Nonglobal convergence of blind recursive identifiers based on gradient descent of continuous cost functions

A blind adaptive identification algorithm uses explicit measurements of the plant output coupled with only implicit knowledge of the input (in the form of its statistical properties) for updating the parameters of a (delayed) plant inverse model. Blind adaptive algorithms are recursive identification schemes used in communication systems, and are often designed via gradient descent minimization of certain memoryless, non-MSE (mean square error) continuous cost functions. It is shown that for these cost functions, there generally exist local minima that do not correspond to the ideal plant inverse identification once the algorithm has converged. The authors demonstrate the general nonglobality of gradient descent blind algorithms and the need for initialization within the regions of attractions of the desired minima in order to achieve the objective of successful identification. Furthermore, it is shown that global convergence in the total (channel plus equalizer) parameter space does not necessarily imply the same behavior in finite equalizer parameter space