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

Abstract

This work deals with adaptive predictive deconvolution of non-stationary channels. In particular, we investigate the use of a cascade of linear predictors in the recovering of sparse and antisparse signals. To do so, we first discuss the behavior of the ℓ Prediction Error Filter (PEF), with ≠ 2, showing that it can better attenuate the effects of non-minimum phase channels in comparison with the classical ℓ 2 PEF, although the ℓ PEF, with ≠ 2, still presents intrinsic limitations in compensating the channel distortions, due to its direct forward structure. Hence, the cascade of linear predictors, i.e., one forward filter followed by a backward one, emerges as a possible solution to circumvent the structure limitation addressed. We apply the proposed cascade structure in the deconvolution of nonstationary channels, with minimum, maximum-, mixedand variable-phase responses, and also in noisy scenarios. From the simulation results we observed that, besides the duality relation between the ℓ 1 and ℓ ∞ norms, they present different algorithmic behavior: a cascade adjusted by minimizing the ℓ 1 norm of the error attains a fast convergence, enhancing the cascade tracking capacity, but is more sensitive to noise. Adjusting the cascade by minimizing the ℓ 4 norm of the error (a smooth approximation of the ℓ ∞ norm), on the other hand, leads to a filter more robust to noise, but presents slower convergence and tracking capability.

New algorithms for sparse multichannel blind deconvolution

“…1: The predictive cascade structures. From [19] Considering the structure in Figure 1a, we can write the error signal e p (n) as…”

“…Also, we reduce the complexity of the problem by using a few combinations of p and q to build the matrix X in Equation (19). In our proposal, instead of using all the possible combinations for traces p and q, we combine trace p with the neighboring traces p − 2, p − 1, p + 1, and p + 2.…”

“…The size of X, which has an impact of the complexity of the algorithm, depends on the number of combinations used in (19). In the original problem there are J(J + 1)/2 combinations, while the number of combinations considering only the neighboring traces is reduced to 2J − 3.…”

“…Earlier works on seismic deconvolution [7], [8] proposed the use of a cascade of linear forward and backwards predictors to deal with the case of non-minimum-phase seismic wavelet. The first algorithm presented in this paper combines the strategies of [7], [8], [17], [18], [19], using a cascade of linear forward and backward predictors that seek to minimize the ℓ 1norm of the prediction error. In [19] we have shown that such a cascade, in its adaptive form, was very effective for equalizing sparse signals.…”

“…The first algorithm presented in this paper combines the strategies of [7], [8], [17], [18], [19], using a cascade of linear forward and backward predictors that seek to minimize the ℓ 1norm of the prediction error. In [19] we have shown that such a cascade, in its adaptive form, was very effective for equalizing sparse signals. In this paper, we present a batch algorithm for the sparse multichannel blind deconvolution problem.…”