On Fast-Decodable Space–Time Block Codes

We focus on full-rate, fast-decodable space–time block codes (STBCs) for $2times2$ and $4times2$ multiple-input multiple-output (MIMO) transmission. We first derive conditions and design criteria for reduced-complexity maximum-likelihood (ML) decodable $2times2$ STBCs, and we apply them to two families of codes that were recently discovered. Next, we derive a novel reduced-complexity $4times2$ STBC, and show that it outperforms all previously known codes with certain constellations.      

Algebraic Number Precoding for Space–Time Block Codes

We propose a space–time block coding framework based on linear precoding. The codes for $P$ transmit antennas are formed by transmitting the information vector (with $P$ independent information symbols) $L$ times where each time it is rotated by a distinct precoding matrix. The framework generalizes conventional spatial multiplexing techniques and facilitates tradeoff between rate and diversity. We propose a simple construction for precoding matrices whose parameters are chosen to guarantee maximal diversity using algebraic number theory. Our codes exhibit circular structure, which greatly simplifies the performance analysis and facilitates linear decoding. Theoretical analysis and numerical simulations demonstrated excellent performance of the proposed algebraic precoding framework.      

Space–Time Block Codes Achieving Full Diversity With Linear Receivers

In most of the existing space–time code designs, achieving full diversity is based on maximum-likelihood (ML) decoding at the receiver that is usually computationally expensive and may not have soft outputs. Recently, Zhang–Liu–Wong introduced Toeplitz codes and showed that Toeplitz codes achieve full diversity when a linear receiver, zero-forcing (ZF) or minimum mean square error (MMSE) receiver, is used. Motivated from Zhang–Liu–Wong's results on Toeplitz codes, in this paper, we propose a design criterion for space–time block codes (STBC), in which information symbols and their complex conjugates are linearly embedded, to achieve full diversity when ZF or MMSE receiver is used. The (complex) orthogonal STBC (OSTBC) satisfy the criterion as one may expect. We also show that the symbol rates of STBC under this criterion are upper bounded by 1. Subsequently, we propose a novel family of STBC that satisfy the criterion and thus achieve full diversity with ZF or MMSE receiver. Our newly proposed STBC are constructed by overlapping the $2,times,2$ Alamouti code and hence named overlapped Alamouti codes in this paper. The new codes are close to orthogonal and their symbol rates can approach 1 for any number of transmit antennas. Simulation results show that overlapped Alamouti codes significantly outperform Toeplitz codes for all numbers of transmit antennas and also outperform OSTBC when the number of transmit antennas is above $4$.      

Diversity Embedded Space–Time Codes

Rate and diversity impose a fundamental tradeoff in wireless communication. High-rate space-time codes come at a cost of lower reliability (diversity), and high reliability (diversity) implies a lower rate. However, wireless networks need to support applications with very different quality-of-service (QoS) requirements, and it is natural to ask what characteristics should be built into the physical layer link in order to accommodate them. In this paper, we design high-rate space-time codes that have a high-diversity code embedded within them. This allows a form of communication where the high-rate code opportunistically takes advantage of good channel realizations while the embedded high-diversity code provides guarantees that at least part of the information is received reliably. We provide constructions of linear and nonlinear codes for a fixed transmit alphabet constraint. The nonlinear constructions are a natural generalization to wireless channels of multilevel codes developed for the additive white Gaussian noise (AWGN) channel that are matched to binary partitions of quadrature amplitude modulation (QAM) and phase-shift keying (PSK) constellations. The importance of set-partitioning to code design for the wireless channel is that it provides a mechanism for translating constraints in the binary domain into lower bounds on diversity protection in the complex domain. We investigate the systems implications of embedded diversity codes by examining value to unequal error protection, rate opportunism, and packet delay optimization. These applications demonstrate that diversity-embedded codes have the potential to outperform traditional single-layer codes in moderate signal-to-noise (SNR) regimes.     

Space–Time Codes From Structured Lattices

We present constructions of space–time (ST) codes based on lattice coset coding. First, we focus on ST code constructions for the short block-length case, i.e., when the block length is equal to or slightly larger than the number of transmit antennas. We present constructions based on dense lattice packings and nested lattice (Voronoi) shaping. Our codes achieve the optimal diversity–multiplexing tradeoff (DMT) of quasi-static multiple-input multiple-output (MIMO) fading channels for any fading statistics, and perform very well also at practical, moderate values of signal-to-noise ratios (SNR). Then, we extend the construction to the case of large block lengths, by using trellis coset coding. We provide constructions of trellis coded modulation (TCM) schemes that are endowed with good packing and shaping properties. Both short-block and trellis constructions allow for a reduced complexity decoding algorithm based on minimum mean-squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoding and a combination of this with a Viterbi TCM decoder for the TCM case. Beyond the interesting algebraic structure, we exhibit codes whose performance is among the state-of-the art considering codes with similar encoding/decoding complexity.      

Design of Spherical Lattice Space–Time Codes

In this paper, we propose a systematic procedure for designing spherical lattice (space–time) codes. By employing stochastic optimization techniques we design lattice codes which are well matched to the fading statistics as well as to the decoder used at the receiver. The decoders we consider here include the optimal albeit of highest decoding complexity maximum-likelihood (ML) decoder, the suboptimal lattice decoders, as well as the suboptimal lattice-reduction-aided (LRA) decoders having the lowest decoding complexity. For each decoder, our design methodology can be tailored to obtain low error-rate lattice codes for arbitrary fading statistics and signal-to-noise ratios (SNRs) of interest. Further, we obtain fundamental lower bounds on the error probabilities yielded by lattice and LRA decoders and characterize their asymptotic behavior.      

A Quasi-Random Approach to Space–Time Codes

This paper presents a quasi-random approach to space–time (ST) codes. The basic principle is to transmit randomly interleaved versions of forward error correction (FEC)-coded sequences simultaneously from all antennas in a multilayer structure. This is conceptually simple, yet still very effective. It is also flexible regarding the transmission rate, antenna numbers, and channel conditions (e.g., with intersymbol interference). It provides a unified solution to various applications where the traditional ST codes may encounter difficulties. We outline turbo-type iterative joint detection and equalization algorithms with complexity (per FEC-coded bit) growing linearly with the transmit antenna number and independently of the layer number. We develop a signal-to-noise-ratio (SNR) evolution technique and a bounding technique to assess the performance of the proposed code in fixed and quasi-static fading channels, respectively. These performance assessment techniques are very simple and reasonably accurate. Using these techniques as a searching tool, efficient power allocation strategies are examined, which can greatly enhance the system performance. Simulation results show that the proposed code can achieve near-capacity performance with both low and high rates at low decoding complexity.      

On the Blind Identifiability of Orthogonal Space–Time Block Codes From Second-Order Statistics

In this paper, the conditions for blind identifiability from second-order statistics (SOS) of multiple-input multiple-output (MIMO) channels under orthogonal space-time block coded (OSTBC) transmissions are studied. The main contribution of the paper consists in the proof that, assuming more than one receive antenna, any OSTBC with a transmission rate higher than a given threshold, which is inversely proportional to the number of transmit antennas, permits the blind identification of the MIMO channel from SOS. Additionally, it has been proven that any real OSTBC with an odd number of transmit antennas is identifiable, and that any OSTBC transmitting an odd number of real symbols permits the blind identification of the MIMO channel regardless of the number of receive antennas, which extends previous identifiability results and suggests that any nonidentifiable OSTBC can be made identifiable by slightly reducing its code rate. The implications of these theoretical results include the explanation of previous simulation examples and, from a practical point of view, they show that the only nonidentifiable OSTBCs with practical interest are the Alamouti codes and the real square orthogonal design with four transmit antennas. Simulation examples and further discussion are also provided.     

The Minimum Decoding Delay of Maximum Rate Complex Orthogonal Space–Time Block Codes

The growing demand for efficient wireless transmissions over fading channels motivated the development of space-time block codes. Space-time block codes built from generalized complex orthogonal designs are particularly attractive because the orthogonality permits a simple decoupled maximum-likelihood decoding algorithm while achieving full transmit diversity. The two main research problems for these complex orthogonal space-time block codes (COSTBCs) have been to determine for any number of antennas the maximum rate and the minimum decoding delay for a maximum rate code. The maximum rate for COSTBCs was determined by Liang in 2003. This paper addresses the second fundamental problem by providing a tight lower bound on the decoding delay for maximum rate codes. It is shown that for a maximum rate COSTBC for 2m - 1 or 2m antennas, a tight lower bound on decoding delay is r = (_m-1 ^2m) . This lower bound on decoding delay is achievable when the number of antennas is congruent to 0, 1, or 3 modulo 4. This paper also derives a tight lower bound on the number of variables required to construct a maximum rate COSTBC for any given number of antennas. Furthermore, it is shown that if a maximum rate COSTBC has a decoding delay of r where r < r les 2r, then r=2r. This is used to provide evidence that when the number of antennas is congruent to 2 modulo 4, the best achievable decoding delay is 2(_m-1 ^2m_).     

On the Existence of Perfect Space–Time Codes

Perfect space-time codes are codes for the coherent multiple-input multiple-output (MIMO) channel. They have been called so since they satisfy a large number of design criteria that makes their performances outmatch many other codes. In this correspondence, we discuss the existence of such codes (or more precisely, the existence of perfect codes with optimal signal complexity).     

Diagonal Lattice Space–Time Codes From Number Fields and Asymptotic Bounds

In this paper, we reformulate some constructions of real and complex diagonal lattice space-time codes from number fields which have been given explicitly or implicitly by other researchers. These constructions establish a connection between good diagonal lattice space-time codes and number fields with small absolute values of discriminants. We present two tables for diversity products of some lattice space-time codes from these constructions. The maximal rank of diagonal lattice space-time codes with positive diversity product is determined. We also discuss the asymptotic problem of lattice space-time codes. By using an infinite tower of Hilbert class field and a tamely ramified class field tower, we obtain asymptotically good sequences of lattice space-time codes. Some asymptotic upper bounds are given in the paper as well.     

Performance of Space–Time Codes: Gallager Bounds and Weight Enumeration

Since the standard union bound for space-time codes may diverge in quasi-static fading channels, the limit-before-average (LBA) technique has been exploited to derive tight performance bounds. However, it suffers from the computational burden arising from a multidimensional integral. In this paper, efficient bounding techniques for space-time codes are developed in the framework of Gallager bounds. Two closed-form upper bounds, the ellipsoidal bound and the spherical bound, are proposed that come close to simulation results within a few tenths of a decibel. In addition, two novel methods of weight enumeration operating on a further reduced state diagram are presented, which, in conjunction with the bounding techniques, give a thorough treatment of performance bounds for space-time codes.     

Some Designs of Full Rate Space–Time Codes With Nonvanishing Determinant

In this correspondence, we first present a transformation technique to improve the normalized diversity product for a full rate algebraic space-time block code (STBC) by balancing the signal mean powers at different transmit antennas. After rewriting a cyclic division algebra structure into a multilayer structure for a full rate code, we show that the normalized diversity product of the transformed code with the multilayer structure is better than the one of the transformed code with the cyclic division algebra structure. We then present a new full rate algebraic STBC with multilayer structure with nonvanishing determinant (NVD) for three transmit antennas when signal constellation is carved from QAM. We show that the new code has larger normalized diversity product than the existing 3 times 3 NVD full rate STBC for quadrature amplitude modulation (QAM) signals, and we also show that it has the largest normalized diversity product in a family of full rate STBC.     

A Unified SNR-Dependent Analysis of Space–Time Codes in Fading Channels

The pairwise error probability (PEP) for multiple- input multiple-output (MIMO) radio interfaces is investigated by means of a novel formulation based on compound matrices. The proposed approach is suitable for any MIMO system having average upper-bounded PEP written as [det( I + gamma A)]^-zeta, where A is a Hermitian matrix, zeta an integer number, and gamma the signal-to-noise ratio (SNR); that bound frequently results in MIMO single- and multicarrier transmissions. It is shown that the minimization of the bounded PEP should consider the whole set of nonzero compound matrices of A. In particular, the SNR of interest marks the compound matrix that mainly drives the system performance. Both diversity advantage and coding gain are given as continuous functions of the variable gamma, hence, their asymptotic behaviors are taken as important case of studies. The interaction effects between channel code and propagation environment are also discussed. It is shown how the eigenvectors and eigenvalues of the autocorrelation channel matrix may be considered for code design. It is also proved the maximization of the code rank is not always a necessary requirement for performance improvement being its optimal value fixed by the channel structure and SNR of interest. Finally, the analysis is applied to space-time trellis-coded transmissions over spatially correlated slow Rayleigh-fading channels.