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## What is Bandlimited Interpolation?

Bandlimited interpolationof discrete-time signals is a basic tool having extensive application in digital signal processing. In general, the problem is to correctly compute signal values at arbitrary continuous times from a set of discrete-time samples of the signal amplitude. In other words, we must be able to interpolate the signal between samples. Since the original signal is always assumed to bebandlimitedto half the sampling rate, (otherwisealiasing distortion would occur upon sampling),Shannon's sampling theoremtells us the signal can be exactly and uniquely reconstructed for all time from its samples by bandlimited interpolation.There are many methods for interpolating discrete points. For example,

Lagrange interpolationis the classical technique of finding an order polynomial which passes through given points.The technique known as

cubic splinesfits a third-order polynomial through two points so as to achieve a certain slope at one of the points. (This allows for a smooth chain of third-order polynomial passing through a set of points.)You may also have heard of

Bezier splineswhich interpolate a set of points using smooth curves which don't necessarily pass through the points. (Bezier curves are commonly used in graphics and drawing programs, such as Adobe Illustrator.)The above methods are suitable for graphics and other uses, but they are not ideal for digital audio. In digital audio, what matters is the

audibilityof interpolation error between samples. Since Shannon's sampling theorem says that it is possible to restore an audio signalexactlyfrom its samples, it makes sense that the best digital audio interpolators would be based on that theory. Such ``ideal'' interpolation is calledbandlimited interpolation.A bandlimited interpolation algorithm designed along these lines is described in the theory of operation tutorial. There is also open-source softwareavailable in the C programming language.