Dispersive tapped delay lines are attractive structures for altering the frequency content of a signal. In previous papers we showed that in the case of a homogeneous line with first order all-pass sections the signal formed by the output samples of the chain of delays at a given time is equivalent to compute the Laguerre transform of the input signal. However, most musical signals require a time-varying frequency modification in order to be properly processed. Vibrato in musical instruments or voice intonation in the case of vocal sounds may be modeled as small and slow pitch variations. Simulations of these effects require techniques for time- varying pitch and/or brightness modification that are very useful for sound processing. In our experiments the basis for time-varying frequency warping is a time-varying version of the Laguerre transformation. The corre- sponding implementation structure is obtained as a dispersive tapped delay line, where each of the frequency dependent delay element has its own phase response. Thus, time-varying warping results in a space-varying, inhomogeneous, propagation structure. We show that time-varying frequency warping may be associated to expansion over biorthogonal sets generalizing the discrete Laguerre basis. Slow time-varying characteristics lead to slowly varying parameter sequences. The corresponding sound transformation does not suffer from discontinuities typical of delay lines based on unit delays.

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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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