Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


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. Note, incidentally, that a lowpass filter having maximally flat group-delay at dc is called a Bessel filter [365, pp. 228-230].



Subsections
Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA