30.tex

Video notes: vid18.mp4
\subsection*{Complex Spectrum (application: conversion to min phase)}
\begin{itemize}
\item{
A transfer function in factored form. Zeros over the poles:
\begin{align*}
H(z) = g\frac{
    (1-q_1z^{-1}) \cdots
    (1-q_mz^{-1})
}{
    (1-p_1z^{-1}) \cdots
    (1-p_nz^{-1})
}
\end{align*}
}
\item{
What happens when we take the log of this? 
\begin{align*}
\ln{H} & = \ln{g} + \ln{1 - q_1 z^{-1}} + \cdots \ln{1 - q_m z^{-1}} \\
& - \ln{1 - p_1 z^{-1}} - \cdots - \ln{1 - p_n z^{-1}}\\
& = \ln{g} + 
\sum\limits_{i = 1}^{N} ( \frac{ 1 } { 1 - P_i z^{-1}})
- \sum\limits_{k = 1}^{N} ( \frac{ 1 } { 1 - q_k z^{-1}})
\end{align*}
}
\item{
Next thing: take a Taylor series about zero (aka Maclaurin series).
\begin{align*}
\ln (\frac{1}{1 - x}) = x + 
\frac{x^2}{2} + \frac{x^{3}{3}} + \cdots \frac{x^{k}}{k} + \cdots
\end{align*}
where $\vert x \vert < 1$
}
\item{ 
Note that there are no factorials, so it makes it very easy to plug in. 
}
\end{itemize}