Audio Effects Based on Biorthogonal Time-Varying Frequency Warping
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udio Effects Based on Biorthogonal Time-Varying Frequency Warping Gianpaolo Evangelista École Polytechnique Fédérale de Lausanne, Switzerland Email: gianpaolo.evangelista@epfl.ch
Sergio Cavaliere Department of Physical Sciences, University of Naples, Italy Email: [email protected] Received 31 March 2000 and in revised form 23 January 2001 We illustrate the mathematical background and musical use of a class of audio effects based on frequency warping. These effects alter the frequency content of a signal via spectral mapping. They can be implemented in dispersive tapped delay lines based on a chain of all-pass filters. In a homogeneous line with first-order all-pass sections, the signal formed by the output samples at a given time is related to the input via the Laguerre transform. 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. Simulation of these effects requires techniques for time-varying pitch and/or brightness modification that are very useful for sound processing. The basis for time-varying frequency warping is a time-varying version of the Laguerre transformation. The corresponding 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 is associated to an 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. Keywords and phrases: signal transformations, frequency warping, Laguerre transform, Kautz functions.
1. INTRODUCTION Frequency warping is an interesting technique in sound processing. Given a map of the frequency axis into itself, one can transform the spectral content of any sound by mapping the original set of frequencies into other frequencies. A simple example is given by transforming a periodic sound via a linear map. In that case the original set of harmonics is scaled by the angular coefficient. One obtains another set of harmonic frequencies multiple of the transformed fundamental frequency. When warping discrete-time signals care must be taken since the map may alter the periodicity of the Fourier transform, i.e., the bandwidth of the signal. Classical upsampling and downsampling operators may be written in terms of warping operators. By frequency warping a periodic sound via a nonlinear map, one obtains an inharmonic set of partials as shown in Figure 1. Voiced sounds of natural instruments do not always possess a harmonic structure. For example, piano tones in the low register show inharmonicity. This is due to the stiffness of the string, which results in disp
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