Convergence analysis of self-adaptive equalizers

A theoretical analysis of self-adaptive equalization for data-transmission is carried out starting from known convergence results for the corresponding trained adaptive filter. The development relies on a suitable ergodicity model for the sequence of observations at the output of the transmission channel. Thanks to the boundedness of the decision function used for data recovery, it can be proved that the algorithm is bounded. Strong convergence results can be reached when a perfect (noiseless) equalizer exists: the algorithm will converge to it if the eye pattern is initially open. Otherwise convergence may take place towards certain other stationary points of the algorithm for which domains of attraction have been defined. Some of them will result in a poor error rate. The case of a noisy channel exhibits limit points for the algorithm that differ from those of the classical (trained) algorithm. The stronger the noise, the greater the difference is. One of the principal results of this study is the proof of the stability of the usual decision feedback algorithms once the learning period is over.