General Loop-Filter Design

For general loop-filter design in vibrating string models (as well as in woodwind and brass instrument bore models), we wish to minimize [437, pp. 182-184]:

1. Errors in partial overtone decay times
2. Errors in partial overtone tuning

The partial decay times are controlled by the filter's amplitude response, while the partial tunings are determined by the filter's phase response. There also may be other filters in the loop (such as a delay-line interpolation filter) which need to be considered when designing the main loop filter.

There are numerous methods for designing the string loop filter based on measurements of real string behavior. In [437], a variety of methods for system identification [294] were explored for this purpose, including ``periodic linear prediction'' in which a linear combination of a small group of samples is used to predict a sample one period away from the midpoint of the group. An approach based on a genetic algorithm is described in [382]; in that work, the error measure used with the genetic algorithm is based on properties of human perception of short-time spectra, as is now standard practice in digital audio coding [63]. Overviews of other approaches appear in [29] and [518].

Below is an outline of a simple and effective method used (ca. 1995) to design loop filters for some of the Sondius sound examples:

• Estimate the