2015 International Conference on Advanced Technologies for Communications (ATC) High-Rate Space-Time Block Coded Spatial Modulation Binh T Vo∗ , Ha H Nguyen∗ and Nguyen Quoc-Tuan† ∗ University of Saskatchewan, Saskatoon, SK, Canada National University, Hanoi, Vietnam binh.vo@usask.ca, ha.nguyen@usask.ca, tuannq@vnu.edu.vn † Vietnam it offers both increased spatial diversity as well as higher transmission rate The scheme proposed in [3], called spacetime block coded spatial modulation (STBC-SM), makes use of the famous Alamouti STBC as a core In contrast, our proposed scheme can increase the data rate and achieve a transmit diversity order of two by making use of the high-rate STBC in [5] To distinguish it from the STBC-SM scheme in [3], the scheme proposed here shall be referred to as highrate space-time block coded spatial modulation (HR-STBCSM) In addition to the coding gain analysis of the proposed HR-STBC-SM scheme, a simplified ML detection is also developed Simulation results shall demonstrate that the HRSTBC-SM scheme outperforms the STBC-SM scheme at high spectral efficiency It also outperforms the scheme recently proposed in [4] that is based on an error-correcting code The remaining of this paper is organized as follows Section II presents our proposed HR-STBC-SM scheme In Section III, a simplified ML detection is obtained to reduce the decoding complexity at the receiver and performance analysis of the HR-STBC-SM scheme is carried out Simulation results and performance comparisons are presented in Section IV Finally, Section V concludes the paper Notation: Bold letters are used for column vectors, while capital bold letters are for matrices The operators (·)∗ , (·)T and (·)H denote complex conjugation, transposition and Hermitian transposition, respectively · , tr(·) and det(·) stand for the Frobenius norm, trace and determinant of a matrix Pr(·) and E{·} denote the probability of an event and expectation The Hermitian inner product of two complex column vectors a and b is denoted by a, b aT b∗ (nk ), x , x denote the binomial coefficient, the largest integer less than or equal to x, and the smallest integer larger than or equal to x, respectively x 2p is the largest integer less than or equal to x and is an integer power of Ψ denotes a complex signal constellation of size M Abstract—Combining the Alamouti space-time block code with spatial modulation (STBC-SM) was recently demonstrated as an effective way to increase the spectral efficiency and achieve a transmit diversity order of two as compared to the original spatial modulation (SM) This paper investigates a new transmission scheme that is based on a high-rate space-time block code rather than the Alamouti STBC A simplified maximum likelihood (ML) detection is also developed for the proposed scheme Analysis of coding gains and simulation results demonstrate that the proposed scheme outperforms previously-proposed spatial modulation schemes at high data transmission rates Index Terms—Spatial modulation, space-time block codes, ML detection, multiple-input multiple-output (MIMO) I I NTRODUCTION Multiple-input multiple-output (MIMO) systems have now become very popular in wireless communications In a typical MIMO system, multiple antennas are set up at the transceivers and multiple bit streams are sent simultaneously to increase the data rate Many strategies have been investigated in order to multiplex the bit streams to multiple antennas For example, in the V-BLAST (vertical Bell Lab layered space-time) strategy, all antennas are active at any given time and bit streams are multiplexed to achieve the highest data rate However, the major disadvantage of this strategy is that inter-channel interference (ICI) exits due to the simultaneous transmission on the same frequency band from multiple transmit antennas To obtain good system performance under the presence of such ICI requires a complex receiver structure, like the maximum likelihood (ML) receiver To completely eliminate ICI, Mesleh et al [1] proposed a technique, called spatial modulation (SM), in which only one antenna is active at any transmission time With this strategy, the antenna index involves in the process of sending data to the receiver Because only one antenna is active at a symbol time, no ICI appears at the receiver and detection can be performed with very low complexity Although the term “spatial modulation” was first used in [1], various researchers independently investigated this strategy since 2001 (see [2] for a comprehensive survey of research activities concerning SM) Focusing on the case that two antennas are active among available transmitted antennas, this paper proposes a SM technique that is better than the state-of-the-art schemes introduced by Basar et al in [3] and Wang in [4] The case of having two active antennas (i.e., two RF chains) is of great practical interest since it is only slightly more complex than the original SM scheme while 978-1-4673-8374-5/15/$31.00 ©2015 IEEE II H IGH -R ATE S PACE -T IME B LOCK C ODED S PATIAL M ODULATION (HR-STBC-SM) Recall that the rate of the Alamouti STBC is one symbol per one time slot, i.e., symbol per channel use (pcu) In contrast, the high-rate STBC proposed in [5] transmits two symbols over one time slot, i.e., its rate is symbols pcu The 2015 International Conference on Advanced Technologies for Communications (ATC) E{|xi |2 } = 1, the optimal angle φ is found to be 1.13 radian and the corresponding minimum coding gain is 0.1846, where the minimum coding gain is defined as transmission matrix of such a high-rate code is as follows: ax1 + bx3 −cx∗2 − dx∗4 X(x1 , x2 , x3 , x4 ) = ax2 + bx4 cx∗1 + dx∗3 , (1) Δ = det(Xi − Xj )(Xi − Xj )H where {xi }4i=1 are information symbols belonging to a standard M -ary constellation Ψ The rows of the above × matrix correspond to the symbol times, while the columns correspond to the transmit antennas In fact, this high-rate code is constructed as a linear combination of two Alamouti spacetime matrices and the parameters a, b, c and d can be optimized to maximize the minimum coding gain It was shown in [5] √ √ 7) √ , c = √12 and d = −ib are that a = √12 , b = (1− 7)+i(1+ the optimal values This high-rate code is chosen to replace the Alamouti code in the construction of the STBC-SM scheme because it achieves a higher coding gain than the Alamouti code for the same transmission rate measured in bits pcu, i.e., bits/s/Hz This is because for the same transmission rate in bits/s/Hz, the constellation used in the high-rate code can have a lower order when compared to the constellation used in the Alamouti code In the following, the operation of the proposed HR-STBCSM scheme is described with an example of available transmit antennas With available transmit antennas, the maximum number of different antenna pairs is 42 = This means that only bits can be used to index antenna pairs The high-rate code is applied for the selected antennas pairs as follows: ax1 + bx3 −cx∗2 − dx∗4 X1 (x1 , x2 , x3 , x4 ) = 0 X2 (x1 , x2 , x3 , x4 ) = 0 ax2 + bx4 cx∗1 + dx∗3 ax1 + bx3 −cx∗2 − dx∗4 X3 (x1 , x2 , x3 , x4 ) = 0 X4 (x1 , x2 , x3 , x4 ) = ax2 + bx4 cx∗1 + dx∗3 ax1 + bx3 −cx∗2 − dx∗4 0 (2) Similar to [3], the general framework of the proposed HR-STBC-SM scheme for an arbitrary number of transmit antennas is described as follows: 1) Determine the number of codewords in each codebook as n = N2t , where Nt is the number of available transmit antenna 2) Determine the total number of codewords as q = Nt 2p 3) Determine the number of codebooks as nq The number of codebooks is also the number of rotation angles that need to be optimized in order to maximize the minimum coding gain The larger the number of needed rotation angles is, the smaller the minimum coding gain becomes Given the number of codewords q, the spectral efficiency of the HR-STBC-SM scheme is m = 12 log2 q + 2log2 M (bits/s/Hz) Table I shows the minimum coding gains and optimized angles for various numbers of available transmit antennas In calculating the minimum coding gains, both BPSK and QPSK constellations are normalized to have unit average energy 0 TABLE I M INIMUM CODING GAINS AND OPTIMIZED ANGLES FOR THE CASES OF 4, AND AVAILABLE TRANSMIT ANTENNAS ax2 + bx4 cx∗1 + dx∗3 ax2 + bx4 cx∗1 + dx∗3 0 0 Xi ,Xj ∈Θ Xi =Xj 0 ejφ ax1 + bx3 −cx∗2 − dx∗4 ejφ As can be seen from the above × matrices, there are only two non-zero columns, which guarantees that only two antennas are active at each transmission time The high-rate code itself conveys information symbols for each time slots and these symbols are drawn from M -ary constellation Ψ If the same constellation is used in both the HR-STBCSM and STBC-SM schemes, then the rate of the former is always higher than the rate of the latter For example, if the constellation is QPSK, the spectral efficiency of the HRSTBC-SM scheme is bits/s/Hz, while that of the STBCSM is only bits/s/Hz The above four transmission matrices are grouped into two different codebooks Ω1 and Ω2 as Θ = {(X1 , X2 ) ∈ Ω1 , (X3 , X4 ) ∈ Ω2 } A rotation is applied for codewords in Ω2 in order to preserve the diversity gain of the system If such a rotation is not implemented, the difference matrix between X1 and X3 will not be a full rank, which reduces the diversity gain The rotation angle φ needs to be optimized to maximize the coding gain For QPSK with Nt BPSK 1.5 0.5858 Angles φ=1.7 φ2 = π3 φ3 = 2π φ2 = π4 π φ3 = φ4 = 3π QPSK 0.1846 0.1497 0.1015 Angles φ=1.13 φ2 = π6 φ3 = π3 φ2 = π8 φ3 = π4 φ4 = 3π III L OW-C OMPLEXITY ML D ETECTION A LGORITHM Let H be a Nt × nR channel gain matrix corresponding to a flat-fading MIMO system with Nt transmit and nR receive antennas For Rayleigh fading, the entries of H are modelled as independent and identically distributed (i.i.d) complex Gaussian random variables with zero mean and unit variance It is further assumed that the fading is such that H varies independently from one codeword to another and is invariant during the transmission of a codeword, i.e., block fading The channel matrix H is perfectly estimated at the receiver, but unknown at the transmitter With X ∈ Θ being the × Nt HR-STBC-SM transmission matrix, the × nR received signal matrix Y is given as Y= ρ XH + N, μ (3) where μ is a normalization factor to ensure that ρ is the average SNR at each receive antenna, N is a 2×nR matrix representing 2015 International Conference on Advanced Technologies for Communications (ATC) ⎛ ah2,1 ϕ ⎜ ah2,2 ϕ ⎜ ⎜ ⎜ ⎜ ⎜ ah2,nR ϕ H3 = ⎜ ∗ ∗ ∗ ⎜ c h3,1 ϕ ⎜ c∗ h∗ ϕ∗ 3,2 ⎜ ⎜ ⎝ ∗ ∗ c h3,nR ϕ∗ ⎛ ah4,1 ϕ ⎜ ah4,2 ϕ ⎜ ⎜ ⎜ ⎜ ⎜ ah R ϕ H4 = ⎜ ∗ 4,n ∗ ∗ ⎜ c h1,1 ϕ ⎜ c∗ h∗ ϕ∗ 1,2 ⎜ ⎜ ⎝ c∗ h∗1,nR ϕ∗ AWGN, whose elements are i.i.d complex Gaussian random variables with zero mean and unit variance The ML detection chooses a codeword that minimizes the following decision metric: ˆ = arg Y − X X∈Θ ρ XH μ (4) Since the HR-STBC-SM transmission matrix X contains information symbols, the ML detection needs to search over qM candidates to find the minimum of the above metric To reduce the computational complexity of the ML detection, (3) can be rewritten in the following form: ⎛ y= ⎞ x1 ρ ⎜ x2 ⎟ ⎟ + n, H ⎜ ⎝ x3 ⎠ μ x4 where y and n are 2nR -length column vectors obtained by vectorizing matrices Y and N as [Y(1, 1), , Y(1, nR ), Y∗ (2, 1), , Y∗ (2, nR )]T n [N(1, 1), , N(1, nR ), N∗ (2, 1), , N∗ (2, nR )] T (6) (7) In (5), H is the 2nR × equivalent channel matrix corresponding to the transmitted codeword X , = 1, 2, · · · , q An example of equivalent channel matrices for the case of Nt = is as follows: ⎛ ah1,1 ⎜ ah1,2 ⎜ ⎜ ⎜ ⎜ ⎜ ah1,nR H1 = ⎜ ∗ ∗ ⎜ c h2,1 ⎜ c∗ h∗ 2,2 ⎜ ⎜ ⎝ c∗ h∗2,nR ⎛ ah3,1 ⎜ ah3,2 ⎜ ⎜ ⎜ ⎜ ⎜ ah3,nR H2 = ⎜ ∗ ∗ ⎜ c h4,1 ⎜ c∗ h∗ 4,2 ⎜ ⎜ ⎝ c∗ h∗4,nR ah2,1 ah2,2 ah2,nR −c∗ h∗1,1 −c∗ h∗1,2 ∗ ∗ −c h1,nR bh1,1 bh1,2 bh1,nR d∗ h∗2,1 d∗ h∗2,2 ∗ ∗ d h2,nR ⎞ bh2,1 bh2,2 ⎟ ⎟ ⎟ ⎟ ⎟ bh2,nR ⎟ ∗ ∗ −d h1,1 ⎟ ⎟ −d∗ h∗1,2 ⎟ ⎟ ⎟ ⎠ −d∗ h∗1,nR ah4,1 ah4,2 ah4,nR −c∗ h∗3,1 −c∗ h∗3,2 −c∗ h∗3,nR bh3,1 bh3,2 bh3,nR d∗ h∗4,1 d∗ h∗4,2 d∗ h∗4,nR ⎞ bh4,1 bh4,2 ⎟ ⎟ ⎟ ⎟ ⎟ bh4,nR ⎟ −d∗ h∗3,1 ⎟ ⎟ −d∗ h∗3,2 ⎟ ⎟ ⎟ ⎠ ∗ ∗ −d h3,nR bh2,1 ϕ bh2,2 ϕ bh2,nR ϕ d∗ h∗3,1 ϕ∗ d∗ h∗3,2 ϕ∗ ∗ ∗ d h3,nR ϕ∗ bh4,1 ϕ bh4,2 ϕ bh4,nR ϕ d∗ h∗1,1 ϕ∗ d∗ h∗1,2 ϕ∗ d∗ h∗1,nR ϕ∗ ⎞ bh3,1 ϕ bh3,2 ϕ ⎟ ⎟ ⎟ ⎟ ⎟ bh3,nR ϕ ⎟ ∗ ∗ ∗ ⎟ −d h2,1 ϕ ⎟ −d∗ h∗2,2 ϕ∗ ⎟ ⎟ ⎟ ⎠ −d∗ h∗2,nR ϕ∗ ⎞ bh1,1 ϕ bh1,2 ϕ ⎟ ⎟ ⎟ ⎟ ⎟ bh1,nR ϕ ⎟ ∗ ∗ ∗ ⎟, −d h4,1 ϕ ⎟ −d∗ h∗4,2 ϕ∗ ⎟ ⎟ ⎟ ⎠ ∗ ∗ ∗ −d h4,nR ϕ where hi,j is the channel fading coefficient between the ith transmit antenna and the jth receive antenna, and ϕ = ejφ Let h1, , h2, , h3, and h4, denote the columns of H Since the high-rate STBC is constructed from a linear combination of two Alamouti codes, the orthogonal property exists for two pairs of the columns of H , namely h1, , h2, = h3, , h4, = Based on this property, the ML detection can be simplified Specifically, for a specific H , the ML detection in (4) can be rewritten as ⎛ ⎞ x1 ρ ⎜ x2 ⎟ ⎟ H ⎜ (ˆ x1, , x ˆ2, , x ˆ3, , xˆ4, ) = arg y − ⎝ x3 ⎠ xi ∈Ψ μ x4 (8) Because the column vectors h1, and h2, are orthogonal, for given values of (x3 , x4 ), the ML estimates of x1 and x2 can be performed independently as follows: (5) y ah3,1 ϕ ah3,2 ϕ ah3,nR ϕ −c∗ h∗2,1 ϕ∗ −c∗ h∗2,2 ϕ∗ ∗ ∗ −c h2,nR ϕ∗ ah1,1 ϕ ah1,2 ϕ ah1,nR ϕ −c∗ h∗4,1 ϕ∗ −c∗ h∗4,2 ϕ∗ −c∗ h∗4,nR ϕ∗ ρ (h1, x1 + h3, x3 + h4, x4 ) x1 ∈Ψ μ (9) ρ (h2, x2 + h3, x3 + h4, x4 ) (ˇ x2, |x3 ,x4 ) = arg y − x2 ∈Ψ μ (10) After collecting the results from (9) and (10), which are expressed as (ˇ x1, , x ˇ2, |x3 ,x4 ), the ML estimates (ˆ x1, , xˆ2, , x ˆ3, , x ˆ4, ) will be then determined by (8) over ˇ2, |x3 ,x4 ), x3 , x4 ) values1 Since the above ML all ((ˇ x1, , x estimations are performed for a particular H , the receiver makes a final decision by choosing the minimum antenna combination metric ˆ = arg m , = 1, 2, · · · , q, where m is the value of the minimum metric in (8) Compared to the ML decoding of the STBC-SM scheme that has a complexity of 2qM , the decoding complexity2 of (ˇ x1, |x3 ,x4 ) = arg y − Alternatively, because of the orthogonal property of h 3, and h4, , the ML detection can be performed the other way round with the same complexity where the receiver first estimates x3 and x4 independently for each known pair (x1 , x2 ) Note that the detections of x ˆ1, , and x ˆ2, in (9) and (10) only require simple threshold circuits for given (x3 , x4 ) 2015 International Conference on Advanced Technologies for Communications (ATC) 10 10 10 10 −1 10 −2 10 −3 −1 −2 −3 BER 10 BER 10 10 10 10 10 −4 −5 −6 −7 10 Nt Nt Nt Nt Nt Nt Nt Nt = = = = = = = = 4, 6, 4, 6, 4, 6, 4, 6, BPSK, simulation BPSK, simulation QPSK, simulation QPSK, simulation BPSK, bound BPSK, bound QPSK, bound QPSK, bound 10 10 10 SNR (dB) 12 14 16 10 18 Fig Performance comparison between theoretical upper bound and simulation results of the HR-STBC-SM scheme Pr (Xi → Xj ) = π 1+ ρλ1 sin2 ϑ nR 1+ ρλ2 sin2 ϑ 10 10 10 2k 2k i=1 j=1 Pr (Xi → Xj ) χ (Xi , Xj ) k −6 −7 STBC-SM, Nt = 4, 16-QAM HR-STBC-SM, Nt = 4, QPSK STBC-SM, Nt = 6, 16-QAM HR-STBC-SM, Nt = 6, QPSK STBC-SM, Nt = 8, 16-QAM HR-STBC-SM, Nt = 8, QPSK 10 SNR (dB) 12 14 16 18 −1 −2 −3 BER 10 10 10 nR dϑ 10 −4 −5 −6 (11) where λ1 and λ2 are the eigenvalues of matrix (Xi −Xj )(Xi − Xj )H under the normalization μ = and E{tr(XH X)} = Assume that k bits are transmitted over two consecutive symbol intervals, the union bound on the bit error probability is P (error) ≤ k −5 Fig BER comparison between HR-STBC-SM and STBC-SM schemes at bits/s/Hz, 5.5 bits/s/Hz and bits/s/Hz the HR-STBC-SM scheme is 2qM , which is higher for the same value of M Fortunately, for the same spectral efficiency in terms of bits/s/Hz the HR-STBC-SM uses a lower-order constellation and it turns out that the ML detection complexity of the HR-STBC-SM scheme could be comparable to that of the STBC-SM scheme Before closing this section, an upper bound on the error probability is given as it shall be used to gauge the performance obtained by computer simulation First, the pairwise error probability for deciding on codeword Xj given that Xi was transmitted is given by [3] π −4 10 −7 STBC-SM, Nt = 4, QPSK HR-STBC-SM, Nt = 4, BPSK STBC-SM, Nt = 6, QPSK HR-STBC-SM, Nt = 6, BPSK STBC-SM, Nt = 8, QPSK HR-STBC-SM, Nt = 8, BPSK SNR (dB) 10 12 14 Fig BER comparison between HR-STBC-SM and STBC-SM schemes at bits/s/Hz, 3.5 bits/s/Hz and bits/s/Hz (12) SNR, which makes it useful to study the error performance behavior of the proposed HR-STBC-SM scheme with different system setups Figure shows the BER performance comparison between the HR-STBC-SM and STBC-SM schemes at spectral efficiencies of 5, 5.5 and bits/s/Hz, which correspond to systems with 4, and antennas To deliver such spectral efficiencies, QPSK is used for the HR-STBC-SM scheme, whereas 16-QAM is used for the STBC-SM scheme More importantly, at bits/s/Hz and BER level of 10−5 , the HRSTBC-SM scheme provides a 1.8dB SNR gain over the STBCSM scheme Similarly, for the cases of and antennas (5.5 and bits/s/Hz), the SNR gains are 1.6dB and 0.8dB, respectively Such SNR gains are predicted by the analysis and comparison of coding gains in Table I where χ (Xi , Xj ) is the number of bits in error when comparing matrices Xi and Xj IV S IMULATION R ESULTS AND C OMPARISON In this section, the BER simulation results of the HR-STBCSM and STBC-SM schemes are presented and compared for various numbers of transmit antennas and spectral efficiency values versus the average SNR per receive antenna (ρ) In all simulations, four receive antennas are employed First, Figure compares the upper bound in (12) and the BER performance obtained by simulation for the cases of and antennas with BPSK and QPSK constellations The figure clearly illustrates the tightness of the union bound at high 2015 International Conference on Advanced Technologies for Communications (ATC) scheme proposed by Wang et al in [4] are evaluated at and bits/s/Hz Wang’s scheme uses a (4, 3) error-correcting code together with transmit antennas to create 32 codewords, which is larger than 16 codewords of HR-STBC-SM As can be seen from the figure, at bits/s/Hz, Wang’s scheme has about 1dB SNR gain as compared to the proposed HRSTBC-SM However, at bits/s/Hz, the HR-STBC-SM scheme achieves a dB gain over Wang’s scheme 10 −1 10 −2 bits/s/Hz BER 10 −3 10 bits/s/Hz V C ONCLUSIONS −4 In this paper, a novel transmission scheme for a MIMO system is developed by combining spatial modulation and a high-rate space time block code Aiming at a system implementation that requires only active transmit antennas, i.e., two RF chains, and operating at high spectral efficiencies, it was demonstrated that the proposed scheme performs better than previously-proposed schemes that are based on either Alamouti STBC or block error-control coding A simplified ML detection of the proposed scheme was also presented 10 −5 10 −6 10 Wang, Nt = 8, BPSK+QPSK HR-STBC-SM, Nt = 8, BPSK Wang, Nt = 8, 8-QAM+16-QAM HR-STBC-SM, Nt = 8, QPSK 10 SNR (dB) 12 14 16 18 Fig BER comparison between HR-STBC-SM and Wang’s schemes at bits/s/Hz and bits/s/Hz R EFERENCES In Figure 3, the BER curves of STBC-SM and HR-STBCSM with 4, and antennas schemes are compared at lower spectral efficiencies Specifically, QPSK is used for STBC-SM while BPSK is used for HR-STBC-SM The corresponding spectral efficiencies are 3, 3.5 and bits/s/Hz, respectively It can be seen that the HR-STBC-SM scheme performs quite similar to the STBC-SM scheme at bits/s/Hz At the spectral efficiency of bits/s/Hz (with available antennas), the STBCSM actually outperforms our proposed scheme by dB Again, this can be predicted from Table I, which shows that the coding gain of the STBC-SM scheme is 1.2179, while that of the HRSTBC-SM scheme is 0.5858 In Figure 4, the BER curves of the HR-STBC-SM and the [1] R Mesleh, H Haas, C W Ahn, and S Yun, “Spatial modulation - a new low complexity spectral efficiency enhancing technique,” in Communications and Networking in China, 2006 ChinaCom ’06 First International Conference on, pp 1–5, Oct 2006 [2] M Di Renzo, H Haas, A Ghrayeb, S Sugiura, and L Hanzo, “Spatial modulation for generalized MIMO: Challenges, opportunities, and implementation,” Proceedings of the IEEE, vol 102, pp 56–103, Jan 2014 [3] E Basar, U Aygolu, E Panayirci, and H Poor, “Space-time block coded spatial modulation,” Communications, IEEE Transactions on, vol 59, pp 823–832, March 2011 [4] L Wang, Z Chen, and X Wang, “A space-time block coded spatial modulation from (n, k) error correcting code,” Wireless Communications Letters, IEEE, vol 3, pp 54–57, February 2014 [5] S Sezginer, H Sari, and E Biglieri, “On high-rate full-diversity × space-time codes with low-complexity optimum detection,” Communications, IEEE Transactions on, vol 57, pp 1532–1541, May 2009 ... Poor, Space- time block coded spatial modulation, ” Communications, IEEE Transactions on, vol 59, pp 823–832, March 2011 [4] L Wang, Z Chen, and X Wang, “A space- time block coded spatial modulation. .. combining spatial modulation and a high- rate space time block code Aiming at a system implementation that requires only active transmit antennas, i.e., two RF chains, and operating at high spectral... correspond to the symbol times, while the columns correspond to the transmit antennas In fact, this high- rate code is constructed as a linear combination of two Alamouti spacetime matrices and the