The aim of this paper is to present results on digital processing of sounds by means of both dispersive delay lines and pitch-synchronous transforms in a unified framework. The background on frequency warping is detailed and applications of this technique are pointed out with reference to the existing literature. These include transient extraction, pitch shifting, harmonic detuning and auditory modeling.

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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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Fast browsing of digital collections of music would largely benefit from the availability of representative audio excerpts of the pieces. Similar to their visual counterparts found in digital photo albums, music thumbnails should offer a comprehensive listening experience while requiring a limited storage space or communication data rate. The approach to the generation of musical thumbnails of music proposed in this paper is based on an application of the PitchSynchronous Wavelet Transform, where the “pitch” is tuned to the elementary measure of the piece. The music thumbnail is encoded by the low rate coefficient sequence pertaining to the scaling residue of the transform. The scaling component represents the pseudo-periodic trend of the piece over several measures. Due to pseudo-periodicity, the time duration of the thumbnail can be arbitrarily extended in listening with no audible artifacts.

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Frequency warping is a modifier that acts on sound signals by remapping the frequency axis. Thus, the spectral content of the original sound is displaced to other frequencies. At the same time, the phase relationship among the signal components is altered, nonlinearly with respect to frequency. While this effect is interesting and has several applications, including in the synthesis by physical models, its use has been so far limited by the lack of an accurate and flexible real-time algorithm. In this paper we present methods for frequency warping that are based on local approximations of the warping operators and allow for real-time implementation. Filter bank structures are derived that allow for efficient realization of the approximate technique. An analysis of the error is also presented, which shows that both numerical and perceptual errors are within acceptable limits. Furthermore, the approximate implementation allows for a larger variety of warping maps than that achieved by the classical (non-causal) first-order allpass cascade implementation.

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