From 49de5fe553c2066520162cccd7d5dd586efc934d Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Mon, 16 Oct 2017 21:05:48 -0500
Subject: structure

---
 .../PyCMDS/ideal axis positions/delay steps.tex    | 99 ++++++++++++++++++++++
 1 file changed, 99 insertions(+)
 create mode 100644 software/PyCMDS/ideal axis positions/delay steps.tex

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diff --git a/software/PyCMDS/ideal axis positions/delay steps.tex b/software/PyCMDS/ideal axis positions/delay steps.tex
new file mode 100644
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+% document
+\documentclass[11 pt]{report}
+\usepackage[letterpaper, margin=0.75in]{geometry}  % 1 inch margins required
+\usepackage{setspace}
+\usepackage{afterpage}
+\usepackage{color}
+\usepackage{soul}
+\usepackage{array}
+
+% text
+\usepackage[utf8]{inputenc}
+\setlength\parindent{0pt}
+\setlength{\parskip}{1em}
+\usepackage{enumitem}
+\renewcommand{\familydefault}{\sfdefault}
+\newcommand{\RomanNumeral}[1]{\textrm{\uppercase\expandafter{\romannumeral #1\relax}}}
+\usepackage{etoolbox}
+\AtBeginEnvironment{verse}{\singlespacing}
+\AtBeginEnvironment{tabular}{\singlespacing}
+
+% graphics
+\usepackage{graphics}
+\usepackage{graphicx}
+\usepackage{epsfig}
+\usepackage{epstopdf}
+\usepackage{etoc}
+\usepackage{tikz}
+
+% math
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage[cm]{sfmath}
+\DeclareMathOperator{\me}{e}
+
+% hyperref
+\usepackage[colorlinks=true, linkcolor=black, urlcolor=blue, citecolor=black, anchorcolor=black]{hyperref}
+\usepackage[all]{hypcap}  % helps hyperref work properly
+
+% date (http://tex.stackexchange.com/a/237251)
+\def\twodigits#1{\ifnum#1<10 0\fi\the#1}
+\def\mydate{\leavevmode\hbox{\the\year-\twodigits\month-\twodigits\day}}
+
+\begin{document}
+
+{\Huge{delay space stepping strategy}}
+
+Blaise Thompson \hfill last modified \mydate
+
+\dotfill
+
+Linear stepping is more expensive than it needs to be.
+
+Want to capture the dynamic range of the data as quickly as possible.
+
+Typically have exponential decay dynamics (perhaps multi-exponential)\dots we can capitalize on this. We want to take high resolution data at early delays and low resolution data at late delays.
+
+Of course, we don't want to throw away any information we would otherwise be entitled to.
+
+Conceptually we want to 'linearize' the data, so that each subsequent delay step accounts for the same change in signal.
+
+Signal goes exponentially...
+
+\begin{eqnarray}
+S &=& \me^{-\frac{t}{\tau}} \\
+\log{(S)} &=& -\frac{t}{\tau} \\
+t &=& -\tau\log{(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{figure}[!htb]
+	\centering
+	\includegraphics[scale=0.5]{"out"}
+	\caption{}
+\end{figure}
+
+
+
+\end{document}
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