On the Independence Theory of Equalizer Convergence

High-speed pulse amplitude modulated (pam) data transmission over telephone channels is only possible when adaptive equalization is used to mitigate the linear distortion found on the (initially unknown) channel. At the beginning of the equalization procedure, the tap weights are adjusted to minimize the inter symbol interference between pulses. The “stochastic gradient” algorithm is an iterative procedure commonly used for setting the coefficients in these and other adaptive filters, but a proper understanding of the convergence has never been obtained. It has been common analytical practice to invoke an assumption stating that a certain sequence of random vectors which direct the “hunting” of the equalizer are statistically independent. Everyone acknowledges this assumption to be far from true, just as everyone agrees that the final predictions made using it are in excellent agreement with experiments and simulations. We take the resolution of this question as our main problem When one begins to analyze the performance of the algorithm, one sees that the average mean-square error after the nth iteration requires knowing, as an intermediate step, the mathematical expectation of the product of a sequence of statistically dependent matrices. We transform the latter problem to a space of sufficiently high dimension where the required average may be obtained from a canonical equation V <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</inf> = A(α)V <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> + Here A(α) is a square matrix, depending on the “step-size” α of the original algorithm, and V <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> and F are vectors. The mean-square error is calculable from the solution V <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> .