Thiran Allpass Interpolators

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Thiran Allpass Interpolators

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

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

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

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

where

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

denotes the

th binomial coefficient. Note, incidentally, that a lowpass filter having maximally flat group-delay at DC is called a Bessel filter [371, pp. 228-230].

without further scaling
For sufficiently large $\Delta$ , stability is guaranteed rule of thumb: $\Delta \approx \hbox{order}$
It can be shown that the mean group delay of any stable th-order allpass filter is samples [460].^K.4
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.
Since Thiran allpass filters have maximally flat group-delay at dc, like Lagrange FIR interpolation filters, they can be considered the recursive extension of Lagrange interpolation.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Physical Audio Signal Processing
Higher Order Delay Line Interpolation
Thiran Allpass Interpolators