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Filter


Let S be a nonempty set, then a filter on S is a nonempty collection F of subsets of S having the following properties:

1. emptyset not in F,

2. If A,B in F, then A intersection B in F,

3. If A in F and A subset= B subset= S, then B in F

If S is an infinite set, then the collection F_S={A subset= S:S-A is finite} is a filter called the cofinite (or Fréchet) filter on S.

In signal processing, a filter is a function or procedure which removes unwanted parts of a signal. The concept of filtering and filter functions is particularly useful in engineering. One particularly elegant method of filtering Fourier transforms a signal into frequency space, performs the filtering operation there, then transforms back into the original space (Press et al. 1992).


See also

Cofinite Filter, Kalman Filter, Remez Algorithm, Savitzky-Golay Filter, Ultrafilter, Wiener Filter

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References

Hamming, R. W. Digital Filters. New York: Dover, 1998.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Digital Filtering in the Time Domain." §13.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 551-556, 1992.

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Filter

Cite this as:

Weisstein, Eric W. "Filter." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Filter.html

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