tdehaeze/spectral-analysis
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity
37c44ee0d9
spectral-analysis/index.html
Thomas Dehaeze 37c44ee0d9 Initial Commit
2019-08-17 10:51:53 +02:00

583 lines
25 KiB
HTML
Raw Blame History

<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2019-08-15 jeu. 12:31 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Compute Spectral Densities of signals with Matlab</title>
<meta name="generator" content="Org mode" />
<meta name="author" content="Dehaeze Thomas" />
<style type="text/css">
<!--/*--><![CDATA[/*><!--*/
.title { text-align: center;
margin-bottom: .2em; }
.subtitle { text-align: center;
font-size: medium;
font-weight: bold;
margin-top:0; }
.todo { font-family: monospace; color: red; }
.done { font-family: monospace; color: green; }
.priority { font-family: monospace; color: orange; }
.tag { background-color: #eee; font-family: monospace;
padding: 2px; font-size: 80%; font-weight: normal; }
.timestamp { color: #bebebe; }
.timestamp-kwd { color: #5f9ea0; }
.org-right { margin-left: auto; margin-right: 0px; text-align: right; }
.org-left { margin-left: 0px; margin-right: auto; text-align: left; }
.org-center { margin-left: auto; margin-right: auto; text-align: center; }
.underline { text-decoration: underline; }
#postamble p, #preamble p { font-size: 90%; margin: .2em; }
p.verse { margin-left: 3%; }
pre {
border: 1px solid #ccc;
box-shadow: 3px 3px 3px #eee;
padding: 8pt;
font-family: monospace;
overflow: auto;
margin: 1.2em;
}
pre.src {
position: relative;
overflow: visible;
padding-top: 1.2em;
}
pre.src:before {
display: none;
position: absolute;
background-color: white;
top: -10px;
right: 10px;
padding: 3px;
border: 1px solid black;
}
pre.src:hover:before { display: inline;}
/* Languages per Org manual */
pre.src-asymptote:before { content: 'Asymptote'; }
pre.src-awk:before { content: 'Awk'; }
pre.src-C:before { content: 'C'; }
/* pre.src-C++ doesn't work in CSS */
pre.src-clojure:before { content: 'Clojure'; }
pre.src-css:before { content: 'CSS'; }
pre.src-D:before { content: 'D'; }
pre.src-ditaa:before { content: 'ditaa'; }
pre.src-dot:before { content: 'Graphviz'; }
pre.src-calc:before { content: 'Emacs Calc'; }
pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
pre.src-fortran:before { content: 'Fortran'; }
pre.src-gnuplot:before { content: 'gnuplot'; }
pre.src-haskell:before { content: 'Haskell'; }
pre.src-hledger:before { content: 'hledger'; }
pre.src-java:before { content: 'Java'; }
pre.src-js:before { content: 'Javascript'; }
pre.src-latex:before { content: 'LaTeX'; }
pre.src-ledger:before { content: 'Ledger'; }
pre.src-lisp:before { content: 'Lisp'; }
pre.src-lilypond:before { content: 'Lilypond'; }
pre.src-lua:before { content: 'Lua'; }
pre.src-matlab:before { content: 'MATLAB'; }
pre.src-mscgen:before { content: 'Mscgen'; }
pre.src-ocaml:before { content: 'Objective Caml'; }
pre.src-octave:before { content: 'Octave'; }
pre.src-org:before { content: 'Org mode'; }
pre.src-oz:before { content: 'OZ'; }
pre.src-plantuml:before { content: 'Plantuml'; }
pre.src-processing:before { content: 'Processing.js'; }
pre.src-python:before { content: 'Python'; }
pre.src-R:before { content: 'R'; }
pre.src-ruby:before { content: 'Ruby'; }
pre.src-sass:before { content: 'Sass'; }
pre.src-scheme:before { content: 'Scheme'; }
pre.src-screen:before { content: 'Gnu Screen'; }
pre.src-sed:before { content: 'Sed'; }
pre.src-sh:before { content: 'shell'; }
pre.src-sql:before { content: 'SQL'; }
pre.src-sqlite:before { content: 'SQLite'; }
/* additional languages in org.el's org-babel-load-languages alist */
pre.src-forth:before { content: 'Forth'; }
pre.src-io:before { content: 'IO'; }
pre.src-J:before { content: 'J'; }
pre.src-makefile:before { content: 'Makefile'; }
pre.src-maxima:before { content: 'Maxima'; }
pre.src-perl:before { content: 'Perl'; }
pre.src-picolisp:before { content: 'Pico Lisp'; }
pre.src-scala:before { content: 'Scala'; }
pre.src-shell:before { content: 'Shell Script'; }
pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
/* additional language identifiers per "defun org-babel-execute"
in ob-*.el */
pre.src-cpp:before { content: 'C++'; }
pre.src-abc:before { content: 'ABC'; }
pre.src-coq:before { content: 'Coq'; }
pre.src-groovy:before { content: 'Groovy'; }
/* additional language identifiers from org-babel-shell-names in
ob-shell.el: ob-shell is the only babel language using a lambda to put
the execution function name together. */
pre.src-bash:before { content: 'bash'; }
pre.src-csh:before { content: 'csh'; }
pre.src-ash:before { content: 'ash'; }
pre.src-dash:before { content: 'dash'; }
pre.src-ksh:before { content: 'ksh'; }
pre.src-mksh:before { content: 'mksh'; }
pre.src-posh:before { content: 'posh'; }
/* Additional Emacs modes also supported by the LaTeX listings package */
pre.src-ada:before { content: 'Ada'; }
pre.src-asm:before { content: 'Assembler'; }
pre.src-caml:before { content: 'Caml'; }
pre.src-delphi:before { content: 'Delphi'; }
pre.src-html:before { content: 'HTML'; }
pre.src-idl:before { content: 'IDL'; }
pre.src-mercury:before { content: 'Mercury'; }
pre.src-metapost:before { content: 'MetaPost'; }
pre.src-modula-2:before { content: 'Modula-2'; }
pre.src-pascal:before { content: 'Pascal'; }
pre.src-ps:before { content: 'PostScript'; }
pre.src-prolog:before { content: 'Prolog'; }
pre.src-simula:before { content: 'Simula'; }
pre.src-tcl:before { content: 'tcl'; }
pre.src-tex:before { content: 'TeX'; }
pre.src-plain-tex:before { content: 'Plain TeX'; }
pre.src-verilog:before { content: 'Verilog'; }
pre.src-vhdl:before { content: 'VHDL'; }
pre.src-xml:before { content: 'XML'; }
pre.src-nxml:before { content: 'XML'; }
/* add a generic configuration mode; LaTeX export needs an additional
(add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
pre.src-conf:before { content: 'Configuration File'; }
table { border-collapse:collapse; }
caption.t-above { caption-side: top; }
caption.t-bottom { caption-side: bottom; }
td, th { vertical-align:top; }
th.org-right { text-align: center; }
th.org-left { text-align: center; }
th.org-center { text-align: center; }
td.org-right { text-align: right; }
td.org-left { text-align: left; }
td.org-center { text-align: center; }
dt { font-weight: bold; }
.footpara { display: inline; }
.footdef { margin-bottom: 1em; }
.figure { padding: 1em; }
.figure p { text-align: center; }
.equation-container {
display: table;
text-align: center;
width: 100%;
}
.equation {
vertical-align: middle;
}
.equation-label {
display: table-cell;
text-align: right;
vertical-align: middle;
}
.inlinetask {
padding: 10px;
border: 2px solid gray;
margin: 10px;
background: #ffffcc;
}
#org-div-home-and-up
{ text-align: right; font-size: 70%; white-space: nowrap; }
textarea { overflow-x: auto; }
.linenr { font-size: smaller }
.code-highlighted { background-color: #ffff00; }
.org-info-js_info-navigation { border-style: none; }
#org-info-js_console-label
{ font-size: 10px; font-weight: bold; white-space: nowrap; }
.org-info-js_search-highlight
{ background-color: #ffff00; color: #000000; font-weight: bold; }
.org-svg { width: 90%; }
/*]]>*/-->
</style>
<link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
<link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
<link rel="stylesheet" type="text/css" href="./css/zenburn.css"/>
<script type="text/javascript" src="./js/jquery.min.js"></script>
<script type="text/javascript" src="./js/bootstrap.min.js"></script>
<script type="text/javascript" src="./js/jquery.stickytableheaders.min.js"></script>
<script type="text/javascript" src="./js/readtheorg.js"></script>
<script type="text/javascript">
/*
@licstart The following is the entire license notice for the
JavaScript code in this tag.
Copyright (C) 2012-2019 Free Software Foundation, Inc.
The JavaScript code in this tag is free software: you can
redistribute it and/or modify it under the terms of the GNU
General Public License (GNU GPL) as published by the Free Software
Foundation, either version 3 of the License, or (at your option)
any later version. The code is distributed WITHOUT ANY WARRANTY;
without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.
As additional permission under GNU GPL version 3 section 7, you
may distribute non-source (e.g., minimized or compacted) forms of
that code without the copy of the GNU GPL normally required by
section 4, provided you include this license notice and a URL
through which recipients can access the Corresponding Source.
@licend The above is the entire license notice
for the JavaScript code in this tag.
*/
<!--/*--><![CDATA[/*><!--*/
function CodeHighlightOn(elem, id)
{
var target = document.getElementById(id);
if(null != target) {
elem.cacheClassElem = elem.className;
elem.cacheClassTarget = target.className;
target.className = "code-highlighted";
elem.className = "code-highlighted";
}
}
function CodeHighlightOff(elem, id)
{
var target = document.getElementById(id);
if(elem.cacheClassElem)
elem.className = elem.cacheClassElem;
if(elem.cacheClassTarget)
target.className = elem.cacheClassTarget;
}
/*]]>*///-->
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
displayAlign: "center",
displayIndent: "0em",
"HTML-CSS": { scale: 100,
linebreaks: { automatic: "false" },
webFont: "TeX"
},
SVG: {scale: 100,
linebreaks: { automatic: "false" },
font: "TeX"},
NativeMML: {scale: 100},
TeX: { equationNumbers: {autoNumber: "AMS"},
MultLineWidth: "85%",
TagSide: "right",
TagIndent: ".8em"
}
});
</script>
<script type="text/javascript"
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"></script>
</head>
<body>
<div id="org-div-home-and-up">
<a accesskey="h" href="./index.html"> UP </a>
|
<a accesskey="H" href="./index.html"> HOME </a>
</div><div id="content">
<h1 class="title">Compute Spectral Densities of signals with Matlab</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org7cc41f2">1. Sensitivity of the instrumentation</a></li>
<li><a href="#orgac6792b">2. Convert the time domain from volts to velocity</a></li>
<li><a href="#org01a363c">3. Power Spectral Density and Amplitude Spectral Density</a></li>
<li><a href="#org90edcdd">4. Modification of a signal's Power Spectral Density when going through an LTI system</a></li>
<li><a href="#orgfb2734c">5. From PSD of the velocity to the PSD of the displacement</a></li>
</ul>
</div>
</div>
<p>
This document presents the mathematics as well as the matlab scripts to do the spectral analysis of a measured signal.
</p>
<p>
Typically this signal is coming from an inertial sensor, a force sensor or any other sensor.
</p>
<p>
We here take the example of a signal coming from a Geophone measurement the vertical velocity of the floor at the ESRF.
</p>
<div id="outline-container-org7cc41f2" class="outline-2">
<h2 id="org7cc41f2"><span class="section-number-2">1</span> Sensitivity of the instrumentation</h2>
<div class="outline-text-2" id="text-1">
<p>
The measured signal \(x\) by the ADC is in Volts.
The corresponding real velocity \(v\) in m/s.
</p>
<p>
To obtain the real quantity as measured by the sensor, one have to know the sensitivity of the sensors and electronics used.
</p>
<div id="org0a5ff56" class="figure">
<p><img src="figs/velocity_to_voltage.png" alt="velocity_to_voltage.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the instrumentation used for the measurement</p>
</div>
</div>
</div>
<div id="outline-container-orgac6792b" class="outline-2">
<h2 id="orgac6792b"><span class="section-number-2">2</span> Convert the time domain from volts to velocity</h2>
<div class="outline-text-2" id="text-2">
<p>
Let's say, we know that the sensitivity of the geophone used is
\[ G_g(s) = G_0 \frac{\frac{s}{2\pi f_0}}{1 + \frac{s}{2\pi f_0}} \quad \left[\frac{V}{m/s}\right] \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">G0 = <span class="org-highlight-numbers-number">88</span>; <span class="org-comment">% Sensitivity [V/(m/s)]</span>
f0 = <span class="org-highlight-numbers-number">2</span>; <span class="org-comment">% Cut-off frequency [Hz]</span>
Gg = G0<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>f0<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">+</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>f0<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
And the gain of the amplifier is 1000: \(G_m(s) = 1000\).
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gm = <span class="org-highlight-numbers-number">1000</span>;
</pre>
</div>
<p>
If \({G_m(s)}^{-1} {G_g(s)}^{-1}\) is proper, we can simulate this dynamical system to go from the voltage to the velocity units (figure <a href="#orgd3dfdc8">2</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab">data = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_028.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; data = data.data;
t = data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [s]</span>
x = data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">-</span>mean<span class="org-rainbow-delimiters-depth-1">(</span>data<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% The offset if removed (coming from the voltage amplifier) [v]</span>
dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">-</span>t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>; Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt;
</pre>
</div>
<div id="orgd3dfdc8" class="figure">
<p><img src="figs/voltage_to_velocity.png" alt="voltage_to_velocity.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Schematic of the instrumentation used for the measurement</p>
</div>
<p>
We simulate this system with matlab:
</p>
<div class="org-src-container">
<pre class="src src-matlab">v = lsim<span class="org-rainbow-delimiters-depth-1">(</span>inv<span class="org-rainbow-delimiters-depth-2">(</span>Gg<span class="org-type">*</span>Gm<span class="org-rainbow-delimiters-depth-2">)</span>, v, t<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
And we plot the obtained velocity
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>t, v<span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">"Time [s]"</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">"Velocity [m/s]"</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div id="org4dd86c0" class="figure">
<p><img src="figs/velocity_time.png" alt="velocity_time.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Measured Velocity</p>
</div>
</div>
</div>
<div id="outline-container-org01a363c" class="outline-2">
<h2 id="org01a363c"><span class="section-number-2">3</span> Power Spectral Density and Amplitude Spectral Density</h2>
<div class="outline-text-2" id="text-3">
<p>
We now have the velocity in the time domain:
\[ v(t)\ [m/s] \]
</p>
<p>
To compute the Power Spectral Density (PSD):
\[ S_v(f)\ \left[\frac{(m/s)^2}{Hz}\right] \]
</p>
<p>
To compute that with matlab, we use the <code>pwelch</code> function.
</p>
<p>
We first have to defined a window:
</p>
<div class="org-src-container">
<pre class="src src-matlab">win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">10</span><span class="org-type">*</span>Fs<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; % <span class="org-highlight-numbers-number">10s</span> window
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>Sv, f<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>v, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
loglog<span class="org-rainbow-delimiters-depth-1">(</span>f, Sv<span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylabel<span class="org-rainbow-delimiters-depth-1">(</span>'Power Spectral Density $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span><span class="org-rainbow-delimiters-depth-4">(</span>m<span class="org-type">/</span>s<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">}{</span>Hz<span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
<p>
The Amplitude Spectral Density (ASD) is the square root of the Power Spectral Density:
</p>
\begin{equation}
\Gamma_{vv}(f) = \sqrt{S_{vv}(f)} \quad \left[ \frac{m/s}{\sqrt{Hz}} \right]
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
loglog<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>Sv<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylabel<span class="org-rainbow-delimiters-depth-1">(</span>'Amplitude Spectral Density $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>m<span class="org-type">/</span>s<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org90edcdd" class="outline-2">
<h2 id="org90edcdd"><span class="section-number-2">4</span> Modification of a signal's Power Spectral Density when going through an LTI system</h2>
<div class="outline-text-2" id="text-4">
<div id="org9eca4b8" class="figure">
<p><img src="figs/velocity_to_voltage_psd.png" alt="velocity_to_voltage_psd.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Schematic of the instrumentation used for the measurement</p>
</div>
<p>
We can show that:
</p>
\begin{equation}
S_{yy}(\omega) = \left|G(j\omega)\right|^2 S_{xx}(\omega)
\end{equation}
<p>
And we also have:
</p>
\begin{equation}
\Gamma_{yy}(\omega) = \left|G(j\omega)\right| \Gamma_{xx}(\omega)
\end{equation}
</div>
</div>
<div id="outline-container-orgfb2734c" class="outline-2">
<h2 id="orgfb2734c"><span class="section-number-2">5</span> From PSD of the velocity to the PSD of the displacement</h2>
<div class="outline-text-2" id="text-5">
<div id="orga1a0bc5" class="figure">
<p><img src="figs/velocity_to_displacement_psd.png" alt="velocity_to_displacement_psd.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Schematic of the instrumentation used for the measurement</p>
</div>
<p>
The displacement is the integral of the velocity.
</p>
<p>
We then have that
</p>
\begin{equation}
S_{xx}(\omega) = \left|\frac{1}{j \omega}\right|^2 S_{vv}(\omega)
\end{equation}
<p>
Using a frequency variable in Hz:
</p>
\begin{equation}
S_{xx}(f) = \left| \frac{1}{j 2\pi f} \right|^2 S_{vv}(f)
\end{equation}
<p>
For the Amplitude Spectral Density:
</p>
\begin{equation}
\Gamma_{xx}(f) = \frac{1}{2\pi f} \Gamma_{vv}(f)
\end{equation}
<div class="note">
\begin{equation}
S_{xx}(\omega = 1) = S_{vv}(\omega = 1)
\end{equation}
</div>
<p>
Now if we want to obtain the Power Spectral Density of the Position or Acceleration:
For each frequency:
\[ \left| \frac{d sin(2 \pi f t)}{dt} \right| = | 2 \pi f | \times | \cos(2\pi f t) | \]
</p>
<p>
\[ \left| \int_0^t sin(2 \pi f \tau) d\tau \right| = \left| \frac{1}{2 \pi f} \right| \times | \cos(2\pi f t) | \]
</p>
<p>
\[ ASD_x(f) = \frac{1}{2\pi f} ASD_v(f) \ \left[\frac{m}{\sqrt{Hz}}\right] \]
</p>
<p>
\[ ASD_a(f) = 2\pi f ASD_v(f) \ \left[\frac{m/s^2}{\sqrt{Hz}}\right] \]
And we have
\[ PSD_x(f) = {ASD_x(f)}^2 = \frac{1}{(2 \pi f)^2} {ASD_v(f)}^2 = \frac{1}{(2 \pi f)^2} PSD_v(f) \]
</p>
<p>
Note here that we always have
\[ PSD_x \left(f = \frac{1}{2\pi}\right) = PSD_v \left(f = \frac{1}{2\pi}\right) = PSD_a \left(f = \frac{1}{2\pi}\right), \quad \frac{1}{2\pi} \approx 0.16 [Hz] \]
</p>
<p>
If we want to compute the Cumulative Power Spectrum:
\[ CPS_v(f) = \int_0^f PSD_v(\nu) d\nu \quad [(m/s)^2] \]
</p>
<p>
We can also want to integrate from high frequency to low frequency:
\[ CPS_v(f) = \int_f^\infty PSD_v(\nu) d\nu \quad [(m/s)^2] \]
</p>
<p>
The Cumulative Amplitude Spectrum is then the square root of the Cumulative Power Spectrum:
\[ CAS_v(f) = \sqrt{CPS_v(f)} = \sqrt{\int_f^\infty PSD_v(\nu) d\nu} \quad [m/s] \]
</p>
<p>
Then, we can obtain the Root Mean Square value of the velocity:
\[ v_{\text{rms}} = CAS_v(0) \quad [m/s \ \text{rms}] \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">
</pre>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-08-15 jeu. 12:31</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink