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\chapter{Acquistion}
% TODO: cool quote, if I can think of one
\clearpage
In the Wright Group, \gls{PyCMDS} replaces the old acquisition softwares `ps control', written by
Kent Meyer and `Control for Lots of Research in Spectroscopy' written by Schuyler Kain.
PyCMDS directly addresses the hardware during experiments.
\section{Overview} % =============================================================================
PyCMDS has, through software improvements alone, dramatically lessened scan times...
\begin{itemize}[topsep=-1.5ex, itemsep=0ex, partopsep=0ex, parsep=0ex, label=$\rightarrow$]
\item simultaneous motor motion
\item digital signal processing % TODO: reference section when it exists
\item ideal axis positions \ref{sec:ideal_axis_positions}
\end{itemize}
\section{Future directions} % ====================================================================
\subsection{Ideal Axis Positions}\label{sec:ideal_axis_positions} % ------------------------------
Frequency domain multidimensional spectroscopy is a time-intensive process. %
A typical \gls{pixel} takes between one-half second and three seconds to acquire. %
Depending on the exact hardware being scanned and signal being detected, this time may be mostly
due to hardware motion or signal collection. %
Due to the \gls{curse of dimensionality}, a typical three-dimensional CMDS experiment contains
roughly 100,000 pixels. %
CMDS hardware is transiently-reliable, so speeding up experiments is a crucial component of
unlocking ever larger dimensionalities and higher resolutions. %
One obvious way to decrease the scan-time is to take fewer pixels. %
Traditionally, multidimensional scans are done with linearly arranged points in each axis---this is
the simplest configuration to program into the acquisition software. %
Because signal features are often sparse or slowly varying (especially so in high-dimensional
scans) linear stepping means that \emph{most of the collected pixels} are duplicates or simply
noise. %
A more intelligent choice of axis points can capture the same nonlinear spectrum in a fraction of
the total pixel count. %
An ideal distribution of pixels is linearized in \emph{signal}, not coordinate. %
This means that every signal level (think of a contour in the N-dimensional case) has roughly the
same number of pixels defining it. %
If some generic multidimensional signal goes between 0 and 1, one would want roughly 10\% of the
pixels to be between 0.9 and 1.0, 10\% between 0.8 and 0.9 and so on. %
If the signal is sparse in the space explored (imagine a narrow two-dimensional Lorentzian in the
center of a large 2D-Frequency scan) this would place the majority of the pixels near the narrow
peak feature(s), with only a few of them defining the large (in axis space) low-signal floor. %
In contrast linear stepping would allocate the vast majority of the pixels in the low-signal 0.0 to
0.1 region, with only a few being used to capture the narrow peak feature. %
Of course, linearizing pixels in signal requires prior expectations about the shape of the
multidimensional signal---linear stepping is still an appropriate choice for low-resolution
``survey'' scans. %
CMDS scans often posses correlated features in the multidimensional space. %
In order to capture such features as cheaply as possible, one would want to define regions of
increased pixel density along the correlated (diagonal) lineshape. %
As a concession to reasonable simplicity, our acquisition software (PyCMDS) assumes that all scans
constitute a regular array with-respect-to the scanned axes. %
We can acquire arbitrary points along each axis, but not for the multidimensional scan. %
This means that we cannot achieve strictly ideal pixel distributions for arbitrary datasets. %
Still, we can do much better than linear spacing.
% TODO: refer to PyCMDS/WrightTools 'regularity' requirement when that section exists
Almost all CMDS lineshapes (in frequency and delay) can be described using just a few lineshape
functions:
\begin{ditemize}
\item exponential
\item Gaussian
\item Lorentzian
\item bimolecular
\end{ditemize}
Exponential and bimolecular dynamics fall out of simple first and second-order kinetics (I will
ignore higher-order kinetics here). %
Gaussians come from our Gaussian pulse envelopes or from normally-distributed inhomogeneous
broadening. %
The measured line-shapes are actually convolutions of the above. %
I will ignore the convolution except for a few illustrative special cases. %
More exotic lineshapes are possible in CMDS---quantum beating and breathing modes, for example---I
will also ignore these. %
Derivations of the ideal pixel positions for each of these lineshapes appear below.
% TODO: cite Wright Group quantum beating paper, Kambempati breathing paper
\subsection{Exponential}
Simple exponential decays are typically used to describe population and coherence-level dynamics in
CMDS. %
For some generic exponential signal $S$ with time constant $\tau$,
\begin{equation} \label{eq:simple_exponential_decay}
S(t) = \me^{-\frac{t}{\tau}}.
\end{equation}
We can write the conjugate equation to \ref{eq:simple_exponential_decay}, asking ``what $t$ do I
need to get a cerain signal level?'':
\begin{eqnarray}
\log{(S)} &=& -\frac{t}{\tau} \\
t &=& -\taulog{(S)}.
\end{eqnarray}
So to step linearly in $t$, my step size has to go as $-\tau\log{(S)}$.
We want to go linearly in signal, meaning that we want to divide $S$ into even sections. %
If $S$ goes from 0 to 1 and we choose to acquire $N$ points,
\begin{eqnarray}
t_n &=& -\tau\log{\left(\frac{n}{N}\right)}.
\end{eqnarray}
Note that $t_n$ starts at long times and approaches zero delay. %
So the first $t_1$ is the smallest signal and $t_N$ is the largest. %
Now we can start to consider realistic cases, like where $\tau$ is not quite known and where some
other longer dynamics persist (manifested as a static offset). %
Since these values are not separable in a general system, I'll keep $S$ normalized between 0 and
1. %
\begin{eqnarray}
S &=& (1-c)\me^{-\frac{t}{\tau_{\mathrm{actual}}}} + c \\
S_n &=& (1-c)\me^{-\frac{-\tau_{\mathrm{step}}\log{\left(\frac{n}{N}\right)}}{\tau_{\mathrm{actual}}}} + c \\
S_n &=& (1-c)\me^{-\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}} \log{\left(\frac{N}{n}\right)}} + c \\
S_n &=& (1-c)\left(\frac{N}{n}\right)^{-\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}}} + c \\
S_n &=& (1-c)\left(\frac{n}{N}\right)^{\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}}} + c
\end{eqnarray}
\begin{dfigure}[p!]
\includegraphics[scale=0.5]{"processing/PyCMDS/ideal axis positions/exponential"}
\caption[TODO]{TODO}
\label{fig:exponential_steps}
\end{dfigure}
\subsubsection{Gaussian}
\subsubsection{Lorentzian}
\subsubsection{Bimolecular}
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