From 9d89c09dfe49aba4c68b6911600715add419babd Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Tue, 27 Feb 2018 23:58:32 -0600
Subject: 2018-02-27 23:58

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

(limited to 'software/PyCMDS/ideal axis positions/delay steps.tex')

diff --git a/software/PyCMDS/ideal axis positions/delay steps.tex b/software/PyCMDS/ideal axis positions/delay steps.tex
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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}
\ No newline at end of file
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