Filtering > a Butterworth ARMA filter

Digital filters can operate in the space domain by using the convolution product. A sampled signal can be represented by a polynomial of variable z and/or by a set of coefficients. The convolution product of two signals is the product of the polynomials in z domain of these signals. All linear filters can be presented by the relationship between two polynomials in z.
A filter for which the polynomial is a fraction, whose denominator is equal to 1, is a moving average filter or MA. A filter for which the polynomial is a fraction, whose numerator is equal to 1, is an autoregressive filter or AR. An ARMA filter is the product of an AR filter by an MA filter. The order of the filter is fixed by the number of polynomial terms in z from which it is made up. In this section, we study the behaviour of an ARMA filter, called Butterworth filter. The Butterworth filter is a frequency band pass filter, currently used in seismic processing.