"Self-Wiener" Filtering: Data-Driven Deconvolution of Deterministic Signals

Abstract

We consider the fundamental problem of robust deconvolution, and particularly the recovery of an unknown deterministic signal convolved with a known filter and corrupted by additive noise. We present a novel, non-iterative data-driven approach. Specifically, our algorithm works in the frequencydomain, where it tries to mimic the optimal unrealizable Wienerlike filter as if the unknown deterministic signal were known. This leads to a threshold-type regularized estimator, where the threshold value at each frequency is found in a fully datadriven manner. We provide an analytical performance analysis, and derive approximate closed-form expressions for the residual Mean Squared Error (MSE) of our proposed estimator in the low and high Signal-to-Noise Ratio (SNR) regimes. We show analytically that in the low SNR regime our method provides enhanced noise suppression, and in the high SNR regime it approaches the performance of the optimal unrealizable solution. Further, as we demonstrate in simulations, our solution is highly suitable for (approximately) bandlimited or frequency-domain sparse signals, and provides a significant gain of several dBs relative to other methods in the resulting MSE.