- Lagrange interpolation is just
*polynomial interpolation* - th-order polynomial interpolates points
- First-order case =
*linear interpolation*

**Problem Formulation**

Given a set of
known samples
,
, find the
*unique* order
*polynomial*
which *interpolates* the samples

**Solution (Waring, Lagrange):**

where

- Numerator gives a
*zero*at all samples but the th - Denominator simply
*normalizes*to at - As a result,
- Generalized bandlimited impulse = generalized sinc function:

Each goes through at and zero at all other sample points

I.e., is analogous to sinc - For uniformly spaced samples, Lagrange interpolaton converges to
*sinc*interpolation as - For uniformly spaced samples and
*finite*, Lagrange interpolaton is equivalent to*windowed sinc*interpolation using a*binomial window*(see text for refs) - Can be viewed as a
*linear, spatially varying filter*

(in analogy with linear, time-varying filters)

- Lagrange Interpolation Optimality
- Order 4 Amplitude Response Over a Range of Fractional Delays
- Order 4 Phase Delay Over a Range of Fractional Delays
- Order 5 Amplitude Response Over a Range of Fractional Delays
- Order 5 Phase Delay Over a Range of Fractional Delays
- Explicit Formula for Lagrange Interpolation Coefficients
- Lagrange Interpolation Coefficients

Orders 1, 2, and 3 - Matlab Code For Lagrange Fractional Delay
- Faust Code For Lagrange Fractional Delay
- Faust-Generated C++ Code

Download Interpolation.pdf

Download Interpolation_2up.pdf

Download Interpolation_4up.pdf

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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