Phase delay of First-Order Allpass Filter

The low-frequency phase delay may be approximated by

$$P_c(f) \approx \frac{1-C}{1+C}.$$

The filter coefficient $C$ may be solved as a function of the desired phase delay $P_c(f)$.

$$C \approx \frac{1 - P_c(f)}{1 + P_c(f)}.$$

Notice if $P_c = 0$ then $C = 1$:

• this produces a pole-zero cancellation on the unit circle which can cause an unstable filter due to round-off errors!
• thus, the one-sample delay control is shifted to
  $$\epsilon \le P_c \le (1+\epsilon).$$