Use of the Newton Method for Blind Adaptive Equalization Based on the Constant Modulus Algorithm

We study the applicability of the second-order Newton gradient descent method for blind equalization of complex signals based on the constant modulus algorithm (CMA). The constant modulus (CM) loss function is real with complex valued arguments, and, hence, nonanalytic. We, therefore, use the framework of the Wirtinger calculus to derive a useful and insightful form of the Hessian for noiseless FIR channels and rederive the known fact that the full Hessian of the CM loss function is always singular in a simpler manner. For the implementation of a suboptimum version of Newton algorithm, we give the conditions under which the leading partial Hessian is nonsingular for a noiseless FIR channel model. For this channel model, we show that the perfectly equalizing solutions are stationary points of the CM loss function and also evaluate the leading partial Hessian and the full Hessian at a perfectly equalizing solution. We also discuss regularization of the full Newton method. Finally, some simulation results are given.