2018-02-27 23:58

4 files changed, 182 insertions, 0 deletions

diff --git a/processing/PyCMDS/ideal axis positions/delay steps.pdf b/processing/PyCMDS/ideal axis positions/delay steps.pdf
new file mode 100644
index 0000000..1472e0c
--- /dev/null
+++ b/processing/PyCMDS/ideal axis positions/delay steps.pdf
diff --git a/processing/PyCMDS/ideal axis positions/delay steps.tex b/processing/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
diff --git a/processing/PyCMDS/ideal axis positions/exponential.png b/processing/PyCMDS/ideal axis positions/exponential.png
new file mode 100644
index 0000000..7ad27f3
--- /dev/null
+++ b/processing/PyCMDS/ideal axis positions/exponential.png
diff --git a/processing/PyCMDS/ideal axis positions/steps.py b/processing/PyCMDS/ideal axis positions/steps.py
new file mode 100644
index 0000000..13419c3
--- /dev/null
+++ b/processing/PyCMDS/ideal axis positions/steps.py
@@ -0,0 +1,83 @@
+### import ####################################################################

+

+

+import matplotlib.pyplot as plt

+plt.close('all')

+

+import numpy as np

+

+import WrightTools as wt

+

+

+### define ####################################################################

+

+

+def get_signal(d, tau, pulsewidth=10, offset=0):

+    # pulse

+    pulse = np.exp((-d**2)/(pulsewidth**2))

+    # signal

+    sig = np.zeros(d.shape)

+    sig[d<=0] = np.exp(d[d<=0]/tau)

+    sig[d<=0] += offset

+    sig /= sig.max()

+    # finish

+    #sig = np.convolve(sig, pulse, mode='same')

+    return sig

+

+

+def logarithmic_stepping(p_tau, p_npts, n_tau, n_npts):

+    # positive

+    p_xi = np.arange(0, p_npts)

+    p_delays = p_tau * np.log((p_xi.size+1)/(p_xi+1))

+    # negative

+    n_xi = np.arange(0, n_npts)

+    n_delays = -n_tau * np.log((n_xi.size+1)/(n_xi+1))

+    return np.hstack((n_delays, [0], p_delays))

+

+

+tau = 200

+

+

+d = logarithmic_stepping(50, 3, 200, 15)

+

+

+### workspace #################################################################

+

+

+if True:

+    fig, gs = wt.artists.create_figure(width=13, cols=[1, 1], nrows=1)

+    # delay space

+    ax = plt.subplot(gs[0, 0]) 

+    ds = np.linspace(-1500, 1500, 1000)

+    sig = get_signal(ds, tau)

+    plt.plot(ds, sig, c='b', lw=2, alpha=0.5)

+    sig = get_signal(ds, tau, offset=0.5)

+    plt.plot(ds, sig, c='r', lw=2, alpha=0.5)

+    sig = get_signal(ds, tau*2, offset=0)

+    plt.plot(ds, sig, c='g', lw=2, alpha=0.5)

+    plt.xlim(-1250, 100)

+    plt.ylim(-0.1, 1.1)

+    for x in d:

+        plt.axvline(x, c='k', zorder=0)

+    plt.axvline(0, lw=3, c='k')

+    ax.set_xlabel('delay', fontsize=18)

+    ax.set_ylabel('signal', fontsize=18)

+    plt.grid(ls=':')

+    # index space

+    ax = plt.subplot(gs[0, 1])

+    d = logarithmic_stepping(50, 3, 200, 15)

+    sig = get_signal(d, tau)

+    plt.scatter(np.arange(sig.size), sig, c='b', edgecolor='none', s=50, alpha=0.5)

+    sig = get_signal(d, tau, offset=0.5)

+    plt.scatter(np.arange(sig.size), sig, c='r', edgecolor='none', s=50, alpha=0.5)

+    sig = get_signal(d, tau*2, offset=0)

+    plt.scatter(np.arange(sig.size), sig, c='g', edgecolor='none', s=50, alpha=0.5) 

+    i = np.argmin(np.abs(d))

+    plt.axvline(i, lw=3, c='k')

+    plt.grid(ls=':')

+    plt.ylim(-0.1, 1.1)

+    plt.setp(ax.get_yticklabels(), visible=False)

+    ax.set_xlim(0-1, sig.size)

+    ax.set_xlabel('index', fontsize=18)

+    # finish

+    wt.artists.savefig('exponential.png')