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  <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Generation of noise with given PSD</h1>
  <hr>
  
  <p>
	


<p>
  Generation of model noise is performed with the function <tt>ao/noisegen1D</tt>.
  Details on the algorithm can be found in <a href="ng1D.html">noisegen1D help page</a>.
</p>

<h2> Generation of noise with given PSD</h2>

<p>
  During this exercise we will:
  <ol>
    <li> Load from file an fsdata object with the model (obtained with a fit to the the PSD of test data with zDomainFit)
    <li> Genarate noise from this model
    <li> Compare PSD of the generated noise with original PSD
  </ol>
</p>

<p>
  Let's open a new editor window and load the test data.
</p>

<div class="fragment"><pre>
    tn = ao(plist(<span class="string">'filename'</span>, <span class="string">'topic5/T5_Ex03_TestNoise.xml'</span>));
    tn.setName;
</pre></div>

<p>
  This command will load an Analysis Object containing a test time series
  10000 seconds long, sampled at 1 Hz. The command <tt>setName</tt> sets the name
  of the AO to be the same as the variable name, in this case <tt>tn</tt>.
</p>
<p>
  Now let's calculate the PSD of our data. We apply some averaging, in order to decrease the
  fluctuations in the data.
</p>

<div class="fragment"><pre>
    tnxx = tn.psd(plist(<span class="string">'Nfft'</span>,2000));
</pre></div>

<p>
  Additionally, we load a smooth model that represents well our data. It was obtained,
  as described <a href="ltpda_training_topic_5_2.html#brbme84">here</a>,
  by fitting the target PSD with z-domain fitting.
  We load the data from disk, and plot them against the target PSD. Please note that
  the colouring filters (whose response represents our model) have no units, so we force them to
  be the same as the PSD we compare with:
  <div class="fragment"><pre>
    fmod = ao(plist(<span class="string">'filename'</span>, <span class="string">'topic5/T5_Ex03_ModelNoise.xml'</span>));
    iplot(tnxx, fmod.setYunits(tnxx.yunits))
</pre></div>
</p>

  <p>
  The comparison beyween the target PSD and the model should look like:
  <div align="center">
    <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_4.png" align="center" border="0">
  </div>
</p>

  We can now start the noise generation process. The first step is to
  generate a white time series Analysis Object, with the desired duration and units:
</p>
<div class="fragment"><pre>
    a = ao(plist(<span class="string">'tsfcn'</span>,<span class="string">'randn(size(t))'</span>,<span class="string">'fs'</span>,1,<span class="string">'nsecs'</span>,10000,<span class="string">'yunits'</span>,<span class="string">'m'</span>));
</pre></div>
<p>
  Then we run the noise coloring process calling <tt>noisegen1D</tt>
</p>

<div class="fragment"><pre>
    plng = plist(...
          <span class="string">'model'</span>, fmod, ...      <span class="comment">% model for colored noise psd</span>
          <span class="string">'MaxIter'</span>, 50, ...      <span class="comment">% maximum number of fit iteration per model order</span>
          <span class="string">'PoleType'</span>, 2, ...      <span class="comment">% generates complex poles distributed in the unitary circle</span>
          <span class="string">'MinOrder'</span>, 20, ...     <span class="comment">% minimum model order</span>
          <span class="string">'MaxOrder'</span>, 50, ...     <span class="comment">% maximum model order</span>
          <span class="string">'Weights'</span>, 2, ...       <span class="comment">% weight with 1/abs(model)</span>
          <span class="string">'Plot'</span>, false,...       <span class="comment">% on to show the plot</span>
          <span class="string">'Disp'</span>, false,...       <span class="comment">% on to display fit progress on the command window</span>
          <span class="string">'RMSEVar'</span>, 7,...        <span class="comment">% Root Mean Squared Error Variation</span>
          <span class="string">'FitTolerance'</span>, 2);     <span class="comment">% Residuals log difference</span>

    ac = noisegen1D(a, plng);
</pre></div>

<p>
  Let's check the result. We calculate the PSD of the generated noise and compare it
  with the PSD of the target data.
</p>

<div class="fragment"><pre>
    acxx = ac.psd(plist(<span class="string">'Nfft'</span>,2000));
    iplot(tnxx,acxx)
</pre></div>

<p>
  As can be seen, the result is in quite satisfactory agreement with the original data
  <div align="center">
    <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_3.png" align="center" border="0">
  </div>
</p>

<a name="brbme84"></a><h3 class="title" id="brbme84">Appendix: evaluation of the model for the noise PSD</h3>
<p>
The smooth model for the data, that we used to reproduce the synthesized noise, was actually obtained
by applying the procedure of z-domain fitting that we discussed in <a href="ltpda_training_topic_5_1.html">the previous section</a>.
If you want to practise more with this fitting technique, we repost here the steps.
</p>
<p>
  In order to extract a reliable model from PSD data we need to discard the first frequency bins;
  we do that by means of the <tt>split</tt> method.
</p>

<div class="fragment"><pre>
    tnxxr = split(tnxx,plist(<span class="string">'frequencies'</span>, [2e-3 +inf]));
    iplot(tnxx,tnxxr)
</pre></div>

<p>
  The result should look like:
  <div align="center">
    <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_1.png"align="center" border="0">
  </div>
</p>
<p>
  Now it's the moment to fit our PSD to extract a smooth model to pass to the noise generator.

  First of all we should define a set of proper weights for our fit process.
  We smooth our PSD data and then define the weights as the inverse of the absolute value
  of the smoothed PSD. This should help the fit function to do a good job with noisy data.
  It is worth noting here that weights are not univocally defined and there could be better
  ways to define them.
</p>
<div class="fragment"><pre>
    stnxx = smoother(tnxxr);
    iplot(tnxxr, stnxx)
    wgh = 1./abs(stnxx);
</pre></div>
<p>
  The result of the <tt>smoother</tt> method is shown in the plot below:
</p>
  <div align="center">
    <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_smoother.png" align="center" border="0">
  </div>
  <p>
  Now let's run an automatic search for the proper model and pass the set
  of externally defined weights. The first output of <tt>zDomainFit</tt> is a <tt>miir</tt> filter model;
  the second output is the model response. Note that we are setting
  <span class="string">'ResFlat'</span> parameter to define the exit condition.
  <span class="string">'ResFlat'</span> check the spectral flatness of the
  absolute value of the fit residuals.
</p>
<div class="fragment"><pre>
    plfit = plist(<span class="string">'fs'</span>,1,...
                  <span class="string">'AutoSearch'</span>,<span class="string">'on'</span>,...
                  <span class="string">'StartPolesOpt'</span>,<span class="string">'clog'</span>,...
                  <span class="string">'maxiter'</span>,50,...
                  <span class="string">'minorder'</span>,30,...
                  <span class="string">'maxorder'</span>,45,...
                  <span class="string">'weights'</span>,wgh,... <span class="comment">% assign externally calculated weights</span>
                  <span class="string">'rmse'</span>,5,...
                  <span class="string">'condtype'</span>,<span class="string">'MSE'</span>,...
                  <span class="string">'msevartol'</span>,0.1,...
                  <span class="string">'fittol'</span>,0.01,...
                  <span class="string">'Plot'</span>,<span class="string">'on'</span>,...
                  <span class="string">'ForceStability'</span>,<span class="string">'off'</span>,...
                  <span class="string">'CheckProgress'</span>,<span class="string">'off'</span>);

    <span class="comment">% Do the fit</span>
    fit_results = zDomainFit(tnxxr,plfit);
</pre></div>

<p>
  Fit result should look like:
  <div align="center">
    <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_2_2.png" align="center" border="0">
  </div>
</p>

<p>
  The fit results consist in a <tt>filterbank</tt> object; we can evaluate the absolute values of
  the response of these filters at the frequencies defined by the <tt>x</tt> field of the PSD we want to match.

<div class="fragment"><pre>
<span class="comment">% Evaluate the absolute value of the response of the colouring filter</span>
b = resp(fit_results,plist(<span class="string">'f'</span>,tnxxr.x));
b.abs;

<span class="comment">% Save the model on disk</span>
b.save(plist(<span class="string">'filename'</span>, <span class="string">'topic5/T5_Ex03_ModelNoise.xml'</span>));
</pre></div>












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