Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Optimal Least Squares Bandlimited Interpolation Formulated as a Fractional Delay Filter

Note that BL interpolation is a special case of linear filtering. (Proof: Convolution representation above.)

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)
$

$ L2$ Error Norm

\begin{eqnarray*}
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
\end{eqnarray*}

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.
$


Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download Interpolation.pdf
Download Interpolation_2up.pdf
Download Interpolation_4up.pdf
Visit the online book containing this material.

``Bandlimited Interpolation, Fractional Delay Filtering, and Optimal FIR Filter Design'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-05-12 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]