A gradient search interpretation of the super-exponential algorithm

This correspondence reviews the super-exponential algorithm proposed by Sha lvi and Weinstein for blind channel equalization. The principle of this alg orithm-Hadamard exponentiation, projection over the set of attainable combi ned channel-equalizer impulse responses followed by a normalization-is show n 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 l(2) sequence norms, where p > 1. This family is very r elevant in blind channel equalization, tracing back to Donoho's work on min imum entropy deconvolution and also underlying the Godard (or Constant Modu lus) and the earlier Shalvi-Weinstein algorithms. Using this gradient searc h interpretation, which is more tractable for analytical study, we give a s imple proof of convergence for the super-exponential algorithm. Finally, we show that the gradient step-size choice giving rise to the super-exponenti al algorithm is optimal.