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Odd-Order Lagrange Interpolation Summary

In contrast to even-order Lagrange interpolation, the odd-order case has the following properties (in fractional delay filtering applications):

The usual centered delay range is

$\displaystyle \Delta\in\left(\frac{N}{2}-\frac{1}{2},\frac{N}{2}+\frac{1}{2}\right),
$

which is between integers, and in this range, the amplitude response is observed to be bounded by 1.

To avoid a discontinuous phase-delay jump at high frequencies when crossing the middle delay, the delay range can be shifted to

$\displaystyle \Delta\in\left(\frac{N\pm 1}{2}-\frac{1}{2},\frac{N\pm 1}{2}+\frac{1}{2}\right),
$

but then the gain may exceed 1 at some frequencies.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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