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Supplementary: Optimal Least Squares Bandlimited Interpolation Formulated as a Fractional Delay Filter

Consider a filter which delays its input by $ \Delta$ samples:

The sinc function is an infinite-impulse-response (IIR) digital filter with no recursive form $ \Rightarrow$ non-realizable.

To obtain a finite impulse response (FIR) interpolating filter, let's formulate a least-squares filter-design problem:

Desired Interpolator Frequency Response

$\displaystyle H_\Delta \left(e^{j\omega T}\right)\;=\;e^{-j\omega \Delta T},\quad \Delta \;=\;\hbox{Desired delay in samples}

FIR Frequency Response, Zero-Phase Alignment

$\displaystyle {\hat H}_\Delta \left(e^{j\omega T}\right)\;=\;\sum_{n=-\frac{L-1}{2}}^\frac{L-1}{2} {{\hat h}_\Delta}(n) e^{-j\omega nT}

Error to Minimize

$\displaystyle E\left(e^{j\omega T}\right)= H_\Delta \left(e^{j\omega T}\right)- {\hat H}_\Delta \left(e^{j\omega T}\right)

$ \ensuremath{L_2}$ Error Norm

J(\underline{h}) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\left\Vert\,E\,\right\Vert _2^2
&=& \frac{T}{2\pi}\int_{-\pi/T}^{\pi/T} \left\vert E\left(e^{j\omega T}\right)\right\vert^2 d\omega \\
&=& \frac{T}{2\pi}\int_{-\pi/T}^{\pi/T} \left\vert H_\Delta \left(e^{j\omega T}\right)- {\hat H}_\Delta \left(e^{j\omega T}\right)\right\vert^2 d\omega

By Parseval's Theorem

$\displaystyle J(\underline{h}) = \sum_{n=0}^\infty \left\vert h_\Delta (n) - {{\hat h}_\Delta}(n)\right\vert^2

Optimal Least-Squares FIR Interpolator

$\displaystyle {{\hat h}_\Delta}(n) = \left\{\begin{array}{ll}
\mbox{sinc}(n-\Delta), & \frac{L-1}{2} \leq n \leq \frac{L-1}{2} \\ [5pt]
0, & \hbox{otherwise} \\
\end{array} \right.

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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 © 2022-09-05 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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