On the equivalence between the super-exponential algorithm and a gradient search method

This paper reviews the super-exponential algorithm proposed by Shalvi and Weinstein (1993) for blind channel equalization. We show that the algorithm coincides with a gradient search of a maximum of a cost function, which belongs to a family of functions very relevant in blind channel equalization. This family traces back to Donoho´s (1981) work on minimum entropy deconvolution, and also underlies the Godard (1980) (or constant modulus) and the Shalvi-Weinstein algorithms. Using this gradient search interpretation, 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