A Maximum Entropy inspired model for the convolutional noise PDF

Abstract

In this paper we consider a blind adaptive deconvolution problem in which we observe the output of an unknown linear system (channel) from which we want to recover its input using an adaptive blind equalizer (adaptive linear filter). Since the channel coefficients are unknown, the optimal equalizer's coefficients are also unknown. Thus, the equalizer's coefficients used in the deconvolution process are only approximated values leading to an error signal in addition to the source signal at the output of the deconvolutional process. We define this error signal throughout the paper as the convolutional noise. It is well known that the convolutional noise probability density function (pdf) is not a Gaussian pdf at the early stages of the deconvolutional process and only at the latter stages of the deconvolutional process the convolutional noise pdf tends to be approximately Gaussian. Despite this knowledge, the convolutional noise pdf was modeled up to recently as a Gaussian pdf because it simplifies the Bayesian calculations when carrying out the conditional expectation of the source input given the equalized or deconvolutional output and since no other model was suggested for it. Recently, a new model was suggested by the same author for the convolutional noise pdf based on the Edgeworth expansion series. This new model leads to improved deconvolution performance for the 16 Quadrature Amplitude Modulation (QAM) input and for a signal to noise ratio (SNR) of 30 dB. Thus, the question that arose here was whether we may find another model for the convolutional noise pdf that will also lead the system with improved deconvolutional performance compared to the case when the Gaussian model is applied for the convolutional noise pdf. In this paper, we propose a new model for the convolutional noise pdf inspired by the Maximum Entropy density approximation technique. We derive the relevant Lagrange multipliers and obtain as a by-product new closed-form approximated expressions for the conditional expectation and mean square error (MSE). Simulation results indicate that improved system performance is obtained from the residual ISI point of view for the 16QAM input case with our new proposed model for the convolutional noise pdf compared to the case when the Gaussian model or Edgeworth expansion series are applied for the convolutional noise pdf. For two other chosen input sources, a faster convergence rate is observed with the algorithm using our new proposed model for the convolutional noise pdf compared to the Maximum Entropy and Godard's algorithm.