14. Appendix B: Relation between Sinc and Lagrange Interpolation

GUIDE: Digital Audio Resampling. Appendix B: Relation between Sinc and Lagrange Interpolation

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Appendix B: Relation between Sinc and Lagrange Interpolation

Lagrange interpolation is a well known, classical technique for interpolation [Hildebrand 1974]. Given a set of $n+1$ known samples$f(x_k)$, $k=0,1,2,\, the problem is to find the unique order $n$polynomial $y(x)$ which interpolates the samples. The solution can be expressed as a linear combination of elementary $n$th order polynomials:

\

where

\

From the numerator of the above definition, we see that $l_k(x)$ is an order $n$ polynomial having zeros at all of the samples except the $k$th. The denominator is simply the constant which normalizes its value to $1$ at$x_k$. Thus, we have

\

In other words, the polynomial $l_k$ is the $k$th basis polynomial for constructing a polynomial interpolation of order $n$ over the $n+1$sample points $x_k$.

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

\

where

\

The simplest argument is based on the fact that any analytic function is determined by its zeros and its value at one point. Since $\ is zero on all the integers except $0$, and since $\, 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$x=0$.

A direct proof can be based on the equivalance between Lagrange interpolation and windowed-sinc interpolation using a ``binomial window'' [Kootsookos and Williamson 1996,Välimäki 1995]. That is, for a fractional sample delay of $D$ samples, multiply the shifted-by-$D$, sampled, sinc function

\

by a binomial window

\

and normalize by [Välimäki 1995]

\

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

\

Since the binomial window converges to the Gaussian window as $N\, 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.

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