Filter

Formally, a filter is defined in terms of a Set $X$ and a Set $\Phi$ of Subsets of $X$. Then $\Phi$ is called a filter if

1. $X\in\Phi$,

2. the Empty Set $\emptyset\notin\Phi$,

3. $A\subset B\subset X$ and $A\in\Phi$ Implies $B\in\Phi$,

4. $A,B\in \Phi$ Implies $A\cup B\in\Phi$.

Informally, 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 Savitzky-Golay Filter, Wiener Filter

References

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.