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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Delay-Line and Signal Interpolation
       Lagrange Interpolation
          Proof of Maximum Flatness at DC

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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 _... ...ert _{\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 [329]. 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_... ...ts\\ &&(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 (online) in [502, Ch. 3, Pt. 2, pp. 82-84].

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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