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 [432, 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 ${\hat G}(z)$ based on measurements of real string behavior. In [432], a variety of methods for system identification [290] 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 [381]; 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 [62]. Overviews of other approaches appear in [29] and [512].

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

1. Estimate the fundamental frequency (see §6.11.4 below)
2. Set a Hamming FFT-window length to approximately four periods
3. Compute the short-time Fourier transform (STFT)
4. Perform a sinusoidal modeling analysis [468] to

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   detect peaks in each spectral frame, and

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   connect peaks through time to form amplitude envelopes

5. Fit an exponential to each amplitude envelope
6. Prepare the desired frequency-response, sampled at the harmonic frequencies of the delay-line loop without the loop filter. At each harmonic frequency,

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   the nearest-partial decay-rate gives the desired loop-filter gains,

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   the nearest-partial peak-frequency give the desired loop-filter phase delay.

7. Use a phase-sensitive filter-design method such as invfreqz in matlab to design the desired loop filter from its frequency-response samples (further details below).

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 invfreqz^7.11 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 [289], [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.