In this paper we review the most peculiar and interesting information-theoretic and communications features of fading channels. We first describe the statistical models of fading channels which are frequently used in the analysis and design of communication systems. Next, we focus on the information theory of fading channels, by emphasizing capacity as the most important performance measure. Both single-user and multiuser transmission are examined. Further, we describe how the structure of fading channels impacts code design, and finally overview equalization of fading multipath channels.

In this semi-tutorial paper, a comprehensive survey of closest-point search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closest-point search algorithm, based on the Schnorr-Euchner variation of the Pohst method, is implemented. Given an arbitrary point x 2 R m and a generator matrix for a lattice , the algorithm computes the point of that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan algorithm and an experimental comparison with the Pohst algorithm and its variants, such as the recent Viterbo-Boutros decoder. The improvement increases with the dimension of the lattice. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, compu...

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.

Maximum-likelihood (ML) decoding algorithms for Gaussian multiple-input multiple-output (MIMO) linear channels are considered. Linearity over the field of real numbers facilitates the design of ML decoders using number-theoretic tools for searching the closest lattice point. These decoders are collectively referred to as sphere decoders in the literature. In this paper, a fresh look at this class of decoding algorithms is taken. In particular, two novel algorithms are developed. The first algorithm is inspired by the Pohst enumeration strategy and is shown to offer a significant reduction in complexity compared to the Viterbo--Boutros sphere decoder. The connection between the proposed algorithm and the stack sequential decoding algorithm is then established. This connection is utilized to construct the second algorithm which can also be viewed as an application of the Schnorr--Euchner strategy to ML decoding. Aided with a detailed study of preprocessing algorithms, a variant of the second algorithm is developed and shown to offer significant reductions in the computational complexity compared to all previously proposed sphere decoders with a near-ML detection performance. This claim is supported by intuitive arguments and simulation results in many relevant scenarios.

The increasing need of high data rate transmissions over time or frequency selective fading channels has drawn attention to modulation schemes with high spectral e#ciency such as QAM. With the aim of increasing the `diversity order' of the signal set we consider the multidimensional rotated QAM constellations. Very high diversity orders can be achieved and this results in an almost Gaussian performance over the fading channel. This multidimensional modulation scheme is essentially uncoded and enables to trade diversity for system complexity, at no expense of power or bandwidth. Key Words : QAM Modulation, fading, diversity, number fields, rotation, lattices. I. Introduction The rapidly growing need of high data rate transmissions over fading channels has stimulated the interest for AM-PM modulation schemes with high spectral e#ciency (or throughput) [1], [2], [3]. The e#ectiveness of these transmission schemes basically relies on the good error correcting capabilities of a code. The pr...

A universal framework is developed for constructing full-rate and full-diversity coherent space--time codes for systems with arbitrary numbers of transmit and receive antennas. The proposed framework combines space--time layering concepts with algebraic component codes optimized for single-input--single-output (SISO) channels. Each component code is assigned to a "thread" in the space--time matrix, allowing it thus full access to the channel spatial diversity in the absence of the other threads. Diophantine approximation theory is then used in order to make the different threads "transparent" to each other. Within this framework, a special class of signals which uses algebraic number-theoretic constellations as component codes is thoroughly investigated. The lattice structure of the proposed number-theoretic codes along with their minimal delay allow for polynomial complexity maximum-likelihood (ML) decoding using algorithms from lattice theory. Combining the design framework with the Cayley transform allows to construct full diversity differential and noncoherent space--time codes. The proposed framework subsumes many of the existing codes in the literature, extends naturally to time-selective and frequency -selective channels, and allows for more flexibility in the tradeoff between power efficiency, bandwidth efficiency, and receiver complexity. Simulation results that demonstrate the significant gains offered by the proposed codes are presented in certain representative scenarios.

We construct a new family of linear space-time block codes by the combination of rotated constellations and the Hadamard transform, and we prove them to achieve the full transmit diversity over a quasi-static or fast fading channels. The proposed codes transmit at a normalized rate of 1 symbol/sec. When the number of transmit antennae n =1, 2 or n is a multiple of 4 we spread a rotated version of the information symbol vector by the Hadamard transform and send it over n transmit antennae and n time periods; for other values of n, we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an nn space-time code matrix. The codes maintain their rate, diversity and coding gains for all real and complex constellations carved from the complex integers ring Z[i], and they outperform the codes from orthogonal design when using complex constellations for n > 2. The maximum likelihood decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity. It is shown that using the proposed codes in a multi-antenna system yields good performances with high spectral efficiency and moderate decoding complexity.

We present a unified approach to designing space-time (ST) block codes using linear constellation precoding (LCP). Our designs are based either on parameterizations of unitary matrices, or on algebraic number-theoretic constructions. With an arbitrary number of transmit- and receive-antennas, ST-LCP achieves rate 1 symbol/s/Hz and enjoys diversity gain as high as over (possibly correlated) quasi-static and fast fading channels. As figures of merit, we use diversity and coding gains, as well as mutual information of the underlying multiple-input-multiple-output system. We show that over quadrature-amplitude modulation and pulse-amplitude modulation, our LCP achieves the upper bound on the coding gain of all linear precoders for certain values of and comes close to this upper bound for other values of , in both correlated and independent fading channels. Compared with existing ST block codes adhering to an orthogonal design (ST-OD), ST-LCP offers not only better performance, but also higher mutual information for...

We construct a full data rate space-time block code over M =2 transmit antennas and T =2 symbol periods, and we prove that it achieves a transmit diversity of 2 over all constellations carved from Z[i] . Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti code. It is shown that the new code outperforms the Alamouti code at low and high SNR when the number of receive antennas N>1. The performance improvement is further enhanced when N or the size of the constellation increases. We relate the problem of space-time diversity gain to algebraic number theory, and the coding gain optimization to the theory of simultaneous Diophantine approximation in the geometry of numbers. We find that the coding gain optimization is equivalent to find irrational numbers "the furthest" from any simultaneous rational approximations.

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