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Interpolation of Uniformly Spaced Samples

In the uniformly sampled case ($ x_k=kT$ for some sampling interval $ T$ ), a Lagrange interpolator can be viewed as a Finite Impulse Response (FIR) filter [452]. Such filters are often called fractional delay filters [269], since they are filters providing a non-integer time delay, in general. Let $ h(n)$ denote the impulse response of such a fractional-delay filter. That is, assume the interpolation at point $ x$ is given by

\begin{eqnarray*}
y(x) &=& h(0)\,f(x_N) + h(1)\,f(x_{N-1}) + \cdots h(N)\,f(x_0)\\
&=& h(0)\,y(N) + h(1)\,y(N-1) + \cdots h(N)\,y(0).
\end{eqnarray*}

where we have set $ T=1$ for simplicity, and used the fact that $ y(x_k)=f(x_k)$ for $ k=0,1,\ldots,N$ in the case of ``true interpolators'' that pass through the given samples exactly. For best results, $ y(x)$ should be evaluated in a one-sample range centered about $ x=N/2$ . For delays outside the central one-sample range, the coefficients can be shifted to translate the desired delay into that range.


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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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