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Ideal Bandlimited (Sinc) Interpolation

Ideal interpolation for digital audio is bandlimited interpolation, i.e., samples are uniquely interpolated based on the assumption of zero spectral energy for $ \left\vert f\right\vert\geq f_s/2$ .

Ideal bandlimited interpolation is sinc interpolation:

$\displaystyle y(t) = (y\ast h_s)(t) = \sum_{n=-\infty}^\infty y(nT) h_s(t-nT)
$

where

\begin{eqnarray*}
h_s(t) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \mbox{sinc}(f_st) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\mbox{sinc}\left(\frac{t}{T}\right) \\ [10pt]
\mbox{sinc}(x) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\sin(\pi x)}{\pi x} \qquad\mbox{(sinc function)}
\end{eqnarray*}

(Proof: sampling theorem)



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``Bandlimited Interpolation, Fractional Delay Filtering, and Optimal FIR Filter Design'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-08-17 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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