Joint recursive implementation of time - frequency representations and their modified version by the reassignment method

Cohen's class time-frequency distributions (CTFDs) have significant potential for the analysis of non-stationary signals, even if the poor readability of their representations makes visual interpretations difficult. To concentrate signal components, Auger and Flandrin recently generalized the reassignment method (first applied to the spectrogram in the 1970s) to any bilinear representations. Unfortunately, this process is computationally expensive. In order to reduce computation time and to improve representations readability, we first introduce a new fast algorithm which allows the recursive evaluation of classical spectrograms and spectrograms modified by the reassignment method. In a second step, we show that rectangular, half-sine, Hamming, Hanning and Blackman functions can be used as running 'short-time' windows. Then the previous algorithm is extended to CTFDs. We show that the windows mentioned above can also be used to compute recursively reassigned smoothed pseudo-Wigner-Ville distributions. Finally, we show that the constraints on candidate windows are not very restrictive: any function (assumed periodic) can be used in practice as long as it admits a 'short enough' Fourier series decomposition. 'hinreichend kurze' Fourier-Reihenzerlegung gestattet. R~sum~ Les distributions de la classe de Cohen jouent un r61e primordial dans l'analyse temps-fr&luence des signaux non-stationnaires, m~me si la prrsence de termes d'interfrrences nuit ~ la lisibilit6 des reprrsentations obtenues. Afin d'amrliorer cette situation, Auger et Flandrin ont rrcemment 61argi l'emploi de la technique de r~allocation, d'abord propos6e pour le spectrogramme, h l'ensemble des reprrsentations bilinraires. Nranmoins, rutilisation de cette m&hode reste drlicate en raison de la lourdeur des calculs mis en oeuvre. Afin de rrduire le temps de calcul et d'amrliorer conjointement la lisibilit6 des repr6sentations, nous proposons dans un premier temps un algorithme rapide, bas6 sur une approche rrcursive, permettant l'rvaluation simultanre de spectrogrammes classiques et r6allours. Nous montrons que les fen~tres d'analyse rectangulaire, sinuso'idale, de Hamming, Hanning et Blackman vrrifient les contraintes imposres par l'implrmentation rrcursive. Cette approche est ensuite 6tendue ~i l'rvaluation rrcursive des distributions de pseudo-Wigner-Ville lissres classiques et rralloures. Nous vrrifions alors que les fonctions enum6rres ci-dessus peuvent jouer le r61e de fenrtres de lissage temporel. Finalement, l'accent est port6 sur le fair que les contraintes imposres aux fen~tres sont peu restrictives: toute fonction peut 6tre utilisre si toutefois elle admet une drcomposition en srrie de Fourier de faible dimension.

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