This paper extends a previously proposed method for designing filters simulating the dispersion phenomenon occurring in string instruments. In digital waveguide synthesis, the phenomenon is traditionally modeled by inserting an allpass filter to the string model feedback loop. In this paper, the concept of tunable dispersion filter design, which provides a closed-form formula to design a dispersion filter, is applied to a cascade of first-order allpass filters. Moreover, the method is extended to design a filter cascade including an arbitrary number of first-order filters. In addition, it is shown how the designed dispersion filter can be used in a waveguide piano synthesis model.

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