Convolutional Noise PDF at the Convergence State of a Blind Adaptive Equalizer

In the literature, the convolutional noise obtained at the output of a blind adaptive equalizer, is often modeled as a Gaussian process during the latter stages of the deconvolution process where the process is close to optimality. However, up to now, no strong mathematical basis was given supporting this phenomenon. Furthermore, no closed-form or closed-form approximated expression is given that shows what are the constraints on the system's parameters (equalizer's tap-length, input signal statistics, channel power, chosen equalization method and step-size parameter) for which the assumption of a Gaussian model for the convolutional noise holds. In this paper, we consider the two independent quadrature carrier input case and type of blind adaptive equalizers where the error that is fed into the adaptive mechanism which updates the equalizer's taps can be expressed as a polynomial function of the equalized output up to order three. We show based on strong mathematical basis that the convolutional noise pdf at the latter stages of the deconvolution process where the process is close to optimality, is approximately Gaussian if complying on some constraints depending on the step-size parameter, input constellation statistics, channel power, chosen equalization method and equalizer's tap-length. Simulation results confirm our findings.