Vector Modulus Algorithms for Blind Equalization of Shaped Modulation

The vector constant modulus algorithm (VCMA) was recently introduced as an extension of CMA which can equalize data from shaped sources having nearly Gaussian marginal distributions. VCMA was developed with the shaping technique of shell mapping in mind, but it also equalizes data from other shaping methods, such as trellis shaping. Some simple changes in the structure of VCMA permit its use in fractionally spaced equalizers, the bene¯ts of which include shorter equalizer lengths, built-in matched ¯ltering, and the possibility of global convergence on FIR channels. The use of a vector modulus in the cost function has general applicability and is used to obtain the vector reduced constellation algorithm (VRCA) and the vector multimodulus algorithm (VMMA), both based on constant modulus criteria and derived from their scalar modulus counterparts, RCA and MMA. Although the global convergence of VCMA has not yet been fully proved, some theoretical results and the close parallels with CMA suggest that it inherits the desirable properties of CMA. Simulations verify the superior performance of T=2-spaced VCMA compared to baud-spaced VCMA on various short FIR channels, the ability of VCMA to equalize trellis-shaped data, and the performance of one of the related vector modulus algorithms