Convergence analysis of finite length blind adaptive equalizers

The paper presents some new analytical results on the convergence of two finite length blind adaptive channel equalizers, namely, the Godard equalizer and the Shalvi-Weinstein equalizer. First, a one-to-one correspondence in local minima is shown to exist between the Godard and Shalvi-Weinstein equalizers, hence establishing the equivalent relationship between the two algorithms. Convergence behaviors of finite length Godard and Shalvi-Weinstein equalizers are analyzed, and the potential stable equilibrium points are identified. The existence of undesirable stable equilibria for the finite length Shalvi-Weinstein equalizer is demonstrated through a simple example. It is proven that the points of convergence for both finite length equalizers depend on an initial kurtosis condition. It is also proven that when the length of equalizer is long enough and the initial equalizer setting satisfies the kurtosis condition, the equalizer will converge to a stable equilibrium near a desired global minimum. When the kurtosis condition is not satisfied, generally the equalizer will take longer to converge to a desired equilibrium given sufficiently many parameters and adequate initialization. The convergence analysis of the equalizers in PAM communication systems can be easily extended to the equalizers in QAM communication systems