k ΩRk D γ ,

m2k −1

m

αk 1k Γ(m1k )

m

βk 2k Γ(m1k )

···

v=0

m1n1 −1 m1nl −1



i1 =0

j1 =0

···

βku u!

K  l=0

(−1)l l!

K 

···

n1 =1

il =0 αv k v!

K  l=0

jl =0

nl =1

(−1)l l!

t=1 K  n1 =1

K  nl =1



 n1 =···=nl =l  m1nl −1 m2nl −1 l αit β jt  

nt nt il =0

jl =0

t=1

(4)

it !jt !

···

 π/2 A1 2q(I) Mγsc − 2 dφ π sin φ 0 π/2

+

−

 A2 (8) Mγsc − 2 dφ sin φ 0 √ tan−1 ( Aκ2 /Aκ1 )   Aκ 2q(I)q(J)  Mγsc − 22 dφ π sin φ

2q(J) π

κ∈P2 (2)

K 



 n1 =···=nl =l  m1nl −1 m2nl −1 l  

αintt βnjtt

m1k −1



k=1

=

(3)

(βBk − s)  where αk = βk = βAk = βBk = αk + βk +  k t l m1k − 1 + u + t=1 (γnt + βnt ), t=1 (αnt ) + βnt , n1 = t and nk2 = m2k − 1 + v + t=1 (γnt + βnt ). Furthermore, Ak and Bk are calculated as follows: k=1

(βAk − s)

+

K 

∞ Ps = 0 Ps (e|γ)pγsc (γ)dγ. However, the Ps can be reexpressed using Craig’s alternative expression for the Q 1 π/2 x2 function, Q(x)= π 0 exp(− 2 sin 2 θ )dθ. Thus, the average SEP can be written as

(5)

it !jt !

III. AVERAGE S YMBOL E RROR P ROBABILITY Let us consider the transmitted signal correspond to an arbitrary I × J rectangular QAM constellation whose signal waveform can be written as [5], [6]

0

where A1 = g(I, J, ζ), A2 = g(I, J, ζ)ζ and κ = (κ1 , κ2 ) is a permutation to P2 (2) = {(1, 2), (2, 1)}. By incorporating (3) in (8), it yields the average SEP of arbitrary rectangular QAM over i.n.i.d Nakagami-m fading channels as given by (9) at the top of the next page. Furthermore, the Ps can be obtained in closed form expression by evaluating the integrals I1 , I2 in (9). By making change of variable t = cos2 φ, the integral I1 can be obtained in closed form expression as nk1 +1  π/2 k 1 1 dφ = (βAk + A1 )−n1 −1 A1 2 βAk + sin2 (φ) 0 −nk1 −1  1 βAk −1/2 nk +1/2 1 1− t (1 − t) t dt βAk + A1 0  βAk 1 3 k k 2 F1 n1 + 1, 2 ; n1 + 2 ; βA +A1 k 1 (10) = −1 2 k +1 1 3 k n 1 (βAk + A1 ) B 2 , n1 + 2

where 2 F1 (a, b; c; x) is Gauss’s hypergeometric function and B(a, b) = Γ(a)Γ(b) Γ(a+b) denotes the Beta function [7]. For the xi,j (t) = Ai,I s(t) cos(2πfc t) − Aj,J s(t) sin(2πfc t), integral I2 in (9), we carry out change of variable t = 0 ≤ t ≤ T ; i = 1, . . . , I, j = 1, . . . , J. (6) 1 − Aκ2 tan2 φ, dt = −2 tan φ sec2 φdφ, sin2 φ = 1−t , then Aκ1 2−t where T is the symbol interval, fc is the carrier frequency, after further manipulation, the integral I2 can be written in and s(t) is the signal pulse shape with energy Es . Note that terms of hypergeometric series n+1  η the I × J rectangular QAM constellation is equivalent to sin2 φ two independent I-ary and J-ary pulse amplitude modulation I2 (a, Aκ1 , Aκ2 , n) = dφ asin2 φ + Aκ2 0 (PAM) signals. {Ai,I = (2i − 1 − I)dI }Ii=1 and {Aj,J = −n−1 1 (2j − 1 − J)dJ }Jj=1 denote the in-phase and quadrature Aκ2 t Aκ1 n+ 12 1− (1 − t) = information-bearing signal amplitudes, respectively, where dI n+1 a + 2Aκ2 Aκ2 (a + 2b) and dJ are arbitrary constants. The conditional SEP for general 0   Aκ2 rectangular QAM in AWGN is [5], [6] −1 Aκ1 F1 1, 1, n + 1, n + 52 ; 12 ; a+2A t κ2  1− dt = n+1 Ps (e|γ) = 2q(I)Q( 2g(I, J, ζ)γ) −1 2 2A (a + 2A ) B(n + 1/2, 1) κ2 κ2  (11) + 2q(J)Q( 2g(I, J, ζ)ζγ) (7)     − 4q(I)q(J)Q( 2g(I, J, ζ)γ)Q( 2g(I, J, ζ)ζγ) Aκ2 where η = tan−1 and F1 (a, b1 ; b2 ; c; x1 ; x2 ) is Aκ1  2 2 where  q(c) = 1 − 1/c, g(I, J, ζ) = 3/ (I − 1) + (J − Appell’s hypergeometric function [7]. By incorporating the 1)ζ . The parameter ζ = d2J /d2I is defined as the squared corresponding results of integral I1 and I2 into (9), it yields quadrature to in-phase decision distances ratio. The average the final closed form expression of the average SEP of SEP of selection decode-and-forward relay over Nakagami selection DF relay over (i.n.i.d) Nakagami-m fading channels distribution with general rectangular QAM is written as for general rectangular QAM as given by (12) at the top of

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Ps

K K   2q(J)   2q(I)  k k k k Ak n1 !I1 (βAk , A1 , n1 )+Bk n2 !I1 (βBk , A1 , n2 ) + Ak nk1 !I1 (βAk , A2 , nk1 )+Bk nk2 !I1 (βBk , A2 , nk2 ) = π π k=1

−

k=1

2q(I)q(J) π

K     Ak nk1 !I2 (βAk , Aκ1 , Aκ2 , nk1 ) + Bk nk2 !I2 (βBk , Aκ1 , Aκ2 , nk2 )

K

Ps =

(9)

κ∈P2 (2) k=1

q(I)  π

Ak nk1 !2 F1

nk1

+ 1,

1 k 2 ; n1

+

βAk 3 2 ; βAk +A1



K

+

q(J)  π

Bk nk2 !2 F1

nk2

+ 1,

1 k 2 ; n2

+

βBk 3 2 ; βBk +A2



−1 nk1 +1 −1 k k=1 (βBk + A2 )n2 +1 B 12 , nk2 + 32 βAk + A1 B 12 , nk1 + 32   βAk βBk 1 3 1 3 k k k k n n F + 1, ; n + ; F + 1, ; n + ; K K 2 1 2 1 1 1 2 2 2 2 βAk +A1 2 2 βBk +A2 q(I)  q(J)  + Ak nk1 ! Bk nk2 ! −1 + k  −1 n2 +1 π π k k=1 k=1 1 3 k (βAk + A2 )n1 +1 B 12 , nk1 + 32 βBk + A2 B 2 , n2 + 2  Aκ2 F1 1, 1, n1k + 1, n1 k + 52 ; 12 ; βA +2A K K κ2 k q(I)q(J)   A q(I)q(J)   Aκ κ − Bk nk1 ! 1 − Bk nk2 ! 1 k   n2 +1 −1 π Aκ2 π Aκ2 κ∈P2 (2) k=1 κ∈P2 (2) k=1 βAk + 2Aκ2 B nk1 + 12 , 1 k=1



 −nk2 −1  Aκ2 1 5 1 k k k ×B n2 + , 1 βBk + 2Aκ2 F1 1, 1, n2 + 1, n2 + ; ; 2 2 2 βBk + 2Aκ2

(12)

K

Ps

=

 2q(I)   Ak nk1 !I1 (βAk , 3/(2M − 2), nk1 ) + Bk nk2 !I1 (βBk , 3/(2M − 2), nk2 ) π

+

 2q(J)   Ak nk1 !I1 (βAk , 3/(2M − 2), nk1 ) + Bk nk2 !I1 (βBk , 3/(2M − 2), nk2 ) π

k=1 K

(13)

k=1

−

K  2q(I)q(J)   Ak nk1 !I2 (βAk , 3/(2M − 2), 3/(2M − 2), nk1 ) + Bk nk2 !I2 (βBk , 3/(2M − 2), 3/(2M − 2), nk2 ) π κ∈P2 (2) k=1

the next page. The SEP expression given in (12) involves finite summations of functions which can be efficiently evaluated using popular mathematical software such as MATLAB, MATHEMATICA and MAPLE. It is worthwhile to mention that when in-phase and quadrature distance between dI and dJ are equal, i.e., ζ = 1 then we have A1 = A2 = g(I, J, ζ) = 3/ (I 2 − 1) + (J 2 − 1) = A yielding a simplified expression for a particular case of (12). For square M -ary √ M implies QAM with √ dI = dJ ,√we have I = J = that g(I = M , J = M , ζ) = 3/(2M − 2). Substituting appropriate parameters into (9) yields SEP for square QAM shown in (13) at the bottom of the next page. IV. N UMERICAL R ESULTS AND D ISCUSSION In this section, we provide few numerical results to illustrate mathematical analysis in previous section. The analytical results are validated with Monte-Carlo simulations. A general case of Nakagami-m fading channels is considered with similar/dissimilar fading parameters. A selection DF relay

networks with three relays (K = 3) is assumed for two cases configured as: 1) Similar case: {ΩSRk }3k=1 = {ΩRk D }3k=1 = 3 and {m1k }3k=1 = {m2k }3k=1 = 2 and 2) Dissimilar case: {ΩSRk }3k=1 = 1, 2, 3, {ΩRk D }3k=1 = 3, 2, 1 and {m1k }3k=1 = 1, 2, 3, {m2k }3k=1 = 3, 2, 1. Fig. 1 shows the SEP of 8 × 4 QAM with ζ = 21/5, computed using (12), versus average SNR. As can be observed from this figure, analytical results are in good agreement with those of Monte-Carlo simulations. Furthermore, to quantify the effect of in-phase and quadrature decision distance of QAM, i.e., ζ, on the SEP performance, we present  numerical results of SEP with different values of ζ = 1, 30/7, and 30/7 in Fig. 2. It is evident that the SEP performance improves with the decrease of ζ i.e., Square QAM performs the  best. At moderate-to-large SNR regime, the cases with ζ = 30/7 and ζ = 30/7 lose 0.3 dB and 2 dB respectively relative to the the case of ζ = 1. V. C ONCLUSIONS In this paper, closed-form expressions of SEP with arbitrary M -ary rectangular QAM for selection DF relay networks over

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[4] T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick, “On the performance of selection decode-and-forward relay networks over Nakagami-m fading channels,” IEEE Commun. Lett., vol. 13, no. 3, pp. 172–174, Mar. 2009. [5] N. C. Beaulieu, “A useful integral for wireless communication theory and its application to rectangular signaling constellation error rates,” IEEE Trans. Commun., vol. 54, no. 5, pp. 802–805, May 2006. [6] A. Maaref and S. A¨ıssa, “Exact error probability analysis of rectangular QAM for single- and multichannel reception in Nakagami-m fading channels,” IEEE Trans. Commun., vol. 57, no. 1, pp. 214–221, Jan. 2009. [7] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.

Fig. 1. Symbol error probability of selection DF relay networks in Nakagamim fading channels versus SNR.

Fig. 2. Symbol error probability of selection DF relay networks in Nakagamim fading channels versus SNR.

i.n.i.d Nakagami-m fading channels are derived. The analytical expressions are in terms of finite summation and hypergeometric series which can be efficiently computed in standard software packages. Special cases can be easily derived, square QAM for example. R EFERENCES [1] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 3062–3080, Dec. 2004. [2] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple cooperative diversity method basedon network path selection,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 659–672, Mar. 2006. [3] T. Q. Duong and V. N. Q. Bao, “Performance analysis of selection decodeand-forward relay networks,” IET Electronics Letters, vol. 44, no. 20, pp. 1206–1207, Sep. 2008.

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