Analysis and Synthesis of Pseudo-Periodic -Like Noise by Means of Wavelets with Applications to Digital Audio
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nalysis and Synthesis of Pseudo-Periodic 1/f-Like Noise by Means of Wavelets with Applications to Digital Audio Pietro Polotti Laboratoire de Communications Audiovisuelles (LCAV) École Polytechnique Fédérale de Lausanne, Switzerland Email: pietro.polotti@epfl.ch
Gianpaolo Evangelista Laboratoire de Communications Audiovisuelles (LCAV) École Polytechnique Fédérale de Lausanne, Switzerland Email: gianpaolo.evangelista@epfl.ch Received 11 April 2000 and in revised form 23 January 2001 Voiced musical sounds have nonzero energy in sidebands of the frequency partials. Our work is based on the assumption, often experimentally verified, that the energy distribution of the sidebands is shaped as powers of the inverse of the distance from the closest partial. The power spectrum of these pseudo-periodic processes is modeled by means of a superposition of modulated 1/f components, that is, by a pseudo-periodic 1/f -like process. Due to the fundamental selfsimilar character of the wavelet transform, 1/f processes can be fruitfully analyzed and synthesized by means of wavelets. We obtain a set of very loosely correlated coefficients at each scale level that can be well approximated by white noise in the synthesis process. Our computational scheme is based on an orthogonal P -band filter bank and a dyadic wavelet transform per channel. The P channels are tuned to the left and right sidebands of the harmonics so that sidebands are mutually independent. The structure computes the expansion coefficients of a new orthogonal and complete set of harmonic-band wavelets. The main point of our scheme is that we need only two parameters per harmonic in order to model the stochastic fluctuations of sounds from a pure periodic behavior. Keywords and phrases: wavelets, 1/f -noise, spectral modeling.
1. INTRODUCTION The purpose of this work is to introduce a technique for the analysis and synthesis of pseudo-periodic signals based on a special kind of multiwavelet transform: the harmonic-band wavelet transform. Long term correlation is detectable in a large class of pseudo-periodic signals such as voiced sounds in speech and music. These signals exhibit an approximate 1/f behavior in the neighborhood of each harmonic partial fn = nf0 (n integer), that is, a 1/|f − fn | behavior. The power spectrum contains peaks, centered on the harmonics, whose shape is influenced by the long-term correlation of the stochastic fluctuations from the periodic behavior of the signal itself. From a perceptual point of view, these chaotic but correlated microfluctuations are relevant if one needs to emulate naturalness and dynamics of sounds with a detectable pitch. Our idea is strongly inspired by the fact that 1/f processes arise not only in musical signals but also in many physical and biological systems as well as in man-made phenomena
such as variations in traffic flow, economic data and fluctuation of pitch in music [1, 2]. These processes are significantly correlated at large time lags. Fractal models, such as fractional Brownian motion (fBm) [3], discrete fractional Ga
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