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 magnitudesquared 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 circularlystationary 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 autoregressivemoving average (ARMA) model.
Baggenstoss
20170519