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Deriving Linear Interpolation from Taylor Series

Truncate a Taylor series expansion to first order and plug in a first-order derivative approximation:

\begin{eqnarray*}
y(n+\alpha)
&=& y(n) + \alpha\,\dot{y}(n) + \alpha^2\,\frac{\ddot{y}(n)}{2!} + \alpha^3\,\frac{\dddot{y}(n)}{3!} + \cdots\\ [10pt]
&\approx& y(n) + \alpha\,\dot{y}(n)
\end{eqnarray*}

\begin{eqnarray*}
\Rightarrow\quad{\hat y}(n+\alpha) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& y(n) + \alpha\,\frac{y(n+1)-y(n)}{1}\\ [10pt]
&=& \alpha\,y(n+1) + (1-\alpha)\,y(n)
\end{eqnarray*}

where $ \alpha=-\eta = $ fractional advance desired
(interpolation time between samples $ n$ and $ n+1$ )


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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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