Odd-Order Lagrange Interpolation Summary

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

Improved phase-delay accuracy at the expense of decreased amplitude-response accuracy (low-order examples in Fig.)
Optimal (centered) delay range lies between two integers

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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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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