Transformée de Fourier à court terme

Résumé

La transformée de Fourier à court terme (TFCT), ou transformée de Fourier locale (en anglais Short-Time Fourier Transform (STFT)) ou encore transformée de Fourier à fenêtre glissante est une transformation liée aux transformées de Fourier utilisée pour déterminer la fréquence sinusoïdale et la phase d'une section locale d'un signal. Le carré de son module donne le spectrogramme. Transformée de Fourier locale Transformée de Fourier locale continue Dans le cas continu, la fonction à transformer est multipliée par une autre fonction qui n'est pas nulle seulement pour une petite période de temps (une fonction à support compact ou a décroissance rapide dite fenêtre). Mathématiquement elle s'écrit : : \mathbf{STFT} \left { x( t ) \right } \equiv X(\tau, \omega) = \int_{-\infty}^{\infty} x(t) w(t-\tau) \mathrm{e}^{-j \omega t} ,\mathrm{d}t où w est la fonction de fenêtrage. Ici, j désigne l'unité imaginaire. Lorsque la fonction de fen

Source officielle

https://fr.wikipedia.org/wiki/Transform%C3%A9e_de_Fourier_%C3%A0_court_terme

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When designing an audio processing system, the target tasks often influence the choice of a data representation or transformation. Low-level time-frequency representations such as the short-time Fourier transform (STFT) are popular, because they offer a meaningful insight on sound properties for a low computational cost. Conversely, when higher level semantics, such as pitch, timbre or phoneme, are sought after, representations usually tend to enhance their discriminative characteristics, at the expense of their invertibility. They become so-called mid-level representations. In this paper, a source/filter signal model which provides a mid-level representation is proposed. This representation makes the pitch content of the signal as well as some timbre information available, hence keeping as much information from the raw data as possible. This model is successfully used within a main melody extraction system and a lead instrument/accompaniment separation system. Both frameworks obtained top results at several international evaluation campaigns.

2011

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This paper presents a signal processing tool for analyzing and manipulating digitized acoustic wave fields, based on a spatio-temporal extension of the time-frequency representation space. The emphasis is on wave fields acquired with a 1-D linear array of equidistant microphones (representing the spatial samples), but the basic formulation is valid for the three dimensions of space. We start by defining a spatio-temporal extension of the short time Fourier transform, in both continuous and discrete space and time, and then focus on applications of the proposed representation. In particular, we show that acoustic scenes with multiple sources can benefit from the use of space-frequency analysis in applications such as source separation (spatial filtering) and spatial audio coding (wave field coding). The experiments suggest that there is a spatial window size for which the performance of filtering and coding is optimal.

Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa2011

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Digital acoustics: processing wave fields in space and time using DSP tools

Mihailo Kolundzija, Francisco Pereira Correia Pinto, Martin Vetterli

Systems with hundreds of microphones for acoustic field acquisition, or hundreds of loudspeakers for rendering, have been proposed and built. To analyze, design, and apply such systems requires a framework that allows us to leverage the vast set of tools available in digital signal processing in order to achieve intuitive and efficient algorithms. We thus propose a discrete space–time framework, grounded in classical acoustics, which addresses the discrete nature of the spatial and temporal sampling. In particular, a short-space/time Fourier transform is introduced, which is the natural extension of the localized or short-time Fourier transform. Processing in this intuitive domain allows us to easily devise algorithms for beam-forming, source separation, and multi-channel compression, among other useful tasks. The essential space band-limitedness of the Fourier spectrum is also used to solve the spatial equalization task required for sound field rendering in a region of interest. Examples of applications are shown.

2014

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