*Bandlimited interpolation* of 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 be *bandlimited *to half the sampling rate, (otherwise
aliasing distortion would occur upon sampling), *Shannon's
sampling theorem* tells 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 interpolation* is the classical technique of finding an
order *N* polynomial which passes through *N*+1 given points.

The technique known as *cubic splines* fits 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 splines* which 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 *audibility* of interpolation error between samples. Since
Shannon's sampling theorem says it is possible to restore an audio
signal *exactly* from its samples, it makes sense that the best
digital audio interpolators would be based on that theory. Such
``ideal'' interpolation is called *bandlimited interpolation*.

A bandlimited interpolation algorithm designed along these lines is described in the theory of operation tutorial. There is also free open-source software available in the C programming language.

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