This paper proposes a new approach to the analysis of the steady-state performance of constant modulus algorithms (CMA), which are among the most popular adaptive schemes for blind equalization. A major feature of the proposed feedback approach is that it bypasses the need for working directly with the weight error covariance matrix. In so doing, approximate expressions for the steady-state mean-square error of several CM algorithms are derived, including CMA2-2, CMA1-2, normalized CMA, and a new normalized CMA variant with less bias. A comparison among the various algorithms is also performed, along with several simulation results. The conclusions confirm the superior performance of CMA2-2.

A feedback approach to the steady-state performance of fractionally spaced blind adaptive equalizers

This paper proposes two modifications of the filtered-x least mean squares (FxLMS) algorithm with improved convergence behavior albeit at the same computational cost of 2M operations per time step as the original FxLMS update The paper further introduces a generalized FxLMS recursion and establishes that the various algorithms are all of filtered-error form A choice of the stepsize parameter that guarantees faster convergence and conditions for robustness are also derived Several simulation results are included to illustrate the discussions

Robust FxLMS algorithms with improved convergence performance

The code-constrained constant modulus algorithm (CCM) implemented with a stochastic gradient (SG) technique is a very effective and efficient blind approach for interference suppression when a communication channel is frequency-selective. In nonstationary wireless environments, users frequently enter and exit the system, making it very difficult for the receiver to compute a predetermined step-size. This suggests the deployment of mechanisms to automatically adjust the step-size in order to ensure good tracking of the interference and the channel. In this paper, the performance of blind CCM adaptive receivers for direct sequence code division multiple access (DS-CDMA) systems that employ stochastic gradient (SG) algorithms with variable step-size mechanisms is investigated. We propose a novel low-complexity variable step-size mechanism for blind CCM CDMA receivers. Convergence and tracking analyses of the proposed adaptation techniques are carried out for multipath channels. Finally, numerical experiments are presented for nonstationary environments, showing that the new mechanism achieves superior performance to previously reported methods at a reduced complexity.

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Low-Complexity Variable Step Size Mechanism for Code-Constrained Constant Modulus Stochastic Gradient Algorithms applied to CDMA Interference Suppression

This paper studies the behavior of the error sequence of stop-anti-go variants of two adaptive blind equalizers, namely CMA2-2 and Sato's (1975) algorithm. It is shown that for transmitted signals with constant modulus /spl gamma/, the equalizer output can be made to lie within the circle of radius /spl gamma//spl radic/c infinitely often, for some value of c that is only slightly larger than one.

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On the convergence of blind adaptive equalizers for constant modulus signals

The minimization of empirical risks over finite sample sizes is an important problem in large-scale machine learning. A variety of algorithms has been proposed in the literature to alleviate the computational burden per iteration at the expense of convergence speed and accuracy. Many of these approaches can be interpreted as stochastic gradient descent algorithms, where data is sampled from particular empirical distributions. In this work, we leverage this interpretation and draw from recent results in the field of online adaptation to derive new tight performance expressions for empirical implementations of stochastic gradient descent, mini-batch gradient descent, and importance sampling. The expressions are exact to first order in the step-size parameter and are tighter than existing bounds. We further quantify the performance gained from employing mini-batch solutions, and propose an optimal importance sampling algorithm to optimize performance.

Stochastic gradient descent with finite samples sizes

An l 2 -stable feedback structure for nonlinear adaptive filtering and identification

Absiruct-This paper provides a time-domain feedback analysis of gradient-based adaptive schemes. A key emphasis is on the robustness performance of the adaptive filters in the presence of disturbances and modeling uncertainties (along the lines of H”-theory and robust filtering). The analysis is carried out in a purely deterministic framework and assumes no prior statistical information or independence conditions. It is shown that an intrinsic feedback structure can be associated with the varied adaptive schemes. The feedback structure is motivated via energy arguments and is shown to consist of two major blocks: a time-variant lossless (i.e., energy preserving) feedforward path and a time-variant feedback path. The configuration is further shown to lend itself to analysis via a socalled small gain theorem, thus leading to stability and robustness conditions that require the contractivity of certain operators. Choices for the step-size parameter in order to guarantee faster rates of convergence are also derived, and simulation results are included to demonstrate the theoretical findings. In addition, the time-domain analysis provided in this paper is shown to extend the so-called transfer function approach to a general time-variant scenario without any approximations.

Markus Rupp

Papers

An l 2 -stable feedback structure for nonlinear adaptive filtering and identification

e-Domain Feedback Analysis of nor Adaptive Gradient Algorithms