## Rational Transfer Function Models

In this chapter, we are concerned with rational transfer function models driven by independent Gaussian noise. Let be a sequence of iid Gaussian noise samples of variance . Squence has a power spectrum (9.1) of

Linear system theory teaches that the power spectrum at the output of a linear system is equal to the input power spectrum times the magnitude-squared of the transfer function. The general form of the rational transfer function is

where we have assumed , . It follows that the power spectrum of is given by

 (9.8)

The corresponding length- circularly-stationary process has circular power spectrum

 (9.9)

where and are the length- DFTs of the numerator and denominator coefficient sequences.

 (9.10)

and

 (9.11)

If and the numerator is 1, the model is said to be autoregressive (AR). If and the denominator is 1, the model is said to me moving average (MA). If and , this is the form of the autoregressive-moving average (ARMA) model.

Baggenstoss 2017-05-19