Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames

We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues. We use this approach to construct computationally efficient, spectrum-adapted, tight vertex-frequency and graph wavelet frames. We give numerous examples of the resulting spectrum-adapted graph filters, and also present an illustrative example of vertex-frequency analysis using the proposed construction

Optimal Window and Lattice in Gabor Transform Application to Audio Analysis

This article deals with the use of optimal lattice and optimal window in Discrete Gabor Transform computation. In the case of a generalized Gaussian window, extending earlier contributions, we introduce an additional local window adaptation technique for non-stationary signals. We illustrate our approach and the earlier one by addressing three time-frequency analysis problems to show the improvements achieved by the use of optimal lattice and window: close frequencies distinction, frequency estimation and SNR estimation. The results are presented, when possible, with real world audio signals

A method for optimizing the ambiguity function concentration

International audienceIn the context of signal analysis and transformation in the time-frequency (TF) domain, controlling the shape of a waveform in this domain is an important issue. Depending on the application, a notion of optimal function may be defined through the properties of the ambiguity function. We present an iterative method for providing such optimal functions under a general concentration constraint of the ambiguity function. At each iteration, it follows a variational approach which maximizes the ambiguity localization via a user-defined weight function F . Under certain assumptions on this latter function, it converges to a waveform which is optimal according to the localization criterion defined by F

An optimally concentrated Gabor transform for localized time-frequency components

Gabor analysis is one of the most common instances of time-frequency signal analysis. Choosing a suitable window for the Gabor transform of a signal is often a challenge for practical applications, in particular in audio signal processing. Many time-frequency (TF) patterns of different shapes may be present in a signal and they can not all be sparsely represented in the same spectrogram. We propose several algorithms, which provide optimal windows for a user-selected TF pattern with respect to different concentration criteria. We base our optimization algorithm on $l^p$-norms as measure of TF spreading. For a given number of sampling points in the TF plane we also propose optimal lattices to be used with the obtained windows. We illustrate the potentiality of the method on selected numerical examples

Fenêtre et grille optimales pour la transformée de Gabor Exemples d'application à l'analyse audio

International audienceThis article deals with the use of optimal lattice and optimal window in Discrete Gabor Transform computation. In the case of a generalized Gaussian window, extending earlier contributions, we introduce an additional local window adaptation technique for non-stationary signals. We illustrate our approach and the earlier one by addressing three time-frequency analysis problems: close frequencies distinction, frequency estimation and Signal to Noise Ratio estimation. The results are presented, when possible, with real world audio signals.Cet article présente l'utilisation d'une grille optimale et d'une fenêtre optimale pour le calcul de la transformée de Gabor discrète. Dans le cas d'une Gaussienne généralisée, nous étendons des travaux précédents et proposons une fenêtre localement optimale pour des si-gnaux non-stationnaires. Nous présentons des résultats sur trois problèmes d'analyse temps-fréquence, sur des signaux réels et synthétiques : la distinction de composantes temps-fréquence proches, l'estimation de fréquence instantané et l'estimation du Rapport Signal à Bruit. Abstract – This article deals with the use of optimal lattice and optimal window in Discrete Gabor Transform computation. In the case of a generalized Gaussian window, extending earlier contributions, we introduce an additional local window adaptation technique for non-stationary signals. We illustrate our approach and the earlier one by addressing three time-frequency analysis problems: close frequencies distinction, frequency estimation and Signal to Noise Ratio estimation. The results are presented, when possible, with real world audio signals

An optimally concentrated Gabor transform for localized time-frequency components

Gabor analysis is one of the most common instances of time-frequency signal analysis. Choosing a suitable window for the Gabor transform of a signal is often a challenge for practical applications, in particular in audio signal processing. Many time-frequency (TF) patterns of different shapes may be present in a signal and they can not all be sparsely represented in the same spectrogram. We propose several algorithms, which provide optimal windows for a user-selected TF pattern with respect to different concentration criteria. We base our optimization algorithm on l p -norms as measure of TF spreading. For a given number of sampling points in the TF plane we also propose optimal lattices to be used with the obtained windows. We illustrate the potentiality of the method on selected numerical examples