A Projected Gradient Method for Automatic Equalization in the Discrete Frequency Domain

Abstract

In a recent paper [1] we reported on a new mean-square error automatic equalizer utilizing Rosen's gradient projection theorem to optimize parameters in the discrete frequency domain. Here we develop another projection method to optimize the discrete frequency parameters. The algorithm converges (in the mean) for any channel, even in the presence of noise. It is shown that for the channels considered, convergence is equivalent to comparable time domain equalizers. The method makes use of fast Fourier transform (FFT) algorithms for computation of the iteration matrix, the gradient, and the projection operation. Use of the FFT for parameter iterations reduces the necessary computations per parameter to a number proportional to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">log_{2} M</tex> compared to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> for a time domain equalizer, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> is the number of equalizer parameters. The method results in fewer computations per parameter for each iteration, but a somewhat slower rate of convergence than the method employing Rosen's gradient projection.