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Thiran Allpass Interpolators
Given a desired delay
samples, an order
allpass filter
can be designed having maximally flat group delay equal to
at dc using the formula
where
denotes the
th binomial coefficient [497]. Note, incidentally,
that a lowpass filter having maximally flat groupdelay at dc is
called a Bessel filter [365, pp. 228230].

without further scaling
 For sufficiently large
, stability is guaranteed
Rule of thumb:
 It can be shown that the mean group delay of any stable
thorder allpass filter is
samples
[452].^{5.8}
 Only known closedform case for allpass interpolators of arbitrary order
 Effective for delayline interpolation needed for tuning since
pitch perception is most acute at low frequencies.
 Since Thiran allpass filters have maximally flat groupdelay at
dc, like Lagrange FIR interpolation filters, they can be considered
the recursive extension of Lagrange interpolation.
Subsections
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