Tuning

For large $N$ (low pitches) the difference between $N$ and $N+1$ is slight, but becomes increasingly noticeable for small $N$ (high pitches). Recall, the fundamental frequency (which is inversely proportional to the period) is given by

$$f_1 \triangleq \frac{1}{(N+1/2)T_s} = \frac{f_s}{N+1/2}$$

which may be expressed more generally in terms of phase delay of our feedback filter:

$$f_1 = \frac{f_s}{N + P_a(f_1)}.$$

We need to introduce a filter into the feedback loop that can contribute a small delay without alterning the loop gain. What kind of filters can introduce a frequency-dependent delay without having an effect on gain?