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Variable FIR Interpolating Filter

Basic idea: Each FIR filter coefficient $ h_n$ becomes a polynomial in the delay parameter $ \Delta$ :

\begin{eqnarray*}
h_\Delta(n) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{m=0}^P c_n(m)\Delta^m, \quad n=0,1,2,\ldots,N \\
\Leftrightarrow \;
H_\Delta(z) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{n=0}^N h_\Delta(n)z^{-n} \\
&=& \sum_{n=0}^N \left[\sum_{m=0}^P c_n(m)\Delta^m\right]z^{-n}\\
&=& \sum_{m=0}^P \left[\sum_{n=0}^N c_n(m) z^{-n}\right]\Delta^m \\
&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{m=0}^P C_m(z) \Delta^m
\end{eqnarray*}


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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 © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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