Abstract
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 coeffcients 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.
Highlights
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]
Due to the complexity of the real physical systems—from the classic design of musical instruments to the molecular structure of extended objects— solutions of these equations cannot be generally found in an analytic way and one should resort to numerical methods or approximations
These equations give the necessary background to the physical modeling of stiff strings. We show that their frequency domain solution provides the link between continuous-time and discrete-time models, useful for the derivation of the digital waveguide and suitable for their simulation
Summary
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]. Another entry point is offered if sound samples of an instrument are available. In this case, the parameters of the synthesis model can be determined by finding the warping curve that best fits the data given by the frequencies of the partials, together with the length of the dispersive delay line. The parameters of the synthesis model can be determined by finding the warping curve that best fits the data given by the frequencies of the partials, together with the length of the dispersive delay line In this case, the synthesis is limited to existing sources, some variations can be obtained in terms of the warping parameters, which are related to, but do not directly represent, physical factors
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