Applying Balanced Model Truncation to Sound Analysis/Synthesis Models

Balanced Model Truncation (BMT) is a powerful technique that can be used to greatly reduce the order of certain digital filters with little distortion to their frequency and phase responses. Since digital filters are commonly used in computer music applications, BMT may be a tool of considerable practical use. To demonstrate this potential, BMT is applied to an autoregressive (AR) drum sound model, showing that its order may be reduced considerably with little change to the accuracy of the model. 1. INTRODUCTION The intention of this paper is to introduce the technique of Balanced Model Truncation (BMT) as a powerful tool for use in computer music applications. BMT is a means by which the order of a digital filter may be significantly reduced while accurately maintaining its input/output response. This is achieved by analysing the filter to quantify and then rid it of system redundancy. In (Beliczynski, B.), for example, it is shown that certain high-order FIR filters may be converted to low-order IIR ones while faithfully reproducing the frequency and phase response of the original. In the field of computer music, digital filters are widely used in the generation and processing of sound where, typically, the demand is for filters that model complex acoustic systems, but that are of low orders for implementational ease and speed. The application discussed in this paper is of an autoregressive (AR) model of percussive and drum sounds which consists of an all-pole IIR digital filter excited by a white noise source. It has been reported in (Sandler, M. 1990) that for such a model to faithfully model a given drum sound, it must be have an order of several hundreds. Such a high order, however, presents significant implementation problems. In this paper we demonstrate how BMT can be used to derive a much lower order autoregressive moving average (ARMA) model with a performance that is nearly indistinguishable from the AR one. Furthermore, this model is of greater accuracy than an ARMA model of the same lower order derived using a conventional method. 2. AR AND ARMA MODELLING Autoregressive (AR) models are widely used in computer music as part of powerful sound analysis/synthesis schemes-see for example (Moorer J.A.) and (Lansky P.). Using linear prediction to estimate the model parameters it is possible to directly find an all-pole filter with frequency response that is closest, for a given model order in a least-squared-error sense, to …