For general loop-filter design in vibrating string models (as well as in woodwind and brass instrument bore models), we wish to minimize [432, pp. 182-184]:
There are numerous methods for designing the string loop filter based on measurements of real string behavior. In , a variety of methods for system identification  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 ; 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 . Overviews of other approaches appear in  and .
Below is an outline of a simple and effective method used (ca. 1995) to design loop filters for some of the Sondius sound examples:
Physically, amplitude envelopes are expected to decay exponentially, although coupling phenomena may obscure the overall exponential trend. On a dB scale, exponential decay is a straight line. Therefore, a simple method of estimating the exponential decay time-constant for each overtone frequency is to fit a straight line to its amplitude envelope and use the slope of the fitted line to compute the decay time-constant. For example, the matlab function polyfit can be used for this purpose (where the requested polynomial order is set to 1). Since ``beating'' is typical in the amplitude envelopes, a refinement is to replace the raw amplitude envelope by a piecewise linear envelope that connects the upper local maxima in the raw amplitude envelope. The estimated decay-rate for each overtone determines a sample of the loop-filter amplitude response at the overtone frequency. Similarly, the overtone frequency determines a sample of the loop-filter phase response.
Taken together, the measured overtone decay rates and tunings determine samples of the complex frequency response of the desired loop filter. The matlab function invfreqz7.11 can be used to convert these complex samples into recursive filter coefficients (see §8.6.4 for a related example). A weighting function inversely proportional to frequency is recommended. Additionally, Steiglitz-McBride iterations can improve the results , [432, pp. 101-103]. Matlab's version of invfreqz has an iteration-count argument for specifying the number of Steiglitz-McBride iterations. The maximum filter-gain versus frequency should be computed, and the filter should be renormalized, if necessary, to ensure that its gain does not exceed 1 at any frequency; one good setting is that which matches the overall decay rate of the original recording.