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:

• Ideal impulse response = bandlimited delayed impulse = delayed sinc
  $$h_\Delta(t) =$$   sinc$$(t-\Delta) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{\sin\left[\pi (t-\Delta)\right]}{\pi (t-\Delta)}$$

• Ideal frequency response = ``brick wall'' lowpass response, cutting off at $f_s/2$ and having linear phase $e^{-j\omega \Delta T}$
  $$\begin{eqnarray*} H_\Delta(e^{j\omega})&\mathrel{\stackrel{\mathrm{\Delta}}{=}}&... ...ightarrow& \mbox{sinc}(n-\Delta), \quad n=0,\pm 1, \pm 2, \ldots \end{eqnarray*}$$
  

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

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

FIR Filter Frequency Response

$${\hat H}_\Delta \left(e^{j\omega T}\right)= \sum_{n=0}^{L-1} {{\hat h}_\Delta}(n) e^{-j\omega nT}$$

Error to Minimize

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

$L2$ Error Norm

$$\begin{eqnarray*} J(\underline{h}) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\left\... ... {\hat H}_\Delta \left(e^{j\omega T}\right)\right\vert^2 d\omega \end{eqnarray*}$$

By Parseval's Theorem

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

Optimal Least-Squares FIR Interpolator

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