Download Time Varying Frequency Warping: Results And Experiments
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.