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Explicit Lagrange Coefficient Formulas

Given a desired fractional delay of $ \Delta$ samples, the Lagrange fraction-delay impulse response can be written in closed form as

$\displaystyle h_\Delta(n) = \prod_{\stackrel{k=0}{k\neq n}}^N \frac{\Delta-k}{n-k}, \quad n=0,1,2,\ldots,N. \protect$ (5.7)

The following table gives specific examples for orders 1, 2, and 3:

\begin{displaymath}
{\small
\begin{array}{\vert\vert r\vert\vert c\vert c\vert c\vert c\vert}
\hline
h_\Delta{\mbox{ Order}} & h_\Delta(0) & h_\Delta(1) & h_\Delta(2) & h_\Delta(3) \\
\hline
\hline
N=1 & 1-\Delta & \Delta & 0 & 0 \\
\hline
N=2 & \frac{(\Delta-1)(\Delta-2)}{2} & -\Delta(\Delta-2) & \frac{\Delta(\Delta-1)}{2} & 0 \\
\hline
N=3 & -\frac{(\Delta-1)(\Delta-2)(\Delta-3)}{6} & \frac{\Delta(\Delta-2)(\Delta-3)}{2} & -\frac{\Delta(\Delta-1)(\Delta-3)}{2} &
\frac{\Delta(\Delta-1)(\Delta-2)}{6} \\
\hline
\end{array}}
\end{displaymath}

Note that, for $ N=1$ , Lagrange interpolation reduces to linear interpolation, i.e., the interpolator impulse response is $ h = [1-\Delta,\Delta]$ . Also, remember that, for order $ N$ , the desired delay should be in a one-sample range centered about $ \Delta=N/2$ .


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``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
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