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| author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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| date | Wed, 23 Nov 2011 19:22:13 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/m-toolbox/html_help/help/ug/ltpda_training_topic_5_1.html Wed Nov 23 19:22:13 2011 +0100 @@ -0,0 +1,307 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" + "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd"> + +<html lang="en"> +<head> + <meta name="generator" content= + "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org"> + <meta http-equiv="Content-Type" content= + "text/html; charset=us-ascii"> + + <title>System identification in z-domain (LTPDA Toolbox)</title> + <link rel="stylesheet" href="docstyle.css" type="text/css"> + <meta name="generator" content="DocBook XSL Stylesheets V1.52.2"> + <meta name="description" content= + "Presents an overview of the features, system requirements, and starting the toolbox."> + </head> + +<body> + <a name="top_of_page" id="top_of_page"></a> + + <p style="font-size:1px;"> </p> + + <table class="nav" summary="Navigation aid" border="0" width= + "100%" cellpadding="0" cellspacing="0"> + <tr> + <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td> + + <td valign="baseline" align="right"><a href= + "ltpda_training_topic_5.html"><img src="b_prev.gif" border="0" align= + "bottom" alt="Topic 5 - Model fitting"></a> <a href= + "ltpda_training_topic_5_2.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Generation of noise with given PSD"></a></td> + </tr> + </table> + + <h1 class="title"><a name="f3-12899" id="f3-12899"></a>System identification in z-domain</h1> + <hr> + + <p> + <p> + System identification in Z-domain is performed with the function <tt>ao/zDomainFit</tt>. + It is based on a modified version of the vector fitting algorithm that was + adapted to fit in the Z-domain. Details of the agorithm can be found in the + <a href="zdomainfit.html">Z-domain fit documentation page</a></li>. +</p> + +<h2> System identification in Z-domain </h2> + +<p> + During this exercise we will: + <ol> + <li> Generate white noise + <li> Filter white noise with a <tt>miir</tt> filter generated by a <tt> pzmodel</tt> + <li> Extract the transfer function from data + <li> Fit the transfer function with <tt>ao/zDomainFit</tt> + <li> Check results + </ol> +</p> + +<p> + Let's start by generating some white noise. +</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> + This command generates a time series of gaussian distributed random noise + with a sampling frequency (<span class="string">'fs'</span>) of 1 Hz, + 10000 seconds long (<span class="string">'nsecs'</span>) and with + <span class="string">'yunits'</span> set to meters (<span class="string">'m'</span>). +</p> +<p> + Now we are ready to move on the second step where we will: + <ul> + <li> Build a pole-zero model (<tt>pzmodel</tt>) + <li> Construct a <tt>miir</tt> filter from the <tt>pzmodel</tt> + <li> filter white noise data with the <tt>miir</tt> filter in order to obtain a + colored noise time series. + </ul> + +</p> +<div class="fragment"><pre> + pzm = pzmodel(1, [0.005 2], [0.05 4]); + + filt = miir(pzm, plist(<span class="string">'fs'</span>, 1, <span class="string">'name'</span>, <span class="string">'None'</span>)); + filt.setIunits(<span class="string">'m'</span>); + filt.setOunits(<span class="string">'V'</span>); + + <span class="comment">% Filter the data</span> + ac = filter(a,filt); + ac.simplifyYunits; +</pre></div> +<p> + We can calculate the PSD of the data streams in order to check the difference between the coloured noise and the + white noise. Let's choose the log-scale estimation method. +</p> +<div class="fragment"><pre> + axx = lpsd(a); + acxx = lpsd(ac); + iplot(axx,acxx) +</pre></div> +<p> + You should obtain a plot similar to this: + <div align="center"> + <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_1_1.png" align="center" border="0"> + </div> +</p> +<p> + Let us move to the third step. We will generate the transfer function from the data + and split it in order to remove the first 3 bins. The last operation is useful + for the fitting process. +</p> +<div class="fragment"><pre> + tf = ltfe(a,ac); + tfsp = split(tf,plist(<span class="string">'frequencies'</span>, [5e-4 5e-1])); + + iplot(tf,tfsp) +</pre></div> +<p> + The plot should look like the following: + <div align="center"> + <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_1_2.png" align="center" border="0"> + </div> +</p> +<p> + It is now the moment to start fitting with <tt>zDomainFit</tt>. As reported in the + function help page we can run an automatic search loop to identify proper model order. + In such a case we have to define a set of conditions to check fit accuracy and to exit the fitting loop.<br/> + We can start checking Mean Squared Error and variation (CONDTYPE = 'MSE'). + It checks if the normalized mean squared error is lower than the value specified in the parameter FITTOL + and if the relative variation of the mean squared error is lower than the value specified in the parameter MSEVARTOL. + <p><!-- Table of the set: Default --> + <a name="1"/> + <table cellspacing="0" class="body" cellpadding="4" summary="" width="100%" border="2"> + <colgroup> + <col width="15%"/> + <col width="20%"/> + <col width="20%"/> + <col width="45%"/> + </colgroup> + <thead> + <tr valign="top"> + <th bgcolor="#B9C6DD" colspan="4"><h3>Default</h3></th> + </tr> + <tr valign="top"> + <th bgcolor="#D7D7D7">Key</th> + <th bgcolor="#D7D7D7">Default Value</th> + <th bgcolor="#D7D7D7">Options</th> + <th bgcolor="#D7D7D7">Description</th> + </tr> + </thead> + <tbody> + <tr valign="top"> + <td bgcolor="#F2F2F2">CONDTYPE</td> + <td bgcolor="#F2F2F2">'MSE'</td> + <td bgcolor="#F2F2F2"> + <ul> + <li><font color="#1111FF">'MSE'</font></li> + <li><font color="#1111FF">'RLD'</font></li> + <li><font color="#1111FF">'RSF'</font></li> + </ul> + </td> + <td bgcolor="#F2F2F2">Fit conditioning type. Admitted values are: + <ul> + <li>'MSE' Mean Squared Error and variation</li> + <li>'RLD' Log residuals difference and mean squared error variation</li> + <li>'RSF' Residuals spectral flatness and mean squared error variation</li> + </ul> + </td> + </tr> + <tr valign="top"> + <td bgcolor="#F2F2F2">FITTOL</td> + <td bgcolor="#F2F2F2">0.001</td> + <td bgcolor="#F2F2F2"><i>none</i></td> + <td bgcolor="#F2F2F2">Fit tolerance.</td> + </tr> + <tr valign="top"> + <td bgcolor="#F2F2F2">MSEVARTOL</td> + <td bgcolor="#F2F2F2">0.01</td> + <td bgcolor="#F2F2F2"><i>none</i></td> + <td bgcolor="#F2F2F2">Mean Squared Error Variation - Check if the<br/>relative variation of the mean squared error is<br/>smaller than the value specified. This<br/>option is useful for finding the minimum of the Chi-squared.</td> + </tr> + </tbody> + </table> + </p> + <p>You will find a list of all Parameters of zDomainFit here: <a href="matlab:web(ao.getInfo('zDomainFit').tohtml, '-helpbrowser')">Parameter of zDomainFit</a></p> + + Now let's run the fit: +</p> +<div class="fragment"><pre> + <span class="comment">% Set up the parameters</span> + plfit = plist(<span class="string">'fs'</span>,1,... <span class="comment">% Sampling frequency for the model filters</span> + <span class="string">'AutoSearch'</span>,<span class="string">'on'</span>,... <span class="comment">% Automatically search for a good model</span> + <span class="string">'StartPolesOpt'</span>,<span class="string">'clog'</span>,... <span class="comment">% Define the properties of the starting poles - complex distributed in the unitary circle</span> + <span class="string">'maxiter'</span>,50,... <span class="comment">% Maximum number of iteration per model order</span> + <span class="string">'minorder'</span>,2,... <span class="comment">% Minimum model order</span> + <span class="string">'maxorder'</span>,9,... <span class="comment">% Maximum model order</span> + <span class="string">'weightparam'</span>,<span class="string">'abs'</span>,... <span class="comment">% Assign weights as 1./abs(data)</span> + <span class="string">'condtype'</span>,<span class="string">'MSE'</span>,... <span class="comment">% Mean Squared Error and variation</span> + <span class="string">'fittol'</span>,1e-2,... <span class="comment">% Fit tolerance</span> + <span class="string">'msevartol'</span>,1e-1,... <span class="comment">% Mean Squared Error Variation tolerance</span> + <span class="string">'Plot'</span>,<span class="string">'on'</span>,... <span class="comment">% Set the plot on or off</span> + <span class="string">'ForceStability'</span>,<span class="string">'on'</span>,... <span class="comment">% Force to output a stable poles model</span> + <span class="string">'CheckProgress'</span>,<span class="string">'off'</span>); <span class="comment">% Display fitting progress on the command window</span> + + <span class="comment">% Do the fit</span> + fobj = zDomainFit(tfsp,plfit); + + <span class="comment">% Set the input and output units for fitted model</span> + fobj.setIunits(<span class="string">'m'</span>); + fobj.setOunits(<span class="string">'V'</span>); +</pre></div> +<p> + When <span class="string">'Plot'</span> parameter is set to <span class="string">'on'</span> + the function plots the fit progress. + <div align="center"> + <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_1_3.png" align="center" border="0"> + </div> +</p> + + <p> + We can now check the result of our fitting procedures. + We calculate the frequency response of the fitted models (filters) and compare them + with the starting IIR filter response, then we will plot the percentage error on the filters magnitudes. + </p> + <p> + Note that the result of the fitting procedure is a <tt>matrix</tt> object containing a <tt>filterbank</tt> object, + which itself contains a parallel bank of 3 IIR filters. +</p> + <p> + Note that at the moment we need to access the individual filter objects inside the <tt>matrix</tt> object that was the result of the fitting procedure.</p> +<div class="fragment"><pre> + <span class="comment">% set plist for filter response</span> + plrsp = plist(<span class="string">'bank'</span>,<span class="string">'parallel'</span>,<span class="string">'f1'</span>,1e-5,<span class="string">'f2'</span>,0.5,<span class="string">'nf'</span>,100,<span class="string">'scale'</span>,<span class="string">'log'</span>); + + <span class="comment">% compute the response of the original noise-shape filter</span> + rfilt = resp(filt,plrsp); + rfilt.setName; + + <span class="comment">% compute the response of our fitted filter bank</span> + rfobj = resp(fobj.filters,plrsp); + rfobj.setName; + + <span class="comment">% compare the responses</span> + iplot(rfilt,rfobj) + + <span class="comment">% and the percentage error on the magnitude</span> + pdiff = 100.*abs((rfobj-rfilt)./rfilt); + pdiff.simplifyYunits; + iplot(pdiff,plist(<span class="string">'YRanges'</span>,[1e-2 100])) +</pre></div> +<p> + The first plot shows the response of the original filter and the fitted filter bank, + whereas the second plot reports the percentage difference between fitted model and target filter magnitude. <br/> + As can be seen, the difference between filters magnitude is at most 10%. + <div align="center"> + <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_1_4.png" align="center" border="0"> + </div> + <div align="center"> + <IMG src="images/ltpda_training_1/topic5/ltpda_training_5_1_5.png" align="center" border="0"> + </div> +</p> + + + + + + + + + + + + + + + + + + + </p> + + <br> + <br> + <table class="nav" summary="Navigation aid" border="0" width= + "100%" cellpadding="0" cellspacing="0"> + <tr valign="top"> + <td align="left" width="20"><a href="ltpda_training_topic_5.html"><img src= + "b_prev.gif" border="0" align="bottom" alt= + "Topic 5 - Model fitting"></a> </td> + + <td align="left">Topic 5 - Model fitting</td> + + <td> </td> + + <td align="right">Generation of noise with given PSD</td> + + <td align="right" width="20"><a href= + "ltpda_training_topic_5_2.html"><img src="b_next.gif" border="0" align= + "bottom" alt="Generation of noise with given PSD"></a></td> + </tr> + </table><br> + + <p class="copy">©LTP Team</p> +</body> +</html>