Performance of reliable communication over a coherent slow fading channel at high SNR is succinctly captured as a fundamental tradeoff between diversity and multiplexing gains. We study the problem of designing codes that optimally tradeoff the diversity and multiplexing gains. Our main contribution is a precise characterization of codes that are universally tradeoff-optimal, i.e., they optimally tradeoff the diversity and multiplexing gains for every statistical characterization of the fading channel. We denote this characterization as one of approximate universality where the approximation is in the connection between error probability and outage capacity with diversity and multiplexing gains, respectively. The characterization of approximate universality is then used to construct new coding schemes as well as to show optimality of several schemes proposed in the space-time coding literature.

Approximately Universal Codes over Slow Fading Channels

Two new rate-one full-diversity space-time block codes (STBCs) are proposed. They are characterized by the lowest decoding complexity among the known rate-one STBC, arising due to the complete separability of the transmitted symbols into four groups for maximum likelihood detection. The first and the second codes are delay-optimal if the number of transmit antennas is a power of 2 and even, respectively. The exact pairwise error probability is derived to allow for the performance optimization of the two codes. Compared with existing low-decoding complexity STBC, the two new codes offer several advantages such as higher code rate, lower encoding/decoding delay and complexity, lower peak-to-average power ratio, and better performance.

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Four-Group Decodable Space–Time Block Codes

This paper proposes a multiple-input multiple-output (MIMO) transmission scheme for M-ary modulations, called Spatially Modulated Orthogonal Space-Time Block Coding (SM-OSTBC), based on the concept of Spatial Constellation (SC) codewords introduced by Le at el. . In the proposed scheme, transmit codeword matrices are generated by multiplying SC matrices with codewords constructed from Orthogonal Space-time Block Codes (O-STBC). The maximum spectral efficiency of the proposed scheme is equal to (n_T-2+ log_2 M) bpcu, where n_T is the number of transmit antennas and M is the modulation order. The SC matrices provide a means of carrying information bits together with the O-STBC codewords and allow the SM-OSTBC scheme to achieve second-order transmit diversity by satisfying the non-vanishing determinant property. A systematic method to design the SC codewords for an even number of transmit antennas greater than 3 is presented. A single-stream maximum-likelihood (ML) decoder, which requires a low computational complexity thanks to the structure of the SM-OSTBC codewords and to the orthogonality of the O-STBCs, and a sphere decoder with further reduced signal processing complexity are developed. The bit error rate (BER) performance of the proposed scheme is studied by using both theoretical union bound analysis and computer simulations. Finally, simulation results are presented in order to compare BER performance, energy efficiency and decoding complexity of the proposed scheme with those of several existing MIMO transmission schemes.

Spatially Modulated Orthogonal Space-Time Block Codes with Non-Vanishing Determinants

Multiple antennas have played an essential role in spatial multiplexing and diversity transmission for a wide range of communication applications. Most advances in the design of high-speed wireless multiple-input-multiple-output (MIMO) systems have been based on information-theoretic principles that demonstrate how to efficiently transmit signals conforming to Gaussian distribution. However, although the Gaussian signal is capacity-achieving, practical systems transmit signals belonging to finite and discrete constellations. Therefore, capacity-achieving transceiver processing based on a Gaussian input signal can be quite suboptimal for practical MIMO systems with discrete constellation input signals. To address this shortcoming, this paper aims to provide a comprehensive overview of MIMO transmission design with finite input signals. It first summarizes existing fundamental results for MIMO systems with finite input signals. Next, focusing on basic point-to-point MIMO systems, it examines transmission schemes based on the three most important criteria for communication systems: mutual-information-driven designs, mean-square-error-driven designs, and diversity-driven designs. In particular, a unified framework is developed for the design of low-complexity transmission schemes applicable to massive MIMO systems in forthcoming 5G wireless networks for the first time. Furthermore, adaptive transmission designs are proposed that switch among these criteria based on channel conditions to formulate the best transmission strategy. A survey is then given of transmission designs with finite input signals for multiuser MIMO scenarios, including MIMO uplink transmission, MIMO downlink transmission, MIMO interference channel, and MIMO wiretap channel. Additionally, transmission designs with finite input signals are discussed for other multi-antenna systems. Finally, a number of technical challenges that remain unresolved at the time of writing are highlighted, and future trends in transmission design with finite input signals are discussed.

A Survey on MIMO Transmission With Finite Input Signals: Technical Challenges, Advances, and Future Trends

To make the implementation of high-rate space-time block code (STBC) realistic in practical systems, fast-decodable (FD) and group-decodable (GD) code structures have been separately introduced into STBC to reduce the sphere decoding complexity. However, no STBC has both the two code structures until now. In this paper, high-rate fast-group-decodable STBC (FGD-STBC, which has both FD and GD code structures) is proposed for the first time. We first derive the condition for FD code structure with the lowest sphere decoding complexity, then we prove that such FD code structure can be integrated with the GD-STBC for FGD-STBC construction. Analysis and simulation show that the proposed FGD-STBC has much lower decoding complexity and comparable performance with the existing FDSTBC.

Fast-group-decodable space-time block code

Full-rate space time codes (STCs) with rate number of transmit antennas have high multiplexing gain, but high decoding complexity even when decoded using reduced-complexity decoders such as sphere or QRDM decoders. In this paper, we introduce a new code property of STC called block-orthogonal property, which can be exploited by QR-decomposition-based decoders to achieve significant decoding complexity reduction without performance loss. We show that such complexity reduction principle can benefit the existing algebraic codes such as Perfect and DjABBA codes due to their inherent (but previously undiscovered) block-orthogonal property. In addition, we construct and optimize new full-rate block-orthogonal STC (BOSTC) that further maximize the QRDM complexity reduction potential. Simulation results of bit error rate (BER) performance against decoding complexity show that the new BOSTC outperforms all previously known codes as long as the QRDM decoder operates in reduced-complexity mode, and the code exhibits a desirable complexity saturation property.

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Block-Orthogonal Space–Time Code Structure and Its Impact on QRDM Decoding Complexity Reduction

High-rate space-time block codes (STBC with code rate >; 1) in multi-input multi-output (MIMO) systems are able to provide both spatial multiplexing gain and diversity gain, but have high maximum likelihood (ML) decoding complexity. Since group-decodable (quasi-orthogonal) code structure can reduce the decoding complexity, we present in this paper systematic methods to construct group-decodable high-rate STBC with full symbol-wise diversity gain for arbitrary transmit antenna number and code length. We show that the proposed group-decodable STBC can achieve high code rate that increases almost linearly with the transmit antenna number, and the slope of this near-linear dependence increases with the code length. Comparisons with existing low-rate and high-rate codes (such as orthogonal STBC and algebraic STBC) are conducted to show the decoding complexity reduction and good code performance achieved by the proposed codes.

Group-Decodable Space-Time Block Codes with Code Rate > 1

Full-rate STBC (space-time block codes) with non-vanishing determinants achieve the optimal diversity-multiplexing tradeoff but incur high decoding complexity. To permit fast decoding, Sezginer, Sari and Biglieri proposed an STBC structure with special QR decomposition characteristics. In this paper, we adopt a simplified form of this fast-decodable code structure and present a new way to optimize the code analytically. We show that the signal constellation topology (such as QAM, APSK, or PSK) has a critical impact on the existence of non-vanishing determinants of the full-rate STBC. In particular, we show for the first time that, in order for APSK-STBC to achieve non-vanishing determinant, an APSK constellation topology with constellation points lying on square grid and ring radius √m2+n2 (m,n integers) needs to be used. For signal constellations with vanishing determinants, we present a methodology to analytically optimize the full-rate STBC at specific constellation dimension.

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Optimization of Fast-Decodable Full-Rate STBC with Non-Vanishing Determinants

Spatial multiplexing code (SMC) has high transmis- sion code rate but also high maximum likelihood (ML) decoding complexity. To overcome this disadvantage, we propose 2-group decodable SMC with full symbol-wise diversity for arbitrary number of transmit antennas. In this paper, unbalanced 2- group decodable SMC is first constructed and used to con- struct balanced 2-group decodable SMC. The rates of both the unbalanced/balanced 2-group decodable SMCs approach Nt asymptotically when TNt where Nt is the number of transmit antennas and T is the code length. To our best knowledge, this is the first time that 2-group decodable SMCs are constructed systematically.

Unbalanced and Balanced 2-Group Decodable Spatial Multiplexing Code

In this paper, we present a new class of spacetime block code, called shift-orthogonal space-time block codes (SOSTBC), which can maintain orthogonality (hence symbolwise linear decoding) in asynchronous wireless relay networks, under both amplify-and-forward (AF) and decode-and-forward (DF) relaying schemes. The definitions, systematic constructions and properties of SOSTBC are presented. We also present some simulation results to illustrate its effectiveness in achieving full asynchronous cooperative diversity.

Er Yang Zhang

Papers

Block-Orthogonal Space–Time Code Structure and Its Impact on QRDM Decoding Complexity Reduction

Group-Decodable Space-Time Block Codes with Code Rate > 1

Optimization of Fast-Decodable Full-Rate STBC with Non-Vanishing Determinants

Unbalanced and Balanced 2-Group Decodable Spatial Multiplexing Code

Shift-orthogonal space-time block codes