Novel Symbol-Wise ML Decodable STBC for IEEE e/m Standard

Transcription

1 Novel Symbol-Wise ML Decodable STBC for IEEE e/m Standard Tian Peng Ren 1 Chau Yuen 2 Yong Liang Guan 3 and Rong Jun Shen 4 1 National University of Defense Technology Changsha China 2 Institute for Information Research Singapore Nanyang Technological University Singapore General Equipment Department of PLA Beijing China tpren@nudt.edu.cn cyuen@i2r.a-star.edu.sg and eylguan@ntu.edu.sg Abstract In this paper novel real symbol-wise ML decodable STBCs with balanced information distribution per codeword are constructed systemically for arbitrary number of transmit antennas. Compared with the existing real symbol-wise ML decodable STBCs in IEEE e standard the proposed STBCs also have lower peak-to-average power ratio (PAPR) and better performance even without adaptive antenna grouping based on channel state information(csi) feedback. I. INTRODUCTION Symbol-wise maximum likelihood (ML) decodable spacetime block codes (STBCs) provide spatial diversity gain from multi-input multi-output (MIMO) channel and low decoding complexity these codes have thus been adopted by various standards 1 2. In IEEE e standard 1 the symbolwise ML decodable STBCs for 3 and 4 transmit antennas are proposed as X 3 and X 4 with diversity level 2 and code rate 1: X 3 = X 4 = s 1 s s 2 s 1 s 3 s s 4 s 3 s 1 s s 2 s s 3 s s 4 s 3 It is easy to see that for X 3 the transmit power among different antennas is unbalanced and the maximum peak-to-average power ratio (PAPR) on X 3 and X 4 caused by constant envelope modulation based space-time coding is 3 db due to the 0 symbols. The regular transmission of 0 implies turning off the transmit antennas at regular intervals. This leads to an Part of the work was done when the author C. Yuen was visiting Hong Kong Polytechnic University. undesirable low-frequency interference and some difficulties in the front-end power amplifier design 3 4. Furthermore for X 3 and X 4 the information distribution is unbalanced and then the feedback channel state information (CSI) is needed for the transmitter to maintain the robustness of the channel condition 5 6. Several CSI closed-loop studies have been conducted on the limited feedback problem in 7 10 called adaptive antenna grouping. Adaptive antenna grouping exhibits good system performance in low mobility environments but the performance degraded significantly in high mobility environments due to outdated feedback information 10. Although a simple feedback strategy that offers better link reliability in a wide range of mobility environments is proposed in 10 the solution is still not desirable because the mobility in the coming IEEE m will up to 350 km/h. This motivates us to design new robust codes that do not require any CSIfeedback for both low and high mobility environments. In this paper novel real symbol-wise ML decodable STBCs with balanced power distribution and low PAPR are constructed systematically for arbitrary number of transmit antennas. Moreover compared with X 3 and X 4 these STBCs have balanced information distribution and thus can work robustly without CSI-feedback based adaptive antenna grouping which also increases the band efficiency and reduces the system implementation complexity. Due to the systematic construction all the novel codes have robust structure and are of great value for the coming IEEE standards. The rest of this paper is organized as follows. In section II the system model and the construction of these novel codes are described. Theoretical analysis and Monte Carlo simulation on the bit error ratio () performance are presented in section III. This paper is concluded in section IV.

2 In what follows bold upper case and lower case letters denote matrices (sets) and vectors respectively; H and denote the complex conjugate the complex conjugate transpose and the Frobenius norm of a matrix respectively. II. NOVEL SYMBOL-WISE ML DECODABLE STBC A. STBC Model We consider a space-time block coding system employing N transmit antennas and M receive antennas. The transmitted signal sequences are partitioned into independent time block for transmission over T symbol durations which is STBC matrix X of size T N. Following the signal model in 11 X can be denoted as: X T N = L c l C l (1) where L is the number of transmitted real symbols c l are L real-value transmitted symbols. Thus the code rate is 2T considering complex symbol transmission. C l called as the dispersion matrices are of size T N and normalized by the power distribution constrain tr(c H l C l )=E/L where E is the average transmit power 11. At the receiver the signal y tm received by antenna m and time t is given by r tm = l=1 N h nm x tn + z tm (2) n=1 where z tm is the additive white Gaussian channel noise with variance /2 per dimension at time t. The coefficient h nm is the path gain from transmit antenna n to receive antenna m. As the most case we assume that the coefficients h nm are independent samples of a zero-mean complex Gaussian random variable with variance E( h 2 nm). The received signals r tm can be arranged in a matrix R = r 1 r 2 r M = r tm of size T M. Thus the transmitreceive signal relation can be represented as: R = X H + Z (3) where H = h 1 h 2 h M = h nm N M Z = z 1 z 2 z M = z tm T M. The received signal can also be shown as 11: r = ρhs + z (4) where r = r R 1 r I 1... r R M r I M h = h R 1 h I 1.. h R M h I M s = s 1 s 2.. s L z = H =h 1 h 2 h L = C 1 h C 2 h C L h C l C l 0 C l = C l M M C l = C R l C I l z R 1 z I 1... z R M z I M C I l C R l 2 2 and l =1 2 L. The ML decoding of STBC is to find the solution ŝ so that ŝ = argmin r Hs 2 (5) s For symbol-wise ML decodable STBC it is required that H T H is a scaled identity matrix 12. Note that symbolwise means real symbol-wise in this paper. In this case the ML decoding metric (5) is converted into i.e. ŝ = L l=1 argmin s l (ˆr l h l 2 s l ) 2 (6) ŝ l = argmin(ˆr l h l 2 s l ) 2 (7) s l where ˆr =ˆr 1 ˆr 2 ˆr L T = H T r ŝ =ŝ 1 ŝ 2 ŝ L and l =1 2 L. B. Construction It has been proved that the necessary and sufficient condition for X to be symbol-wise ML decodable is that the dispersion matrices must satisfy Quasi-Orthogonal Constraint (QOC) 13 with each other i.e. C H l 1 C l2 = C H l 2 C l1 l 1 l 2 =1 2 L and l 1 l 2. (8) In the following one theorem is derived to construct a symbolwise ML decodable STBC based on another symbol-wise ML decodable STBC systematically which is represented as Theorem 1. Suppose that Ĉl(l =1 2 L) are the dispersion matrices of a symbol-wise ML decodable STBC X 1 of size T N and code rate L 2T then Ĉl Ĉ l Ĉl Ĉl C 2l 1 = C 2l = Ĉ l Ĉ l Ĉl Ĉ l

3 can be regarded as the dispersion matrices of one symbolwise ML decodable STBC X 2 of size 2T 2N which is with the same diversity level and code rate as X 1. Proof: Because Ĉl are the dispersion matrices of a symbol-wise ML decodable STBC that satisfy the constraint Ĉ H l 1 = ĈH l 2 where l l 1 l 2 =1 2 L and l 1 l 2 we have C H 2l 1 1C 2l2 1 = = Ĉl1 H Ĉl2 = 2ĈH l 1 2ĈH l 1 2ĈH l 1 2ĈH l 1 = 2ĈH l 2 2ĈH l 2 2ĈH l 2 2ĈH l 2 H Ĉl2 Ĉl1 = C H 2l 2 1C 2l1 1 Similarly it is easy to prove that C H 2l 1 1C 2l1 = C H 2l 1 C 2l1 1 (9) C H 2l 1 1C 2l2 = C H 2l 2 C 2l1 1 (10) C H 2l 1 C 2l2 = C H 2l 2 C 2l1 Therefore C 2l 1 and C 2l (l = 1 2 L) can be applied as the dispersion matrices of the symbol-wise ML decodable STBC X 2. Since rank(ĉl) =rank(c 2l 1 )=rank(c 2l ) the diversity levels of X 1 and X 2 are the same. Clearly the code rate 2L of X 2 is 2 2T = L 2T as the same as that of X 1. C. STBC for IEEE e/m standard Following the construction in Theorem 1 the symbol-wise ML decodable STBC with diversity level 2 and code rate 1 for arbitrary number of transmit antennas can be constructed based on Alamouti code 15 where the dispersion matrices are presented as follows: 1 0 j j j 1 0 j 0 where j = 1. For example the symbol-wise ML decodable STBC for 4 transmit antennas can be obtained as: s 1 + s 3 s 2 s 4 s 1 s 3 s 2 + s 4 s 2 + s 4 s 1 + s 3 s 2 s 4 s 1 s 3 X 4new = s 1 s 3 s 2 + s 4 s 1 + s 3 s 2 s 4 s 2 s 4 s 1 s 3 s 2 + s 4 s 1 + s 3 The symbol-wise ML decodable STBC for 3 transmit antennas can be built by the first 3 columns in X 4new i.e. X 3new = s 1 + s 3 s 2 s 4 s 1 s 3 s 2 + s 4 s 2 + s 4 s 1 + s 3 s 2 s 4 s 1 s 3 s 1 s 3 s 2 + s 4 s 1 + s 3 s 2 s 4 Clearly X 3new (X 4new ) is with the same diversity level and code rate as X 3 (X 4 ). Moreover the power distributions of X 3new /X 4new are balanced. To estimate the PAPR caused by space-time coding we only consider constant envelope modulation e.g. M-ary phase-shift keying(mpsk). When MPSK is applied a proposition on PAPR of X 3new and X 4new can be derived as follows: Proposition 1. The PAPR of X 3new (X 4new ) is 10 log(2 cos 2 π 2M ) db when s 1s 2 are obtained from MPSK and s 3 s 4 are obtained from π M -rotated MPSK. Proof: Assuming that the complex symbols transmitted are obtained from MPSK with unit power the average transmission power is 2 due to 2 complex symbols transmitted at the same space-time position. If s 1 and s 2 are obtained from MPSK while s 3 s 4 are obtained from φ-rotated MPSK where φ 0 π M wehave that the peak transmission power is 1+exp(jφ) 2. Thus the PAPR(dB) of X 3new (X 4new ) can be expressed as 1+exp(jφ) 2 10 log( ) 2 = 10 log( (1 + cos φ)2 +sin 2 φ ) 2 = 10 log(2 cos 2 φ 2 ) (11) Since that cos 2 φ 2 is monotonically decreasing function when φ 0 π M PAPR(dB)ofX 3new(X 4new ) reaches its minimum 10 log(2 cos 2 π 2M ) when φ = π M. Clearly 10 log(2 cos 2 π 2M ) db is lower than 3 db. Specially PAPR is 0 db when s 1 s 2 are obtained from binary PSK and s 3 s 4 are got from π 2 -rotated binary PSK. III. PERFORMANCE EVALUATION In this section we analyze the performance of the symbol-wise ML decodable STBC proposed in IEEE e standard and this paper. Suppose that there is 1 receive antenna and the Rayleigh fading channel is quasi-static in the sense that the channel coefficients do not change during one codeword transmission.

4 A. Performance Analysis Since it is difficult to get the closed performance form for STBC when E( h 2 n1)(n = ) are different 14 for simplification we assume that E( h 2 11) = E( h 2 21) and E( h 2 31) =E( h 2 41) when X 4 is applied. Consider uncoded binary PSK signals it is well known that the bit error probability of s 1 and s 2 is 16 P b (SNR) 3 16 SNR 2 (12) when the signal to noise ratio SNR = E(x2 tn )E( h 2 11 ) = E( h 2 11 ) is sufficiently large. Similar s 3 and s 4 have the same bit error probability when E( h 2 31 ) is sufficiently large. Therefore the system bit error probability is P b 3 ( E( h 2 11) ) 2 +( E( h 2 31) ) 2 (13) 32 When the proposed STBC X 4new is applied the transmission of s i (i = ) implemented by 4 transmit antennas is different from that of X 4 where the transmission of s i (i = ) is implemented by 2 transmit antennas only. In other words the information in X 4new spreads through all paths and then adaptive antenna grouping operation is not necessary. Under the same condition the system bit error probability should be 2 P bnew 3 E( h 2 11)+E( h 2 31) (14) 16 2 Based on Cauchy-Schwartz inequality we can prove that P b P bnew and the equality holds if and only if E( h 2 11) = E( h 2 31). For more general results the comparisons on performance will be shown in the following section. B. Monte Carlo Simulation For comparison the Monte Carlo simulations of three schemes: X 3 /X 4 without adaptive antenna grouping X 3 /X 4 with adaptive antenna grouping and X 3new /X 4new without adaptive antenna grouping are proposed. Here without loss of generality the relative path attenuation vector is adopted as db i.e. 10 log E( h 2 21 ) = 1 db and so on. E( h 2 11 ) We plot against the signal to noise ratio (SNR) in Fig. 1 and 2 where quadrature PSK (QPSK) is applied. Clearly the performance of X 3new /X 4new without adaptive antenna grouping is better than that of X 3 /X 4 no matter with or without adaptive antenna grouping although both of them have the similar diversity gain of 2. Furthermore we consider an extreme case: one path is unavailable suddenly. Then the minimum diversity level of Fig. 1. performance of different coding schemes(qpsk and 3 transmit antennas are applied the relative path attenuation vector is db) X 3 /X 4 must reduce to 1 while the minimum diversity level of X 4new remains 2 as Fig. 3 where X 4 and X 4new are applied and the third path is unavailable suddenly. It can be seen that the performance gain of the novel codes is more impressive in a deep-fading environment. If adaptive antenna grouping is applied for X 3new the minimum diversity level of X 3new also remains 2 in this caes. These results show that X 3new /X 4new have higher robustness than that of X 3 /X 4. Meanwhile there is no CSI-feedback based adaptive antenna grouping for X 3new /X 4new. Therefore the bandwidth efficiency is improved and the system implementation is also simpler. We would like to emphasize that all the codes constructed by Theorem 1 have the same high robustness as X 3new /X 4new for arbitrary number of transmit antennas. IV. CONCLUSION In this paper a scalable construction of real symbol-wise ML decodable STBCs with robust structure for arbitrary number of transmit antennas are presented. The novel codes are with balanced transmit power and low PAPR. Compared with the existing real symbol-wise ML decodable STBCs in IEEE e the proposed STBCs have higher robustness and better performance even without CSI-feedback based adaptive antenna grouping hence they are competitive for the forthcoming standards e.g. IEEE m.

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