Autoregressive, Moving Average and ARMA filters

Let
be the output of a digital
filter with input data
. A
moving average (MA) algorithm is a linear function of a finite number of
input data:
where m is called order of the filter. If then the filter is purely causal. An
autoregressive (AR) algorithm is a function of the output data:
Again,
the filter is causal if
. The combination of the two algorithms is called ARMA. The main feature of a digital implementation of a filter as an AR, MA or ARMA is that the order of the filter depends only on the order of the differential equations that define the filter in the continuous time domain. In particular the number of operations do not increase with the number of input data (like for the FFT algorithm).
