We explore in this letter the lattice sphere packing representation of a multi-antenna system and the algebraic space-time (ST) codes. We apply the sphere decoding (SD) algorithm to the resulted lattice code. For the uncoded system, SD yields, with small increase in complexity, a huge improvement over the well-known V-BLAST detection algorithm. SD of algebraic ST codes exploits the full diversity of the coded multi-antenna system, and makes the proposed scheme very appealing to take advantage of the richness of the multi-antenna environment. The fact that the SD does not depend on the constellation size, gives rise to systems with very high spectral efficiency, maximum-likelihood performance, and low decoding complexity.

In this paper, we propose practical yet effective statistically-aided codebook-based hybrid precoding schemes for massive multiple-input multiple-output systems in millimeter wave bands. Particularly, we develop novel low-overhead hybrid precoding algorithms for selecting the baseband digital and radio frequency analog precoders from statistically skewed DFT-based codebooks. The proposed algorithms aim at maximizing the spectral efficiency based on minimizing the chordal distance between the optimal unconstrained precoder and the hybrid beamformer and maximizing the signal to the interference noise ratio for the single-user and multiuser cases, respectively. We investigate the performance of the proposed algorithms by considering the mutual information of the analog beamforming procedure (the common stage among the proposed algorithms) as a performance evaluation metric. We derive lower and upper bounds on the mutual information of the channel given the proposed algorithms. Moreover, we show that the performance gap between the lower and upper bounds depends heavily on how many DFT columns are aligned to the largest eigenvectors of the transmit antenna array response of the millimeter wave channel or equivalently the transmit channel covariance matrix when only statistical channel knowledge is available at the transmitter. Then, we show that the proposed algorithms are asymptotically optimal as the number of transmit antennas M goes to infinity and the millimeter wave channel has a limited number of paths P, i.e., P

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We explore in this paper the lattice sphere packing representation of a multi-antenna system and the algebraic space-time (ST) codes. We apply the sphere decoding (SD) algorithm to the resulted lattice code. For the uncoded system, SD yields, with small increase in complexity, a huge improvement over the well-known V-BLAST detection algorithm. SD of algebraic ST codes exploits the full diversity of the coded multi-antenna system, and makes the proposed scheme very appealing to take advantage of the richness of the multi-antenna environment. The fact that the SD does not depend on the constellation size, gives rise to systems with very high spectral efficiency, maximum likelihood (ML) performance, and low decoding complexity. I . INTRODUCTIONRecently, the field of multi-antenna processing and space-time (ST) coding has attracted large interest in the communication community due to the huge capacity of the multi-antenna environment [l]. Because of the maximum likelihood (ML) detection high complexity sub-optimal detection like the V-BLAST have been proposed for the uncoded system [2].In this paper, we prove that one can reach the ML performance of the uncoded system with low complexity by applying the sphere decoder [3] on the lattice sphere packing representation of a multi-antenna system. Moreover, it is shown that one can achieve the full diversity of the multi-antenna system by using full spatial diversity rotated constellations without adding redundancy [4], and still reach the ML performance with reasonable complexity. SIMULATION RESULTSIn simulations we use the constellation q-QAM, with q = 4 , 16. The average energy per bit is fixed to E b = 1. We consider a multi-antenna system with M transmitters and N = M receivers. The algebraic coding is done over 1 periods by using rotated constellations of dimension MZ. The channel transfer matrix is modeled by independent complex Gaussian random variables of variance 0.5 per real dimension. The curves are plotted as a function of SNR (the signal-to-noise ratio per bit), and the variance g 2 of the complex AWGN per real dimension is adjusted by the formula m2 = ,zf;p) 10-SNR''o , where E, is the average symbol energy of the q-QAM when E b = 1 and equals v. In figures 1 and 2 we applied the SD on both uncoded data streams and algebraic ST codes over 1 periods with M = N transmit/receive antennas [4]. Comparisons are done with the V-BLAST detection algorithm [2]. It is shown that at the expense of a moderate increase in complexity, a huge improvement in performance is achieved. 10 E c B 10' B 2 10' 10 1 1 12 13 14 1s 16 17 in SNR per D 1 Fig. 1: SD of V-BLAST architecture, M = N = 8 , averagesymbol error rate of the 16-QAM modulation, 32 bits/s/Hz. b 0 2 4 6 6 10 12 14 18 18 SNR per bn 10 Fig. 2: SD of algebraic ST codes, M = N = 4, average symbol error rate of the 4-QAM modulation, 8 bits/s/Hz, 1 = 1 , 2 , 3 . REFERENCES V. Tarokh, N. Seshadri and A. Calderbank, "Space-time codes for high data rate wireless communications: performance criterion and code construction,"...