Low Complexity Decoding in DF MIMO Relaying System - IITd

Manuscript revised for IEEE Transactions on Vehicular Technology, AUGUST 24, 2012

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Low Complexity Decoding in DF MIMO Relaying System Ankur Bansal, Member, IEEE, Manav R. Bhatnagar† , Member, IEEE, Are Hjørungnes, Senior Member, IEEE, and Zhu Han, Senior Member, IEEE

Abstract We derive a maximum-likelihood (ML) decoder for decode-and-forward (DF) based multiple-input multipleoutput (MIMO) cooperative system having equal number of antennas at the source and the relay node, and utilizing an arbitrary complex-valued M -ary constellation. The DF based MIMO cooperative system considered in this paper utilizes orthogonal space-time block codes (OSTBCs) for the transmission of data of the source to the destination. In order to reduce the decoding complexity, a sub-optimal piece-wise linear (PL) decoder is also derived, which performs close to the ML decoder. The proposed ML and PL decoders are applicable to arbitrary complex-valued M -ary constellations and require the destination node to possess knowledge of the channel statistics of the source-relay links; whereas, the existing decoder of multi-antenna based DF cooperative system needs to know the exact channel coefficients of the source-relay link at the destination. The proposed decoders outperform an amplify-and-forward (AF) protocol based multi-antenna cooperative system. We obtain an expression of the average probability of error of the proposed PL decoder using M -PSK constellation and a single MIMO relay. The approximate symbol error rate (SER) of the proposed PL decoder is derived with multiple MIMO relays, M -PSK constellation, and asymptotically high signal-to-noise ratio (SNR) of the sourcerelay links. It is analytically shown that the proposed ML and PL decoders achieve maximum possible diversity in the multi-antenna cooperative system with a single relay.

I. I NTRODUCTION A multiple-input multiple-output (MIMO) communication system can be realized by installing multiple antennas at the transmitter and receiver. The multi-antenna systems provide benefits like Ankur Bansal and Manav R. Bhatnagar are with Department of Electrical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India. (E-mail: [email protected], [email protected]). Are Hjørungnes was with UNIK - University Graduate Center, University od Oslo, NO-2027, Kjeller, Norway. (E-mail: [email protected]). Zhu Han is with Electrical and Computer Engineering Department, University of Houston, Houston, TX, USA 77004. (E-mail: [email protected]). †

Corresponding author.

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diversity gain and improved capacity over single-input single-output (SISO) systems [1], [2]. It is shown in literature [3]–[5] that cooperation among users or nodes in a communication system can be used to realize a distributed MIMO system. A relaying node can either decode-and-forward (DF) or amplify-and-forward (AF) [6] the data of the source. In the simplest form of the DF protocol, the relay can restrict itself to a low complexity operation, in which the relay merely demodulates and forwards the uncoded data of the source in a symbol-by-symbol manner [7], [8]. However, in a complex form of the DF based cooperative communication, the source transmits data by using a forward error correction (FEC) code, and the relay decodes the source transmission and re-encodes it through an FEC code before forwarding it to the destination [9], [10]. In AF protocols, the relaying node scales the received data before transmitting it to the destination node in order to satisfy a power constraint over the total transmit power. The destination requires knowledge of all links involved in the cooperation for decoding the data of the source. Whereas, in the DF protocol, the destination requires knowledge of the channel gains of the source-destination and relay-destination links for decoding the data of the source. Since the relay cannot decode the data perfectly, erroneous relaying causes significant error floor in the performance of the destination receiver in the DF protocol. Therefore, the AF based cooperation has been explored in much more detail as compared to the DF based systems [11], [12]. In [7], [13], a DF based coherent single antenna based cooperative system is considered by utilizing symbol-by-symbol demodulation and forwarding of uncoded data of the source by a single relay. It is shown in [7] that by using a maximum-likelihood (ML) decoder in the destination receiver, the performance can be improved for the uncoded binary data based DF cooperative communication system with a single pair of source and destination nodes and a single relay. The ML decoder is obtained by considering the possibility of erroneous transmissions by the relay terminal and by maximizing the probability density function (p.d.f.) of the data received at the destination [7]. In [13], the decoders of M -PAM and M 2 -QAM constellations in the DF based cooperative system are obtained to achieve maximum possible diversity. An ML decoder and a low complexity piecewise linear (PL) decoder of the DF based single antenna cooperative system are derived in [14], which require the average error probability of source-relay links for the decoding of the source data at the destination and are applicable to arbitrary complex-valued constellation. Also, the conditional SER is derived for the low complexity decoder in [14]. Further, the diversity analysis of the DF based low complexity decoder is given in [15] for the SISO cooperative relaying system.

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An ML decoder for binary signaling in a DF-based cooperative communication system is considered in [16], where each node contains multiple antennas. The multiple antenna based set-up considered in [16] is an extension of the single antenna set-up of [7], where multiple relays utilize orthogonal transmissions. In [17, Eq. (4)], an approximate ML decoder of multiple antenna based uncoded DF cooperative system is obtained by utilizing the exact channel state information (CSI) of the source-relay, relay-destination, and source-destination links for a complex-valued M -point constellation. Moreover, pairwise error probability (PEP) analysis of the multiple antenna based DF cooperative system is also performed in [17]. In order to achieve capacity advantages and better coverage, cooperative relay protocols can be extended to heterogeneous networks. In [18], hierarchical cooperative relay based heterogeneous networks are presented, where cost effective and improved coverage is obtained using cooperative relay nodes in heterogeneous radio networks. Moreover, network coding can be used in multi-user cooperative networks to overcome the inter-user interference. In [19], it is shown that network coding based AF protocol provides significant improvement in the overall throughput of the cooperative system. Similarly, the network coding based DF protocol can be applied to a multiple source/destination based cooperative system to improve the overall data rate. In this paper, the DF based MIMO cooperative relaying system using orthogonal space-time block code (OSTBC) is presented for the transmission of the data of the source to the destination. We have considered equal number of antennas at the source and the relay node throughout the analysis presented in the paper. Our main contributions in this paper are as follows: 1) We derive an ML decoder for the multiple antenna based DF cooperative system utilizing an arbitrary complex-valued M -point constellation, which does not require knowledge of the channel gains of the source-relay links. 2) A sub-optimal PL decoder of the complex-valued unitary and non-unitary constellations is also derived, which significantly reduces the computational complexity in decoding of the data in the multiple antenna based DF cooperative system. 3) A closed-form expression of the average symbol error rate (SER) of the proposed PL decoder is obtained with M -PSK constellation. 4) It is analytically shown that the proposed PL decoder achieves maximum possible diversity in the DF cooperative system with a single MIMO relay. 5) We derive an expression of the SER of the DF cooperative system with N orthogonal MIMO relays under the asymptotic condition of error free source-relay links with M -PSK constellation.

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The main differences between this paper and [15] are as follows. 1) A very simple cooperative set-up with single antenna nodes is considered in [15]; whereas, in this paper, we consider a multiple antenna based DF cooperative system which utilizes OSTBC for transmission and relaying of the data of the source. 2) In [15], the conditional SER of a low complexity PL decoder is derived for a single antenna based cooperative system; whereas, a closed form expression of the average SER of a multiple antenna based cooperative relaying system with OSTBCs is derived in this paper. 3) We also derive an asymptotic SER of the DF based MIMO cooperative system with N orthogonal MIMO relays assuming error free source-relay links; however, no such analysis is given in [15]. 4) For the diversity analysis, we derive fresh upper bounds of the SER of the multiple antenna based cooperative system, which are different from those given in [15]. The rest of this paper is organized as follows: In Section II, the system model of the DF based MIMO cooperative system is introduced. The ML and PL decoders of the DF MIMO relay system utilizing OSTBCs are derived in Section III, and a closed form expression of the average SER of the proposed PL decoder using M -PSK constellation is obtained in Section IV. The diversity analysis of the proposed PL decoder with a single MIMO relay is given in Section V. In Section VI, asymptotic performance analysis of the PL decoder of DF based multi-antenna cooperative system having multiple MIMO relays is performed assuming that the S-R links are error free. The simulation results are discussed in Section VII. Section VIII concludes the article. The article contains an appendix. Notations: CN (0, σ 2 ) denotes the circular symmetric complex Gaussian distribution with zero mean and σ 2 variance; z ∗ , |z|, and Re{z} denote the complex conjugate, absolute value, and real part, respectively, of a complex number z; [·]T denotes the transpose of a matrix; Diag [·] represents a diagonal matrix, |Q| denotes the determinant of a matrix Q, and QH denotes the conjugate transpose of a complex matrix Q, q represents a vector, and q and Q denote scalar quantities. II. S YSTEM M ODEL We consider a multiple antenna based cooperative communication system consisting of a single source-destination pair and one relay as shown in Fig. 1. The source (S), the relay (R), and the destination (D) are equipped with Ns , Nr , and Nd antennas, respectively. The OSTBC matrices are used for transmission of the data in the multiple antenna environment. The transmission of OSTBC data from S to D is performed in two orthogonal phases. In the first phase, an OSTBC is broadcasted by S over S-R and S-D links. In the second phase, R decodes the transmitted symbols, encodes these August 24, 2012

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symbols into an OSTBC (having the same structure as used by S), and transmits it over the R-D links. The source remains silent in this phase. Let Z be an Ns × T code matrix broadcasted by the source in the first phase of data transmission, where T is the number of time slots used for the transmission. The received data at the relay and the destination in a block will be Y0 = H0 Z + E0 , Y1 = H1 Z + E1 , [ ] [ ] where H0 = h0,i,j Nr ×Ns and H1 = h1,m,j N

d ×Ns

(1)

represent the complex-valued random channel

matrices of the S-R and S-D links, respectively. The channel gain between the i-th, i = 1, .., Nr , receive antenna of R and the j-th, j = 1, 2, .., Ns , transmit antenna of S is represented by h0,i,j , whereas, h1,m,j represents the channel gain between the m-th, m = 1, 2, .., Nd , receive antenna of D and the j-th transmit antenna of S. The matrices E0 ∈ CNr ×T and E1 ∈ CNd ×T represent the additive white Gaussian noise (AWGN) matrices of S-R and S-D links, respectively. It is assumed that the elements of the channel matrices H0 and H1 , i.e., h0,i,j ∼ CN (0, Ω0 ), and h1,m,j ∼ CN (0, Ω1 ), respectively, are independent and identically distributed (i.i.d.). The elements of the noise matrices E0 and E1 are also complex Gaussian random variables with zero mean and variances σ02 and σ12 , respectively. ˆ be the Nr × T OSTBC matrix forwarded by the relay after employing coherent decoding of Let Z Z. The received data at the destination in second phase will be ˆ + E2 , Y2 = H2 Z [ ] where H2 = h2,m,i N

d ×Nr

(2)

is the channel matrix of the R-D link containing complex-valued i.i.d.

Gaussian random variables h2,m,i ∼ CN (0, Ω2 ) and E2 ∈ CNd ×T contains the AWGN noise with zero mean and σ22 variance. All MIMO channels are assumed to be Rayleigh block fading, which remain constant over the transmission of the OSTBC matrices. The system model explained here is considered throughout the paper unless otherwise stated. III. ML

AND

PL D ECODERS IN DF MIMO R ELAY S YSTEM

In this section, we derive an ML decoder of the DF based MIMO cooperative system using OSTBCs for an arbitrary complex-valued M -ary constellation. A sub-optimal PL decoder is also derived for this cooperative set-up, which has significantly lower decoding complexity as compared to the ML August 24, 2012

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decoder. For simplicity, let us consider an OSTBC based MIMO cooperative system in which the source and the relay have equal number of antennas, i.e., Ns = Nr = K, and the destination has an arbitrary number of receive antennas. Let the OSTBC transmitted by S (i.e., Z) contains J data symbols {z1 , z2 , ..., zJ } ∈ AJ , where A is an arbitrary complex-valued M -ary constellation. The rate of the OSTBC Z will be J/T bits per second per hertz (b/s/Hz). Full rate orthogonal design is available for two transmit antennas only. The rate of other OSTBCs available for more than two transmit antennas is less than 1 b/s/Hz. A. ML Decoder in the Relay If R has perfect knowledge of the channel gains of the S-R links, the transmitted symbols can be decoded by applying the ML decoder as follows: ( K ) ∑ zˆk = arg max Re{w0,i,k z˜∗ } , z˜∈A

(3)

i=1

where w0,i,k , k = 1, ..., J is an intermediate variable corresponding to the decoupled decoding of symbol zk transmitted by the source. We can obtain the variable w0,i,k , k = 1, ..., J as follows. By using the property of complex orthogonal designs [20], we can rewrite the first relation in (1) as        y0,1 A0,1 z1 q0,1                y0,2   A0,2   z2   q0,2   =   +   (4)  ..   ..   ..   ..  ,  .   .  .   .         y0,K A0,K zJ q0,K where y0,i , i = 1, 2, ..., K represents the T ×1 vector containing the signals received in the i-th receive antenna in R, q0,i is the T × 1 AWGN noise vector containing the elements with zero mean and σ02 variance. The matrix A0,i denotes the T × J orthogonal matrix which contains elements of the i-th row of H0 and can be uniquely obtained for any complex OSTBC [20]. For example, if S utilizes a 2 × 2 Alamouti OSTBC for transmission of two data symbols z1 and z2 , then the orthogonal matrix A0,i , i = 1, 2 will be

A0,i =  and for a 3 × 4 OSTBC given as

 z  1  Z =  z2  z3

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

 h0,i,1

h0,i,2

h∗0,i,2 −h∗0,i,1

, 

z3∗

0

z1∗

0

−z3∗

0

−z1∗

z2∗

−z2∗

(5)

  , 

(6)

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the corresponding orthogonal matrix A0,i , i = 1, 2, 3 will be   h0,i,2 h0,i,3   h0,i,1  ∗   h0,i,2 −h∗0,i,1 0  . A0,i =     −h∗0,i,3 0 h∗0,i,1    ∗ ∗ 0 h0,i,3 −h0,i,2

(7)

Similarly, we can obtain the orthogonal matrix A0,i for an arbitrary complex OSTBC. It can be 2 seen from (5) and (7) that A0,i is proportional to the unitary matrix as AH 0,i A0,i = ∥a0,i ∥F IJ , where

a0,i = [h0,i,1 , h0,i,2 , ..., h0,i,K ], ∥ · ∥F represents the Frobenius norm, and IJ denotes the J × J identity matrix. We can now find the intermediate variable w0,i,k , k = 1, ..., J, at the i-th receiving antenna in the relay as

w0,i = AH 0,i y0,i ,

(8)

where w0,i = [w0,i,1 , w0,i,2 , ..., w0,i,J ]T , y0,i , i = 1, 2, ..., K is the received signal vector at the i-th receiving antenna in R and can be independently obtained from (4) as y0,i = A0,i z + q0,i , where z = [z1 , z2 , ..., zJ ]T . It can be deduced from (4) and (8) that w0,i,k ∼ CN (µ0,i zk , µ0,i σ02 ), where µ0,i = ∥a0,i ∥2F . Therefore, the variables obtained in (8) can be used to find the decoded symbols zˆk , k = 1, ..., J in R by employing the ML decoder in (3). Further, R encodes these decoded symbols into a K × T orthogonal design and forwards to D in the second phase of transmission. B. ML Decoder in the Destination The destination contains the signals received from the source and relay in the two orthogonal phases. Following the similar approach as given in Subsection III-A, we can find the intermediate variables required for decoding of the data symbols zk , k = 1, ..., J, at the m-th receiving antenna in D as wℓ,m = AH ℓ,m yℓ,m ,

(9)

where wℓ,m = [wℓ,m,1 , wℓ,m,2 , ..., wℓ,m,J ]T . The suffix ℓ = 1, 2 denotes the S-D and the R-D links, respectively. The orthogonal matrix Aℓ,m , ℓ = 1, 2 contains the elements of m-th row of the channel matrix Hℓ . It can be concluded from the examples given in Subsection III-A that the orthogonal matrix Aℓ,m can be obtained uniquely for any complex OSTBC and satisfies the property AH ℓ,m Aℓ,m = ∥aℓ,m ∥2F IJ , where aℓ,m = [hℓ,m,1 , hℓ,m,2 , ..., hℓ,m,K ]. The vectors y1,m and y2,m contain the signals received at the m-th receiving antenna in the destination over S-D and R-D links, respectively, and can be calculated as August 24, 2012

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y1,m = A1,m z + q1,m , y2,m = A2,m zˆ + q2,m ,

(10)

where zˆ = [ˆ z1 , zˆ2 , ..., zˆJ ]T , q1,m and q2,m contain zero mean AWGN noise elements with σ12 and σ22 variances, respectively. Moreover, we can see from (9) and (10) that the intermediate variables w1,m,k ∼ CN (µ1,m zk , µ1,m σ12 ), and w2,m,k ∼ CN (µ2,m zˆk , µ2,m σ22 ), where µℓ,m = ∥aℓ,m ∥2F , ℓ = 1, 2. Now we will derive an ML decoder in the destination of a multiple antenna based DF cooperative system that requires calculation of a log-likelihood ratio (LLR) for decoding of the symbol transmitted by S. The ML decoder maximizes the joint p.d.f. of the data received in the destination in two orthogonal phases. The ML decoding rule corresponding to the decoding of symbol zk , k = 1, 2, ..., J, can be given as

( ) zkd = arg max pw|H1 ,H2 ,zk ∈A,ˆzk ∈A , zk ∈A

(11)

where zkd denotes the decoded symbol at the destination corresponding to the symbol zk transmitted by the source, w = [w1 , w2 ] ∈ C1×2Nd , with w1 = [w1,m,k ]1×Nd and w2 = [w2,m,k ]1×Nd containing the intermediate variables corresponding to symbol zk , obtained from the signals received over the S-D and the R-D links, respectively, as given in (9). It can be shown, from the analysis given in [21, Section 2.3], that the destination needs to calculate the following LLR for ML decoding of a complex (

symbol zk : Λdp,q

= ln

pw|H1 ,H2 ,zk =xp ,ˆzk ∈A pw|H1 ,H2 ,zk =xq ,ˆzk ∈A

) ,

(12)

where xp , xq ∈ A, p, q = 1, 2, ..., M, p ̸= q, and pw|H1 ,H2 ,zk =xp ,ˆzk ∈A denotes the conditional joint p.d.f. of w given that the channel matrices H1 , H2 , and the symbols transmitted by the source (zk ) and the relay (ˆ zk ) are perfectly known in the destination. For example, let A = {x1 , x2 , x3 , x4 } be a complex-valued 4-point constellation. The destination will take a decision corresponding to a symbol zk ∈ A, transmitted from the source, by    x1 , if      x , if 2 zkd =   x3 , if      x4 , if

following the LLR of (12) as Λd1,2 > 0, Λd1,3 > 0, and Λd1,4 > 0, Λd2,1 > 0, Λd2,3 > 0, and Λd2,4 > 0, Λd3,1 > 0, Λd3,2 > 0, and Λd3,4 > 0,

(13)

Λd4,1 > 0, Λd4,2 > 0, and Λd4,3 > 0.

Since the AWGN noises in the S-D and R-D links are independent of each other, we can write pw|H1 ,H2 ,zk =xp ,ˆzk ∈A = pw1 |H1 ,zk =xp pw2 |H2 ,zk =xp ,ˆzk ∈A . August 24, 2012

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Assuming the elements of w1 to be independent, we can write the conditional p.d.f. pw1 |H1 ,zk =xp as follows: pw1 |H1 ,zk =xp

( N ) d ∑ 1 |w1,m,k − µ1,m xp |2 = N exp − , π d |K1 | µ1,m σ12 m=1

(15)

where K1 = σ12Nd Diag[µ1,1 , µ1,2 , ..., µ1,Nd ] is the covariance matrix of w1 . If ϵ is the average probability of error of the S-R links for an arbitrary complex-valued M -point constellation, then we can write pw2 |H2 ,zk =xp ,ˆzk ∈A = ϵpw2 |H2 ,ˆzk ̸=xp + (1 − ϵ)pw2 |H2 ,ˆzk =xp .

(16)

The conditional p.d.f. pw2 |H2 ,ˆzk =xp can also be written similar to (15) as ( N ) 2 d ∑ |w2,m,k − µ2,m xp | 1 pw2 |H2 ,ˆzk =xp = N exp − , d π |K2 | µ2,m σ22 m=1

(17)

where K2 = σ22Nd Diag[µ2,1 , µ2,2 , ..., µ2,Nd ] denotes the covariance matrix of w2 . From [22, Section III], it follows that pw2 |H2 ,ˆzk ̸=xp represents the p.d.f. of a Gaussian mixture random variable given by ( N ) M d ∑ ∑ 1 |w2,m,k − µ2,m xs |2 pw2 |H2 ,ˆzk ̸=xp = N exp − . (18) π d |K2 |(M − 1) s=1 µ2,m σ22 m=1 s̸=p

From (14)- (18), the LLR in (12) will get simplified into ] Nd [ ∑ ) µ1,m ( 2 2 d 2 ∗ Λp,q = |xq | − |xp | + 2 Re{w1,m,k (xp − xq ) } 2 σ σ1 1 m=1  ( ( ) ) N N ∑d µ2,m ∑d µ2,m 2 − 2 Re{w ∗} 2 − 2 Re{w ∗} M − |x | x − |x | x ∑ s p 2,m,k 2,m,k s p 2 2 2 2 σ2 σ2 σ2 m=1  ϵ +(1 − ϵ)e m=1 σ2  M −1 s=1 e  s̸=p + ln ( ) ) ( N N  ∑d µ2,m ∑d µ2,m 2 − 2 Re{w ∗} 2 − 2 Re{w ∗} M −  |x | x − |x | x ∑ u q 2,m,k u 2,m,k q 2 2 2 2 σ2 σ2  ϵ e m=1 σ2 +(1 − ϵ)e m=1 σ2 M −1

    .   

(19)

u=1 u̸=q

Thus, the ML decoding rule in the destination for arbitrary M -point constellation is given in the form of LLR by (19).

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C. Low Complexity PL Decoder The LLR of (19) can be rewritten as ] Nd [ ∑ ) µ1,m ( 2 2 d 2 ∗ Λp,q = |xq | − |xp | + 2 Re{w1,m,k (xp − xq ) } σ12 σ1 m=1  ) N ( N ( ∑d

−

µ2,m 2 2 ∗ 2 |xq | − σ 2 Re{w2,m,k xq } σ1 2

−

∑d

)

µ2,m 2 2 ∗ 2 |xp | − σ 2 Re{w2,m,k xp } σ2 2

 P + ϵ e m=1 +(1 − ϵ)e m=1 M −1  + ln ( ) ( ) N N ∑d µ2,m ∑d µ2,m  2 − 2 Re{w ∗} 2 − 2 Re{w ∗} − |x | x − |x | x p q 2,m,k 2,m,k p q 2 2 2 2 σ2 σ2 P + Mϵ−1 e m=1 σ2 +(1 − ϵ)e m=1 σ2 where P =

ϵ M −1

M ∑

e

−

N ∑d

(

m=1

µ2,m 2 2 ∗ 2 |xs | − σ 2 Re{w2,m,k xs } σ2 2

   , 

(20)

)

. An approximate LLR decoder can be obtained

s=1 s̸=p,q

from (20) by neglecting P and after some algebraic manipulations as Λdp,q ≈ t0 + ψ(t1 ), (21) ] Nd [ Nd [ ∑ ∑ µ1,m µ2,m 2 2 2 ∗ where t0 = (|x | −|x | )+ Re{w (x −x ) } , t = (|xq |2−|xp |2 )+ σ22 Re{w2,m,k (xp q p 1,m,k p q 1 σ12 σ12 σ22 2 m=1 ] m=1 ( ) ϵ+(M −1)(1−ϵ)et1 ∗ −xq ) } , and ψ(t1 ) , ln ϵet1 +(M −1)(1−ϵ) . It follows from [14] that we can approximate ψ(t1 ) by a PL function as follows: ψP L (t1 ) ≈

   −T1 , if      

t1 ,

if

T1 , if

t1 < −T1 , −T1 ≤ t1 ≤ T1 ,

(22)

t 1 > T1 ,

where T1 = ± ln [(M − 1)(1 − ϵ)/ϵ]. The mapping function ψ(t1 ) essentially clips its input to the values ±T1 and is approximately linear between these extreme values for small values of t1 . Hence, we can rewrite (21) as Λdp,q ≈ t0 + ψP L (t1 ).

(23)

The approximate low complexity PL decoder can be implemented using (23). Remark 1. The intuitive reason for working of the proposed PL decoder is as follows. Since the relay assumes all the symbols of an M -ary constellation to be equiprobable, the average probability of conceding an error in the relay is ϵ/(M − 1). Therefore, while calculating the approximate LLR for symbol pair {xp , xq } in the destination in (21), the proposed PL decoder only considers the error in decoding of xp as xq in the relay. Hence, the PL decoder in the destination neglects the remaining M − 2 possibilities (other than xq ) of the error in the relay, which are included in the term P. August 24, 2012

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Remark 2. Since the destination node possesses the exact information of channel gains of the S-D and R-D links, and average probability of error of the S-R link, we can define the PL decoder as a module which considers the received signals over the S-D and R-D links as inputs and outputs the detected transmitted symbol. Remark 3. It follows from (19) and (23) that the proposed ML and PL decoders do not require the destination to have perfect knowledge of channel gains of the S-R links contrary to the decoder of [17]. Since the channel statistics vary far more slower than the instantaneous channel coefficients, it is practical that the relay can estimate the channel variance (rather than the instantaneous channel coefficients) of the S-R channel and forward the estimation to the destination. D. Computational Complexity of the ML and PL decoders In addition, we have found the computational complexity of the ML and PL decoders by manually calculating and summing the total number of real additions, real multiplications, logarithms and exponentials required for evaluating the LLR of (19) and (23), respectively. The variables µℓ,m , wℓ,m,k , ℓ = 1, 2 in the LLR of (19) and (23) depend on K, and both ML and PL decoders require exactly the same number of computations in the destination to calculate these variables. So, for the comparative analysis of decoding complexity of the ML and PL decoders, we assume that these variables are known as a real or complex constant in the destination. Therefore, we have not incorporated the computations required to calculate these variables in the overall computational complexity of the decoders. Hence, the complexity analysis given here is valid for any value of K. The decoding complexity of the proposed ML and PL decoders using 8-PSK constellation, a single relay, and Nd = 2 is compared in Table I. It can be observed from Table I that the proposed PL decoder requires a significantly less number of computations as compared to the ML decoder (19). For an arbitrary M -PSK constellation, a single MIMO relay, and Nd antennas at the destination, the proposed ML decoder requires (8Nd − 1)M 2 + 5Nd M + 17M/2 real additions, 5Nd M 2 + 9Nd M/2 + 23M/2 real multiplications, M/2 logarithms, and M 2 exponential operations to calculate the LLR in (19) for decoding an M -PSK symbol. Therefore, the overall computational complexity of the ML decoder (CM L ) is obtained by summing all required mathematical operations as, CM L = (26Nd M + 19Nd + 41)M/2. Similarly, we find that the overall computational complexity of the proposed PL decoder can be given as CP L = (17Nd − 1)M + 1. It can be seen from the expressions of CM L and CP L that the August 24, 2012

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12

computational complexity of ML decoder grows exponentially; whereas, that of the PL decoder grows linearly with the constellation size. IV. P ERFORMANCE A NALYSIS OF THE PL D ECODER In this section, we derive an expression of approximate average SER of the proposed PL decoder utilizing an arbitrary OSTBC and M -PSK constellation. It can be noticed from Section III that the proposed PL decoder decodes the transmitted OSTBC of the source in a symbol-wise manner. Therefore, the average probability of error of the proposed PL decoder can be obtained by deriving the average probability of error in decoding of a symbol encoded in the OSTBC matrix. Let A be a complex-valued M -PSK constellation and xp ∈ A, p = 1, 2, ..., M , be a symbol transmitted by the source. Consider that the destination wrongly decodes the transmitted symbol xp as another symbol xq ∈ A, q = 1, 2, ..., M , q ̸= p. Therefore, the average PEP of the proposed PL decoder given in (23) can be expressed by Pe = Pr{t0 −T1 < 0|t1 < −T1 , zk = xp }Pr{t1 < −T1 |zk = xp } + Pr{t0 +T1 < 0|t1 > T1 , zk = xp }Pr{t1 > T1 |zk = xp } + Pr{t0 +t1 < 0, −T1 ≤ t1 ≤ T1 |zk = xp }, (24) where Pr{·} represents the probability. Since the relay can decode the data of the source erroneously, we can write Pr{t1 < −T1 |zk = xp } = (1 − ϵ)Pr{t1 < −T1 , zk = xp , zˆk = xp }+ϵPr{t1 < −T1 , zk = xp , zˆk ̸= xp }, Pr{t1 > T1 |zk = xp } = (1−ϵ)Pr{t1 > T1 , zk = xp , zˆk = xp }+ϵPr{t1 > T1 , zk = xp , zˆk ̸= xp }.

(25)

The p.d.f. of a Hermitian quadratic random variable [23] t0 can be calculated for the Rayleigh fading MIMO channel as

 ( )D 1 D∑ D −1 1 −1 v D1 −λ−1 Lλ 1 (0)  b0 −a0 v  , v > 0, e  aa00+b (a0 +b0 )λ (D1 −λ−1)! 0 λ=0 pt0 (v) = ( )D 1 D∑ D −1 1 −1  (−v)D1 −λ−1 Lλ 1 (0) a0 b0   a0 +b0 eb0 v , v ≤ 0, λ (a0 +b0 ) (D1 −λ−1)!

(26)

λ=0

where D1 = KNd , LM λ (x) is the associated Laguerre polynomial [24], and )1/2 ( 4σ12 + 1, a0 = 1 + Ω1 |xp − xq |2 ( )1/2 4σ12 b0 = 1 + − 1. Ω1 |xp − xq |2 August 24, 2012

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13

Similarly, we can find the conditional p.d.f. (conditioned on decoding in the relay) of Hermitian quadratic random variable t1 . When the relay performs correct decoding, the p.d.f. of t1 will be  ( )D 1 D∑ 1 −1 D −1  a b uD1 −τ −1 Lτ 1 (0)  e−ap u , u > 0,  app+bpp (ap +bp )τ (D1 −τ −1)! τ =0 pt1 |zk =xp ,ˆzk =xp (u) = (28) )D 1 ( D∑ 1 −1 D −1  ap bp (−u)D1 −τ −1 Lτ 1 (0) bp u   ap +bp e , u ≤ 0, (ap +bp )τ (D1 −τ −1)! τ =0

( ap = 1 +

where

( bp = 1 +

4σ22 Ω2 |xp − xq |2

)1/2

4σ22 Ω2 |xp − xq |2

+ 1, )1/2 − 1.

(29)

When relay decodes erroneously, the conditional p.d.f. of t1 can be given by [22] 1 ∑ pt |z =x ,ˆz =x (u), M − 1 s=1 1 k p k s M

pt1 |zk =xp ,ˆzk ̸=xp (u) =

(30)

s̸=p

 ( )D1 D∑ 1 −1 D −1  uD1 −τ −1 Lτ 1 (0) bs −as u  , u > 0, e  aass+b (as +bs )τ (D1 −τ −1)! s τ =0 pt1 |zk =xp ,ˆzk =xs (u) = ( )D1 D∑ 1 −1 D −1  (−u)D1 −τ −1 Lτ 1 (0) bs bs u   aass+b u ≤ 0. e τ (D −τ −1)! , (a +b ) s s s 1

where

(31)

τ =0

We can calculate the terms as and bs corresponding to symbol xs , s ̸= p, as ( )1/2 4σ22 2 as = rs + − rs , Ω2 |xp − xq |2 ( )1/2 4σ22 2 bs = rs + + rs , Ω2 |xp − xq |2 where rs =

|xp |2 − |xq |2 − 2Re{xs (xp − xq )∗ } . |xp − xq |2

From the p.d.f. of t0 in (26), we have

(32)

(33)

)D1 D∑ 1 −1 b0 Γ(D1 − λ, a0 T1 )LλD1 −1 (0) Pr{t0 − T1 < 0|t1 < −T1 , zk = xp } = 1 − , b0 λ a0 + b0 (1 + ) (D − λ − 1)! 1 a λ=0 0 )D1 D∑ ( 1 −1 Γ(D1 − λ, b0 T1 )LλD1 −1 (0) a0 , (34) Pr{t0 + T1 < 0|t1 > T1 , zk = xp } = a0 + b0 (1 + ab00 )λ (D1 − λ − 1)! λ=0 ∫∞ where Γ(ω, ξ) is the upper incomplete Gamma function defined as Γ(ω, ξ) = ξ tω−1 e−t dt. Using the p.d.f. of (28), we get Pr{t1 < −T1 , zk = xp , zˆk = xp } =

August 24, 2012

(

(

ap ap + bp

)D1 D∑ 1 −1 τ =0

Γ(D1 − τ, bp T1 )LτD1 −1 (0) , (1 + abpp )τ (D1 − τ − 1)!

(35)

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Manuscript revised for IEEE Transactions on Vehicular Technology, AUGUST 24, 2012

and from (30), it follows that

1 ∑ Pr{t1 < −T1 , zk = xp , zˆk = ̸ xp } = M − 1 s=1 M

(

as as + bs

)D1 D∑ 1 −1

s̸=p

From (25), (35), and (36), we get

14

τ =0

Γ(D1 − τ, bs T1 )LτD1 −1 (0) . (1 + abss )τ (D1 − τ − 1)!

)D1 D∑ 1 −1 ap Γ(D1 − τ, bp T1 )LτD1 −1 (0) Pr{t1 < −T1 |zk = xp } = (1 − ϵ) a p + bp (1 + abpp )τ (D1 − τ − 1)! τ =0 )D1 D∑ M ( 1 −1 ϵ ∑ as Γ(D1 − τ, bs T1 )LτD1 −1 (0) + . as τ M − 1 s=1 as + bs (1 + ) (D − τ − 1)! 1 b s τ =0

(36)

(

(37)

s̸=p

Similarly, we can write

(

Pr{t1 > T1 |zk = xp } = (1 − ϵ) ϵ ∑ + M − 1 s=1 M

(

bp ap + bp

bs as + bs

)D1 D∑ 1 −1 τ =0

)D1 D∑ 1 −1 τ =0

s̸=p

Γ(D1 − τ, ap T1 )LτD1 −1 (0) (1 +

bp τ ) (D1 ap

− τ − 1)!

Γ(D1 − τ, as T1 )LτD1 −1 (0) . (1 + abss )τ (D1 − τ − 1)!

(38)

Using the fact that the relay can perform erroneous decoding, we can write the third term of (24) as Pr{t0 + t1 < 0, −T1 ≤ t1 ≤ T1 |zk = xp } , (1 − ϵ)Ip + ϵ I ′ p ,

(39)

where Ip = Pr{t0 + t1 < 0,−T1 ≤ t1 ≤ T1 |zk = xp , zˆk = xp } and I ′ p = Pr{t0 + t1 < 0,−T1 ≤ t1 ≤ T1 |zk = xp , zˆk ̸= xp }. After some algebra, we get Ip = I1,p + I2,p , where I1,p and I2,p are calculated in Appendix A. Using the conditional p.d.f. of t1 from (30), we get 1 ∑ Ip= (I1,s + I2,s ), M − 1 s=1 M

′

(40)

s̸=p

where I1,s and I2,s can be evaluated similar to the calculations given in Appendix A with change of parameters (ap , bp ) by (as , bs ) defined in (32). Therefore, the average PEP of the proposed PL decoder can be calculated by using (24), (34), (37), (38), and (39). Considering the equiprobability of M -PSK constellation points, and by using the nearest neighbor approach [25], it can be shown that the average probability of error of the proposed PL decoder for M -PSK constellation will be approximately equal to 2Pe , where Pe denotes the average PEP. However, it can be observed from Fig. 4 and 5 that the derived approximate average SER is very close to the exact SER.

August 24, 2012

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15

V. D IVERSITY A NALYSIS OF THE P ROPOSED PL D ECODER The diversity order of the proposed PL decoder of the DF based multi-antenna cooperative system with single erroneous relay is derived in this Section. Let us define the instantaneous SNR of the link △

between the j-th transmitting antenna of S and the i-th receiving antenna of R as γ0,i,j = |h0,i,j |2 /σ02 , instantaneous SNR of the link between the j-th transmitting antenna of S and the m-th receiving △

antenna of D as γ1,m,j = |h1,m,j |2 /σ12 , and the instantaneous SNR of the link between the i-th △

transmitting antenna of R and m-th receiving antenna of D by γ2,m,i = |h2,m,i |2 /σ22 . The average values of γ0,i,j , γ1,m,j , and γ2,m,i are γ¯0,i,j = Ω0 /σ02 , γ¯1,m,j = Ω1 /σ12 , and γ¯2,m,i = Ω2 /σ22 , respectively. Since all the channel coefficients are complex Gaussian distributed, the instantaneous SNRs γ0,i,j , γ1,m,j , and γ2,m,i are exponential distributed with average values γ¯0,i,j , γ¯1,m,j and γ¯2,m,i , respectively. Since diversity is a high SNR phenomenon, we assume that all links approach infinity with the same rate, i.e., γ¯0,i,j = γ¯1,m,j = γ¯2,m,i = γ¯ → ∞. If the channel gain matrices H1 and H2 are perfectly known at the destination, the conditional probability of error of the PL decoder can be expressed as Pe (H1 , H2 ) = Pr{t0 −T1 < 0|t1 < −T1 , zk = xp }Pr{t1 < −T1 |zk = xp } + Pr{t0 +T1 < 0|t1 > T1 , zk = xp }Pr{t1 > T1 |zk = xp } + Pr{t0 +t1 < 0,−T1 ≤ t1 ≤ T1 |zk = xp }.

(41)

It follows from the definition of t0 and (9) that for M -PSK constellation, if channel gains of all the links are known, variable t0 represents a real Gaussian distributed random variable with conditional p.d.f given as (v−Gp γ1 )2 1 − pt0 |H1 ,zk =xp (v) = √ e 2Jp γ1 , 2πJp γ1 △

where Gp = 2Re{x∗p (xp − xq )}, Jp = 2|xp − xq |2 , and γ1 =

Nd K ∑ ∑

(42)

γ1,m,j . Similarly, we can write the

j=1 m=1

conditional p.d.f.’s of t1 as (u−Gp γ2 )2 1 − pt1 |H2 ,ˆzk =xp (u) = √ e 2Jp γ2 , 2πJp γ2 M ∑ (u−Gs γ2 )2 1 − √ pt1 |H2 ,ˆzk ̸=xp (u) = e 2Jp γ2 , (M − 1) 2πJp γ2 s=1

(43)

s̸=p

△

where Gs = 2Re{x∗s (xp − xq )} and γ2 =

Nd K ∑ ∑

γ2,m,i . It can be noticed from the definition of γ1

i=1 m=1

and γ2 that we can obtain these random variables by summing KNd exponential random variables each having a mean γ¯ . It is well known that the sum of k, k ≥ 0, i.i.d. exponential random variables

August 24, 2012

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16

represents a Gamma distributed random variable. Therefore, we can write the p.d.f. of γ1 and γ2 as pγℓ (γℓ ) =

γ γℓD1 −1 − γ¯ℓ e , D Γ(D1 )¯ γ 1

for ℓ = 1, 2,

(44)

where D1 = KNd . From (41), the average pairwise probability of error of the PL decoder will be Pe = Eγ1 [Pr{t0 −T1 < 0|t1 T1 |zk = xp }] +Eγ1 ,γ2 [Pr{t0 +t1 < 0,−T1 ≤ t1 ≤ T1 |zk = xp }], ≈ 2F1 (¯ γ , T1 ) + F2 (¯ γ , T1 ),

(45)

where E[·] denotes the expectation. The terms F1 (¯ γ , T1 ) and F2 (¯ γ , T1 ) in (45) can be obtained by using the p.d.f.’s in (42), (43), and (44), and after some algebra as ( ) [ ∫ ∞ ∫ ∞ γ1 1 −T + G γ (1 − ϵ) 1 p 1 − γ¯ D1 −1 √ F1 (¯ γ , T1 ) = γ Q e dγ1 γ2D1 −1 D1 Γ(D1 )¯ γ D1 0 1 Γ(D )¯ γ Jp γ1 1 0 ) ) ( ( )] ( ∫ ∞ M ∑ γ γ T1 + Gp γ2 ϵ T1 + Gs γ2 − γ¯2 − γ¯2 D1 −1 √ √ ×Q e dγ2 + e dγ2 , γ2 Q (M − 1)Γ(D1 )¯ γ D1 s=1 Jp γ2 Jp γ2 0 s̸=p

[ △

=

(1) F1 (¯ γ , T1 )

(1 −

and

∫

F2 (¯ γ , T1 ) = k0 (1− ϵ)

(2) ϵ)F1 (¯ γ , T1 )

∞

γ1D1 −1 e−

γ1 γ ¯

0

k0 ϵ + M −1

∫

] ϵ (3) + F (¯ γ , T1 ) , (M − 1) 1

∫

[

∞

∞

γ1D1 −1 e−

γ1 γ ¯

∫

∞

γ2

γ2D1 −1 e− γ¯

0

0

) ] 2 p γ2 ) w+Gp γ1 − (w−G Q √ e 2Jp γ2 dw dγ2 dγ1 Jp γ1 −T1   ) M ∫ T1 ( 2 s γ2 ) 1 w+Gp γ1 ∑ − (w−G   e 2Jp γ2 dw dγ2 dγ1 , Q √ √ 2πJp γ2 −T1 Jp γ1 s=1

γ2D1 −1 e−

0

(46)

γ2 γ ¯

1 √ 2πJp γ2

∫

T1

(

s̸=p △

(1)

(2)

= F2 (¯ γ , T1 ) + F2 (¯ γ , T1 ),

(47)

where k0 = 1/[Γ2 (D1 )¯ γ 2D1 ]. Proposition 1. For a multichannel source-relay link with L channels and M -PSK constellation, the average error probability at high value of γ¯ is proportional to the γ¯ −L , where γ¯ is the average SNR of the channel. Proof: It can be shown from [26, Section 9.2.3] that the average error probability of a multichannel link having L channels and utilizing M -PSK constellation can be approximated as ∫ γ k0 ∞ √ Q(a γ)γ L−1 e− γ¯ dγ, ϵ≈ L γ¯ 0 August 24, 2012

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Manuscript revised for IEEE Transactions on Vehicular Technology, AUGUST 24, 2012

△

where k0 > 0, γ =

L ∑

17

γl , where γl represents the instantaneous SNR of the l-th channel of the link.

l=1

Using the Chernov bound and evaluating the integral for high value of γ¯ , we get ϵ ∝ γ¯ −L . It follows from Proposition 1 that for L = Ns Nr = K 2 and large values of γ¯ , ϵ ∝ γ¯ −K and 2

T1 ∝ ±K 2 ln γ¯ . Lemma 1. The integral of Q-function can always be upper bounded as ) ∫ ∞ ( ∫ ∞ (δ+c γ)2 γ γ 0 1 δ + c0 γ 1 − L−1 − γ¯ γ e dγ ≤ L Q √ e 2c1 γ γ L−1 e− γ¯ dγ, L γ¯ Γ(L) 0 c1 γ γ¯ Γ(L) 0

(49)

where δ, c0 , c1 > 0. 2 Proof: Lemma 1 can be proved by using the Chernov bound Q(x) ≤ e−x /2 , x ≥ 0. Lemma 2. If δ > 0, c0 = −c2 , c2 > 0, c1 > 0, and γ¯ → ∞, then ) ∫ ∞ ( γ 1 δ − c2 γ L−1 − γ¯ dγ ≈ 1 − φ(¯ γ ), (50) γ e Q √ γ¯ L Γ(L) 0 c1 γ )L ( where φ(¯ γ ) = 2c12c+c1 2 γ¯ . 2 Proof: On substituting γ = x¯ γ , using Q(−x) = 1 − Q(x), and evaluating the integral for γ¯ → ∞, we get (50). Corollary 1. If δ < 0, let δ = −β, β > 0. By using the series expansion of γ(L, β/c0 ) =

∞ ∑ n=0

(−1)n (β/¯ γ )L+n n!(L+n)cL+n 0

[27, Eq. (8.354.1)], Chernov bound, and some algebraic manipulations in (49), we get ) ∫ ∞ ( ∫ ∞ (β−c γ)2 ∞ ∑ γ γ 1 (−1)n (β/¯ γ )L+n −β + c0 γ 1 − 2c 0γ L−1 − γ¯ L−1 − γ¯ 1 dγ ≤ dγ. γ e + γ e Q e √ γ¯ L Γ(L) 0 c1 γ n!(L + n)cL+n Γ(L) γ¯ L Γ(L) 0 0 n=0 (51) Lemma 3. If δ ≫ 0, c1 > 0 and c0 is an arbitrary real-valued constant, then we have ∫ ∞ (δ−c γ)2 γ 0 1 ϕ1 (¯ γ , δ) − e c1 γ γ L−1 e− γ¯ dγ ≈ , L γ¯ Γ(L) 0 γ¯ L ∫ ∞ 1 − (δ−cc 0γγ)2 L−1 − γγ¯ 1 ϕ2 (¯ γ , δ) 1 , γ e dγ ≈ √ e L γ¯ Γ(L) 0 γ γ¯ L √ (  −L−1/2 [ )] where )L−1/2 √ 2 c2 √ ( 2c0 1 0+1 −2 δ c1 c1 c1 γ ¯ π δ  c0 + 1  ϕ1 (¯ γ , δ) = e , √ Γ(L) c1 c1 γ¯ [

)L−1 [ 2 ]−L/2 √ ( c0 1 π δ ϕ2 (¯ γ , δ) = + e √ Γ(L) c1 c1 γ¯ August 24, 2012

√ 2c0 −2 c1

( 1 c1

c2 0+1 c1 γ ¯

(52) (53)

(54)

)] δ

.

(55)

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18

Proof: This Lemma can be proved by applying [27, Eq. (3.471.9)] in the L.H.S. of (52) and (53), √ followed by the use of an approximation Kv (x) ≈ π/(2x) e−x , x ≫ 0 [27, Eq. (8.451.6)], where Kv (·) represents the modified Bessel function [28, Section 9.6.1]. Corollary 2. For c0 = 0, γ¯ → ∞, and using an approximation Kv (x) ≈ 12 Γ(v)( 12 x)−v , x → 0 [28, Eq. (9.6.9)], (52) reduces to

1 γ¯ L Γ(L)

∫

∞

2

− cδ γ

e

1

γ

γ L−1 e− γ¯ dγ ≈ 1.

(56)

0 ln γ ¯ ¯d γ ¯ →∞ γ

By using the L’Hôpital’s rule [28, Section 3.4.1], we can show that lim lim

γ ¯ →∞

(ln γ ¯ )ρ−1/2 γ ¯d

= d′ lim

√1 ¯ d ln γ ¯ γ ¯ →∞ γ

=

1 lim ¯1d d γ ¯ →∞ γ

= 0, and

= 0, where d, ρ, and d′ are positive constants. Using these observations

with the results of Lemmas 1, 3, and 2, Corollaries 1, 2, and Proposition 1 in (46) for high value of γ¯ , it can be deduced that

k1 + g1 (¯ γ ), γ ¯ →∞ γ¯ D1 k2 (2) lim F1 (¯ γ , T1 ) ∝ D1 + g2 (¯ γ ), γ ¯ →∞ γ¯ (1)

lim F1 (¯ γ , T1 ) ∝

(3)

lim F1 (¯ γ , T1 ) ∝ 1

γ ¯ →∞

(57)

where k1 , k2 > 0 are constants; g1 (¯ γ ) and g2 (¯ γ ) are functions containing summation terms, each decaying at rate higher than γ¯ −D1 for γ¯ → ∞. Similarly, it can also be shown that by using the L’Hôpital’s rule in (47), with the results of Proposition 1, (49)-(56), and the fact that Q(x) is a decaying function of x, we get (1)

lim F2 (¯ γ , T1 ) ∝

γ ¯ →∞

k3 + g3 (¯ γ ), γ¯ 2D1

(2)

lim F2 (¯ γ , T1 ) ∝ g4 (¯ γ ),

γ ¯ →∞

(58)

where k3 is a positive constant. The functions g3 (¯ γ ) and g4 (¯ γ ) contain summation terms, each decaying at rate more than γ¯ −2D1 for high values of γ¯ . It can be concluded from (45), (48), (57), and (58) that the average probability of error of the proposed PL decoder of multiple antenna based DF cooperative system decays as γ¯ −D1 −min{D1 ,K

2}

at γ¯ → ∞, where D1 = KNd . Hence, the PL decoder achieves the

diversity of D1 + min{D1 , K 2 }. VI. A SYMPTOTIC A NALYSIS WITH M ULTIPLE MIMO R ELAYS Let us consider a generalized DF based MIMO cooperative system with N , N ≥ 1, orthogonal relays, each equipped with Nr antennas and a single pair of multi-antenna source and destination August 24, 2012

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nodes. The source and the relays utilize OSTBCs for the transmission and relaying of data of the source to the destination. A. PL Decoder for DF System with Multiple MIMO Relays It can be shown after some algebra that for this multiple relay based DF MIMO cooperative system using the M -PSK constellation, the PL decoder will be Λdp,q ≈ t0 +

N ∑

ψP L (tn ),

(59)

n=1

where t0 =

2 σ12

Nd ∑

[Re{w1,m,k (xp − xq )∗ }], tn =

m=1

2 2 σn ˜

Nd ∑

[Re{wn˜ ,m,k (xp − xq )∗ }], n ˜ = n + 1, n =

m=1

1, 2, ..., N , σn2˜ denotes variance of the AWGN noise of the MIMO wireless channel between the n-th relay and the destination. The intermediate variable w1,m,k can be directly obtained from (9) for ℓ = 1. The variable wn˜ ,m,k , n ˜ = 2, 3, ..., N + 1 corresponding to the n-th relay-destination link can also be T obtained similar to (9) as wn˜ ,m = AH ˜ ,m , where wn ˜ ,m = [wn ˜ ,m,1 , wn ˜ ,m,2 , ..., wn ˜ ,m,J ] , and yn ˜ ,m n ˜ ,m yn

contains the signals received at the m-th receiving antenna in the destination over the n-th relaydestination link. The matrix An˜ ,m , n ˜ = 2, 3, ..., N + 1 represents the equivalent orthogonal matrix [ ] containing the elements of the m-th row of the channel matrix Hn˜ = hn˜ ,m,i N ×Nr , between the d

n-th relay and the destination. The ψP L (tn ) is given in (22) with t1 and T1 replaced by tn and Tn , [ ] n) respectively, where Tn = ± ln (M −1)(1−ϵ and ϵn is the average probability of error of the links ϵn between S and the n-th relay. B. Asymptotic SER of the Proposed PL Decoder with Multiple MIMO Relays If the average SNR of the S-R links is considered to be asymptotically high, the average probability of error of the S-R links approaches to zero, and therefore, Tn → ∞, ∀n. It can be observed from (22) that under the asymptotically high SNR condition, ψP L (tn ) = tn , −∞ ≤ tn ≤ ∞. Hence, the asymptotic PL decoder of DF based cooperative system with N orthogonal MIMO relays can be written as Λdp,q

≈ t0 +

N ∑

tn ,

(60)

n=1

and the PEP of the asymptotic PL decoder in (60) will be Pe = Pr{t0 +

N ∑

tn < 0|zk = xp }.

(61)

n=1 August 24, 2012

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20

Since the relay will always forward the correct symbol under the asymptotic condition, we can write the p.d.f. of Hermitian quadratic variable tn similar to (28) as  ( )D 1 D∑ D −λ−1 D1 −1 1 −1 un 1 Lλ (0)  an bn  e−an un , un > 0,  an +bn (an +bn )λ (D1 −λ−1)! λ=0 ptn (un ) = ( )D 1 D∑ D −1 1 −1  (−un )D1 −λ−1 Lλ 1 (0) bn bn un   aann+b e , un ≤ 0, λ (an +bn ) (D1 −λ−1)! n

(62)

λ=0

where

( an = 1 +

4σn2˜ Ωn˜ |xp −xq |2

)1/2 + 1,

)1/2 4σn2˜ − 1, (63) Ωn˜ |xp −xq |2 where Ωn˜ is the average value of the channel coefficient of the links between the n-th relay and D. ( bn = 1 +

Let us define the instantaneous SNR of the link between the i-th transmitting antenna of the n-th relay △

and m-th receiving antenna of D by γn˜ ,m,i = |hn˜ ,m,i |2 /σn2˜ with the average value γ¯n˜ ,m,i = Ωn˜ /σn2˜ . We also assume that all links experience the same average SNR, i.e., γ¯1,m,j = γ¯n˜ ,m,i = γ¯ , ∀ m, i, j, n ˜. Therefore, it can be observed from (27) and (63) that a0 = an = a and b0 = bn = b such that ( )1/2 4 a= 1+ + 1, γ¯ |xp −xq |2 )1/2 ( 4 − 1. (64) b= 1+ γ¯ |xp −xq |2 Using (61), (62), and the above observations, we can evaluate the asymptotic average PEP of the proposed PL decoder with multiple relays utilizing M -PSK constellation as ] ( )D2 [D∑ 2 −1 ( )−λ a a 2 −1 Pe = LD (0) 1 + , λ a+b b λ=0

(65)

where D2 = (N +1)KNd and a and b can be calculated from (64). It follows from [25] that the average probability of error of the proposed PL decoder with M -PSK constellation will be approximately 2Pe . It can be noticed from (63) and (65) that for the high values of average SNR of all links, the error probability of the asymptotic system decays as γ¯ −D2 . Therefore, the asymptotic cooperative system with multiple MIMO relays achieves maximum possible diversity of D2 . VII. N UMERICAL R ESULTS In this section, the simulation and analytical results of the proposed ML and PL decoders of the DF based MIMO cooperative system are given for various complex-valued M -ary constellations over Rayleigh fading MIMO channels. It is assumed that the average SNR of all links is same, i.e., γ¯0,i,j = γ¯1,m,j = γ¯2,m,i . The X-axis in each figure denotes the average SNR of all links. August 24, 2012

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A. Simulated SER Performance of the Proposed ML and PL Decoders The simulation results of error performance of the proposed ML decoder (19) and the PL decoder (23) are shown in Fig. 2 for different M -PSK constellations and Ns = Nr = Nd = 2. The performance of the optimal ML decoder is also plotted in Fig. 2. The optimal ML decoder requires perfect knowledge of the instantaneous channel coefficients of the S-R, S-D, and the R-D links in the destination; whereas, the proposed ML and PL decoders require the knowledge of the average SNR of the S-R links and the instantaneous channel coefficients of the S-D and R-D links in the destination. It can be observed from Fig. 2 that the proposed ML and PL decoders perform very close to the optimal ML decoder at all SNRs considered for simulations. Moreover, the proposed PL decoder performs approximately similar to the proposed ML decoder for the given M -PSK constellation and all SNR values shown in Fig. 2. Fig. 3 shows the simulated SER versus SNR performance of the proposed ML and PL decoders using 16-QAM constellation, Nd = 1, and K = 2, 3. It can be seen from Fig. 3 that the proposed ML and PL decoders of multi-antenna based DF cooperative system work close to each other for M -QAM constellation as well. B. Comparison of Analytical and Simulated SER of the PL Decoder The analytical and simulation results of average SER of the proposed PL decoder are plotted with average SNR using Ns = Nr = 2, Nd = 1, a single relay, and different M -PSK constellations in Fig. 4. Moreover, Fig. 5 shows the analytical and simulated error performance of the average SER of the proposed PL decoder utilizing a 3 × 4 OSTBC transmission from the source, Ns = Nr = 3, and Nd = 1. It can be observed from Fig. 4 and Fig. 5 that the analytical values of the approximate SER of the proposed PL decoder, obtained as twice of the PEP in (24), are close to the exact SER values for each SNR value considered in the figures. This demonstrates the accuracy of the approximation used for the derivation of the analytical SER. C. Asymptotic Performance of the PL Decoder with Multiple MIMO Relays In this subsection, we present the simulation and analytical results of the asymptotic SER of the proposed PL decoder of the DF based MIMO cooperative system with N , N ≥ 1, orthogonal relays utilizing OSTBCs and M -PSK constellation. In the asymptotic condition, the average error probability August 24, 2012

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of the S-R links approaches to zero (i.e., ϵn → 0, ∀n), and the asymptotic PL decoder requires only the exact information of S-D and R-D links in the destination. Fig. 6 shows the simulated SER versus SNR performance of the asymptotic PL decoder (60) in a cooperative system with a single MIMO relay with equal average SNR of the S-D and R-D links. The simulation results of the average SER of the proposed PL decoder (23) of DF based cooperative system having a single erroneous MIMO relay is also shown in Fig. 6 with equal average SNR of the S-R, R-D, and S-D links. The proposed PL decoder (23) considers the possibility of error in the source-relay links and uses the average probability of error of S-R links for decoding of the data of the source in D; whereas the asymptotic PL decoder (60) assumes the error-free S-R links to decode an M -PSK symbol in the destination. The curves in Fig. 6 are plotted for Ns = Nr = 2, Nd = 1, 2, 3, and 8-PSK constellations. It can be observed from Fig. 6 that for Nd ≤ Ns = Nr , the simulated SER curves of the proposed PL decoder for a single relay cooperative MIMO set-up approximately follow the asymptotic SER curves at high SNR. However, for Nd > Ns = Nr , the proposed PL decoder provides less diversity gain than the asymptotic diversity order. Further, we have compared the analytical error performances of the proposed PL decoder and asymptotic PL decoder with single relay, Ns = Nr = 2, Nd = 3, and 8-PSK constellation in Fig. 7. It can be observed from Fig. 7 that the proposed PL decoder provides an SER of 1.58 × 10−20 at 30 dB and 1.66 × 10−30 at 40 dB SNR. Therefore, the diversity of 10 (i.e., D1 + min{D1 , K 2 }) is achieved by the proposed PL decoder. Whereas, in Fig. 7, the SER values of asymptotic PL decoder at 30 dB and 40 dB SNRs are 1.53×10−27 and 1.64×10−39 , respectively. Hence, it concludes from these numerical results that the asymptotic PL decoder achieves the full diversity of 12 (i.e., 2D1 ) with single MIMO relay, Ns = Nr = 2, Nd = 3. The analytical and simulated values of the asymptotic SER are plotted for an arbitrary number of MIMO relays, Ns = Nr = 2, Nd = 1, and QPSK constellation in Fig. 8. The analytical values of asymptotic SER are obtained from Subsection VI-B. It can be observed from the plots in Fig. 8 that analytical asymptotic SER closely follows the exact (simulated) asymptotic SER. Moreover, we can see from Fig. 8 that asymptotic PL decoder of the DF based multi-antenna cooperative system with N MIMO relays provides full diversity of 2(N + 1). D. Comparison of the Proposed PL decoder with Existing DF Based Decoder and AF protocol In Fig. 9, the simulated error performance of the proposed PL decoder is compared with the same rate AF based MIMO cooperative system and the conventional decoder of the DF cooperative protocol August 24, 2012

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23

[5] using 8-PSK constellation, a single MIMO relay, and Ns = Nr = Nd = 2. The conventional decoder of the DF cooperative system [5] wrongly assumes that the source-relay links are error-free. Hence, it requires neither instantaneous nor average information of the channel of the S-R links in the destination. We have also plotted the SER of AF based MIMO cooperative system in Fig. 9. It can be observed from Fig. 9 that the proposed PL decoder outperforms the AF protocol and the conventional decoder for all SNRs considered in the figure. We have considered the following assumptions for the comparison: 1) All the cooperative links are assumed to have the same average SNR. 2) We assume constant transmit power per time interval for all transmitting nodes in the AF based decoder. 3) The proposed DF based decoder utilizes the exact information of the channel of the S-D and R-D links, and the average error probability of the S-R links in the destination. 4) It is also assumed that the AF based decoder requires the destination to possess perfect knowledge of channel gains of all the links involved in the cooperation. Moreover, we compare the performance of the proposed PL decoder with the ML decoder in the DF based MIMO relay system, where the destination has full channel state information (CSI) of all links involved in cooperation. It can be seen from Fig. 9 that the proposed PL decoder performs very close to the ML decoder with full CSI at all SNRs considered in the figure. VIII. C ONCLUSION We have derived an ML decoder of the DF based MIMO cooperative communication system, which requires information of the average SNR of the S-R links and performs close to an optimal ML decoder. A sub-optimal PL decoder is also derived which provides a significant reduction in the decoding complexity. The proposed ML and PL decoders are applicable to arbitrary complex-valued M -ary constellations and provide the maximum possible diversity gains in DF MIMO relay system. Moreover, we have derived the average SER expression of the proposed PL decoder using M -PSK constellations. The asymptotic SER of the DF cooperative system with multiple MIMO relays utilizing M -PSK constellations is also derived. It is analytically shown that the proposed PL decoder with a single MIMO relay achieves the maximum possible diversity.

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24

A PPENDIX A D ERIVATION OF I1,p

AND

I2,p

It can be shown by using the p.d.f.’s of t0 and t1 from (26) and (28), respectively, that (∫ −u ) ∫ T1 ∫ T1 ∫ 0 pt1 (u)Ft0 (−v)du, Ip = pt1 (u) pt0 (v)dv du = pt1 (u)Ft0 (−v)du + 0 −T1 −∞ −T1 {z } | {z } | I1,p

(66)

I2,p

where Ft0 (v) is the cumulative distribution function (c.d.f.) of t0 and can be calculated from (26). ω−1 ∑ ξκ , Using the p.d.f. of t1 from (28) and employing a series expansion of Γ(ω, ξ) = (ω − 1)!e−ξ κ! κ=0

[27, Eq. 8.352.2] we get ( )D1 D∑ ( )D1 ( )D1 1 −1 1 −1 ap ap bp γ(D1 − τ, bp T1 )LD (0) b0 τ I1,p = − ap τ ap + bp (1 + ) (D − τ − 1)! ap + bp a0 + b0 1 bp τ =0 ] [D −1 D −1 D −l−1 1 1 1 ∑ ∑ ∑ LτD1 −1 (0)LλD1 −1 (0)(bp + a0 )−(D1 −τ +κ) aκ0 γ(D1 − τ + κ, (bp + a0 )T1 ) × , (1 + ab00 )λ (ap + bp )τ (D1 − τ − 1)!κ! κ=0 τ =0 λ=0 [D −1 D −1 D −l−1 1 1 1 ∑ ∑ ∑

] LτD1 −1 (0)LλD1 −1 (0)(ap + b0 )−(D1 −τ +κ) bκ0 I2,p = γ(D1 − τ + κ, (ap + b0 )T1 ) (1 + ab00 )λ (ap + bp )τ (D1 − τ − 1)!κ! τ =0 λ=0 κ=0 ( )D1 ( )D1 ap bp a0 × , (67) ap + bp a 0 + b0 ∫ξ where γ(ω, ξ) is the lower incomplete Gamma function defined as γ(ω, ξ) = 0 tω−1 e−t dt. R EFERENCES [1] S. M. Alamouti, “A simple transmit diversity techniques for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1451-1458, Oct. 1998. [2] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit diversity,” IEEE J. Select. Areas Commun., vol. 18, no. 7, pp. 1169-1174, Jul. 2000. [3] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperative diversity Part-I: System description,” IEEE Trans. commun., vol. 51, no. 11, pp. 1927-1938, Nov. 2003. [4] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun. Mag., vol. 42, no. 10, pp. 74-80, Oct. 2004. [5] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behaviour,” IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 389-400, Dec. 2004. [6] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2415-2425, Oct. 2003. [7] ——, “Eenergy-efficient antenna sharing and relaying for wireless networks,” IEEE Wireless Communications and Networking Conference (WCNC), pp. 7-12, Sep. 2000, Chicago, IL, USA. August 24, 2012

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[8] D. Chen and J. N. Laneman, “Modulation and demodulation for cooperative diversity in wireless systems,” IEEE Trans. Wireless Commun., vol. 5, no. 7, pp. 1785-1794, Jul. 2006. [9] B. Zhao and M. C. Valenti, “Distributed turbo coded diversity for the realy channel,” IEE Electronics Letters., vol. 39, no. 10, pp. 786-787, May 2003. [10] M. Janani, A. Hedayat, T. E. Hunter, and A. Nosratinia, “Coded cooperation in wireless communications: Space-time transmission and iterative decoding,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 362-371, Feb. 2004. [11] Y. Jing and B. Hassibi, “Distributed space-time coding in wireless relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524-3536, Dec. 2006. [12] Y. Jing and H. Jafarkhani, “Using orthogonal and quasi-orthogonal designs in wireless relay networks,” IEEE Trans. Inform. Theory, vol. 53, no. 11, pp. 4106-4118, Nov. 2007. [13] M. Ju and I.-M. Kim, “ML performance analysis of the decode-and-forward protocol in cooperative diversity networks,” IEEE Trans. Wireless Commun., vol. 8, no. 7, pp. 3855-3867, Jul. 2009. [14] M. R. Bhatnagar and A. Hjørungnes, “ML decoding in decode-and-forward based cooperative communication system,” IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2812-2815, May 2011, Prague, Czech Republic. [15] M. R. Bhatnagar and A. Hjørungnes, “ML decoder for decode-and-forward based cooperative communication system,” IEEE Trans. Wireless Commun., vol. 10, no. 12, pp. 4080-4090, Dec. 2011. [16] G. V. V. Sharma, V. Ganwani, U. B. Desai, and S. N. Merchant, “Performance analysis of maximum likelihood detection for decode and forward MIMO relay channels in Rayleigh fading,” IEEE Trans. Wireless Commun., vol. 9, no. 9, pp. 2880-2889, Sep. 2010. [17] X. Jin, D. -S. Jin, J. -S. No, and D. -J. Shin, “On the diversity analysis of decode-and-forward protocol with multiple antennas,” IEEE International Symposium on Information Theory (ISIT), pp. 513-516, June-Jul. 2009, Seoul, Korea. [18] M. Peng, Y. Liu, D. Wei, and W. Wang, “Hierarchical cooperative relay based heterogeneous networks,” IEEE Wireless Communications, vol. 18, no. 3, pp. 48-56, June 2011. [19] Z. Zhao, Z. Ding, and et. al., “A Special Case of Multi-Way Relay Channel: When Beamforming is not Applicable,” IEEE Trans. Wireless Commun., vol.10, no.7, pp. 2046-2051, July 2011. [20] E. G. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications. Cambridge University Press, Cambridge, UK, 2003. [21] H. L. V. Trees, Detection, Estimation, and Modulation Theory: Part I. Detection, Estimation, and Linear Modulation Theory. New York, USA: John Willey & Sons, Inc., 2001. [22] L. Trailovic and L.Y. Pao, “Variance estimation and ranking of target tracking position errors modeled using Gaussian mixture distributions,” Automatica, vol. 41, no. 8, pp. 1433-1438, Aug. 2005. [23] K. H. Biyari and W. C. Lindsey, “Statistical distributions of Hermitian quadratic forms in complex Gaussian variables,” IEEE Trans. Inform. Theory, vol. 39, no. 3, pp. 1076-1082, May 1993. [24] W. W. Bell, Special functions for Scientists and Engineers. London: Van Nostrand, 1968. [25] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2005. [26] M. K. Simon and M. -S. Alouini, Digital Communication over Fading Channels, New Jersey, USA: John Wiley & Sons, Inc., 2005. [27] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA, USA: Academic Press, 2000. [28] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, New York, USA: Dover Publications, Inc., 1972.

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Fig. 1.

26

System model of the DF based MIMO cooperative system.

TABLE I COMPARISON OF COMPUTATIONAL COMPLEXITY OF THE PROPOSED ML DECODER (19) AND THE PL DECODER (23) UTILIZING 8-PSK CONSTELLATION, SINGLE RELAY, AND Nd = 2.

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Computations

Real Summation

Real Multiplication

Logarithm

Exponential

ML Decoder

1108

804

4

64

PL Decoder

120

144

1

0

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27

0

10

16−PSK

−1

10

−2

10

8−PSK

−3

SER

10

−4

10

QPSK −5

10

−6

10

PL Decoder Proposed ML Decoder Optimal ML Decoder

−7

10

0

2

4

6 8 Average SNR [dB]

10

12

Fig. 2. SER versus SNR performance of the proposed ML decoder, PL decoder, and an optimal ML decoder using M -PSK constellations for Ns = Nr = Nd = 2.

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0

10

N =N =2, N =1

−1

10

s

r

d

−2

SER

10

Ns=Nr=3, Nd=1

−3

10

ML decoder PL decoder −4

10

−5

10

Fig. 3.

0

5

10 Average SNR [dB]

15

SER versus SNR performance of the proposed ML and PL decoders utilizing 16-QAM constellation with Nd = 1 and

Ns = Nr = 2, 3.

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0

10

16−PSK

−1

10

−2

10

8−PSK SER

−3

10

−4

10

QPKS −5

Analytical Simulated

10

−6

10

−7

10

Fig. 4.

0

2

4

6

8 10 12 Average SNR [dB]

14

16

18

Analytical and simulated error performance of the proposed PL decoder for M -PSK constellations using Nd = 1 and

Ns = Nr = 2.

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30

0

10

16−PSK −2

10

SER

8−PSK

−4

10

QPKS −6

10

Simulated Analytical

−8

10

Fig. 5.

0

5

10 Average SNR [dB]

15

Analytical and simulated error performance of the proposed PL decoder for M -PSK constellations using Nd = 1 and

Ns = Nr = 3.

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0

10

−1

10

Nd=1 −2

10

SER

−3

10

−4

10

Nd=3

Nd=2

−5

10

−6

10

0

Proposed PL decoder Asymptotic PL decoder

5

10 Average SNR [dB]

15

Fig. 6. Simulated SER versus SNR performance of the proposed PL decoder (23) and asymptotic PL decoder (60) of DF based MIMO cooperative system with a single relay, Ns = Nr = 2, and 8-PSK constellation.

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0

10

−5

10

−10

10

−15

SER

10

−20

10

−25

10

−30

10

−35

10

Proposed PL decoder (Simulation) Proposed PL decoder (Analysis) Asymptotic PL decoder (Simulation) Asymptotic PL decoder (Analysis)

−40

10

0

5

10

15 20 25 Average SNR [dB]

30

35

40

Fig. 7. Diversity comparison of the proposed PL decoder (23) and asymptotic PL decoder (60) of DF based MIMO cooperative system with a single relay, Ns = Nr = 2, Nd = 3, and 8-PSK constellation.

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0

10

N=1

−5

Asymptotic SER

10

N=2

−10

10

Analytical Simulated

−15

10

N=3

−20

10

0

5

10 15 Average SNR [dB]

20

25

Fig. 8. Analytical and simulated asymptotic SER versus SNR performance of the proposed PL decoder for different number of relays and QPSK constellation using Nd = 1 and Ns = Nr = 2.

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−1

10

−2

SER

10

−3

10

−4

10

−5

10

0

Fig. 9.

Conventional decoder AF based decoder Proposed PL decoder DF based ML decoder with full CSI 3

6

9 12 Average SNR [dB]

15

18

Comparison of the proposed PL decoder with the same rate AF based decoder, a conventional decoder for the DF protocol

[5], and the ML decoder in the DF system with full CSI using 8-PSK constellation and Ns = Nr = Nd = 2.

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