Setting the Allpass Phase Delay to a Desired Frequency

A fundamental frequency $f_1$ has a corresponding period of

$$P_1=f_s/f_1$$    samples$$.$$

The model should then have a phase delay of

$$N + P_a(f_1) + P_c(f_1) = P_1$$    samples$$.$$

where $P_a(f_1) = 1/2$ for the two-point averager.

The delay line length $N$ becomes

$$N \triangleq$$   Floor$$(P_1 - P_a(f_1) - \epsilon),$$

where $\epsilon$ is a number much less than 1, that was used to shift $P_c(f_1)$'s one-sampe delay range above 0 to 1.

The fractional phase delay (in samples) for the allpass interpolator becomes

$$P_c(f_1) \triangleq P_1 - N - P_a(f_1).$$