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  <title>Empirical Transfer Function estimation (LTPDA Toolbox)</title>
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  <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Empirical Transfer Function estimation</h1>
  <hr>
  
  <p>
	<p>
Let's run this exercise on empirical estimation of Transfer
Functions on the Matlab terminal.</p>

<p>The idea of the exercise is the following:
<ol>
<li>simulate some white noise <em>x(t)</em></li>
<li>build a band-pass filter <em>F</em></li>
<li>pass the input noise <em>x(t)</em> through the filter and add some more noise
 <em>yn(t)</em> at the output so to have <em>y = F*x(t) + yn(t)</em></li>
<li>evaluate and plot the transfer function <em>x -> y</em> </li>
</ol>
In a flow diagram, the representation is as follows:
</p>
<img src="images/ltpda_training_1/topic3/TFE_1_flowchart.png"
alt="Dataflow for the 1St example of ao/tfe" width="600px" border="1">

<p>The command-line sequence is the following:
<div class="fragment"><pre>
<span class="comment">%% General definitions</span>
nsecs = 10000;
fs = 1;
<span class="comment">%% Input noise</span>
x = ao(plist(<span class="string">'waveform'</span>, <span class="string">'noise'</span>, <span class="string">'sigma'</span>, 3, <span class="string">'fs'</span>, fs, <span class="string"> 'nsecs'</span>, nsecs, <span class="string">'yunits'</span>, <span class="string">'V'</span>))

<span class="comment">%% Filter</span> 
bp_filter = miir(plist(<span class="string">'type'</span>, <span class="string">'bandpass'</span>, <span class="string">'fc'</span>, [0.01 0.1], <span class="string">'fs'</span>, 1, <span class="string">'order'</span>, 3,<span class="string">'iunits'</span>, <span class="string">'V'</span>, <span class="string">'ounits'</span>, <span class="string">'A'</span>))
xf = simplifyYunits(filter(x, bp_filter))

<span class="comment">%% Output noise</span>
yn = ao(plist(<span class="string">'waveform'</span>,<span class="string">'noise'</span>, <span class="string">'sigma'</span>, 1, <span class="string">'fs'</span>, fs, <span class="string">'nsecs'</span>, xf.nsecs, <span class="string">'yunits'</span>, <span class="string">'A'</span>))
y = xf + yn

<span class="comment">%% Plotting input and output noise</span>
xx = psd(x, plist(<span class="string">'scale'</span>,...
                  <span class="string">'ASD'</span>,...
                  <span class="string">'nfft'</span>, 1000))
yy = psd(y, plist(<span class="string">'scale'</span>,<span class="string">'ASD'</span>, ...
                  <span class="string">'nfft'</span>, 1000))
iplot(xx, yy, plist(<span class="string">'Arrangement'</span>, <span class="string">'subplots'</span>, <span class="string">'YRanges'</span>, {[1e-1 1e1], [1e-2 1e2]}));

</pre></div>
<img src="images/ltpda_training_1/topic3/TFE_in_out.png" alt="Ohmic admittance between y and x" border="1">
<br>
<p>Now we can proceed with the call to the <tt>ao/tfe</tt> method. The
parameter list is very similar to the one employed for the other spectral
estimators:</p>

<table cellspacing="0" class="body" cellpadding="2" border="0" width="80%">
    <colgroup>
      <col width="15%"/>
      <col width="20%"/>
      <col width="65%"/>
    </colgroup>
    <thead>
      <tr valign="top">
        <th class="categorylist">Key</th>
        <th class="categorylist">Value</th>
        <th class="categorylist">Description</th>
      </tr>
    </thead>
    <tbody>
      <!-- Key 'NFFT' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>NFFT</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p><span class="string">1000</span></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>The number of samples defining the length of the window to apply</p>
        </td>
      </tr>
       <!-- Key 'WIN' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>WIN</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p><span class="string">'BH92'</span></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>Or a different one, if you want.</p>
        </td>
      </tr>
       <!-- Key 'OLAP' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>OLAP</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p><span class="string">-1</span></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>Overlap will be chosen based on the window properties</p>
        </td>
      </tr>
      <!-- Key 'ORDER' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>ORDER</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p><span class="string">0</span></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>Segment-wise detrending up to order 0</p>
        </td>
      </tr>      
    </tbody>
  </table>
</p>
  <p>The command line is the following:</p>
  <div class="fragment"><pre>
<span class="comment">%% Estimate the x->y transfer function</span>
tfxy = tfe(x, y, plist(<span class="string">'nfft'</span>, 1000, <span class="string">'win'</span>, <span class="string">'BH92'</span>, <span class="string">'olap'</span>, -1, <span class="string">'order'</span>, 0));
</pre></div>
<p>We also would like to evaluate the expected transfer function x->y, which is obviously the filter transfer function, or response. This can be calculated by means of the
  <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
    <tr width="90%">
      <td>
       <tt>miir/resp</tt>
      </td>
    </tr>
  </table> method. A detailed description of digital filtering is available in the User Manual dedicated <a href="sigproc_dfilt.html" >section</a> and will be touched upon in <a href="ltpda_training_topic_4_4_2.html" >this</a> topic; here let's just use the simplest form, where the needed parameter is a list of the frequency to evaluate the response at:</p>
 <table cellspacing="0" class="body" cellpadding="2" border="0" width="80%">
    <colgroup>
      <col width="15%"/>
      <col width="35%"/>
      <col width="50%"/>
    </colgroup>
    <thead>
      <tr valign="top">
        <th class="categorylist">Key</th>
        <th class="categorylist">Value</th>
        <th class="categorylist">Description</th>
      </tr>
    </thead>
    <tbody>
      <!-- Key 'f' -->
      <tr valign="top">
        <td bgcolor="#f3f4f5">
          <p><tt>F</tt></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p><span class="string">tfxy.x</span></p>
        </td>
        <td bgcolor="#f3f4f5">
          <p>a vector of frequency values or an <tt>ao</tt> whereby the x-axis is taken for the frequency values</p>
        </td>
      </tr>
    </tbody>
  </table>
  <p>
    So we can just pass the x field of the fsdata <tt>ao</tt> containing the transfer function estimate. However,
    we can also just pass the AO itself. In which case, the <tt>resp</tt> function will take the X values from 
    the AO.
  </p>
  <p>
  The command line is the following:</p>
  <div class="fragment"><pre>
<span class="comment">%% Evaluate the expected x->y transfer function</span>
rf = resp(bp_filter, plist(<span class="string">'f'</span>, tfxy))
  </pre></div>
  
  <p>Eventually let's look at the results:
  <div class="fragment"><pre>
<span class="comment">%% Plotting estimated and expected transfer functions</span>
iplot(tfxy, rf, plist(<span class="string">'colors'</span>,{[1 0 0],[0 0 0]},<span class="string">'YRanges'</span>, {[1e-2 1e2], [-200 200]}))
</pre></div>
  </p>
<img src="images/ltpda_training_1/topic3/TFE_result_1.png" alt="TFE of x into y" border="1">
<br>


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