Monika Dörfler

Theory, implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.

Social Sparsity! Neighborhood Systems Enrich Structured Shrinkage Operators

Sparse and structured signal expansions on dictionaries can be obtained through explicit modeling in the coefficient domain. The originality of the present article lies in the construction and the study of generalized shrinkage operators, whose goal is to identify structured significance maps and give rise to structured thresholding. These generalize Group-Lasso and the previously introduced Elitist Lasso by introducing more flexibility in the coefficient domain modeling, and lead to the notion of social sparsity. The proposed operators are studied theoretically and embedded in iterative thresholding algorithms. Moreover, a link between these operators and a convex functional is established. Numerical studies on both simulated and real signals confirm the benefits of such an approach.

A Framework for Invertible, Real-Time Constant-Q Transforms

Audio signal processing frequently requires time-frequency representations and in many applications, a non-linear spacing of frequency bands is preferable. This paper introduces a framework for efficient implementation of invertible signal transforms allowing for non-uniform frequency resolution. Non-uniformity in frequency is realized by applying nonstationary Gabor frames with adaptivity in the frequency domain. The realization of a perfectly invertible constant-Q transform is described in detail. To achieve real-time processing, independent of signal length, slice-wise processing of the full input signal is proposed and referred to as sliCQ transform. By applying frame theory and FFT-based processing, the presented approach overcomes computational inefficiency and lack of invertibility of classical constant-Q transform implementations. Numerical simulations evaluate the efficiency of the proposed algorithm and the methods applicability is illustrated by experiments on real-life audio signals .

A time-frequency method for increasing the signal-to-noise ratio in system identification with exponential sweeps

Exponential sweeps are widely used to measure impulse responses of electro-acoustic systems. Measurements are often contaminated by environmental noise and nonlinear distortions. We propose a method to increase the signal-to-noise ratio (SNR) by denoising the recorded signal in the time-frequency plane. In contrast to state-of-the art denoising methods, no assumption about the spectral characteristics of the noise is required. Numerical simulations demonstrate improvements in the SNR under low-SNR conditions even for measurements contaminated by colored noise.

Audio declipping with social sparsity

We consider the audio declipping problem by using iterative thresholding algorithms and the principle of social sparsity. This recently introduced approach features thresholding/shrinkage operators which allow to model dependencies between neighboring coefficients in expansions with time-frequency dictionaries. A new unconstrained convex formulation of the audio declipping problem is introduced. The chosen structured thresholding operators are the so called windowed group-Lasso and the persistent empirical Wiener. The usage of these operators significantly improves the quality of the reconstruction, compared to simple soft-thresholding. The resulting algorithm is fast, simple to implement, and it outperforms the state of the art in terms of signal to noise ratio.

Nonstationary Gabor frames — Existence and construction

Nonstationary Gabor frames were recently introduced in adaptive signal analysis. They represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. In this paper, we show a general existence result for this family of frames. Then, we give a perturbation result for nonstationary Gabor frames and construct nonstationary Gabor frames with non-compactly supported windows from a related painless nonorthogonal expansion. Finally, the theoretical results are illustrated by two examples of practical relevance.

An inverse problem for localization operators

A classical result of time–frequency analysis, obtained by Daubechies in 1988, states that the eigenfunctions of a time–frequency localization operator with circular localization domain and Gaussian analysis window are the Hermite functions. In this contribution, a converse of Daubechies’ theorem is proved. More precisely, it is shown that, for simply connected localization domains, if one of the eigenfunctions of a time–frequency localization operator with Gaussian window is a Hermite function, then its localization domain is a disc. The general problem of obtaining, from some knowledge of its eigenfunctions, information about the symbol of a time–frequency localization operator is denoted as the inverse problem, and the problem studied by Daubechies as the direct problem of time–frequency analysis. Here, we also solve the corresponding problem for wavelet localization, providing the inverse problem analogue of the direct problem studied by Daubechies and Paul.

Nonstationary Gabor frames - approximately dual frames and reconstruction errors

Nonstationary Gabor frames, recently introduced in adaptive signal analysis, represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. Due to the lack of a complete lattice structure, perfect reconstruction is in general not feasible from coefficients obtained from nonstationary Gabor frames. In this paper it is shown that for nonstationary Gabor frames that are related to some known frames for which dual frames can be computed, good approximate reconstruction can be achieved by resorting to approximately dual frames. In particular, we give constructive examples for so-called almost painless nonstationary frames, that is, frames that are closely related to nonstationary frames with compactly supported windows. The theoretical results are illustrated by concrete computational and numerical examples.

Multi-window weaving frames

In this work we deal with the recently introduced concept of weaving frames. We extend the concept to include multi-window frames and present the first sufficient criteria for a family of multi-window Gabor frames to be woven. We give a Hilbert space norm criterion and a pointwise criterion in phase space. The key ingredient are localization operators in phase space and we give examples of woven multi-window Gabor frames consisting of Hermite functions.

Inside the spectrogram: Convolutional Neural Networks in audio processing

Convolutional Neural Networks have established a new standard in many machine learning applications not only in image but also in audio processing. In this contribution we investigate the interplay between the primary representation mapping a raw audio signal to some kind of image (feature) and the convolutional layers of an ensuing neural network. We introduce a new notion of equivalence of feature-network pairs and show the relation of feature and networks for the example of mel-spectrogram input on the one hand and varying analysis windows on the other hand.