Filter

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

1. ,

2. If , then ,

3. If  and  then 

If  is an infinite set, then the collection  is a filter called the cofinite (or Fréchet) filter on .

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).

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.