IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 5, MAY 2013 1847 Toward a Preferred 4 × 4 Space-Time Block Code: A Performance-Versus-Complexity Sweet Spot with Linear-Filter Decoding Sandipan Kundu, Dimitris A. Pados, Member IEEE, Weifeng Su, Member IEEE, and Rohan Grover, Member IEEE Abstract—We develop a new 4 × 4 Hadamard-precoded quasi- orthogonal space-time block code (QO-STBC) that enables highly effective near-maximum-likelihood (near-ML) reliability-based prioritized symbol detection using linear filters. Approximate block-error-rate minimization is being used to optimize the code rotation angle. Detailed computational complexity evaluation of the decoder in terms of real multiplications and additions shows significant complexity reduction for symbol alphabet sizes of interest. Numerical and simulation studies demonstrate negligible bit-error-rate degradation compared to the state-of-the-art in bit- error-rate by ML decoded 4 × 4 codewords. Index Terms—Hadamard precoding, matched-filter (MF), maximum-likelihood (ML) detection, minimum-mean-square- error (MMSE) filter, multi-input multi-output (MIMO) commu- nications, quasi-orthogonal space-time block codes (QO-STBC). I. I NTRODUCTION O RTHOGONAL space-time block codes (O-STBC) [1] offer full transmit diversity and allow disjoint single- complex-symbol maximum-likelihood (ML) decoding. The first-in-line, celebrated 2-transmit-antenna Alamouti O-STBC [2] has also rate one and ushered wireless communications in the era of space-time coded transmissions. Full-diversity, rate-one O-STBCs for complex symbols drawn from arbitrary constellations do not exist, however, for systems with more than two transmit antennas [1], [3]. For systems with four transmit antennas, a case of great practical importance, the rate limitation of O-STBCs was overcome by quasi-orthogonal space-time block codes (QO-STBCs) [4]-[8] at the expense of diversity loss and increased decoding complexity. Rotation of few symbols in the QO-STBC can, however, reinstate full- diversity. In particular, full-diversity rate-one QO-STBCs for 4-transmit-antennas were described in [9]-[14]. On the other hand, ML decoding of the full-diversity QO-STBCs in [9], [10] requires joint detection of two complex symbols (four real symbols), which is computationally expensive for high-order signal constellations. In [11], interleaving real and imaginary Manuscript received May 17, 2012; revised October 14 and December 13, 2012. The associate editor coordinating the review of this paper and approving it for publication was M. Matthaiou. S. Kundu, D. A. Pados, and W. Su are with the COMSENS Research Center, Dept. of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY, 14260, USA (e-mail: {skundu, pados, weifeng}@buffalo.edu). R. Grover is with Mirics, Needham, MA, 02494, USA (e-mail: rohan.grover@mirics.com). This paper was presented in part at the IEEE Wireless Communications and Networking Conference (WCNC), Paris, France, April 2012. Digital Object Identifier 10.1109/TCOMM.2013.021913.120338 parts of rotated symbols enables single-symbol decoding of full-rate full-diversity QO-STBCs for the 4-transmit-antenna case. Linear-transformation-based full-diversity QO-STBCs with joint two-real-symbol ML decoding were presented in [12], [13]. Grouping of dispersion matrices leading to joint two-real-symbol ML decodable full-diversity codewords was described in [14]. Noise prewhitening followed by subgroup ML decoding was suggested in [15]. Arguably, the practical significance of a high-performing low-computational-complexity 4 × 4 space-time code warrants further research. Recently, low complexity block-orthogonal [16], [17] and several other general proposals appeared in the literature on linear (or partial-interference-cancelation (PIC)) receivers of space-time codes [18]-[20], but have not provided yet an appealing definitive solution for the 4 transmit antenna case. In this paper, we develop a new (open-loop) Hadamard precoded 4 × 4 QO-STBC codeword that, unlike code- words from [11]-[15], facilitates high-performance reliability- based linear-complexity decoding using minimum-mean- square-error (MMSE) and matched filters (MF). Precoded codewords and linear receivers were also pursued earlier on by Sezgin and Oechtering [21] with resulting error rates that may be deemed, however, unsatisfactory. For the de- veloped codeword herein 1 and the MMSE-MF decoder, a semi-closed-form approximate block-error-rate expression is derived, which is then minimized to find the optimal codeword rotation angle. Simulation and numerical studies are, then, provided to compare the error rate of the proposed code under ML and MMSE-MF decoding and existing works in the literature. The tradeoff performance-versus-complexity point becomes, arguably, the most appealing to date in view of the minimal computational cost per decoding measured in number of real multiplications and additions executed by the described scheme. The rest of the paper is organized as follows. In Section II, the system model and necessary notation for the proposed codeword is introduced. In Section III, we derive the proposed associated linear-filter (MMSE-MF) detector. Performance op- timization for the MMSE-MF detector is carried out in Sec- tion IV. Complexity and error rate performance comparisons are presented in Section V. A few concluding remarks are drawn in Section VI. 1 The proposed precoded codeword enjoys full-diversity under joint two-real symbol ML decoding as in the works of [11]-[14]. 0090-6778/13$31.00 c  2013 IEEE