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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Delay-Line and Signal Interpolation
       Thiran Allpass Interpolators

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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... ...^{-(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 [362, pp. 228-230].

  • $ a_0=1$ 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 $ N$th-order allpass filter is $ N$ samples [449].5.7
  • 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.


Subsections
Previous: Relation of Lagrange to Sinc Interpolation
Next: Thiran Allpass Interpolation in Matlab

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About the Author: 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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