# Sinc filter

In signal processing, the sinc filter strips high-frequency data from a signal. It is based upon the sinc function.

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

Low-frequency data is defined as the restriction of a signal to [itex]\left[ -W, W \right][itex]. In many physical problems, the low-frequency information of a signal is the most important portion of the signal. In some cases, high-frequency data is deemed invalid/unwanted because the underlying physical process is unable to generate such frequencies. Therefore, we would wish to have a version of a signal that would be stripped of all high frequencies but whose low frequency data is preserved. If the frequency spectrum is that of a rectangular function then the low-frequencies can be preserved with higher frequencies removed.

[itex]\mathbf{rect}(x) = \left \{ \begin{matrix}

0 & \mbox{if } |x| < \frac{1}{2} \\[3pt] \frac{1}{2} & \mbox{if } |x| = \frac{1}{2} \\[3pt] 1 & \mbox{if } |x| > \frac{1}{2} \end{matrix} \right.[itex]

By taking the inverse Fourier transform of the rectangular function the result is the sinc function

[itex]\mathcal{F}^{-1} \left[ \mathbf{rect} \left( \frac{\omega}{2 W} \right) \right] = \frac{W}{\pi} \operatorname{sinc}(W t)[itex]

If the signal is convolved with the sinc filter then the original signal can be restored. This is precisely the Nyquist-Shannon interpolation formula.

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