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A simple and general form of threaded algebraic space-time (TAST) codes and constellations for arbitrary numbers of transmit and receive antennas and arbitrary input alphabets is given and analyzed. This new form gives revealing insights on the TAST framework, elucidates the connection between space-time constellation expansion and the peak-to-average power ratio (PAR), and establishes the equivalence between a certain class of TAST constellations and constellations derived from division algebras. Index Terms--Codes from division algebra, constellations, multiple-input multiple-output (MIMO) systems, threaded algebraic space-time (TAST) framework.

A simple and systematic approach for constructing full diversity m-dimensional constellations, carved from lattices over a number ring R, is proposed for an arbitrary dimension m. When R = Z[[w.sub.n]], the nth cyclotamic number ring, all the possible dimensions that allow for achieving the optimal minimum product distances using the proposed approach are determined. It turns out that one can construct optimal unitary transformations using our construction if and only if m factors into a power of 2 and powers of the primes dividing n. For m not satisfying these conditions, a method based on Diophantine approximation theory is proposed to "optimize" the minimum product distance. A lower bound on the product distance is given in this case, thus ensuring full diversity with "good" minimum product distances. Furthermore, the proposed approach subsumes the optimal unitary transformations proposed by Giraud et al. over R = Z[[w.sub.4]] and R. = Z[[w.sub.3]], while giving optimal unitary transformations for infinitely many new values of n and m. Index Terms--Constellations, Diophantine approximation, diversity methods, number fields, number rings.

Space-time (ST) constellations that are linear over the field of complex numbers are considered. Relevant design criteria for these constellations are summarized and some fundamental limits to their achievable performances are established. The fundamental tradeoff between rate and diversity is investigated under different constraints on the peak power, receiver complexity, and rate scaling with the signal-to-noise ratio (SNR). A new family of constellations that achieve optimal or near-optimal performance with respect to the different criteria is presented. The proposed constellations belong to the threaded algebraic ST (TAST) signaling framework, and achieve the optimal minimum squared Euclidean distance and the optimal delay. For systems with one receive antenna, these constellations also achieve the optimal peak-to-average power ratio for quadrature amplitude modulation (QAM) and phase-shift keying (PSK) input constellations, as well as optimal coding gains in certain scenarios. The framework is general for any number of transmit and receive antennas and allows for realizing the optimal tradeoff between rate and diversity under different constraints. Simulation results demonstrate the performance gaines offered by the proposed designs in average power and peak power limited systems. Index Terms--Constant modulus, diversity-versus-rate tradeoff, maximum-likelihood (ML) decoding, multiple-input multiple-output (MIMO) channels, space-time constellations.

Some M x T modulation matrices for M transmit antennas and T symbol periods, with M = 2, 3, 4 and T = 2, and M = T = 4 are studied. A transmission rate of M symbols per channel use and a transmit diversity order of min(M, T) are achieved over a quasi-static fading channel when using rotated versions of a multidimensional quadratic amplitude modulation with spectral efficiency 2 bits/symbol. Extension to input constellations with higher spectral efficiencies is then considered. The modulations are then generalized to any number of transmit antennas M and any number of symbol periods T, such that a transmission rate of M symbols per channel use, and a transmit diversity of T are achieved under fast fading (ergodic scenario). By means of signal space diversity, the proposed modulations exploit the degrees of freedom of multi-antenna channels and have moderate detection complexity at moderate and large signal-to-noise ratios (SNRs). Index Terms--Constellations, diversity methods, maximum-likelihood (ML) detection, modulation, multiple-input multiple-output (MIMO) systems.

We examine some algebraic properties of two high-rate linear space-time block codes over M = 2, 3 transmit antennas. Although these high-rate codes have positive coding gain, the gain decreases when increasing the constellation size. We give tight upper and lower bounds on the achieved coding gains as functions of the size of the constellations used. We show that when using the irrational numbers [square root of 3] and [square root of 2], the coding gains express the approximation of these numbers by continued fractions depending on the constellations used. The poor approximation of these numbers by rational numbers is then shown to make the coding gains decrease slowly when increasing the constellation size. Index Terms--Block codes, continued fractions, diversity methods, multiple-input-multiple-output (MIMO) systems, number theory.

The lack of a well-defined and meaningful measure of the quality of computer-generated correlated random variates complicates the design and comparison of generation algorithms for communications system simulation applications. This paper presents quantitative quality measures for computer-generated random variates having a multivariate Gaussian distribution or having a distribution that derives from the multivariate Gaussian distribution. These measures are, in particular, useful for digital communication system simulation applications. A theoretical basis for the definition of the measures is given and it is shown that the measures also possess useful intuitive interpretations. Examples of the use of these quality measures are provided. Index Terms--Fading channels, random number generation, Rayleigh channels, simulation.

New upper and lower bounds on the mean recovery time of decision feedback equalization (DFE) are derived. The recovery time is defined as the time it takes the decision feedback equalizer (DFEQ) to reach the error-free state after an error has corrupted an error-free DFEQ. The derivations of the bounds assume a causal channel response, independent data symbols, and independent noise samples. The bounds are found to be tighter, especially at large SNR, than previous bounds in a numerical example. Index Terms - Decision feedback equalization, equalizer error statistics.

The exact computation of the symbol error probability of systems using decision feedback equalization is difficult due to the propagation of errors. New upper bounds to the error probability of decision feedback equalization, which take error propagation into account, are developed. The derivations of the bounds assume a causal channel response, independent data symbols, and independent noise samples. The bounds are valid for any noise process that has a symmetric and unimodal probability density function. In some cases, the new bounds are significantly tighter than a well-known upper bound of Duttweiler, Mazo, and Messerschmitt.

New upper and lower bounds on the variances of the recovery times of decision feedback equalizers are derived. The analysis is based on generalizing a difference equation technique, used previously only for first-order moments, to second-order moments. The different error recovery time variances resulting from random channel errors and from the reset or startup condition are both investigated. The cases of noisy and noiseless operating conditions are studied and the influence of the closing of the sampling eye is investigated. Index Terms--Decision feedback equalization, equalizer error statistics, error propagation.