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Thiran Allpass Interpolators

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

$\displaystyle 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

$\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 $ k$ th binomial coefficient



Subsections
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``Bandlimited Interpolation, Fractional Delay Filtering, and Optimal FIR Filter Design'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-05-12 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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