Physically Inspired Models for the Synthesis of Stiff Strings with Dispersive Waveguides
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Physically Inspired Models for the Synthesis of Stiff Strings with Dispersive Waveguides I. Testa Dipartimento di Scienze Fisiche, Universit`a di Napoli “Federico II,” Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy Email: [email protected]
G. Evangelista Dipartimento di Scienze Fisiche, Universit`a di Napoli “Federico II,” Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy Email: [email protected]
S. Cavaliere Dipartimento di Scienze Fisiche, Universit`a di Napoli “Federico II,” Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy Email: [email protected] Received 30 June 2003; Revised 17 November 2003 We review the derivation and design of digital waveguides from physical models of stiff systems, useful for the synthesis of sounds from strings, rods, and similar objects. A transform method approach is proposed to solve the classic fourth-order equations of stiff systems in order to reduce it to two second-order equations. By introducing scattering boundary matrices, the eigenfrequencies are determined and their n2 dependency is discussed for the clamped, hinged, and intermediate cases. On the basis of the frequency-domain physical model, the numerical discretization is carried out, showing how the insertion of an all-pass delay line generalizes the Karplus-Strong algorithm for the synthesis of ideally flexible vibrating strings. Knowing the physical parameters, the synthesis can proceed using the generalized structure. Another point of view is offered by Laguerre expansions and frequency warping, which are introduced in order to show that a stiff system can be treated as a nonstiff one, provided that the solutions are warped. A method to compute the all-pass chain coefficients and the optimum warping curves from sound samples is discussed. Once the optimum warping characteristic is found, the length of the dispersive delay line to be employed in the simulation is simply determined from the requirement of matching the desired fundamental frequency. The regularization of the dispersion curves by means of optimum unwarping is experimentally evaluated. Keywords and phrases: physical models, dispersive waveguides, frequency warping.
1. INTRODUCTION Interest in digital audio synthesis techniques has been reinforced by the possibility of transmitting signals to a wider audience within the structured audio paradigm, in which algorithms and restricted sets of data are exchanged [1]. Among these techniques, the physically inspired models play a privileged role since the data are directly related to physical quantities and can be easily and intuitively manipulated in order to obtain realistic sounds in a flexible framework. Applications are, amongst the others, the simulation of a “physical situation” producing a class of sounds as, for example, a closing door, a car crash, the hiss made by a crawling creature, the human-computer interaction and, of course, the simulation of musical instruments. In the general physical models technique, continuoustime solutions of the
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