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“Self-Wiener” Filtering: Non-Iterative Data-Driven Robust Deconvolution of Deterministic Signals Amir Weiss and Boaz Nadler

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“Self-Wiener” Filtering: Non-Iterative Data-Driven Robust Deconvolution of Deterministic Signals Amir Weiss and Boaz Nadler 1 “Self-Wiener” Filtering: Non-Iterative Data-Driven Robust Deconvolution of Deterministic Signals Amir Weiss and Boaz Nadler Abstract—We consider the fundamental problem of robust GENERATION deconvolution, and particularly the recovery of an unknown deterministic signal convolved with a known filter and corrupted v[n] by additive noise. We present a novel, non-iterative data-driven approach. Specifically, our algorithm works in the frequency- x[n] y[n] domain, where it tries to mimic the optimal unrealizable Wiener- h[n] Σ like filter as if the unknown deterministic signal were known. This leads to a threshold-type regularized estimator, where the RECONSTRUCTION threshold value at each frequency is found in a fully data- driven manner. We provide an analytical performance analysis, y[n] xb[n] and derive approximate closed-form expressions for the residual g[n] 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 Fig. 1: Block diagram of model (1) (“GENERATION”), and the considered enhanced noise suppression, and in the high SNR regime it class of estimators, produced by filtering (“RECONSTRUCTION”). Note that in our framework, g[n] may depend on the measurements fy[n]gN−1. approaches the performance of the optimal unrealizable solution. n=0 Further, as we demonstrate in simulations, our solution is highly suitable for (approximately) bandlimited or frequency-domain and noise are stochastic, with a-priori known upper and lower sparse signals, and provides a significant gain of several dBs bounds on their spectra at each frequency. relative to other methods in the resulting MSE. While the random signal model is suitable in some problems Index Terms—Deconvolution, system identification, Wiener and scenarios, in others the input signal is better modeled as filter. deterministic unknown. Several methods have been proposed for this signal model as well [14]–[16]. One example is the WaveD algorithm proposed by Johnstone et al. [17], which is I. INTRODUCTION based on hard thresholding of a wavelet expansion (see also A ubiquitous task in signal processing is deconvolution [18], Section II, for a concise description of WaveD). While [1], namely inverting the action of some system (or channel) some of these algorithms offer considerable enhancement, on an input, desired signal. A major difficulty arises when their performance may be affected by tuning parameters, the measured convolved signal is contaminated with noise, which either need to be set by the user, or require separate wherein a careful balancing of bandwidth and Signal-to- careful calibration. Another class of deconvolution algorithms Noise Ratio (SNR) is required [2]. This robust deconvolution are iterative [19]–[23]. Some of these methods also require problem appears in a wide variety of applications, such as tuning parameters, such as the λ parameter in [19], controlling communication systems, controllers, image and video process- the balance between noise reduction and filtration errors, ing, audio signal processing and ground-penetrating radar data caused by the deviation from the ideal inverse filter when λ analysis, to name but a few [3]–[9]. is non-zero. For a detailed discussion on this topic, see [24]. A common measure for the quality of the signal recon- In this work, assuming known SOS of the noise, or an structed from the noisy convolution measurements is the Mean estimate thereof, we propose a novel non-iterative, computa- arXiv:2007.10164v1 [eess.SP] 20 Jul 2020 Squared Error (MSE) between the unknown input signal and tionally simple, fully data-driven deconvolution approach for the deconvolved one. When the unknown signal and the deterministic signals. The guiding principle of our approach, noise are both modeled as stochastic stationary processes with termed “Self-Wiener” (SW) filtering, is an attempt to mimic known Second-Order Statistics (SOSs), the optimal solution the optimal Minimum MSE (MMSE) unrealizable Wiener-like within the class of linear estimators is the celebrated Wiener filter, as if the unknown deterministic signal were known. filter [10], [11]. Following Wiener’s work, many other decon- This yields a self-consistent thresholding-type method with volution methods have been proposed, most of them under no tuning parameters. Our resulting threshold-type estimator some a-priori knowledge about the class of the unknown input is reminiscent of other thresholding methods (e.g., [25]), but signals. For example, Berkhout [12] derived the least-squares due to its data-driven nature, it automatically computes a inverse filtering for known noise SOSs, assuming that the input data-dependent threshold value. Moreover, this threshold can signal is white, namely with a constant spectral level. In [13], be intuitively explained from an estimated SNR perspective. Eldar proposed a minimax approach, assuming the input signal We further present an analytical performance analysis of our The authors are with the Department of Computer Science and Applied proposed SW estimator, and derive approximate closed-form Mathematics, Faculty of Mathematics and Computer Science, Weizmann Institute of Science, 234 Herzl Street, Rehovot 7610001 Israel, e-mail: expressions for its predicted MSE. Comparison in simulations [email protected], [email protected] to other approaches, some of which are fully data-driven as 2 N−1 well, show that our method offers highly competitive perfor- fy[n]gn=0 , when N is sufficiently large, the linear convo- mance, and attains an MSE lower by several dBs for various lution coincides or may be well-approximated by circular signals representative of those appearing in applications. convolution (neglecting boundary effects) [29]. Hence, we consider the problem in the frequency domain. II. PROBLEM FORMULATION Recall that the unitary Discrete Fourier Transform (DFT) of a length-N sequence a[n] is defined as Consider the classical discrete-time convolution model de- picted in Fig. 1 (“GENERATION”), N−1 1 X −| 2π nk A[k] , DFTfa[n]g , p a[n]e N 2 C; (4) X N y[n] = h[k]x[n − k] + v[n] 2 R; 8n 2 Z; (1) n=0 k2Z for all k 2 f0;:::;N − 1g. Since circular convolution in where h[n] is a known impulse response of a Linear Time- the discrete-time domain is equivalent to multiplication in Invariant (LTI) system; x[n] is an unknown deterministic the (discrete-)frequency domain, applying the DFT to the N−1 signal; and v[n] is a stationary, zero-mean additive noise with sequence fy[n]gn=0 (1) gives a positive Power Spectral Density (PSD) function, denoted by Y [k] = H[k]X[k] + V [k] 2 C; 8k 2 f0;:::;N − 1g; Sv(!). We assume that the noise PSD Sv(!) is either known or has been estimated a-priori, e.g., from realizations of pure and (3) becomes X“[k] = G[k]Y [k]. Since the DFT is a unitary noise, measured in a “training period” [26]. We further assume transformation, by Parseval’s identity that x[n] is periodic or has finite support, and that h[n] is N−1 N−1 a Finite Impulse Response (FIR) filter, which is compactly X ï 2ò X MSE (x; x) = X[k] − X“[k] MSE[k]; (5) supported by definition, or at least may be well-approximated b E , k=0 k=0 by an FIR filter. The robust deconvolution problem [27] is to recover the where, with a slight abuse in notation, signal x[n] based on the noisy measurements fy[n]gN−1. ï 2ò î ó n=0 MSE[k]= X[k] − X[k] = jX[k] − G[k]Y [k]j2 (6) Switching the roles of x[n] and h[n], this problem is equivalent E “ E to the system identification problem [28]. In that context, denotes the MSE at the k-th frequency component. Obviously, x[n] is known and the problem is to estimate the unknown individually minimizing each term MSE[k] in the sum (5), impulse response h[n], namely to identify the system. Hence, minimizes the MSE (2). However, when x[n], or equiva- while in this work we consider deconvolution, our proposed lently X[k], is assumed deterministic, the resulting optimal SW estimator is applicable to the system identification prob- solution—X [k], to be introduced shortly in (11) below—is lem as well. “opt not a realizable estimator, as we show next. In the robust deconvolution context, the quality of an estimator x[n] of x[n] is often measured by its MSE, b III. THE OPTIMAL DECONVOLUTION MMSE SOLUTION "N−1 # X 2 To motivate our proposed estimator, it is first instructive to MSE (x; x) jx[n] − x[n]j ; (2) b , E b present the structure of the optimal, yet not realizable solution, n=0 which minimizes (2) over all estimators of the form (3). For where the expectation is w.r.t. the noise v[n] in the observa- this, let us begin by introducing the following quantities: tions y[n], the only random component in the problem. In this 2 2 work, we focus on deconvolution methods of the following jX[k]j jH[k]X[k]j SNR[k] , ; SNRout[k] , ; (7) form, as depicted in Fig. 1 (“RECONSTRUCTION”), Sv[k] Sv[k] x[n] = g[n] ⊗ y[n]; 8n 2 f0;:::;N − 1g; (3) where b S [k] S (!) = S 2πk v , v 2πk v N where ⊗ denotes circular convolution. Hence the goal is to != N design a filter g[n] that gives a low MSE. In contrast to the is the noise PSD at the k-th frequency. By definition, SNR[k] classical linear Wiener filter, we allow the filter g[n] to depend and SNRout[k] are the SNRs at the k-th frequency of the on the observed signal y[n], hence leading to a nonlinear noise-free signal x[n] and of its convolution with h[n], at the estimator.
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