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.

A general tool for multichannel and multipath problems is given in FIR matrix algebra. With Finite Impulse Response (FIR) filters (or polynomials) assuming the role played by complex scalars in traditional matrix algebra, we adapt standard eigenvalue routines, factorizations, decompositions, and matrix algorithms for use in multichannel /multipath problems. Using abstract algebra/group theoretic concepts, information theoretic principles, and the Bussgang property, methods of single channel filtering and source separation of multipath mixtures are merged into a general FIR matrix framework. Techniques developed for equalization may be applied to source separation and vice versa. Potential applications of these results lie in neural networks with feed-forward memory connections, wideband array processing, and in problems with a multi-input, multi-output network having channels between each source and sensor, such as source separation. Particular applications of FIR polynomial matrix alg...

The Godard or the constant modulus algorithm (CMA) is an effective technique for blind receiver design in communications. However, due to the complexity of the Constant Modulus (CM) cost function, the performance of CM receivers has primarily been evaluated using simulations. Theoretical analysis is typically based on either the noiseless case or approximations of the cost function. The following question, while resolvable numerically for a specific example, remains unanswered in a generic manner: In the presence of channel noise, where are the CM local minima and what are their mean-squared errors (MSE)? In this paper, a geometrical approach is presented that relates CM to Wiener (or minimum MSE) receivers. Given the MSE and the intersymbol/user interference of a Wiener receiver, a sufficient condition is given for the existence of a CM local minimum in the neighborhood of the Wiener receiver. MSE bounds on CM receiver performance are derived and shown to be tight in simulations. The ...

Convergence properties of the constant modulus (CM) and the Shalvi--Weinstein (SW) algorithms in the presence of noise remain largely unknown. A new geometrical approach to the analysis of constant modulus and Shalvi-Weinstein receivers is proposed by considering a special constrained optimization involving norms of the combined channel-receiver response. This approach provides a unified framework within which various blind and (nonblind) Wiener receivers can all be analyzed by circumscribing an ellipsoid by norm balls of different types. A necessary and sufficient condition for the equivalence among constant modulus, Shalvi-Weinstein, zero forcing, and Wiener receivers is obtained. Answers to open questions with regard to CM and SW receivers, including their locations and their relationship with Wiener receivers, are provided for the special orthogonal channel and the general two-dimensional (2-D) channel-receiver impulse response. It is also shown that in two dimensions, each CM or...

The constant modulus (CM) cost function is analyzed for array signal processing. The analysis includes arbitrary source types. It is shown that CM receivers have the signal space property except in two special cases. Local minima of CM cost function are completely characterized for the noiseless case. In the presence of measurement noise, the existence of CM local minima in the neighborhood of Wiener receivers is established, and their mean squared error (MSE) performance, output power, estimation bias and residual interference are evaluated. Numerical examples are presented to demonstrate the e#ects of signal statistics on the locations of CM local minima. # 1999 Published by Elsevier Science B.V. All rights reserved.

This paper investigates connections between (nonblind) Wiener receivers and blind receivers designed by minimizing the constant modulus (CM) cost. Applicable to both T-spaced and fractionally spaced FIR equalization, the main results include 1) a test for the existence of CM local minima near Wiener receivers; 2) an analytical description of CM receivers in the neighborhood of Wiener receivers; 3) mean square error (MSE) bounds for CM receivers. When the channel matrix is invertible, we also show that the CM receiver is approximately colinear with the Wiener receiver and provide a quantitative measure of the size of neighborhoods that contain the CM receivers and the accuracy of the MSE bounds. Index Terms---Blind equalization, CMA, MSE. I. INTRODUCTION B LIND equalization of intersymbol interference (ISI) in communication channels and blind separation of multiple users are promising signal processing techniques in certain communication system designs. One of the earliest blind ...

Largo quadrature amplitude modulation (QAM) constellations are currently used in throughput efficient high speed communication applications such as digital TV. For such largo signal constellations, carrier phase synchronization is a crucial problem because for efficiency reasons the carrier acquisition must often be performed blindly, without the use of training or pilot sequences. The goal of the present paper is to provide thorough performance analysis of the blind carrier phase estimaters that have boon proposed in the literature and to assess their relative merits.

: This paper presents a theoretical analysis of the static and dynamic convergence behavior for a general class of adaptive blind equalizers. We first study the properties of prediction error functions of blind equalization algorithms, and then we use these properties to analyze the static and dynamic convergence behavior based on the independent assumption. We prove in this paper that with a small step-size, the ensemble average of equalizer coefficients will converge to the minimum of the cost function near the channel inverse. However, the convergence is not consistent. The correlation matrix of equalizer coefficients at equilibrium is determined by a Lyapunov equation. According to our analysis results, for a given channel and step-size, there is an optimal length for an equalizer to minimize the intersymbol interference. This result implies that a longer-length blind equalizer does not necessarily outperform a shorter one, as contrary to what conventionally conjectured. The theore...

INTRODUCTION In a digital communication setting, the transmitted signal is often distorted through a dispersive channel which introduces intersymbol interference (ISI). In a high SNR situation, where ISI is the limiting factor more so than channel noise or interference, a linear equalizer may be used to remove ISI. Successful source estimation from the equalizer output is accomplished by decision device (e.g. a quantizer) if enough of the ISI is removed. A condition on the combined channel-equalizer resulting in source recovery is known as an "open-eye" condition. Due to the fact that the (unknown) channel impulse response may vary over time, an adaptive equalization scheme, with its ability to track time-varying system is considered. A standard adaptive filtering scheme (e.g. LMS) adjusts the tap weights based on an error signal which is the di#erence between the filter (equalizer) output and the desired output (e.g. the transmitted sequence). To di#erentiate our problem from those

The problem of blind equalization is addressed. The approach presented herein examines the possible input sequences directly by using a bank of filters and, in contrast to common approaches, does not try to find an approximative inverse of the channel dynamics. First, the identifiability question of a noise-free FIR model is investigated. A sufficient condition for the input sequence (persistently exciting of a certain order) is given which guarantees that both the channel model and the input sequence can be determined exactly in finite time. A recursive algorithm is given for a time-varying IIR channel model with additive noise, which does not require a training sequence. The estimated sequence is an arbitrary good approximation of the maximum a posteriori estimate. The proposed method is evaluated on a Rayleigh fading communication channel. It shows fast convergence properties and good tracking ability. 1 INTRODUCTION The problem of channel equalization is of considerable interest ...