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Topic: sinc(x)
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Mark Bridger

Posts: 22
Registered: 12/6/04
Posted: Oct 16, 2001 5:23 PM
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One of the (at least theoretical) uses of sinc(x) = sin(x)/x is to recover certain functions from samples.

Suppose f(x) is a periodic bandwidth limited function. this means that its Fourier transform F(s) vanishes outside some finite interval [-s0, s0] i.e. it has no frequency components greater than s0. Suppose we sample f at times that are within 1/(2s0) of each other. Then f(x) can be completely recovered from these *finitely many* samples. The recovery is obtained , essentially, by

** taking the convolution of the sampled function g(x) with sinc(kx) for a certain k **.

(g(x) is obtained from f(x) by mutiplying it by a sum of equally spaced "delta functions" -- i.e. impulses).

This is a "constructive" version of Shannon's "Sampling Theorem." In the case of a function which has a finite Fourier *series*, it makes the geometrically reasonable statement that if the greatest frequency that occurs in this series is, say, k times per second, than if you sample the graph at (at least) 2k times per second you will get enough information to reconstruct the function. Unless you sample this often you get "aliasing" which gives distorted information (e.g. when you try to plot something like sin(61x) over [-10,10] on a graphing calculator and get a sine curve with obviously the wrong -- too low -- frequency).

What is surprising is that Shannon's theorem still holds for functions that have continuous spectra (i.e. are Fourier transforms of continuous functions) as opposed to discrete ones (given by Fourier series).

This is mostly a theoretical result; in practice functions, with high frequencies filtered out, are *partially* reconstructed using Fourier series, and the sampling is usually at a much higher frequency than the absolute minimum described above. CD players do something like this.

You can look this up in books on image or signal processing (e.g. Castleman "Digital Image Processing"). I confess to having forgotten a lot of the details since I last used this stuff about 10 - 15 years ago; I hope I got this about right (I snuck a peek at Castleman).


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