Rational Transfer Function Models

In this chapter, we are concerned with rational transfer function models driven by independent Gaussian noise. Let $ e_t$ be a sequence of iid Gaussian noise samples of variance $ \sigma^2$. Squence $ e_t$ has a power spectrum (9.1) of

$\displaystyle P_e(\omega) = \sigma^2.$

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

$\displaystyle H(\omega) = \frac{1+ b_1 e^{j\omega} + \cdots b_Q e^{Qj\omega} }
...ga}} {\sum_{i=0}^P a_i e^{ij\omega}}
= \frac{A(e^{j\omega})}{B(e^{j\omega})},

where we have assumed $ b_0=1$, $ a_0=1$. It follows that the power spectrum of $ {\bf x}$ is given by

$\displaystyle P_x(\omega) = \sigma^2 \frac{\vert B(e^{i\omega})\vert^2 }{\vert A(e^{i\omega})\vert^2}.$ (9.8)

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

$\displaystyle \rho_k = \sigma^2 \frac{\vert B_k\vert^2 }{\vert A_k\vert^2}, \;\;\;\; 0\leq k \leq N,$ (9.9)

where $ B_k$ and $ A_k$ are the length-$ N$ DFTs of the numerator and denominator coefficient sequences.

$\displaystyle A_k = \sum_{i=0}^Q \; a_i \; e^{-j2\pi i k/N}, \;\;\;\; 0\leq k \leq N,$ (9.10)


$\displaystyle B_k = \sum_{i=0}^Q \; b_i \; e^{-j2\pi i k/N}. \;\;\;\; 0\leq k \leq N,$ (9.11)

If $ Q=0$ and the numerator is 1, the model is said to be autoregressive (AR). If $ P=0$ and the denominator is 1, the model is said to me moving average (MA). If $ Q>0$ and $ P>0$, this is the form of the autoregressive-moving average (ARMA) model.

Baggenstoss 2017-05-19