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 single 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 channel 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 protocol, 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 relaying. 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. Similarly, r N R S < D R S R when nr nt nr , the diversity order is nr nt . Therefore, the nt R ( − R) knt N knt diversity order is Pe,D = p0 + ∑ pk[ Pe,RD + ] (22) k=1 N N d = min(nD,nRnS) (20) r r t Ψ( R D, , ) Ψ( , , ) where Pe,RD= nt nr R D , l X Y is given by psi, 3. PERFORMANCE ANALYSIS OF COOPERATIVE and ρ MIMO SYSTEMS WITH A RELAY EQUIPPED WITH Γ¯ (m,n) = ER (| (m,n)|2)=ER R,D R,D R E hR,D R (23) MULTIPLE TX ANTENNAS IN THE ABSENCE OF A N0nt N0nt DIRECT LINK 3.3 Asymptotic BEP 3.1 System Model The asymptotic BEP at the destination is given by [7]

D R R S nr nt nr nt C D R− C R S− asymp 2nr nt 1 2nr nt 1 P , ≈ + (24) e D (m,n) D R (i, j) R S ( Γ¯ )nr nt ( Γ¯ )nr nt 4 R,D 4 S,R Therefore, the diversity order is = ( D R, R S) d min nr nt nr nt (25) R = For nt 1, we obtain the same result as pedasymptotique which is valid when the relay has a single transmit antenna. 4. PERFORMANCE ANALYSIS OF COOPERATIVE MIMO SYSTEMS IN THE PRESENCE OF A DIRECT LINK 4.1 System Model Figure 2: Cooperative MIMO system with a relay equipped with multiple Tx antennas.

R As shown in Figure 2, the relay has now nt transmit antennas. All transmissions are made over orthogonal channels. The cooperation protocol has 3 phases : • Phase 1 : = ( S, R) The source transmits N lcm nt nt symbols to the relay where lcm is the abbreviation of least common multiple.

• Phase 2 : - If the relay R correctly decodes i symbols with ≤ < R 0 i nt then it remains idle because it can’t form any code matrix. - If the relay R correctly decodes j symbols with R ≤ < ( + ) R knt j k 1 nt then it transmits k matrices of code, thus it transmits knR symbols and the rest of the symbols, Figure 3: Cooperative MIMO system with a single Relay t which supports a single TX antenna in the presence of a di- N − knR, are not emitted. t rect link. - If the relay R correctly decodes the totality of the N N symbols then it transmits R matrices of code. nt In this section and as shown in Figure 3, there are a relaying link (S-R-D) and a direct link (S-D). The destination • Phase 3 : The destination decodes the signals received coherently combines the received signals from the source and from the relay using a ML decoder. the relay using a Maximum Ratio Combaning (MRC).

816 4.2 Conditional BER at the destination Therefore, when the relay has correctly decoded, the BEP at If the relay has correctly decoded, the SNR at the destination D can be written as can be written as : S D D nt nr nr S D D D S D P , | = ∑ a Ψ(n n ,S,D)+ ∑ b Ψ(n ,R,D) (38) nr nt nr e D R l t r k r Γ = Γ(i, j) + Γ(k,1) l=1 k=1 D|R ∑ ∑ S,D ∑ R,D (26) i=1 j=1 k=1 f and Ψ were previously defined in f and psi. = ΓS,D + ΓR,D (27) The BEP at D is written as = × +( − ) × where Pe,D Pe,SR Pe,SD 1 Pe,SR Pe,D|R (39) Γ(i, j) = ES | (i, j)|2 S,D S hS,D (28) nt N0 where = Ψ( S R, , ) Pe,SR nt nr S R (40) Γ(k,1) = ER | (k,1)|2 R,D hR,D (29) N0 and = Ψ( S D, , ) The BEP at the destination is given by Pe,SD nt nr S D (41) 4.3 Asymptotic BEP = ( γ) Γ (γ) γ Pe,D|R A Q B p D|R d (30) At high SNR, Pe,SR 1 and PeD can be approximated by (γ) where pΓ | is the PDF of the received SNR at the desti- S R D S D R nt nr nr nt C S R− C D S− nation. To derive the expression of the PDF, we use the MGF asymp 2nt nr 1 2nr nt 1 P , ≈ × (42) of the SNR. e D (i, j) S R (k, j) D S (4Γ¯ )nt nr (4Γ¯ )nr nt Γ ( )= ( Γ ( )) S,R S,D M D|R s LT p D|R s (31) D( S+ ) nr nt 1 where LT is the Laplace transform. The MGF of the SNR at C ( S D+ D)− 1 1 + 2 nt nr nr 1 × × the destination can be written as S ( , ) ( , ) nD(n +1) k j nSnD k 1 nD 4 r t (Γ¯ , ) t r (Γ¯ , ) r −sΓ S D R D Γ ( )= ( D|R ) M D|R s E e D S D The diversity order is : nr nt 1 nr 1 = ∏∏ ∏ (32) S R D D S (i, j) (k,1) d = min(n (n + n ),n (1 + n )) (43) = = + Γ = + Γ t r r r t i 1 j 1 1 s S,D k 1 1 s R,D 5. SIMULATION AND THEORETICAL RESULTS where Γ (i, j) S,D In this section, we compare the derived theoretical results to Γ , = (33) S D S D nt nr simulation ones. We plot the BEP at the destination versus E /N (dB) where E is the transmitted energy per bit for and b 0 b a dual-hop DF MIMO systems using a BPSK modulation. (k,1) ΓR,D Γ , = (34) We have allocated the same power to the source and relay: R D nD r ER = ES = Eb/2. In order to take into account the path loss, Thus the average power of the channel coefficient between the i-th Rx antenna of Y and the j-th Tx antenna of X is given by 1 1 Γ ( )= M D|R s S D D (35) nt nr Γ nr ( , ) β ΓS,D R,D i j 2 (1 + s ) (1 + s ) E(|h , | )= α (44) nSnD nD X Y t r r dXY Using a fraction decomposition, we have where dXY is the normalized distance between X and Y and α eff S D D = dXY eff nt nr nr is the path loss exponent. Note that dXY , d is the ef- al bk d0 XY MΓ (s)= ∑ + ∑ (36) D|R Γ l Γ k fective distance in meters between X and Y, d0 is an arbitrary l=1 ( + S,D ) k=1 ( + R,D ) reference distance and β is the path loss at the reference dis- 1 s S D 1 s nD nt nr r tance. We have used the following parameters β = 1, α = 3, = . = . where al and bk are the residues which were computed using dSR 0 4 and dRD 0 6. the software Mathematica. The PDF of the SNR can be cal- Figure 4 shows the exact, asymptotic BEP and simulation culated using the inverse Laplace Transform of the MGF as results for a cooperative MIMO systems using DF relaying follows for the case where the relay has a single transmit antenna in the absence of a direct link. For this scenario the diversity −1 Γ ( )= ( Γ ( )) ( D, R S) S = p D|R s LT M D|R s order is: min nr nr nt . We notice that the 3113 (nt 3, R R D S D n = 1, n = 1, n = 3) offers better performance than the nt nr r t r ΓS,D S = R = R = D = S = = ∑ a f (x, ,l) 2112 (nt 2, nr 1, nt 1, nr 2) and 2111 (nt 2, l S D R = R = D = l=1 nt nr nr 1, nt 1, nr 1). In fact, the diversity order is equal D to three for the first system however the two others have re- nr Γ + ( , R,D , ) spectively a diversity order equal to two and one.Thus, if the ∑ bk f x D k (37) k=1 nr diversity increases, the performances of the system improve.

817 0 2 10 10 Sim:2111 with direct link Sim:2111 without direct link −2 0 Theory:2111 with direct link 10 10 Asymp:2111 with direct link Theory:2111 without direct link −4 Asymp:2111 without direct link 10 −2 10

−6 10 Sim:2111 −4 BEP Theory:2111 10 BEP Asymp:2111 −8 10 Sim:2112 −6 Theory:2112 10 Asymp:2112 −10 10 Sim:3113 −8 Theory:3113 10 Asymp:3113 −12 10 0 5 10 15 20 25 30 −10 E /N (dB) 10 b 0 0 5 10 15 20 25 30 E /N (dB) b 0 Figure 4: Simulation results, exact and asymptotic BEP of cooper- S = Figure 6: BEP of Cooperative MIMO system in the presence and ative MIMO system in the absence of a direct link for 2111 (nt 2, R = R = D = S = R = R = D = absence of a direct link. nr 1, nt 1, nr 1); 2112 (nt 2, nr 1, nt 1, nr 2) and S = R = R = D = 3113 (nt 3, nr 1, nt 1, nr 3). 6. CONCLUSION Figure 5 shows the BEP plots, computed from the ana- In this paper, considering a source, a relay and a destination lytical results and simulation ones for a MIMO relay system with multiple antennas, we study the end-to-end performance where the relay has multiple transmit antenna in the absence of dual-hop cooperative MIMO systems employing DF re- of a direct link. For this scenario, the diversity order is : laying. We have derived the PDF and MGF of the SNR at ( D R, R S) min nr nt nr nt . We see that the BEP for the scenario 2222 the destination. Numerical results have shown that the BEP S = R = R = D = (nt 2, nr 2, nt 2, nr 2) where we have multiple analysis provided in the paper is in accordance with the sim- transmit antennas at the relay is lower than the same scenario ulation results for various scenarios. S = but with a single transmit antenna at the relay, 2212 (nt 2, R = R = D = nr 2, nt 1, nr 2), and the cooperative SISO system REFERENCES where all nodes have a single antenna. [1] Hasna, Alouini. MO and M.-S., ”End-to-end performance of transmission systems with relays over Rayleigh-fading channels,” Wireless Communications, 0 10 IEEE Transactions on, Vol. 2, no 6, pp. 1126-1131, Nov.2003. −2 10 [2] A. K. Sadek, W. Su and K. J. Ray Liu, ”Multinode Co-

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