Soft-Decision-and-Forward Protocol for Cooperative Communication Networks with Multiple Antennas

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1 Soft-Decision-and-Forward Protocol for Cooperative Communication Networks with Multiple Antenn Jae-Dong Yang, Kyoung-Young Song Department of EECS, INMC Seoul National University {yjdong, Jong-Seon No Department of EECS, INMC Seoul National University Dong-Joon Shin Department of ECE Hanyang University Abstract A cooperative protocol called soft-decision-andforward (SDF) is introduced, which exploits the soft decision values of the received signal at the relay node. Alamouti code is used for orthogonal transmission and distributed spacetime codes are designed for non-orthogonal transmission. The maximum likelihood decoders with low decoding complexity are proposed. From simulations, it can be seen that SDF protocol outperforms AF protocol. S R G H 2 H 1 D I. INTRODUCTION Recently, there have been much interest in cooperative communication networks because the spectral efficiency and reliability of the wireless communication systems can be improved by the cooperation of relay node. Cooperative communication networks consist of three types of nodes which are source nodes, relay nodes, and destination nodes 1. In this paper, we express a source node S, a relay node R, and a destination node D. There are several cooperative protocols including amplify-and-forward (AF) and decode-and-forward (DF) protocols 1 4. AF protocol allows R to amplify the received signals from S and forward them to D. DF protocol permits R to decode, re-encode the received signals, and then forward them to D. In this paper, we introduce a cooperative protocol called soft-decision-and-forward (SDF). Unlike AF and DF protocols, SDF protocol exploits the soft decision values of the received signals at R. Two phe cooperation is sumed. In the first phe, S transmits, and in the second phe, R transmits and S transmits or not. If S does not transmit in the second phe, it is called orthogonal transmission (OT). Otherwise, it is called non-orthogonal transmission (NT). We investigate four schemes, i.e., OT-AF, OT-SDF, NT-AF, and NT-SDF, and their ML decoders are derived simplified forms. A couple of protocols similar to SDF protocol were presented in 5 and 6. In 5, process-and-forward (PF) protocol w proposed, which allows relay to perform spacetime processing on the received signals. It w shown that AF and PF protocols have the same performance for one antenna at R. However, PF protocol w not applied to the multiple-antenna ce. In this paper, we consider multiple antenn at all nodes and verify that SDF protocol shows better performance than AF protocol. In 6, decouple-and-forward Fig. 1. A cooperative communication network with two antenn. (DCF) protocol for dual hop cooperative communications w proposed, which allows R to decouple the received signals. In spite of correlated noises at D, squaring method 7 w used for DCF protocol. However, in this paper, it is sumed that D can hear S and the optimal maximum likelihood (ML) decoders are used. Furthermore, it is shown that the ML decoders can be simplified even when the noises at D are correlated. In this paper, the following notations are used: Capital boldface letters denote matrices, small boldface letters denote vectors, R( ) denotes the real part of a complex number, ( ) denotes the complex conjugate, ( ) T denotes the transpose of a matrix, ( ) H denotes the complex conjugate and transpose of a matrix, denotes the norm of a complex number, denotes the Frobenius norm of a matrix, I n denotes the n n identity matrix, 0 denotes the all zero matrix. II. SYSTEM MODEL A three-node cooperative communication network is considered. It is sumed that all nodes have two antenn. Fig. 1 shows the considered cooperative communication network. Let G be the fading channel matrix between S and R, be the fading channel matrix between S and D, and H 1 H 2 be the fading channel matrix between R and D. Since we sume two antenn, all the fading channel matrices are 2 2 matrices whose entries are independent complex Gaussian random variables with zero mean and unit variance. Note that g ij, h ij 1, and hij 2, for i, j 1, 2 are the fading coefficients from the ith transmit antenna to the jth receive antenna for the corresponding channels. We sume qui-

2 static Rayleigh fading for all channels. Further, perfect channel state information is sumed to be known only at the receiver, that is, R knows g ij and D knows g ij, h ij 1, and hij 2. Half duplex communication is sumed, that is, all nodes can either transmit or receive. Thus, we sume two phe cooperative communications. For NT, source antenna switching (SAS) 4 is considered such that S utilizes two RF chains, where each RF chain h two transmit antenn. In the first phe, one transmit antenna at each RF chain is used and then in the second phe the other at each RF chain is used. The additional diversity gain can be obtained by using SAS for DF protocol 4. Cooperative protocols are clsified by the operation at R. There are two well-known protocols, AF and DF protocols. For AF protocol, R only amplifies the received signals and forwards them to D. Since AF protocol amplifies the received signal, noise component is also amplified and forwarded. For DF protocol, R decodes the received signals, re-encodes, and transmits them to D. Since R performs hard decision in DF protocol, the soft information of the received signal is disappeared. Therefore, we propose the SDF protocol which exploits the soft-decision values at R. In the next section, we investigate four schemes, i.e., OT-AF, OT-SDF, NT-AF, and NT-SDF. III. AF AND SDF PROTOCOLS It is sumed that S transmits Alamouti code 8. For AF protocol, the received signal at the first (second) receive antenna of R is amplified and forwarded through the first (second) transmit antenna of R. For SDF protocol, R decodes the received signals into the soft decision values. R re-encodes the soft decision values and transmits to D. To explain four schemes, i.e., OT-AF, OT-SDF, NT-AF, and NT-SDF, the following definition will be used. Definition: Alamouti operation and complex vectorization: Alamouti operation for two symbols a and b is represented A(a, b) a b b a ; b11 b For 2 2 matrices B 12 and M b 21 b M, let B 0 1 A(b11, b 21 ) ; M A(b 12, b 22 ) Let cv( ) denote complex ( vectorization ) operation for a a b 2 2 matrix, i.e., cv a c c d b d T. A. OT-AF Scheme In the first phe, S transmits an Almouti code which consists of two independent symbols x 1 and x 2, i.e., X A(x 1, x 2 ). The received signals at R and D in the first phe can be expressed Y R P 1 XG + W, Y D1 P 1 XH 1 + N 1 (1) where Y R is the received signal matrix at R, Y D1 is the received signal at D, W and N 1 are additive white Gaussian noise (AWGN) matrices with entries of zero mean and unit variance independent complex Gaussian random variables. The average transmit power per antenna at S is sumed to be P 1. In the second phe, R transmits the signal X R Y R β β1 0 where β is the amplification matrix at R. It can 0 β 2 be eily shown that β 1 P 2 /(P 1 ( g g 21 2 ) + 1) and β 2 P 2 /(P 1 ( g g 22 2 ) + 1) to ensure the average transmit power per antenna at R to be P 2. Since we consider OT, only R transmits in the second phe. The received signal at D in the second phe is given Y D2 Y R βh 2 + N 2 P 1 XF + N (2) where F GβH 2, N WβH 2 + N 2, and N 2 is AWGN matrix with entries of zero mean and unit variance independent complex Gaussian random variables. By using (1), (2), and the Almouti code structure, the equivalent vector model for OT-AF can be expressed cv(yd1 ) cv(y D2 ) where x x 1 x 2 T. B. OT-SDF Scheme P1 H 1 P1 F x + cv(n1 ) cv(n) In the first phe, we have the same situation in Subsection III-A and thus the received signals at R and D are given in (1). For SDF protocol, R decodes the received signals into the soft decision values, re-encodes them, and transmits the re-encoded signals. Since S transmits Alamouti code, soft decision values for x 1 and x 2 at R can be obtained ˆx1 γ(g ) H cv(y ˆx R ) (4) 2 where γ is the power gain at the relay. To guarantee the average transmit power per antenna at R to be P 2, γ should be P2 /( G 2 (1 + P 1 G 2 )). R re-encodes the soft decision values in (4) by using Alamouti code to generate X R A(ˆx 1, ˆx 2 ). The received signal at D in the second phe is given Y D2 X R H 2 + N 2. Using (4) and after some manipulations, we have the following equivalent vector model of OT-SDF cv(yd1 ) P1 H 1 cv(n1 ) x + (5) cv(y D2 ) cv(n) ah 2 where a γ P 1 G 2, cv(n) γh 2(G ) H cv(w) + cv(n 2 ), and x x 1 x 2 T. (3)

3 C. NT-AF Scheme Considering NT, S transmits signals in the second phe. Since S and R transmit signals in the second phe simultaneously, we design a distributed space-time code (DSTC) using a qui-orthogonal space-time block code (QO-STBC) 9. For the use of DSTC in the second phe, it is sumed that S transmits two Alamouti codes, i.e., X 1 A(x 1, x 2 ) and X 2 A(x 3, x 4 ), in the first phe. Then, the received signals at R and D in the first phe are given Y R1 P 1X 1G + W 1, Y R2 P 1X 2G + W 2 Y D1,1 P 1 X 1 H 1 + N 1,1, Y D1,2 P 1 X 2 H 1 + N 1,2 (6) where Y R1 and Y R2 are 2 2 received signal matrices at R, Y D1,1 and Y D1,2 are received signal matrices at D, and W 1, W 2, N 1,1, and N 1,2 are 2 2 AWGN matrices. In the second phe, similarly to Subsection III-A, R amplifies the received signals and forwards them to D. Thus, R transmits R θ Y R2 β at first and then transmits Y R1 β. Note that R θ A(e jθ, 0) and θ is the constellation rotation angle for full diversity in QO-STBC 10. S transmits X 1 at first and then R θ X 2. The received signal at D in the second phe can be given Y D2,1 P 3 X 1 H 1 + R θ Y R2 βh 2 + N 2,1 P 3 X 1 H 1 + P 1 R θ X 2 F + N e1 Y D2,2 P 3 R θ X 2 H 1 + Y R1 βh 2 + N 2,2 P 3 R θ X 2 H 1 + P 1 X 1 F + N e2 (7) where P 2 and P 3 are the average transmit power per antenna at R and S, respectively, N 2,1 and N 2,2 are 2 2 AWGN matrices, F GβH 2, N e1 R θ W 2 βh 2 +N 2,1, and N e2 W 1 βh 2 + N 2,2. From (7), we can see that a QO-STBC is transmitted to D in the second phe. For NT-AF, the equivalent vector model can be expressed cv(y D1,1 ) cv(y D1,2) cv(y D2,1) cv(y D2,2) where x x 1 x 2 x 3 x 4 T. D. NT-SDF Scheme P1 H P1 H 1 P3H 1 P1e jθ F x + P1 F P3 e jθ H 1 cv(n 1,1 ) cv(n 1,2) cv(n e1) cv(n e2) Similarly to Subsection III-C, for NT-SDF, it is sumed that S transmits X 1 and X 2 in the first phe, and a QO-STBC is used in the second phe. Since there is no difference between NT-AF and NT-SDF in the first phe, the received signals at R and D are given in (6). Considering SDF protocol, R performs the similar operation in Subsection III-B. Soft decision values at R can be obtained ˆx1 ˆx 2 γ(g ) H cv(y R1 ), (8) ˆx3 ˆx 4 γ(g ) H cv(y R2 ) where γ is the same in (4). Similarly to Subsection III-C, to construct a QO-STBC in the second phe, R transmits R θ A(ˆx 3, ˆx 4 ) at first and then transmits A(ˆx 1, ˆx 2 ). S transmits X 1 at first and then R θ X 2 in the second phe. The equivalent vector model can be expressed cv(y D1,1 ) cv(y D1,2 ) cv(y D2,1 ) cv(y D2,2 ) P1H P1H 1 P3 H 1 ae jθ H x + 2 ah 2 P3e jθ H 1 cv(n 1,1 ) cv(n 1,2 ) cv(n e1 ) cv(n e2 ) where a is the same in (5), x x 1 x 2 x 3 x 4 T, cv(n e1 ) γe jθ H 2(G ) H cv(w 2 ) + cv(n 2,1 ), and cv(n e2 ) γh 2(G ) H cv(w 1 ) + cv(n 2,2 ). IV. OPTIMAL ML DECODERS In this section, we derive the optimal ML decoder for each scheme presented in the previous section. To this end, we have derived the equivalent vector model. The four equivalent vector models (3), (5), (8), and (9) can be commonly expressed (9) y e H e x + n e. (10) Since the noise at R is transmitted to both receive antenn at D, entries of equivalent noise vector n e are correlated. Let K ne be the covariance matrix of n e. Then, the ML decoder of (10) can be derived ˆx arg min (ye H e x) H K 1 n x e (y e H e x). (11) Thus, we need to derive K ne and calculate the term inside the bracket in (11) for each scheme. We use Alamouti code a building block and thus the following properties for Alamouti operation are useful to derive the ML decoder for each scheme. Properties of Alamouti operation: Properties of Alamouti operation: 1) A H (a, b) A(a, b); 2) A(a, b) + A(c, d) A(a + c, b + d); 3) A(a, b) A(c, d) A(ac bd, ad + bc ); 4) A H (a, b) A(a, b) A(a, b) A H (a, b) A( a 2 + b 2, 0) ( a 2 + b 2 )I 2. Let P K a xi 2 A(u, v) A(u, v) yi 2 P H K a P {p( a 2 + b 2 ) M A(a, b), Q M A(c, d) pi 2 A(q, 0) A(q, 0) ri 2. Then, we have M A(e, f) M A(g, h),, and K s + r( c 2 + d 2 ) + 2R(a c + b d)q}i 2 (12) P H K s P {x( a 2 + b 2 ) + y( c 2 + d 2 ) + 2R(a c + b d)u + (a d b c )v}i 2 (13) P H K a Q + Q H K a P {2pRa e + bf + 2rRc g + dh + 2R(a g + b h + ce + df )q}i 2 (14) P H K s Q + Q H K s P {2xRa e + bf + 2yRc g + dh + 2R(a g + b h + ce + df )u + (a h b g + d e c f )v}i 2. (15)

4 OT-AF (ML decoding) OT-AF (squaring) OT-SDF (ML decoding) OT-SDF (squaring) Fig. 2. Performance comparison of various orthogonal protocols with QPSK. OT-AF (ML decoding) OT-AF (squaring) OT-SDF (ML decoding) OT-SDF (squaring) Fig. 3. Performance comparison of various orthogonal protocols with 16QAM. The above four equations can be eily shown by using properties of Alamouti operation. For AF and SDF protocols, we derive the covariance matrix of cv(n) which corresponds to the equivalent noise at D in the second phe under OT. Considering the AF protocol, the covariance matrix of cv(n) in (3) is given K cv(n) E(cv(N)(cv(N)) H ) pi 2 A(q, 0) A(q, 0) ri 2 (16) where p β1 h β2 h , q β1h ) + β2h (h 22 2 ), and r β1 h β2 h For SDF protocol, the covariance matrix of cv(n) in (5) is given 2 (h 12 K cv(n) E(cv(N)(cv(N)) H P 2 ) 1 + P 1 G 2 H 2(H 2) H + I 4 xi 2 A(u, v) A(u. (17), v) yi 2 Let y H e K 1 n e H e η 1 η 2 for OT and y H e K 1 n e H e η 1 η 2 η 3 η 4 for NT. Using (3), (12), and (16), after some manipulations, the ML decoder for OT-AF can be simplified to symbolwise decoder x i arg min x i (P1 H t a ) x i 2 2Rη i x i, for i 1, 2 where t a P 1 s a /(pr q 2 ) and s a ( f f 22 2 )p + ( f f 21 2 )r 2R(f 11f 12 + f 21f 22 )q. Similarly to OT-AF, using (5), (13), and (17), the ML decoder for OT-SDF can be simplified to symbolwise decoder x i arg min x i (P1 H t s ) x i 2 2Rη i x i, for i 1, 2 where t s a 2 s s /(xy u 2 v 2 ) and s s ( h h )x+( h h )y 2R((h 11 2 ) h (h 21 2 ) h 22 2 )u+ ((h 11 2 ) (h 22 2 ) (h 21 2 ) )v. Using (8), (14), and (16), after some manipulations, the ML decoder for NT-AF can be simplified to pairwise decoder ( x i, x i+2 ) arg min x i,x i+2 αa ( x i 2 + x i+2 2 ) +2Rβ ax i x i+2 η i x i η i+2 x i+2, for i 1, 2 where α a P 1 H (P 3 s a1 + P 1 s a2 )/(pr q 2 ) and β a P 1 P 3 e jθ z a. Note that s a1 ( h h )p + ( h h )r 2R((h (h 21 1 )q, s a2 ( f f 22 2 )p + ( f f 21 2 )r 2R(f11f 12 + f21f 22 )q, and (pr q 2 )z a 2rR(h 11 1 ) f 11 + h 21 1 f21 + 2pR(h 12 1 ) f 12 + h 22 1 f22 2R((h 11 1 ) f 12 + (h 21 1 ) f f21)q. h 12 1 f 11 + h 22 Similarly to NT-AF, using (9), (15), and (17), the ML decoder for NT-SDF can be simplified to pairwise decoder ( x i, x i+2 ) arg min x i,x i+2 αs ( x i 2 + x i+2 2 ) +2Rβ s x i x i+2 η i x i η i+2 x i+2, for i 1, 2 where α s P 1 H (P 3 s s1 + a 2 s s2 )/(xy u 2 v 2 ) and β s a P 3 e jθ z s. Note that s s1 ( h h )x + ( h h )y 2R((h (h 21 1 )u + ((h 11 1 ) (h 22 1 ) (h 12 1 ) (h 21 1 ) )v, s s2 ( h h )x+( h h )y 2R((h 11 2 ) h ) )v, and (h 21 2 ) h 22 2 )u + ((h 11 2 ) (h 22 2 ) (h 21 (xy u 2 v 2 )z s 2yR(h 11 1 ) h h 21 1 (h 21 2 ) + 2xR(h h 22 1 (h 22 2 ) 2R((h (h 21 h 12 1 (h 11 2 ) + h 22 1 (h 21 2 ) )u + ((h 11 1 ) (h 22 2 ) (h 21 1 ) (h 12 2 ) )v. (h 22 1 ) (h 11 1 ) (h 21 V. NUMERICAL RESULTS ) + All the channels are sumed to be qui-static Rayleigh fading channels and QPSK and 16QAM are sumed to be used. The average power of the transmitted symbols is sumed to be 1. For OT, the equal power distribution is sumed, i.e., P 1 P 2. Figs 2 and 3 show the performance of OT-AF and OT-SDF with QPSK and 16QAM, respectively. The horizontal axis represents the total transmit power per antenna, i.e., P 1 + P 2. ML decoding means that D uses the optimal ML decoder derived in the previous section and squaring means that D uses squaring method presented in 6. It is shown that SDF protocol outperforms AF protocol.

5 NT-AF NT-AF (SAS) NT-SDF NT-SDF (SAS) Fig. 4. Performance comparison of various non-orthogonal protocols with QPSK. NT-AF NT-AF (SAS) NT-SDF NT-SDF (SAS) Fig. 5. Performance comparison of various non-orthogonal protocols with 16QAM. Also, optimal decoder shows better performance than squaring decoder, although they have the same decoding complexity. For NT, the power used in the first phe is sumed to be equal to the total power used in the second phe, and S and R are sumed to use the same power in the second phe. Thus, we sume P 1 + P 2 + P 3, P 1 P 2 + P 3, and P 2 P 3. Also, we apply SAS technique to achieve better performance. Although SAS w applied to DF protocol in 4, SAS can be similarly applied to AF and SDF protocols. We use θ π/4 which is optimal angle with respect to diversity product 10. Figs 4 and 5 show the performance of NT-AF and NT-SDF with QPSK and 16QAM, respectively. It can be seen that SDF protocol also shows better performance than AF protocol and SAS technique enhances the diversity order of the cooperative communication networks. 7 X. Li, T. Luo, G. Yue, and C. Yin, A squaring method to simplify the decoding of orthogonal space-time block codes, IEEE Trans. Commun., vol. 49, no. 10, pp , Oct S. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Sel. Are Commun., vol. 16, no. 8, pp , Aug O. Tirkkonen, A. Boariu, and A. Hottinen, Minimal nonorthogonality rate 1 space-time block code for 3+ Tx antenn, IEEE 6th Int. Symp. On Spread-Spectrum Tech. & Appl (ISSSTA 2000), pp , Sep W. Su and X. Xia, Signal constellations for qui-orthogonal spacetime block codes with full diversity, IEEE Trans. Inf. Theory, vol. 50, no. 10, pp , Oct VI. CONCLUSION In this paper, we introduce the SDF protocol which decodes the received signals into the soft-decision values at R. Furthermore, we design DSTCs for the cooperation between S and R. It is shown that the ML decoders for AF and SDF protocols with both OT and NT can be simplified although there are correlated noises at D. From the numerical analysis, it is confirmed that SDF protocol shows better performance than AF protocol. REFERENCES 1 T. M. Cover and A. A. El Gamal, Capacity theorems for the relay channel, IEEE Trans. Inf. Theory, vol. IT-23, no. 5, pp , Sep A. Nosratinia and A. Hedayat, Cooperative communication in wireless networks, IEEE Commun. Mag., vol. 42, no. 10, pp , Oct C. Hucher, G. R. Rekaya, and J. Belfiore, AF and DF protocols bed on Alamouti ST code, IEEE Int. Symp. On Inf. Theory (ISIT 2007), pp , Jun X. Jin, J.-D. Yang, J.-S. No, and D.-J. Shin, Distributed space-time coded non-orthogonal DF protocol with source antenna switching, submitted to IEEE Trans. Wireless Commun., Aug J. Vazifehdan and H. Shafiee, Cooperative diversity in space-time coded wireless networks, ICCS 2004, pp , Sep I.-H. Lee and D.-W. Kim, Decouple-and-forward relaying for dual-hop Alamouti transmissions, IEEE Commun. Lett., vol. 12, no. 2, pp , Feb