Waveform Approximation Using Lagrange Polynomial Interpolation

August 23, 2019

Attempting to reconstruct a waveform 𝗑(𝑑) from its samples 𝗑(π‘›πœ) requires some form of interpolation. One interpolation method (of an infinite number of possible methods) is Lagrange interpolation, which connects n points using an order-n βˆ’ 1 polynomial. Examples of this method follow:

Order-0 (Stair-Step) Polynomial Interpolation

Order-1 (Line) Polynomial Interpolation

In the examples above, the approximation waveform appears to get β€œcloser” to the original waveform as the sample rate π–₯s increases. As a result of B-splines, the original waveform can be approximated with precision by increasing the sample rate toward infinity.

In particular, the order-0 interpolation is equivalent to interpolation using the order-0 B-spline N0(t) and the order-1 interpolation using the order-0 B-spline N1(t). The sequence of shifted B-splines (𝖭0(𝑑 βˆ’ π‘›πœ)) forms an orthonormal basis, and for any π‘š β‰₯ 1, the set of shifted B-splinesΒ (𝖭m(𝑑 βˆ’ π‘›πœ)) forms a Riesz basis (but not an orthonormal basis).