Effect of Decay Stretching on Tuning

Changing $S$ changes the effective loop length as a function of frequency since it changes the phase delay of the overall loop.

• we must therefore compute $P_a(f_1)$ when using the allpass filter fractional delay to tune to the desired frequency.

As shown in the paper by Smith and Jaffe, for low frequencies relative to the sampling rate, we may use the approximation

$$P_a(f,S) \approx S, \quad 0 \le S \le 1.$$

See this in Matlab:

S = .6;
[H, omega] = freqz([1-S S], 1);

% start at index 2 to avoid division by 0
mean(angle(H(2:end))./omega(2:end));

When $S = 1/2$, we have the basic string algorithm.