Sinc filter
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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
Let <math>f</math> be a square integrable function of a real parameter <math>t</math> (a signal, in this context). Then it has a fourier transform <math>\hat f</math>. The sinc filter maps <math>f</math> to <math>{\rm sinc}*f</math>, where <math>{\rm sinc}*f</math> is a version of f whose Fourier coefficients are all zero above a cutoff frequency <math>N</math>. That is, <math>{\rm sinc}*f</math> is defined by
- <math>\widehat{{\rm sinc}*f}(\omega) = \left \{ \begin{matrix}
\hat{f}(\omega) & \mbox{ if } |\omega| \leq N, \\ 0 & \mbox{ otherwise.} \end{matrix} \right.</math>
Since the Lebesgue integral does not "see" individual frequencies, it does not matter what we do to the frequencies <math>N</math> and <math>-N</math>. For instance, we can zero them, leave them untouched, halve them or randomize them.
Another characterization of <math>{\rm sinc}*f</math> is that it is the best approximation to <math>f</math> inside the space of signals whose Fourier transform is supported in the interval <math>[-N,N]</math>. In other words, the choice <math>g={\rm sinc}*f</math> minimizes
- <math>\int_{-\infty}^\infty |f(t)-g(t)|^2 dt</math>
when <math>g</math> is a signal whose Fourier transform <math>\hat g</math> is zero outside the interval <math>[-N,N]</math>.
If we rewrite the sinc filter as
- <math>\widehat{{\rm sinc}*f}(\omega)=1_{[-N,N]}(\omega)\hat{f}(\omega)</math>
and using the convolution theorem, we can see that the sinc filter is given by convolution against the inverse Fourier transform of <math>1_{[-N,N]}</math>, the indicator function of the interval <math>[-N,N]</math>. The inverse Fourier transform of <math>1_{[-N,N]}</math> is called the sinc function and it is given by
- <math>{\rm sinc}_N(x)=\sqrt{2\pi}\,{\sin(Nx) \over Nx}.</math>
Some texts state that the Fourier transform of <math>{\rm sinc}_N(x)</math> is <math>1_{(-N,N)}</math> instead of <math>1_{[-N,N]}</math> and others even claim that its value is <math>1/2</math> at <math>\pm N</math>. The truth is that all these choices are almost everywhere equal and they are all "the" Fourier transform of <math>{\rm sinc}_N(x)</math>.