Proof of Maximum Flatness at DC
The maximumally flat fractional-delay FIR filter is obtained by equating
to zero all leading terms in the Taylor (Maclaurin) expansion of
the frequency-response error at dc:
![\begin{eqnarray*}
0 &=& \left.\frac{d^k}{d\omega^k} E(e^{j\omega}) \right\vert _...
...ert _{\omega=0}\\
&=& (-j\Delta)^k - \sum_{n=0}^N (-jn)^k h(n)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img1060.png)
![$\displaystyle \,\,\Rightarrow\,\,\zbox {\sum_{n=0}^N n^k h(n) = \Delta^k, \; k=0,1,\ldots,N}
$](http://www.dsprelated.com/josimages_new/pasp/img1061.png)
![$ V\underline{h}=\underline{\Delta}$](http://www.dsprelated.com/josimages_new/pasp/img1062.png)
![$ V$](http://www.dsprelated.com/josimages_new/pasp/img239.png)
![$ [p_i^{j-1}]$](http://www.dsprelated.com/josimages_new/pasp/img1063.png)
![$ i,j=1,\ldots,N$](http://www.dsprelated.com/josimages_new/pasp/img1064.png)
![\begin{eqnarray*}
\left\vert\left[p_i^{j-1}\right]\right\vert &=& \prod_{j>i}(p_...
...ts\\
&&(p_{N-1}-p_{N-2})(p_N-p_{N-2})\cdots\\
&&(p_N-p_{N-1}).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img1065.png)
Making this substitution in the solution obtained by Cramer's rule
yields that the impulse response of the order , maximally flat,
fractional-delay FIR filter may be written in closed form as
![$\displaystyle h(n) = \prod_{\stackrel{k=0}{k\ne n}}^N \frac{D-k}{n-k}, \quad n=0,1,\ldots N,
$](http://www.dsprelated.com/josimages_new/pasp/img1066.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![$ N-1$](http://www.dsprelated.com/josimages_new/pasp/img1067.png)
Further details regarding the theory of Lagrange interpolation can be found (online) in [502, Ch. 3, Pt. 2, pp. 82-84].
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