Vector-CM stable equilibrium analysis

The vector-constant modulus (VCM) criterion is an extension of the constant modulus (CM) criterion introduced recently for equalization of channels involving Gaussian sources. In this letter, we analyze the behavior of VCM for arbitrary source distributions and combined channel-receiver impulse responses of finite dimension. We begin by pointing out the difference between the VCM and CM cost functions and showing that the VCM criterion can be expressed as a composite criterion combining the CM cost function and a penalty term involving cross-correlations of the equalizer output. We continue by providing conditions for noise-free channels, under which VCM admits stable minima corresponding to zero-forcing (ZF) channel receivers. We find that for sub-Gaussian sources, the VCM and CM criteria share the same global minima. For Gaussian and super-Gaussian sources, however, it appears that only ZF receivers corresponding to input/output transmission delays at the extremes of the range of possible delays are truly stable equilibria of VCM.