Voiced musical sounds have non-zero 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, i.e., 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, obtaining 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 Wavelets. The main point of our scheme is that we need only one parameter in order to model the stochastic fluctuation of sounds from a pure periodic behavior.

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In previous papers [1], [2] we introduced a model for pseudo-periodic sounds based on Wornell results [3] concerning the synthesis of 1/f noise by means of the Wavelet transform (WT). This method provided a good model for representing not only the harmonic part of reallife sounds but also the stochastic components. The latter are of fundamental importance from a perceptual point of view since they contain all the information related to the natural dynamic of musical timbres. In this paper we introduce a refinement of the method, making the spectralmodel technique more flexible and the resynthesis coefficient model more accurate. In this way we obtain a powerful tool for sound processing and cross-synthesis.

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The goal of this paper is to introduce a complete analysis/resynthesis method for the stationary part of voiced-sounds. The method is based on a new class of wavelets, the Harmonic-Band Wavelets (HBWT). Wavelets have been widely employed in signal processing [1, 2]. In the context of sound processing they provided very interesting results in their first harmonic version: the Pitch Synchronous Wavelets Transform (PSWT) [3]. We introduced the Harmonic-Band Wavelets in a previous edition of the DAFx [4]. The HBWT, with respect to the PSWT allows one to manipulate the analysis coefficients of each harmonic independently. Furthermore one is able to group the analysis coefficients according to a finer subdivision of the spectrum of each harmonic, due to the multiresolution analysis of the wavelets. This allows one to separate the deterministic components of voiced sounds, corresponding to the harmonic peaks, from the noisy/stochastic components. A first result was the development of a parametric representation of the HBWT analysis coefficients corresponding to the stochastic components [5, 7]. In this paper we present the results concerning a parametric representation of the HBWT analysis coefficients of the deterministic components. The method recalls the sinusoidal models, where one models time-varying amplitudes and time varying phases [8, 9]. This method provides a new interesting technique for sound synthesis and sound processing, integrating a parametric representation of both the deterministic and the stochastic components of sounds. At the same time it can be seen as a tool for a parametric representation of sound and data compression.

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In previous editions of the DAFX [1, 2] we presented a method for the analysis and the resynthesis of voiced sounds, i.e., of sounds with well defined pitch and harmonic-peak spectra. In a following paper [3] we called the method Fractal Additive Synthesis (FAS). The main point of the FAS is to provide two different models for representing the deterministic and the stochastic components of voiced-sounds, respectively. This allows one to represent and reproduce voiced-sounds without loosing the noisy components and stochastic elements present in real-life sounds. These components are important in order to perceive a synthetic sound as a natural one. The topic of this paper is the extension of the technique to inharmonic sounds. We can apply the method to sounds produced by percussion instruments as gongs, tympani or tubular bells, as well as to sounds with expanded quasi-harmonic spectrum as piano sounds.

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