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Performance Analysis of Cooperative Mimo Systems Using Decode and Forward Relaying 19th European Signal Processing Conference (EUSIPCO 2011) Barcelona, Spain, August 29 - September 2, 2011 PERFORMANCE ANALYSIS OF COOPERATIVE MIMO SYSTEMS USING DECODE AND FORWARD RELAYING Emna Ben Yahia and Hatem Boujemaa Higher School of Communications of Tunis/TECHTRA Research Unit, University of Carthage, Tunisia [email protected], [email protected] ABSTRACT of cooperative MIMO systems. Then we derive the expres- In this paper, considering a source, a relay and a destina- sion of the diversity order. tion with multiple antennas, we provide the exact Bit Er- The paper is organized as follows. Sections 2, 3 and 4 ror Probability (BEP) of dual-hop cooperative Multiple Input presents the performance analysis of cooperative MIMO sys- Multiple Output (MIMO) systems using Decode and For- tems using DF relaying respectively in the absence and pres- ward (DF) relaying for Binary Phase Shift Keying (BPSK) ence of a direct link. For the case where we don’t have a modulation assuming independent and identically distributed direct link, we study two scenarios. The first concerns sys- (i.i.d.) Rayleigh fading channels. When the source or the re- tems where the relay has a single transmit antenna and the lay has multiple antennas, it employs an Orthogonal Space second is about systems where the relay has multiple trans- Time Block Code (OSTBC) to transmit. The receiver uses mit antennas. Section 5 gives some numerical and simulation a Maximum Likelihood (ML) decoder. We derive the ex- results. Section 6 draws some conclusions. pressions of the Moment Generating Functions (MGF), the Probability Density Functions (PDF) of the Signal-to-Noise Ratio (SNR), the exact and asymptotic BEP at the destination 2. PERFORMANCE ANALYSIS OF COOPERATIVE then we compare theoretical results to simulation ones. The MIMO SYSTEMS WITH A RELAY EQUIPPED WITH analysis is performed in the presence or absence of the direct A SINGLE TX ANTENNA IN THE ABSENCE OF A link. We also derive the diversity order and we study the ef- DIRECT LINK fect of the number of antennas equipped with the source, the relay and the destination on the end-to-end BEP. 2.1 System Model 1. INTRODUCTION MIMO is an emerging technology which consists in using multiple transmit and receive antennas in order to create spa- tial diversity between transmitter and receiver. The success of MIMO technology has led to the concept of cooperative communications where multiple nodes equipped with a sin- gle antenna cooperate together in order to form a virtual MIMO array and offer cooperative diversity. Cooperative system is a promising solution for high data-rate coverage re- quired in future wireless communications systems. Combin- ing a cooperative system with a Multiple Input Multiple Out- put transmission is an interesting way to overcome the chan- nel impairments and to improve the system performances 1. The Symbol Error Probability (SEP) of cooperative systems using Amplify and Forward (AF) or Decode and Forward (DF) relaying was derived in [1]-[2]-[3]. In the AF proto- col, each relay amplifies the received signal using an adap- Figure 1: Cooperative MIMO system with a relay equipped tive gain. In the DF protocol, each relay decodes the received with a single TX antenna. signal from the source and transmits only if it has correctly decoded. In this paper, we are interested only in DF relay- ing. Exact end-to-end BER analysis of dual-hop orthogonal As shown in Figure 1, we consider a dual-hop wireless space-time transmission with multiple antenna is found in communication system in which the source S has nS trans- [4] for AF and DF relaying. Performance analysis of Am- t mit antennas. S is communicating with the destination D plify and Forward based cooperative diversity in MIMO re- D equipped with nr receive antennas through a relay R which lay channels was studied in [5]. Power allocation strategies R in cooperative MIMO networks were investigated in [6]. supports multiple receive antennas, nr , and a single transmit In this paper, we present new simple expressions of the exact antenna.The transmission mode is assumed to be done over and asymptotic BEP at the destination for different scenarios orthogonal channels and is composed of three phases : • Phase 1 : 1This work was supported by Qatar National Research Fund under Grant the source transmits the signal to the relay S R × S NPRP 08 - 577-2-241 through its nt antennas by using an OSTBC. The nr nt © EURASIP, 2011 - ISSN 2076-1465 814 channel matrix of the first hop is and ( , ) ( , ) E ( , ) ⎛ ⎞ Γ¯ i j = E(Γ i j )= S E(|h i j |2) (7) (1,1) (1,nS) S,R S,R S S,R h ··· h t N0nt ⎜ S,R S,R ⎟ ⎜ . ⎟ (i, j) HS,R = . (.) Γ ⎝ . ⎠ where E is the expectation operator. S,R is independent ( R, ) ( R, S) nr 1 ··· nr nt of i and j. hS,R hS,R (i, j) ESρS,R , Γ¯ = We assume that each element of HS R is an i.i.d. com- S,R S (8) plex Gaussian random variable (r.v.) with zero mean and N0nt ρ variance S,R. The received signal at the relay from the The BEP at the relay can be written as source is given by = ( γ) (γ) γ Pe,SR A Q B pΓS,R d ES Y , = H , S + N , (1) S R S S R S R = Ψ( R S, , ) nt nr nt S R (9) where E is the transmitted energy per symbol by the where A and B depends on the considered modulation (A = 1 S = source, HX,Y is the channel matrix of the link between X and B 2 for BPSK modulation). and Y and NX,Y is an additive white complex gaussian +∞ S 1 −u2 noise with a variance equal to N0. is the code matrix Q(x)=√ e 2 du (10) which depends on the number of transmit antennas of 2π x the source. and • Phase 2 : if the relay correctly decodes, it forwards the − μ l−1 + μ Ψ( , , )= [1 X,Y ]l k (1 X,Y )k decoded symbol with symbol energy ER to the destina- l X Y A ∑ Cl−1+k (11) tion; otherwise, it remains idle. The received signal at 2 k=0 2 the destination from the relay, YR,D, can be written as √ where Y = H ˆ+ n! R,D ER R,DS NR,D (2) Ck = (12) n k!(n − k)! where Sˆ = si if symbol si has been correctly decoded, ˆ = D × S 0 otherwise and HR,D is an nr 1 channel vector. BΓ¯ X,Y μ , = X Y Y X • Phase 3 : the destination decodes the received signals 2n n + BΓ¯ X,Y r t from the relay using a ML decoder. Γ¯ (i, j) B X,Y 2.2 Statistical description of the SNR = (13) Γ¯ (i, j) + B X,Y 2 The Signal to Noise Ratio (SNR) at the relay is given by and nR nS ( , ) r t ( , ) Γ = Y X Γ i j Γ = Γ i j X,Y nr nt X,Y (14) S,R ∑ ∑ S,R (3) i=1 j=1 The BEP at the destination can be written as where Pe,D = Pe,SR +(1 − Pe,SR)Pe,RD (15) (i, j) ES (i, j) 2 Γ , = |h , | (4) S R nSN S R t 0 where Pe,RD represents the BEP of the (R-D) link between the Γ(i, j) relay and the destination. S,R is the SNR between the i-th relay’s receive antenna and (i, j) = Ψ( D, , ) the j-th source’s transmit antenna, and hX,Y represents the Pe,RD nr R D (16) D− channel coefficient between the i-th Rx antenna of Y and the nr 1 1 − μR,D D 1 + μR,D j-th Tx antenna of X. =( )nr k ( )k ∑ C D− + nr 1 k The SNR at the relay follows a chi-square distribution with 2 k=0 2 R S Γ 2nr nt degrees of freedom. Therefore, the PDF of S,R is given by where (i, j) R S ( , ) pΓ (γ)= f (γ,Γ¯ ,n n ) (5) Γ¯ k 1 S,R S,R r t B R,D μR,D = (17) (γ, , ) Γ¯ (k,1) + where f a l is the chi-square distribution with 2l degrees B R,D 2 of freedom and average mean a × l : where − ρ γl 1 −γ Γ¯ (k,1) = ER (| (k,1)|2)=ER R,D f (γ,a,l)= e a ,γ ≥ 0 (6) R,D E hR,D (18) al(l − 1)! N0 N0 815 2.3 Asymptotic BEP 3.2 Exact BEP At a high SNR, the relay always correctly decodes the emit- The probability that the relay will transmit k matrices of code ted symbols. Therefore, Pe,SR ≈ 0 and peD becomes [7] is : [( + ) R− , ] min k 1 nt 1 N (N− j) D R S = j ( − ) j nr nr nt pk ∑ CN 1 Pe,SR Pe,SR (21) C D− C R S− asymp 2nr 1 2nr nt 1 j=knR P , ≈ + (19) t e D (k,1) D (i, j) R S ( Γ¯ )nr ( Γ¯ )nr nt 4 R,D 4 S,R where j is the number of symbols correctly received by the relay and Pe,SR is given by (8). D < R S If nr nr nt , the second term of pedasymptotique can be ne- The BEP at the destination is given by glected at high SNR and the diversity order is nD.
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