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 Lp norms, with p different of 2, as optimization criteria. First we demonstrate that the Lp 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 Lp 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.
Highlights
The problem of estimating future values of a time series from its present and past samples is a very challenging problem in the field of signal processing
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. samples
It is well known in the literature that the classical prediction error filter obtained through the mean squared error (MSE) criterion is only effective for the deconvolution of minimum phase systems, since this filter provides decorrelated output samples
Summary
The problem of estimating future values of a time series from its present and past samples is a very challenging problem in the field of signal processing. Models [6], such as in the linear predictive coding of speech signals; in time series forecasting [7]; and in the deconvolution problem, which is the problem of interest in this work [3]. It is well known in the literature that the classical prediction error filter obtained through the mean squared error (MSE) criterion is only effective for the deconvolution of minimum phase systems, since this filter provides decorrelated output samples. For signals that have been distorted by nonminimum phase systems, the classical PEF equalizes only magnitude, but the phase response remains distorted
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