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Proof of Maximum Flatness at DC

The maximumally flat fractional-delay FIR filter is obtained by equating to zero all $ N+1$ 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 _{\omega=0}
= \left.\frac{d^k}{d\omega^k} e^{-j\omega\Delta} - \sum_{n=0}^N h(n)e^{-j\omega n}\right\vert _{\omega=0}\\
&=& (-j\Delta)^k - \sum_{n=0}^N (-jn)^k h(n)
\end{eqnarray*}

$\displaystyle \,\,\Rightarrow\,\,\zbox {\sum_{n=0}^N n^k h(n) = \Delta^k, \; k=0,1,\ldots,N}
$

This is a linear system of equations of the form $ V\underline{h}=\underline{\Delta}$ , where $ V$ is a Vandermonde matrix. The solution can be written as a ratio of Vandermonde determinants using Cramer's rule [332]. As shown by Cauchy (1812), the determinant of a Vandermonde matrix $ [p_i^{j-1}]$ , $ i,j=1,\ldots,N$ can be expressed in closed form as

\begin{eqnarray*}
\left\vert\left[p_i^{j-1}\right]\right\vert &=& \prod_{j>i}(p_j-p_i)\\
&=&
(p_2-p_1)(p_3-p_1)\cdots(p_N-p_1)\cdots\\
&&(p_3-p_2)(p_4-p_2)\cdots(p_N-p_2)\cdots\\
&&(p_{N-1}-p_{N-2})(p_N-p_{N-2})\cdots\\
&&(p_N-p_{N-1}).
\end{eqnarray*}

Making this substitution in the solution obtained by Cramer's rule yields that the impulse response of the order $ N$ , 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,
$

which is the formula for Lagrange-interpolation coefficients (Eq.$ \,$ (4.6)) adapted to this problem (in which abscissae are equally spaced on the integers from 0 to $ N-1$ ).

Further details regarding the theory of Lagrange interpolation can be found in [504, Ch. 3, Pt. 2, pp. 82-84].5.5


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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