Cooperative Communication in Spatially Modulated MIMO systems Multimedia Wireless Networks (MWN) Group, Department Of Electrical Engineering, Indian Institute of Technology, Kanpur, India {neerajv,adityaj}@iitk.ac.in Electrical and Computer Engineering, University of Illinois, Urbana, Illinois agoel10@illinois.edu April 5, 2016
Overview 1 Introduction 2 3 4 5
Introduction Spatial modulation (SM) 1 has been recently gaining popularity due to its ability to enable MIMO communication with fewer transmit RF chains. SM systems can be combined with STBC to obtain a substantial improvement in performance due to the additional transmit diversity and enhanced spectral efficiency 2. In the STBC-SM scheme, both STBC symbols and the indices of the transmit antennas from which these symbols are transmitted, carry information. The end-to-end reliability of the STBC-SM system can be further enhanced by exploiting the cooperative diversity along with spatial diversity. 1 R. Y. Mesleh, H. Haas, S. Sinanovic, C. W. Ahn, and S. Yun, Spatial modulation, IEEE Transactions on Vehicular Technology, vol. 57, no. 4, pp. 2228-2241, 2008. 2 E. Basar, U. Aygolu, E. Panayirci, and H. V. Poor, Space-time block coded spatial modulation, IEEE Transactions on Communications, vol. 59, no. 3, pp. 823-832, 2011.
I Figure: Selective DF based SM MIMO-STBC Cooperative System.
II Kronecker channel model is employed to model MIMO channel matrices with receive/transmit antenna correlation as 3, H SD =R 1 2 d HSD R T 2 s, H SR =R 1 2 r HSR R T 2 s, H RD =R 1 2 d HRD R T 2 r, where R s,r d and R r denote the spatial correlation matrices at the source, destination and relay nodes respectively. The coefficients of H SD, H SR and H RD are distributed as CN(0,δ 2 sd ), CN(0,δ2 sr ) and CN(0,δ2 rd ) respectively. 3 D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, Fading correlation and its effect on the capacity of multielement antenna systems, IEEE Transactions on Communications, vol. 48, no. 3, pp. 502-513, 2000.
III The received codeword matrices Y SD C Nd T,Y SR C Nr T at the destination and relay respectively in the first phase are given as, P0 Y SD = H SD X+W SD, N a P0 Y SR = H SR X+W SR, N a where X C Ns T denotes the transmitted codeword, P 0 is the transmit power at the source, N a denotes number of active antennas at the transmitter. Entries of matrices W SD C N d T and W SR C Nr T are symmetric complex Gaussian with variance η 0. (1)
IV The received codeword matrix Y RD C Nd T at the destination, in the second phase, is given as, P1 Y RD = H RD X+W RD, (2) N a where P 1 is the transmit power at the relay, Entries of matrix W RD C N d T is symmetric complex Gaussian with variance η 0 /2 per dimension.
Pair-wise Error Probability (PEP) I Theorem (Average PEP) The average PEP bound at the destination for the cooperative SM MIMO-STBC system is given by C C P e Pr(X i ) P e (X i X j ), (3) i=1 j=1,j i where the end-to-end PEP for the error event X i X j is P e (X i X j ) =P S D (X i X j ) P e S R (X i) +P S D,R D (X i X j ) (1 P e S R (X i)), and the average error probability at the relay is C PS R e (X i) = P S R (X i X k ). k=1,k i
Pair-wise Error Probability (PEP) II The PEP for the error event corresponding to the transmitted codeword X i C being confused for codeword X k C at the relay, where k i, conditioned on H SR, is given as 4, P P S R (X i X k H SR ) =Q 0 H SR (X i X k ) 2 F, 2N a η 0 = 1 π π/2 0 exp( P0 H SR (X i X k ) 2 F 4N a η 0 sin 2 θ ) dθ, 4 N. Varshney, A. Krishna, and A. K. Jagannatham, Selective DF protocol for MIMO STBC based single/multiple relay cooperative communication: End-to-end performance and optimal power allocation, IEEE Transactions on Communications, vol. 63, no. 7, pp. 2458-2474, 2015.
Pair-wise Error Probability (PEP) III The average PEP at the relay can now be obtained as, π/2 1 P0 H SR (X i X k ) P S R (X i X k )=E HSR exp( 2 ) F π 4N a η 0 sin 2 dθ θ, = 1 π π/2 0 0 φ ik H SR ik 2 F ( ) P 0 4N a η 0 sin 2 dθ, (4) θ where φ ik H SR ik 2 F(s) denotes the moment generating function of H SR (X i X k ) 2 F = H SR ik 2 F, where ik = X i X k.
Pair-wise Error Probability (PEP) IV The moment generating function of H SR ik 2 F obtained as 5, can be φ ik H SR ik 2 F(s) = γa ik a=1b=1 γb (1 sµ ik a λ b δ2 SR ) 1, where µ ik a is the a th eigenvalue of the matrix ik H ik R s, λ b is the bth eigenvalue of the correlation matrix R r, γa ik denotes the rank of the matrix ik H ik R s, γb denotes the rank of the correlation matrix R r. 5 A. Hedayat, H. Shah, and A. Nosratinia, Analysis of space-time coding in correlated fading channels, IEEE Transactions on Wireless Communications, vol. 4, no. 6, pp. 2882-2891, 2005.
Pair-wise Error Probability (PEP) V Using φ ik H SR ik 2 F(s) = γa ik a=1b=1 γb (1 sµ ik a λ b δ2 SR ) 1, the average error probability at the relay can be solved as, π/2 P S R (X i X k ) = 1 π 0 where = G γa ik γ b a=1b=1 γa ik γ b a=1b=1 (1+ P 0µ ik a λ ) 1 b δ2 SR 4N a η 0 sin 2 dθ θ (1+ P 0µ ik a λ ) b δ2 SR, 4N a η 0 sin 2 (5) θ The quantity G(v(θ)) is defined as G(v(θ))= 1 π π/2 1 v(θ) dθ. 0
Pair-wise Error Probability (PEP) VI Following a similar approach, the average PEP of confusion event X i X j at the destination when the relay decodes all the symbols erroneously, can be solved as, γ ij ( ) a γ b P S D (X i X j )=G 1+ P 0µ ij aλ b δsd 2, 4N a η 0 sin 2 (6) θ a=1b=1 where µ ij a is the a th eigenvalue of the matrix ij H ij R s, λ b is the b th eigenvalue of the correlation matrix R d, γa ij denotes the rank of the matrix ij H ij R s, γ b denotes the rank of the correlation matrix R d.
Pair-wise Error Probability (PEP) VII The average PEP of the error event X i X j at the destination when the relay decodes all the symbols correctly, is given as, P S D,R D (X i X j ) = E HSD,H RD { 1 π = G ( γ ij a γ b a=1b=1 π/2 0 exp( P0 H SD ij 2 F P 1 H RD ij 2 F 4N a η 0 sin 2 θ (1+ P 0µ ij aλ b δ 2 SD 4N a η 0 sin 2 θ ) γp ij γ q ( p=1q=1 1+ P 1µ ij pλ q δ 2 RD 4N a η 0 sin 2 θ ) } dθ )) where µ ij p is the pth eigenvalue of the matrix ij H ij R r, λ q represents the q th eigenvalue of the correlation matrix R d, γp ij denotes the rank of the matrix ij H ij R r, γ q denotes the rank of the correlation matrix R d.,
Diversity Order Analysis I Using the fact that at high SNR 1 P e S R (X i) 1, 1+ P0µij a λ bδsd 2 4N aη 0 sin 2 θ P0µij a λ b δsd 2 4N aη 0 sin 2 θ 1+ P1µij p λqδ2 RD 4N aη 0 sin 2 θ C P e i=1,1+ P0µik a λ b δ2 SR 4N aη 0 sin 2 θ P0µik a λ b δ2 SR 1, the asymptotic PEP is given as, [ C C j=1, j i k=1, k i 4N aη 0 sin 2 θ and ( ) (γ ij P a γ b+γ ik a γ b ) ( ) (γ ij P a γ b+γ ij γq)] p A 1 +A 2 η 0 η 0 ( ) (γaγ P b +γ aγb ) ( ) (γaγ P b +γ pγ q) =C 1 +C 2, (7) η 0 η 0 where γ ij a = rank( ij H ij R s) = γ a and γ ij p = rank( ij H ij R r) = γ p. This assumption is true if spatial correlation matrices have full rank and all the codeword difference-matrices have equal rank.
Diversity Order Analysis II The diversity order of the above system is log(p lim ( e) P log P = γ a γ b +min{γ a γb,γ pγ q } η η0) 0 For uncorrelated systems, the diversity order is γ a N d +min{γ a N r,γ p N d }, where γ a = γ p denotes the rank of the matrix ij H ij. This result shows that both the performance of the system and the resulting diversity order depend on the the correlation. If the correlation matrix at any of the nodes is not full rank, the diversity order of the system decreases.
Optimal Power Allocation I The convex optimization problem for optimal source-relay power allocation can be formulated as, { } α β min + a γaγ, b 0 a γpγq 1 (8) where α=c 1 a0 =1 ( P η0 a γaγ b+γ aγ b 0 s.t. a 0 +a 1 = 1, ) (γaγ b +γ aγ b ) and β=c 2 a0 =1,a 1 =1 ( P η0 ) (γaγ b +γ pγ q).
Optimal Power Allocation II Theorem (Optimal Power Allocation) One non-negative zero of the polynomial equation below α(γ a γ b +γ a γ b )(1 a 0) 1+γpγq β(γ a γ b +γ p γ q )a 1+γaγ b 0 +β(γ a γ b )a γaγ b 0 =0, determines the optimal power factor a0 at the source and hence, the optimal powers at the source and relay are P0 = a 0 P and P1 = (1 a 0 )P for a given total cooperative power budget P.
Simulation Result I: PEP Performance 10 0 Pair wise Error Probability (PEP) 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 STBC (Simulated) [6] STBC SM (Simulated) [7] Cooperative STBC (Simulated) [8] Proposed Cooperative STBC SM (Simulated) Proposed Cooperative STBC SM (Analytical) 0 5 10 15 SNR (db) Asymptotic Bound, Diversity Order = 8 Figure: Comparison of the PEP performance of the proposed scheme with that of existing works for unity average channel gains. 678 6 V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block coding for wireless communications: performance results, IEEE Journal on Selected Areas in Communications, vol. 17, no. 3, pp. 451460, 1999. 7 E. Basar, U. Aygolu, E. Panayirci, and H. V. Poor, Space-time block coded spatial modulation, IEEE Transactions on Communications, vol. 59, no. 3, pp. 823832, 2011. 8 N. Varshney, A. Krishna, and A. K. Jagannatham, Selective DF protocol for MIMO STBC based single/multiple relay cooperative communication: End-to-end performance and optimal power allocation, IEEE Transactions on Communications, vol. 63, no. 7, pp. 2458-2474, 2015.
Simulation Result II: PEP Performance 10 0 Pair wise Error Probability (PEP) 10 1 10 2 10 3 10 4 PEP (Simulated), r = 0 PEP (Analytical), r = 0 PEP (Simulated), r = 0.1 10 5 PEP (Analytical), r = 0.1 PEP (Simulated), r = 0.5 10 6 PEP (Analytical), r = 0.5 PEP (Simulated), r = 0.9 PEP (Analytical), r = 0.9 10 7 0 4 8 12 16 SNR (db) Figure: PEP performance of the proposed SM MIMO-STBC based selective DF cooperative scheme over correlated Rayleigh fading links.
Simulation Result III: PEP Performance 10 0 Pair wise Error Probability (PEP) 10 1 10 2 10 3 10 4 10 5 Uniform Power Scenario I Optimal Power Scenario I 10 6 Uniform Power = Optimal Power Scenario II Uniform Power Scenario III Optimal Power Scenario III 10 7 0 2 4 6 8 10 12 SNR(dB) Figure: PEP Performance of the cooperative SM MIMO-STBC system using optimal/equal power allocation for various channel conditions. Scenario I: δ 2 sr =δ2 sd =δ2 rd =1, Scenario II: δ 2 sr =100,δ2 sd =δ2 rd =1, Scenario III: δ 2 sr=δ 2 sd =1,δ2 rd =100.
Closed form analytical expressions have been derived for the end-to-end PEP, diversity order and optimal power allocation for a selective DF based SM MIMO-STBC cooperative system considering receive/transmit antenna correlation. It has been observed that the end-to-end performance of selective DF based SM MIMO-STBC cooperative systems is superior in comparison to that of STBC, STBC-SM and STBC based cooperative systems.
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