Sufficient conditions for the local convergence of constant modulus algorithms

The constant modulus (CM) criterion has become popular in the design of blind linear estimators of sub-Gaussian i.i.d. processes transmitted through unknown linear channels in the presence of unknown additive interference. The existence of multiple CM minima, however, makes it difficult for CM-minimizing schemes to generate estimates of the desired source (as opposed to an interferer) in multiuser environments. In this paper, we present three separate sufficient conditions under which gradient descent (GD) minimization of CM cost will locally converge to an estimator of the desired source at a particular delay. The sufficient conditions are expressed in terms of statistical properties of the initial estimates, specifically, CM cost, kurtosis, and signal-to-interference-plus-noise ratio (SINR). Implications on CM-GD initialization methods are also discussed.