*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.

If *f*(*t*) is a function of the real line into the set of complex numbers, and if *f*^{2} 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))*

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*)