Vector-CM stable equilibrium analysis

Abstract

The vector-constant modulus (VCM) criterion is an extension of the constant modulus (CM) criterion [2] introduced recently for equalization of channels involving Gaussian sources [1], [5]. 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.

Blind equalization in spatial multiplexing MIMO-OFDM systems based on vector CMA and decorrelation criteria

“…We show that under this condition the proposed composite criterion admits truly stable minima corresponding to the zero-forcing equalizer performing both the blind equalization and co-channel signal cancellation. These local minima always exist and they are stable for any input-output delays, unlike the classical VCMA criterion [18] which admits stable zero forcing solutions only for extreme input-output delays [15]. For other input-output delays the VCMA behavior remains unknown.…”

“…Its global convergence has not been established [9,15]. If multiple signals are present, depending on its initialization, VCMA may converge to any of the transmitted signals, usually to the ones that have the strongest power [18].…”

“…In other words, it reduces to a power constraint on the global channel-equalizer impulse response. Consequently, the CMA criterion alone is not able to equalize Gaussian signals [15], the stationary points are any vectors which have a certain norm. 2.…”

“…The VCMA criterion penalizes the deviation of the block energy from a dispersion constant. The VCMA cost function may be decomposed into a constant modulus (CM) cost function [7] and an auto-correlation function of the squared magnitudes of the received signal [15]. Therefore, it is not applicable to signals which have a periodic correlation such as OFDM signal.…”

“…The local convergence properties of VCMA have been investigated in [15]. In our analysis, we assume a composite cost function and we analyze its local convergence properties under more general conditions, as it will be shown later in Sect.…”