Alternative Criteria for Predictive Blind Deconvolution

Abstract

Blind deconvolution is a major theme in signal processing and has been intensely investigated over the last decades. Among its several applications, we can mention the problem of seismic deconvolution and channel equalization in telecommunications. In these two cases, predictive techniques have been studied by different authors, and presented satisfactory results when some suitables conditions were fulfilled. In fact, the predictive deconvolution structure, when associated with the classical mean squared error criterion is only effective when the distortion system is minimum phase. In the case of nonminimum phase systems, it only provides magnitude equalization, but the phase response remains distorted. In order to overcome this problem, we present in this work some interesting results obtained with the use of p norms, with p 2, as optimization criteria. First we demonstrate that the p prediction error filter works as the Maximum Likelihood solution for blind deconvolution when the signal to be recovered has a generalized Gaussian distribution, with i.i.d. (identically and independently distributed) samples. From this, we show how the best p can be chosen according to the signal distribution. Then we further investigate the phase response of the p filter, emphasizing its potential as well as some limitations in dealing with blind deconvolution, even for nonminimum phase distortion system. Finally, some performance simulations results are provided.

Cascade of Linear Predictors for Deconvolution of Non-Stationary Channels in Sparse and Antisparse Scenarios

“…for all odd, nonlinear functions (•) and (•) [1]. We also investigated in [7] the phase-response of the ℓ PEF, with ≠ 2, verifying that it can present a nonminimum phase response. However, we also observed that the ℓ PEF was not able to compensate the distortions of a channel with a generic phase response.…”

“…In [6] and [7] we have shown that the ℓ 1 PEF, with ≠ 2, performs the nonlinear decorrelation, i.e., the error signal produced has the property…”

Adaptive linear predictors in cascade for blind deconvolution of non-stationary and non-minimum-phase channels

“…In [9], we investigated how the p norms, with p = 2, can be used as alternative criteria for deconvolution. To do so, we have the following optimization problem:…”

“…• For sub Gaussian distributions [14], we use p > 2, with p → ∞ for the uniform distribution. In [9] we provide a rather complete study on p criteria for predictive deconvolution, which shows that the p forward prediction error filters are able to perform the blind deconvolution of some non-minimum-phase systems. However, it still presents performance limitations according to the positions of the zeros of the channel.…”

“…An alternative to perform phase equalization is to adopt a criterion different from the classical MSE, which is related to the 2 norm. In [8], [9] it has been shown that the p norms, with p = 2, are able to perform deconvolution in some nonminimum-phase channels, but not in all of them.…”