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# Sinc filter

In signal processing, the sinc filter strips high-frequency data from a wave.

If you find this article confusing, you may want to read about aliasing and signal processing as well as the Fourier transform and convolutions.

Many physical processes are subject to noise. For instance, reception of an ordinary music radio is rarely crystal-clear, telephones don't transmit a perfect sound and old pictures get scratched or lose some of their colors. One method of minimizing such defects is to filter the sounds and images, to remove obvious noises and scratches. At this point it is unfortunately impossible to create filters that restore a signal to its pristine original self, but we have a starting point, and that is the sinc filter.

The sinc filter presumes that noise will be (in audio signals) principally high-pitched. The idea is that most people don't produce very high pitched sounds, so if we remove all high-pitched sounds from a telephone conversation, we are probably removing mostly noise, and the conversation is unhindered and perhaps improved.

## Technical discussion

If f(t) is a function of the real line into the set of complex numbers, and if f2 is integrable then we say that f is a signal. Let Ff be the Fourier transform of f, and Gg be the inverse Fourier transform of g (when g is the Fourier transform of a signal.) Let N be a real number.

Low-frequency data is defined as the restriction of Ff to [-N,N]. In many physical problems, the low-frequency information of a signal is the most important portion of the signal. In fact, in some cases, high-frequency data is considered to be mostly bogus, because the underlying physical process is unlikely to generate such waves. Therefore, we would wish to have a version of f that would be stripped of all such bogus waves, but whose low frequency data is preserved. From this discussion, the sinc filter writes itself:

Sf=G(R(Ff))

where R is the map which takes a signal g and gives a signal h which is equal to g in [-N,N] and 0 elsewhere. We can write R(g) as the pointwise product r(w)g(w) where r is the function which is 1 on [-N,N] and 0 elsewhere.

In view of Fourier analysis and convolutions, we see that Sf is the convolution of f(t) with sinc(t) where sinc(t) is the inverse Fourier transform of r(w). One checks that the inverse Fourier transform of r(w) is in fact:

sinc(t)=sin(Nt)/(Nt)

(with perhaps a scaling constant depending on the definition of the Fourier transform used.)