27.tex

Class notes: 3-1-16
\subsection*{State variable filter: Review}
\begin{itemize}
\item{What is the form of a SVF? It kind of looks like a DF-2 filter.}
\item{What happens when you transpose an integrator chain? It becomes a differentiator chain. }
\item{Numerically, you get better results with integrators because it rolls off the noise floors.
Differentiators ramp up the noise floor.
}
\item{Digitization: We can't use the BLT because it's a delay free loop.}
\item{Simple differentiator:
    \begin{align*}
    H(s) = s \rightarrow
    \frac{z}{T} \frac{1 - z^{-1}}{1 + z^{-1}}
    \end{align*}
}
\item{We looked at backward Euler: still problems}
\item{We looked at forward Euler (half sample delay:
    \begin{align*}
        \frac{z - 1 }{T} \rightarrow \frac{1}{s} = \frac{T}{z - 1}
    \end{align*}
}
\item{FEBE (foward/backward euler): multiply them together
    \begin{align*}
        \frac{z - 2 + z^{-1}}{T^2} 
        \rightarrow 
        \frac{1}{s^2} 
        \rightarrow 
        \frac{T^2 z^{-1}}{1 - 2z^{-1} + z^{-2}}
    \end{align*}
}
\end{itemize}

\subsection*{$S$ to $Z$ mappings}
\begin{itemize}
\item{
\begin{align*}
s &= \frac{1 - z^{-1}}{T}\\
sT &= 1 - z^{-1} \rightarrow z^{-1} = 1 - sT\\
z = \frac{1}{1 - sT}
\end{align*}
}
\item{
Backward Euler: We get artificial damping from backward euler
}
\item{
FE: $s = \frac{z - 1}{T}$, therefore $z = 1 + sT \rightarrow 1 + j\omega T$
} \item{
DC is fine, but everywhere else, the $jw$ axis is outside the unit circle, 
if your poles are lightly damped, they may be unstable after digitization.
}
\item{
If you read up on finite difference schemes, you oftne hear about the stabliiity of the finite
difference scheme... this is an example of what you are looking out for. 
}
\item{
Backward/Forward case: $s^2 = \frac{z - 2 + z^{-1}}{T^2} $,
$z = \frac{-b}{z} \pm \sqrt{b^2 - 4ac} / 2$
}
\item{
Intuitively it's going to be inbetween, BE is inside the unit cirlce, FE is outside the unit circle, 
BFE is somewhere in the middle. (mama bear, papa bear, "just right"
}
\item{
Sooner or later, the computer will round against you, and you will lose...
}
\end{itemize}

\subsection*{Limit cycles}
\begin{itemize}
\item{
$h(t) = e^{-\alpha t}
$
}
\item{
The level that your signal reaches due to rounding.
}
\item{
Fix: magnitude truncation, but it sacrifices accuracy
}
\item{
"Reverberation limit cycle synthesis"
}
\end{itemize}


\subsection*{Digitizing Analogue State Variable filters}
\begin{itemize}
\item{
Piles of poles at DC
}
\item{
2nd order is arbitray, can have any pile of poles.
}
\item{
Unrelated: Operational Transconductance Amplifier (OTA)
}
\item{
FE: $\frac{1}{s} \rightarrow \frac{Tz^{-1}}{1 - Z^{-1}}$
}
\item{
First is FE, second is BE, what are the others? We don't need to add any more delays. 
We can use the BLT for the rest (this is paper-worthy)
}
\end{itemize}
\subsection*{Faust digitizatoin}
\begin{itemize}
\item{
    It's really easy to frequency scale each integrator.  (that's your knob)
}
\item{

}
\end{itemize}