Proof of Maximum Flatness at DC

The maximally 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:

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

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

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

$$h(n) = \prod_{\stackrel{k=0}{k\ne n}}^N \frac{D-k}{n-k}, \quad n=0,1,\ldots N,$$

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 $N$ ).

Further details regarding the theory of Lagrange interpolation can be found in [506, Ch. 3, Pt. 2, pp. 82-84].^5.5