Time-Varying Frequency Warping: Results and Experiments Gianpaolo Evangelista (1) Sergio Cavaliere (2) (1) École Polytechnique Fédérale de Lausanne, Switzerland gianpaolo.evangelista@epfl.ch, (2) Department of Physical Sciences, Univ. of Naples, Italy cavaliere@na.infn.it, http://www.na.infn.it/mfa/acust Abstract 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. Keywords: signal transformations, frequency warping 1 Introduction In recent papers [3-9] the authors considered fre- quency warping by means of orthogonal Laguerre transform as a building block of algorithms for sound manipulation. Frequency warping adds flexibility in the design of orthogonal bases for signal representa- tion and, at the same time, the computational scheme associated with the Laguerre transform has all the prerequisites for digital realizations. The authors showed that the definition of or- thogonal warping set based on a rational filter structure is useful for the construction of wavelet bases with arbitrary frequency band allocation [4]. The new combined transform leads to fine applica- tions such as orthogonal and complete perceptual filter banks [5,7]. The Laguerre transform may also be used for adapting quasi-periodic sounds to pitch-synchronous schemes [6]. In particular, by combining this trans- form with the pitch-synchronous wavelet transform [11,12], one can achieve transient and noise separa- tion from resonant components by means of a unitary transformation where resonant and noise components are projected onto orthogonal subspaces [8,9]. In order to achieve this separation in signals whose partials are not equally spaced in the frequency do- main one needs to determine a warping map bring- ing partials onto harmonics. By combining inhar- monic and harmonic components of different in- struments one can obtain interesting cross-synthesis examples. The authors showed that inharmonic sounds, such as those produced by stiff strings, plates, etc., may be conveniently modeled by means of waveguides based on simple delay lines followed by frequency warping elements [6]. The warping char- acteristic of the Laguerre family is particularly accu- rate in modeling inharmonicity of piano tones, as comparison with the characteristics derived from the physical model and direct analysis of the tones shows [10,4]. When pitch shifting sample piano tones, this concept allows us to take into account stiffness in- crease as we move to the lower tones. Frequency warping generates interesting sound effects such as sound morphing, phasing, chorusing, flanging, pitch-shifting and new effects not yet in the catalogue. However, in order to be able to capture the full range of possibilities one needs to consider dy- namic variations of the warping parameters. In this paper we approach the problem of time- varying frequency warping by means of generalized Laguerre transform. In this context, we show that time-varying frequency warping may be imple- mented in a space-varying sampled delay-line. The