32.tex

\subsection*{Series Expansion of the Log Transfer Function in Factored Form}
\begin{itemize}
\item{
to recap:
\begin{align*}
\ln{H(z)} = \ln{g} 
\sum \limits_{m = 1}^{M} \ln {\frac{1}{1 - q_m z^{-1}}} +
\sum \limits_{n = 1}^{N} \ln {\frac{1}{1 - p_n z^{-1}}} +
\end{align*}
}
\item{
This equation can be expaned using an expansion of $\frac{1}{1 - x}$, eg:
\begin{align*}
q_m z^{-1} + \frac{1}{2}(q_m z^{-1})^2 + \cdots
\end{align*}
}
\item{
\begin{align*}
    \ln(\frac{1}{1 - q_m z^{-1}}) &= 
\sum\limits_{m = 1}^{\infty}  \frac{1_n}{n} z^{-n} \leftrightarrow 
\big[0, q_m, \frac{q_m^2}{2}, \cdots 
\end{align*}
    Where $\vert q_m \vert  < 1 $, aka the zeros are inside the unit circle.
}
\item{
    One way to think about this: the output is a sampled exponential decay divided by a ramp.
}
\item{
If it is outside the unit cirlce, it's a maximum phase zero (non minimum phase zero) 
and you don't get this result. You get a blowing up exponential, or an non-causal or
anti causal expansion..
}
\end{itemize}