Overview

We shall use the term quadrature filter or just filter to denote an operator which convolves and then decimates. We will define filter operators both on sequences and on functions of one real variable. We can also project such actions onto periodic sequences and periodic functions, and define periodized filters.

In all cases where f is summable, the operation of convolution by f can be regarded as multiplication by the bounded, continuous Fourier transform of f. The Weierstrass approximation theorem assures us that we can uniformly approximate an arbitrary continuous one-periodic function by a trigonometric polynomial, in other words by the Fourier transform of a finitely supported sequence f. Thus we can arrange for an operator F which involves only finitely many operations per output value to multiply "on the Fourier transform side" by a function that attenuates certain values ( i.e., multiplies them by zero or a small number) while it preserves or amplifies certain others (multiplies them by one or a large number). Since the Fourier transform of the input is a decomposition of the input into monochromatic waves e ^{2 ?ix?}, the operator F modifies the frequency spectrum of the input to produce the output. Hence the name "filter" which is inherited from electrical engineering.

If the filter sequence is finitely supported, we have a finite impulse response or FIR filter; otherwise we have an IIR or infinite impulse response