On the Rapid Initial Convergence of Least-Squares Equalizer Adjustment Algorithms
Mathias S. Mueller · Bell System Technical Journal · 1981
Adjustment algorithms for transversal equalizers derived from least-squares cost functions are known to converge extremely fast. While various simulation results confirming this fact abound in the literature, a theory explaining the fast convergence has been lacking. This paper reports on steps toward such a theory. For some commonly used start-up data sequences it was found that algebraic properties of the sampled signal vectors play a critical role in the transient behavior of these algorithms; namely, successive signal vectors are linearly independent for a large class of transmission channels in the absence of noise. After N iterations (N being the number of taps), the resulting coefficient vector is found to be related to well-known equalizer coefficient vectors. If a single pulse is used as a training signal, the zero forcing equalizer is obtained; if a pseudo random noise sequence, with a period in symbols equal to the number of coefficients is used, the steady-state solution of the cyclic equalization is obtained. Thus, after only N iterations, the least-squares algorithms yield a coefficient vector which is only asymptotically obtainable by gradient techniques.