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 (10.1) of

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

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

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

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

$$P_x(\omega) = \sigma^2 \frac{\vert B(e^{i\omega})\vert^2 }{\vert A(e^{i\omega})\vert^2}.$$ (10.8)

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

$$\rho_k = \sigma^2 \frac{\vert B_k\vert^2 }{\vert A_k\vert^2}, \;\;\;\; 0\leq k \leq N,$$ (10.9)

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

$$A_k = \sum_{i=0}^Q \; a_i \; e^{-j2\pi i k/N}, \;\;\;\; 0\leq k \leq N,$$ (10.10)

and

$$B_k = \sum_{i=0}^Q \; b_i \; e^{-j2\pi i k/N}. \;\;\;\; 0\leq k \leq N,$$ (10.11)

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.