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:

$$\begin{eqnarray*} 0 &=& \left.\frac{d^k}{d\omega^k} E(e^{j\omega}) \right\vert _... ...ert _{\omega=0}\\ &=& (-j\Delta)^k - \sum_{n=0}^N (-jn)^k h(n) \end{eqnarray*}$$

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

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 [333]. 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

$$\begin{eqnarray*} \left\vert\left[p_i^{j-1}\right]\right\vert &=& \prod_{j>i}(p_... ...dots\\ &&(p_{N-1}-p_{N-2})(p_N-p_{N-2})\cdot\\ &&(p_N-p_{N-1}) \end{eqnarray*}$$

Making this substitution in the solution obtained by Cremer's rule yields that the impulse response of the order $N$ maximally flat fractional-delay FIR filter may be written in closed form as

$$h(n) = \prod_{\stackrel{k=0}{k\ne n}}^N \frac{D-k}{n-k}$$

which coincides with the formula for Lagrange interpolation when the abscissae are equally spaced on the integers from 0 to $N-1$.

Further details regarding the theory of Lagrange interpolation can be found (online) in [511, Ch. 3, Pt. 2, pp. 82-84].