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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Delay-Line and Signal Interpolation
       Lagrange Interpolation
          Relation of Lagrange to Sinc Interpolation

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Relation of Lagrange to Sinc Interpolation

For an infinite number of equally spaced samples, with spacing $ x_{k+1}-x_k = \Delta$, the Lagrangian basis polynomials converge to shifts of the sinc function, i.e.,

$\displaystyle l_k(x) =$   sinc$\displaystyle \left(\frac{x-k\Delta}{\Delta}\right), \quad k=\ldots,-2,-1,0,1,2,\ldots $

where

   sinc$\displaystyle (x) \isdef \frac{\sin(\pi x)}{\pi x} $

A simple argument is based on the fact that any analytic function is determined by its zeros and its value at one point. Since sinc$ (x)$ is zero on all the integers except 0, and since sinc$ (0)=1$, it must coincide with the infinite-order Lagrangian basis polynomial for the sample at $ x=0$ which also has its zeros on the nonzero integers and equals $ 1$ at $ x=0$.

The equivalence of sinc interpolation to Lagrange interpolation was apparently first published by the mathematician Borel in 1899, and has been rediscovered many times since [309, p. 325].

A direct proof can be based on the equivalance between Lagrange interpolation and windowed-sinc interpolation using a ``scaled binomial window'' [262,502]. That is, for a fractional sample delay of $ D$ samples, multiply the shifted-by-$ D$, sampled, sinc function

$\displaystyle h_s(n) =$   sinc$\displaystyle (n-D) = \frac{\sin[\pi(n-D)]}{\pi(n-D)} $

by a binomial window

$\displaystyle w(n) = \left(\begin{array}{c}N\\ n\end{array}\right), \quad n=0,1,2,\ldots N $

and normalize by [502]

$\displaystyle C(D) = (-1)^N\frac{\pi(N+1)}{\sin(\pi D)}\left(\begin{array}{c}D\\ N+1\end{array}\right), $

which scales the interpolating filter to have a unit $ L_2$ norm, to obtain the $ N$th-order Lagrange interpolating filter

$\displaystyle h_D(n)=C(D)w(n)h_s(n), \quad n=0,1,2,\ldots,N $

Since the binomial window converges to the Gaussian window as $ N\to\infty$, and since the window gets wider and wider, approaching a unit constant in the limit, the convergence of Lagrange to sinc interpolation can be seen.

A more recent alternate proof appears in [557].

Previous: Recent Developments in Lagrange Interpolation
Next: Thiran Allpass Interpolators

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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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