This paper analyzes and compares the robustness of the constant modulus algorithm (CMA) and multimodulus algorithm (MMA) in the presence of additive channel noise when the channel satisfies the zero-forcing conditions. A generalized formulation of the relocation of stationary points toward the origin in terms of signal-to-noise ratio (SNR) was developed for both the CMA and the MMA to facilitate the comparison of the sensitivity to channel noise of CMA with that of MMA for various source constellations. Relationships between the relocation of the stationary points for the MMA and CMA were derived. Simulation results appear to confirm the main results of this work, regarding the sensitivity to channel noise when the fractionally spaced CMA (FS-CMA) and fractionally spaced MMA (FS- MMA) criteria were applied with various sources.

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.