An allpass filter design method with application to piano string synthesis

Abstract

A nonparametric allpass filter design method for matching a desired group delay as a function of frequency is presented. The technique is useful in physical modeling synthesis of musical instruments exhibiting dispersive wave propagation in which different frequency bands travel at different speeds. While current group delay filter design methods suffer from numerical difficulties except at low filter orders, the technique presented here is numerically robust, producing an allpass filter in cascaded biquad form, and with the filter poles following a smooth loop within the unit circle. The technique was inspired by the observation that a pole-zero pair arranged in allpass form has 2π total group delay when integrated around the unit circle, regardless of the pole location. To match a given group delay characteristic, the method divides the frequency axis into sections containing 2π total group delay, and assigns a pole-zero allpass pair to each. In this way, the method incorporates an order selection technique, and by adding a pure delay to the desired group delay, allows the trading of increased filter order for improved fit to the frequency-dependent group delay. Results are presented for modeling the group delay of a stiff piano string under several computational constraints.