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<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>
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