A gradient search interpretation of the super-exponential algorithm

This article reviews the super-exponential algorithm proposed by Shalvi and Weinstein (1993) for blind channel equalization. The principle of this algorithm-Hadamard exponentiation, projection over the set of attainable combined channel-equalizer impulse responses followed by a normalization-is shown to coincide with a gradient search of an extremum of a cost function. The cost function belongs to the family of functions given as the ratio of the standard l_2p and l₂ sequence norms, where p>1. This family is very relevant in blind channel equalization, tracing back to Donoho´s (1981) work on minimum entropy deconvolution and also underlying the Godard (1980) (or constant modulus) and the earlier Shalvi-Weinstein algorithms. Using this gradient search interpretation, which is more tractable for analytical study, we give a simple proof of convergence for the super-exponential algorithm. Finally, we show that the gradient step-size choice giving rise to the super-exponential algorithm is optimal