- 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 is the

- 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) - Nonuniformly spaced sample locations, such as along the zeros of
a Chebyshev polynomial, generally do better than uniform spacing,
when applicable

- 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
- Faust Test Program for Generating Above Frequency-Response Examples
- Relation of Lagrange Interpolation to Windowed Sinc Interpolation
- Variable FIR Interpolating Filter
- Farrow Structure for Variable Delay FIR Filters
- Farrow Structure Design Procedure
- Farrow Interpolation Features
- Lagrange Interpolation by Taylor Expansion
- Taylor Series Expansion of
- Recursive Term Computation
- Efficient Time-Invariant Lagrange Interpolation
- Time-Varying Lagrange Interpolation
- A More Elegant Derivation
- Features of Truncated-Taylor Lagrange-Interpolation
- References
- Lagrange Interpolation in Faust, Fixed Delay
- Lagrange Interpolation in Faust, Variable Delay
- Summary of Lagrange Interpolators Considered

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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