Thiran Allpass Interpolators

Given a desired delay $\Delta = N+\delta$ samples, an order $N$ allpass filter

$$H(z) = \frac{z^{-N}A\left(z^{-1}\right)}{A(z)} = \frac{a_N + a_{N-1}z^{-1} + \cdots + a_1 z^{-(N-1)} + z^{-N}}{1 + a_1 z^{-1} + \cdots + a_{N-1} z^{-(N-1)} + a_N z^{-N}}$$

can be designed having maximally flat group delay equal to $\Delta$ at DC using the formula

$$a_k=(-1)^k\left(\begin{array}{c} N \\ [2pt] k \end{array}\right)\prod_{n=0}^N\frac{\Delta-N+n}{\Delta-N+k+n}, \; k=0,1,2,\ldots,N$$

where

$$\left(\begin{array}{c} N \\ [2pt] k \end{array}\right) = \frac{N!}{k!(N-k)!}$$

denotes the $k$ th binomial coefficient

• $a_0=1$ without further scaling
• For sufficiently large $\Delta$ , stability is guaranteed
  rule of thumb: $\Delta \approx \hbox{order}$
• Mean group delay is always $N$ samples
  (for any stable $N$ th-order allpass filter):
  $$\frac{1}{2\pi}\int_0^{2\pi} D(\omega)d\omega \mathrel{\stackrel{\mathrm{\Delta}}{=}} -\frac{1}{2\pi}\int_0^{2\pi} \Theta^\prime(\omega)d\omega = -\frac{1}{2\pi}\left[\Theta(2\pi) - \Theta(0)\right] = N$$

• Only known closed-form case for allpass interpolators of arbitrary order
• Effective for delay-line interpolation needed for tuning since pitch perception is most acute at low frequencies.