Effect of Channel Noise on Blind Equalization and Carrier Phase Recovery of CMA and MMA

This work investigates the effect of channel noise on the performances of the constant modulus algorithm (CMA), the multimodulus algorithm (MMA), and a phase tracker (PT), following the CMA for carrier phase recovery, when various symmetrical two-dimensional complex-valued source constellations are used. The contributions of this work are two-fold. First, a generalized formulation of the relocation of stationary points toward the origin owing to channel noise is derived for both the CMA and the MMA, allowing their sensitivity to channel noise to be explicitly compared when various constellations are used. Second, the curvatures of the error surface at the stationary points in both the radial and the angular directions are reduced in a way that is linearly proportional to the square of the distances between the stationary points and the origin as the stationary points of the CMA and the MMA relocate toward the origin in the presence of additive channel noise. The reduction of the error surface curvature in the radial direction reduces the rate of blind equalization convergence, whereas the reduction in the angular direction degrades the capacity for carrier phase recovery.