The Core of FSE-CMA Behavior Theory 1

This chapter presents the basics of the current theory regarding the behavior of blind fractionally-spaced and/or spatial-diversity equalizers (FSE) adapted via the constant modulus algorithm (CMA). The constant modulus algorithm, which was developed in the late 1970s and disclosed in the early 1980s, performs a stochastic gradient descent of a cost function that penalizes the dispersion of the equalizer output from a constant value. The constant modulus (CM) cost function leads to a blind algorithm because evaluation of the CM cost at the receiver does not rely on access to a replica of the transmitted source, as in so-called “trained” scenarios. The capability for blind start-up makes certain communication systems feasible in circumstances that do not admit training. The analytically convenient feature of the fractionally-spaced realization of a linear equalizer is the potential for perfect equalization in the absence of channel noise given a finite impulse response equalizer of time span matching that of the finite impulse response channel. The conditions for perfect equalization coupled with some mild conditions on the source can be used to establish convergence to perfect performance with FSE parameter adaptation by CMA from any equalizer parameter initialization. The FSE-CMA behavior theory presented here merges the taxonomy of the behavior theory of trained adaptive equalization and recent robustness analysis of FSE-CMA with violation of the conditions leading to perfect equalization and global asymptotic optimality of FSE-CMA.

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