Basics of Digital Filtering

Abstract

Analog signals can be filtered directly using electrical circuits. However, analog signals are normally converted to digital signals and filtered numerically. In the time domain, a digital filter is characterized by a discrete impulse response function (h _n ) and in the frequency domain by its discrete Fourier transform, the discrete frequency response function (H _k ). The input-output relationships are described in the time domain by the discrete convolution

$$ {y_{n}} = \sum\limits_{{j = 0}}^{M} {{h_{j}}{x_{{n - j}}},n = 0,1, \ldots ,} $$

(15.1)

and, according to the convolution theorem, in the frequency domain by the product

$$ {Y_{k}} = {H_{k}}{X_{{k.}}} $$

(15.2)