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Recursive Term Computation

Our Lagrange interpolation filter is again

$\displaystyle \hat{H}_\Delta(q^{-1}) \;=\;\sum_{n=0}^{N} \hbar _\Delta(n)(q-1)^n
$

where the coefficients of $ (q-1)^n=(z^{-1}-1)^n$ are again

$\displaystyle \hbar _\Delta(n) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\frac{1}{n!}\prod_{k=0}^{n-1}(\Delta-k) \;=\;\frac{\Delta(\Delta-1)(\Delta-2)\cdots(\Delta-n+1)}{n!}
$

Note that we can recursively compute the terms in the sum from left to right:

\begin{eqnarray*}
\hbar _\Delta(n) &=& \hbar _\Delta(n-1)\cdot \frac{\Delta-n+1}{n}\\
(q-1)^{n} &=& (q-1)^{n-1}\cdot (q-1)
\end{eqnarray*}

Thus, we can crank out the terms in series and sum the intermediate signals:

\begin{center}
\epsfig{file=eps/DepalleTassartQ.eps,width=\textwidth } \\
\end{center}


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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 © 2017-05-12 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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