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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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  <title>IFO/Temperature Example - Simulation (LTPDA Toolbox)</title>
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    <tr>
      <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>

      <td valign="baseline" align="right"><a href=
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  <h1 class="title"><a name="f3-12899" id="f3-12899"></a>IFO/Temperature Example - Simulation</h1>
  <hr>
  
  <p>
	<p>
  We now come back to the IFO/Temperature working example. Our interest here is not
  to add more tools to the data analysis chain that you've been developing but
  to create a simple toy model that allows you to reproduce the steps done up
  to now but with synthetic data.
</p>
<p>
  The problem is shown schematically in the figure below. We will generate temperature and interferometer
  noisy data by applying a filter (<b>TMP</b> and <b>IFO</b>) to white noise data
  (<b>WN1</b> and <b>WN2</b>). We will then add a temperature coupling (<b>K2RAD</b>) to the
  interferometer and from the measured interferometer and temperature data we will then estimate
  the temperature to interferometer coupling.
</p>

<p  align="center">
  <img src="images/ltpda_training_1/topic4/IFO2TSimulation.png" alt="temperature substraction scheme" border="1" >
</p>
<p>
  Since we've been through the same steps that you need to apply here in the previous
  section we will give here the step by step description of the task and let you
  play with the models.
</p>

<h2> Build the models<h2>
<p>
  We need three models: one to generate temperature-like data, another modelling the
  interferometer and a third one acting as the K-to-rad transfer function.
</p>

<h3> STEP 1: Build a temperature noise PZMODEL with the following properties<h4>

<table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
  <colgroup>
    <col width="15%"/>
    <col width="35%"/>
  </colgroup>
  <thead>
    <tr valign="top">
      <th class="categorylist">Key</th>
      <th class="categorylist">Value</th>
    </tr>
  </thead>
  <tbody>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'name'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'TMP'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'ounits'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'K'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt>GAIN</tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        10
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
    <td bgcolor="#f3f4f5">
      <p><tt>POLE 1</tt></p>
    </td>
    <td bgcolor="#f3f4f5">
      1e-5
    </td>
  </tbody>
</table>
<p>
  For example, this few lines would do the job
</p>
<div class="fragment"><pre>
    TMP = pzmodel(10,1e-5,[]);
    TMP.setOunits(<span class="string">'K'</span>)
    TMP.setName(<span class="string">'TMP'</span>)
</pre></div>

<h3> STEP 2: Build a interferometer noise PZMODEL with the following properties<h4>

<table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
  <colgroup>
    <col width="15%"/>
    <col width="35%"/>
  </colgroup>
  <thead>
    <tr valign="top">
      <th class="categorylist">Key</th>
      <th class="categorylist">Value</th>
    </tr>
  </thead>
  <tbody>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'name'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'IFO'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'ounits'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'rad'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt>GAIN</tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        1e-3
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt>POLE 1</tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        0.4
      </td>
    </tr>
  </tbody>
</table>


<h3> STEP 3: Build temperature to interferometer coupling PZMODEL with the following properties<h4>

<table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
  <colgroup>
    <col width="15%"/>
    <col width="35%"/>
  </colgroup>
  <thead>
    <tr valign="top">
      <th class="categorylist">Key</th>
      <th class="categorylist">Value</th>
    </tr>
  </thead>
  <tbody>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'name'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'K2RAD'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'iunits'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'K'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'ounits'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'rad'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt>GAIN</tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        1e-1
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt>POLE 1</tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        5e-4
      </td>
    </tr>
  </tbody>
</table>

<p>
  You can take a look at your models. Since we are interested in the projection of temperature
  into interferometric data, we can plot the response of TMP*K2RAD against the IFO
</p>
<div class="fragment"><pre>
    pl = plist(<span class="string">'f1'</span>,1e-5,<span class="string">'f2'</span>,0.01)
    resp(K2RAD*TMP,IFO,pl)
</pre></div>
<p>
  <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_1.png" alt="inteferometer data" border="1" >
</p>

<h2>Discretize the models<h2>
<p>
  Now discretize the models at fs = 1Hz using the <tt>miir</tt> constructor.
  After that you will obtain three digital filters
</p>
<h3> STEP 4: Discretize the three transfer (<b>TMP</b>,<b>IFO</b>,<b>K2RAD</b>) with the
MIIR constructor<h3>
<p>
  For example, the model related to temperature noise would be discretized like this:
</p>
<div class="fragment"><pre>
    TMPd = miir(TMP,plist(<span class="string">'fs'</span>,1));
</pre></div>
<h2> Generate white noise data<h2>
<p>
  We will need two initial white noise time series, <b>WN1</b> and <b>WN2</b>, that we will use as a seed
  to apply our filters and get noise shaped time series.
</p>
<h3> STEP 5: Generate white noise with the AO constructor<h3>
<p>
  You will need the ao constructor for that. You could use the following settings
</p>
<table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
  <colgroup>
    <col width="15%"/>
    <col width="35%"/>
  </colgroup>
  <thead>
    <tr valign="top">
      <th class="categorylist">Key</th>
      <th class="categorylist">Value</th>
    </tr>
  </thead>
  <tbody>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'name'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'WN1'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'tsfcn'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'randn(size(t))'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'fs'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        1
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'nsecs'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        250000
      </td>
    </tr>
  </tbody>
</table>

<table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
  <colgroup>
    <col width="15%"/>
    <col width="35%"/>
  </colgroup>
  <thead>
    <tr valign="top">
      <th class="categorylist">Key</th>
      <th class="categorylist">Value</th>
    </tr>
  </thead>
  <tbody>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'name'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'WN2'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'tsfcn'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        <span class="string">'randn(size(t))'</span>
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'fs'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        1
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'nsecs'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        250000
      </td>
    </tr>
  </tbody>
</table>


<h2>Generate the noise data streams<h2>
<p>
  For each noise source you will need to apply the filter that you have designed to
  the white noise data:
</p>

<h3> STEP 6: Filter white noise WN1 with the TMPd filter </h3>
<p>
  For example, following our notation:
</p>
<div class="fragment"><pre>
    T = filter(WN1,TMPd);
</pre></div>
<h3> STEP 7: Filter white noise WN2 with the IFOd filter</h3>
<p>
  Temperature and interferometric noise are uncorrelated, so we need to use here the
  second noise time series <b>WN2</b>
</p>
<p>
  For example, following our notation:
</p>
<div class="fragment"><pre>
    IFO = filter(WN2,IFOd);
</pre></div>
<h3> STEP 8: Filter white noise WN1 with the TMPd and the K2RADd filter</h3>
<p>
  In this case you need to apply both filters in serial, you can do this in one command
  by using the 'bank' property of the <tt>filter</tt> method.
</p>
<p>
  <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
    <tr width="90%">
      <td>
        Hint: you can input a vector of filters into the <tt>filter</tt> method and
        ask it to filter the data in 'parallel' or in 'serial' (the one we are interested
        here) by doing the following
        <div class="fragment"><pre>
            b = filter(<b>WN1</b>,[<b>TMPd</b> <b>K2RADd</b>],plist(<span class="string">'bank'</span>,<span class="string">'serial'</span>));
        </pre></div>

      </td>
    </tr>
  </table>
</p>
<h3> STEP 9: Add the IFO noise to the K2RAD noise</h3>
<p>
  At this point the IFO represents the purely interferometric noise and the K2RAD the
  contribution to interferometric noise coming from temperature. You need to add both to
  get the final interferometric data. This only requires to add both AOs.
</p>

<h2>Perform the noise projection<h3>
<p>
  Here we will reproduce the main steps performed in topic 3 analysis: power spectral and
  transfer function estimation.
</p>

<h3> STEP 10: Split the data streams  </h3>
<p>
  We will do the analysis with data in the region going from 1e5 to 2e5 seconds to avoid
  initial transients. You must then split your two data streams introducing the following
  parameters in the <tt>split</tt> method.
</p>

<table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
  <colgroup>
    <col width="15%"/>
    <col width="35%"/>
  </colgroup>
  <thead>
    <tr valign="top">
      <th class="categorylist">Key</th>
      <th class="categorylist">Value</th>
    </tr>
  </thead>
  <tbody>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt><span class="string">'times'</span></tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        [1e5 2e5]
      </td>
    </tr>
  </tbody>
</table>
<p>
After the splitting you must have two data streams that plot together should look like the
ones below. The code should be similar to one in the following lines

<div class="fragment"><pre>
    Ts = split(T,plist(<span class="string">'times'</span>,[1e5 2e5]))
    Ts.setName(<span class="string">'Temperature'</span>)
    IFOs = split(IFO_all,plist(<span class="string">'times'</span>,[1e5 2e5]))
    IFOs.setName(<span class="string">'Interferometer'</span>)
    <span class="comment">% Plot in different panels with 'subplots' options</span>
    iplot(Ts,IFOs,plist(<span class="string">'arrangement'</span>,<span class="string">'subplots'</span>))
</pre></div>

<p>
  <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_3.png" alt="inteferometer data" border="1" >
</p>

<h3> STEP 11: Compute power spectral estimates for the temperature and interferometric data </h3>
<p>
  Here you need to apply <tt>lpsd</tt> or <tt>psd</tt> methods. For example:
</p>
<div class="fragment"><pre>
    pl = plist(<span class="string">'order'</span>,1,<span class="string">'scale'</span>,<span class="string">'ASD'</span>)
    psd_T = lpsd(Ts,pl)
    psd_IFO = lpsd(IFOs,pl)
</pre></div>
<p>
  The resulting spectrum should look like this
</p>
<p>
  <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_4.png" alt="inteferometer data" width="800px" border="1" >
</p>

<h3> STEP 12: Compute transfer function estimate for the temperature and interferometric data </h3>
<p>
  Here you need to apply <tt>ltfe</tt> or <tt>tfe</tt> methods. For example:
</p>
<div class="fragment"><pre>
    T2IFO = ltfe(Ts,IFOs)
    T2IFO.setName(<span class="string">'Transfer function'</span>)
</pre></div>
<p>
  You can now compare the transfer function model with the estimation obtained from the data:
</p>
<div class="fragment"><pre>
    pl = plist(<span class="string">'f1'</span>,1e-5,<span class="string">'f2'</span>,1)
    iplot(T2IFO,resp(K2RAD,pl))
</pre></div>
<p>
  <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_5.png" alt="inteferometer data" width="800px" border="1" >
</p>
<h3> STEP 13: Project the temperature noise </h3>
<p>
  Reproducing the analysis performed in topic 3 you will be able to project the temperature noise contribution
  into interferometric noise. The result obtained should be the one in the figure below and the code you will need
  for that should be similar to this:
</p>

<div class="fragment"><pre>
    <span class="comment">% Compute projection</span>
    Projection = abs(T2IFO).*psd_T;
    Projection.simplifyYunits
    Projection.setName;
    <span class="comment">% Plot against interferometer noise</span>
    iplot(psd_IFO,Projection)
</pre></div>

<p>
  <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_6.png" alt="inteferometer data" width="800px" border="1" >
</p>

















  </p>

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