Revisiting Synthesis Model in Sparse Audio Declipper

Abstract

The state of the art in audio declipping has currently been achieved by SPADE (SParse Audio DEclipper) algorithm by Kitić et al. Until now, the synthesis/sparse variant, S-SPADE, has been considered significantly slower than its analysis/cosparse counterpart, A-SPADE. It turns out that the opposite is true: by exploiting a recent projection lemma, individual iterations of both algorithms can be made equally computationally expensive, while S-SPADE tends to require considerably fewer iterations to converge. In this paper, the two algorithms are compared across a range of parameters such as the window length, window overlap and redundancy of the transform. The experiments show that although S-SPADE typically converges faster, the average performance in terms of restoration quality is not superior to A-SPADE.

A Proper Version of Synthesis-based Sparse Audio Declipper

“…The report [10] shows in detail that the subproblem (8a) is, in fact, a projection of (A * (z (i) + u (i) )) onto Γ, efficiently implemented as an elementwise mapping in the time domain [8,9]. Furthermore, the solution of (8b) is obtained by applying the hard-thresholding operator H k to (Ax (i+1) + u (i) ), setting all but k its largest elements to zero, taking into account the complex conjugate coefficients.…”

“…Unlike the original variant of S-SPADE in [8], where the projection in the frequency domain was required and a special projection lemma had to be used [9], the projection step (12b) in the proposed S-SPADE algorithm is a simple elementwise mapping as is the case of the analysis variant.…”

“…As far as the authors know, [8] presented the current stateof-the-art, a heuristic declipping algorithm for both the analysis and the synthesis models (the SPADE algorithm). Until recently, the synthesis variant was considered significantly slower due to the difficult projection step, but [9] has shown that the opposite is true-the acceleration makes the synthesis model require even fewer iterations to converge, making it faster than the analysis variant. Unfortunately, the restoration quality of the synthesis variant has been shown to be substantially worse.…”

Introducing SPAIN (SParse Audio INpainter)

“…The A-SPADE algorithm (Alg. 1) was originally designed for the declipping task, where the set of feasible solutions Γ is the (convex) set of time-domain signals whose samples are identical to the observed ones in the reliable part, while the restored samples are required to lie above or below the upper or the lower clipping thresholds, respectively; see [12] or [13] for more details.…”

Psychoacoustically Motivated Audio Declipping Based on Weighted l1 Minimization

“…where D : C P → R N is the synthesis operator of a Parseval tight frame, with P ≥ N [24], [17], [18]. More specifically, throughout this paper, the Discrete Gabor Transform (DGT) is used in place of the time-frequency transform.…”