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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/zdomainfit.html Wed Nov 23 19:22:13 2011 +0100 @@ -0,0 +1,329 @@ +<!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>Z-Domain Fit (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= + "sigproc_example_matrix_linfitsvd_ssm.html"><img src="b_prev.gif" border="0" align= + "bottom" alt="Iterative linear parameter estimation for multichannel systems - ssm system model in time domain"></a> <a href= + "sdomainfit.html"><img src="b_next.gif" border="0" align= + "bottom" alt="S-Domain Fit"></a></td> + </tr> + </table> + + <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Z-Domain Fit</h1> + <hr> + + <p> + + <!-- ================================================== --> + <!-- BEGIN CONTENT FILE --> + <!-- ================================================== --> + <!-- ===== link box: Begin ===== --> + <p> + <table border="1" width="80%"> + <tr> + <td> + <table border="0" cellpadding="5" class="categorylist" width="100%"> + <colgroup> + <col width="37%"/> + <col width="63%"/> + </colgroup> + <tbody> + <tr valign="top"> + <td> + <a href="#description">Description</a> + </td> + <td>Z-domain system identification in LTPDA.</td> + </tr> + <tr valign="top"> + <td> + <a href="#algorithm">Algorithm</a> + </td> + <td>Fit Algorithm.</td> + </tr> + <tr valign="top"> + <td> + <a href="#examples">Examples</a> + </td> + <td>Usage example of z-domain system identification tool.</td> + </tr> + <tr valign="top"> + <td> + <a href="#references">References</a> + </td> + <td>Bibliographic references.</td> + </tr> + </tbody> + </table> + </td> + </tr> + </table> + </p> + <!-- ===== link box: End ====== --> + + + +<h2><a name="description">Z-domain system identification in LTPDA</a></h2> +<p> + System identification in z-domain is performed with the function + <a href="matlab:doc('ao/zDomainFit')">zDomainFit</a>. + It is based on a modeified version of the vector fitting algorithm that was + adapted to fit in z-domain. Details on the core agorithm can be found in [1 - 3]. +</p> +<p> + If you provide more than one AO as input, they will be fitted + together with a common set of poles. + Only frequency domain (<a href="matlab:doc('fsdata')">fsdata</a>) data can be + fitted. Each non fsdata object is ignored. Input + objects must have the same number of elements. +</p> + + +<h2><a name="algorithm">Fit algorithm</a></h2> + +<p> + The function performs a fitting loop to automatically identify model + order and parameters in z-domain. Output is a z-domain model expanded + in partial fractions: +</p> +<p> + <div> + <IMG src="images/zdomainfit_1.gif" border="0"> + </div> +</p> +<p> + Each element of the partial fraction expansion can be seen as a + <a href="sigproc_iir.html">miir</a> filter. Therefore the complete expansion + is simply a parallel <a href="sigproc_filterbanks.html">filterbank</a> of + <a href="sigproc_iir.html">miir</a> filters. + Since the function can fit more than one input analysis object at a time + with a common set of poles, output filterbank are embedded in a + <a href="class_desc_matrix.html">matrix</a> (note that this characteristic + will be probably changed becausse of the introduction of the + <a href="class_desc_collection.html">collection</a> class). +</p> +<p> + Identification loop stops when the stop condition is reached. + Stop criterion is based on three different approaches: + <ol> + <li> Mean Squared Error and variation <br> + Check if the normalized mean squared error is lower than the value specified in + <tt>FITTOL</tt> and if the relative variation of the mean squared error is lower + than the value specified in <tt>MSEVARTOL</tt>. + E.g. <tt>FITTOL = 1e-3</tt>, <tt>MSEVARTOL = 1e-2</tt> search for a fit with + normalized meam square error lower than <tt>1e-3</tt> and <tt>MSE</tt> relative + variation lower than <tt>1e-2</tt>. + </li> + <li> Log residuals difference and root mean squared error + <ul> + <li> Log Residuals difference <br> + Check if the minimum of the logarithmic difference between data and + residuals is larger than a specified value. ie. if the conditioning + value is <tt>2</tt>, the function ensures that the difference between data and + residuals is at lest two order of magnitude lower than data itsleves. + <li> Root Mean Squared Error <br> + Check that the variation of the root mean squared error is lower than + <tt>10^(-1*value)</tt>. + </ul> + </li> + <li> Residuals spectral flatness and root mean squared error + <ul> + <li> Residuals Spectral Flatness <br> + In case of a fit on noisy data, the residuals from a good fit are + expected to be as much as possible similar to a white noise. This + property can be used to test the accuracy of a fit procedure. In + particular it can be tested that the spectral flatness coefficient of + the residuals is larger than a certain qiantity sf such that <tt>0 < sf < 1</tt>. + <li> Root Mean Squared Error <br> + Check that the variation of the root mean squared error is lower than + <tt>10^(-1*value)</tt>. + </ul> + </li> + </ol> + Fitting loop stops when the two stopping conditions are satisfied, in both cases. +</p> +<p> + The function can also perform a single loop without taking care of + the stop conditions. This happens when <span class="string">'AUTOSEARCH'</span> parameter is + set to <span class="string">'OFF'</span>. +</p> + + + +<h2><a name="examples">Usage example of z-domain system identification tool</a></h2> +<p> + In this example we fit a given frequency response to get a stable <tt>miir</tt> filter. + For the meaning of any parameter please refer to + <a href="matlab:doc('ao')">ao</a> and + <a href="matlab:doc('ao/zDomainFit')">zDomainFit</a> + documentation pages. +</p> + +<div class="fragment"><pre> + pl = plist(... + <span class="string">'fsfcn'</span>, <span class="string">'(1e-3./(2.*pi.*1i.*f).^2 + 1e3./(0.001+2.*pi.*1i.*f) + 1e5.*(2.*pi.*1i.*f).^2).*1e-10'</span>,... + <span class="string">'f1'</span>, 1e-6,... + <span class="string">'f2'</span>, 5,... + <span class="string">'nf'</span>, 100); + + a = ao(pl); + a.setName; + + <span class="comment">% Fit parameter list</span> + pl_fit = plist(<span class="string">'FS'</span>,10,... + <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>,15,... + <span class="string">'maxorder'</span>,30,... + <span class="string">'weightparam'</span>,<span class="string">'abs'</span>,... + <span class="string">'CONDTYPE'</span>,<span class="string">'MSE'</span>,... + <span class="string">'FITTOL'</span>,1e-2,... + <span class="string">'MSEVARTOL'</span>,1e-1,... + <span class="string">'Plot'</span>,<span class="string">'on'</span>,... + <span class="string">'ForceStability'</span>,<span class="string">'on'</span>); + + <span class="comment">% Do fit</span> + mod = zDomainFit(a, pl_fit); +</pre></div> + +<p> + <tt>mod</tt> is a <tt>matrix</tt> object containing a <tt>filterbank</tt> object. +</p> + +<div class="fragment"><pre> + >> mod + ---- matrix 1 ---- + name: fit(a) + size: 1x1 + 01: filterbank | filterbank(fit(a)(fs=10.00, ntaps=2.00, a=[-1.19e+005 0], b=[1 0.0223]), fit(a)(fs=10.00, ntaps=2.00, a=[1.67e+005 0], b=[1 0.137]), fit(a)(fs=10.00, ntaps=2.00, a=[-5.41e+004 0], b=[1 0.348]), fit(a)(fs=10.00, ntaps=2.00, a=[1.15e+004 0], b=[1 0.603]), fit(a)(fs=10.00, ntaps=2.00, a=[-1.69e+005 0], b=[1 0.639]), fit(a)(fs=10.00, ntaps=2.00, a=[1.6e+005 0], b=[1 0.64]), fit(a)(fs=10.00, ntaps=2.00, a=[9.99e-009 0], b=[1 -1]), fit(a)(fs=10.00, ntaps=2.00, a=[-4.95e-010 0], b=[1 1]), fit(a)(fs=10.00, ntaps=2.00, a=[9.4e+003-i*3.7e+003 0], b=[1 -0.0528-i*0.0424]), fit(a)(fs=10.00, ntaps=2.00, a=[9.4e+003+i*3.7e+003 0], b=[1 -0.0528+i*0.0424]), fit(a)(fs=10.00, ntaps=2.00, a=[1.66e+003-i*1.45e+004 0], b=[1 0.0233-i*0.112]), fit(a)(fs=10.00, ntaps=2.00, a=[1.66e+003+i*1.45e+004 0], b=[1 0.0233+i*0.112]), fit(a)(fs=10.00, ntaps=2.00, a=[-1.67e+004+i*432 0], b=[1 0.171-i*0.14]), fit(a)(fs=10.00, ntaps=2.00, a=[-1.67e+004-i*432 0], b=[1 0.171+i*0.14]), fit(a)(fs=10.00, ntaps=2.00, a=[7.61e+003+i*7.36e+003 0], b=[1 0.378-i*0.112]), fit(a)(fs=10.00, ntaps=2.00, a=[7.61e+003-i*7.36e+003 0], b=[1 0.378+i*0.112]), fit(a)(fs=10.00, ntaps=2.00, a=[3.67e-015-i*4.61e-006 0], b=[1 -1-i*1.08e-010]), fit(a)(fs=10.00, ntaps=2.00, a=[3.67e-015+i*4.61e-006 0], b=[1 -1+i*1.08e-010])) + description: + UUID: 9274455a-68e8-4bf1-b1ad-db81551f3cd6 + ------------------ +</pre></div> + +<p> + The <tt>filterbank</tt> object contains a parallel bank of 18 filters. +</p> + +<div class="fragment"><pre> + >> mod.objs + ---- filterbank 1 ---- + name: fit(a) + type: parallel + 01: fit(a)(fs=10.00, ntaps=2.00, a=[-1.19e+005 0], b=[1 0.0223]) + 02: fit(a)(fs=10.00, ntaps=2.00, a=[1.67e+005 0], b=[1 0.137]) + 03: fit(a)(fs=10.00, ntaps=2.00, a=[-5.41e+004 0], b=[1 0.348]) + 04: fit(a)(fs=10.00, ntaps=2.00, a=[1.15e+004 0], b=[1 0.603]) + 05: fit(a)(fs=10.00, ntaps=2.00, a=[-1.69e+005 0], b=[1 0.639]) + 06: fit(a)(fs=10.00, ntaps=2.00, a=[1.6e+005 0], b=[1 0.64]) + 07: fit(a)(fs=10.00, ntaps=2.00, a=[9.99e-009 0], b=[1 -1]) + 08: fit(a)(fs=10.00, ntaps=2.00, a=[-4.95e-010 0], b=[1 1]) + 09: fit(a)(fs=10.00, ntaps=2.00, a=[9.4e+003-i*3.7e+003 0], b=[1 -0.0528-i*0.0424]) + 10: fit(a)(fs=10.00, ntaps=2.00, a=[9.4e+003+i*3.7e+003 0], b=[1 -0.0528+i*0.0424]) + 11: fit(a)(fs=10.00, ntaps=2.00, a=[1.66e+003-i*1.45e+004 0], b=[1 0.0233-i*0.112]) + 12: fit(a)(fs=10.00, ntaps=2.00, a=[1.66e+003+i*1.45e+004 0], b=[1 0.0233+i*0.112]) + 13: fit(a)(fs=10.00, ntaps=2.00, a=[-1.67e+004+i*432 0], b=[1 0.171-i*0.14]) + 14: fit(a)(fs=10.00, ntaps=2.00, a=[-1.67e+004-i*432 0], b=[1 0.171+i*0.14]) + 15: fit(a)(fs=10.00, ntaps=2.00, a=[7.61e+003+i*7.36e+003 0], b=[1 0.378-i*0.112]) + 16: fit(a)(fs=10.00, ntaps=2.00, a=[7.61e+003-i*7.36e+003 0], b=[1 0.378+i*0.112]) + 17: fit(a)(fs=10.00, ntaps=2.00, a=[3.67e-015-i*4.61e-006 0], b=[1 -1-i*1.08e-010]) + 18: fit(a)(fs=10.00, ntaps=2.00, a=[3.67e-015+i*4.61e-006 0], b=[1 -1+i*1.08e-010]) + description: + UUID: 21af6960-61a8-4351-b504-e6f2b5e55b06 + ---------------------- +</pre></div> + +<p> + Each object of the <tt>filterbank</tt> is a <tt>miir</tt> filter. +</p> + +<div class="fragment"><pre> + filt = mod.objs.filters.index(3) + ------ miir/1 ------- + b: [1 0.348484501572296] + histin: 0 + version: $Id: zdomainfit_content.html,v 1.6 2009/08/27 11:38:58 luigi Exp $ + ntaps: 2 + fs: 10 + infile: + a: [-54055.7700068032 0] + histout: 0 + iunits: [] [1x1 unit] + ounits: [] [1x1 unit] + hist: miir.hist [1x1 history] + procinfo: (empty-plist) [1x1 plist] + plotinfo: (empty-plist) [1x1 plist] + name: (fit(a)(3,1))(3) + description: + mdlfile: + UUID: 6e2a1cd8-f17d-4c9d-aea9-4d9a96e41e68 + --------------------- +</pre></div> + + +<h2><a name="references">References</a></h2> +<p> + <ol> + <li> B. Gustavsen and A. Semlyen, "Rational approximation of frequency + domain responses by Vector Fitting", IEEE Trans. Power Delivery + vol. 14, no. 3, pp. 1052-1061, July 1999. + <li> B. Gustavsen, "Improving the Pole Relocating Properties of Vector + Fitting", IEEE Trans. Power Delivery vol. 21, no. 3, pp. + 1587-1592, July 2006. + <li> Y. S. Mekonnen and J. E. Schutt-Aine, "Fast broadband + macromodeling technique of sampled time/frequency data using + z-domain vector-fitting method", Electronic Components and + Technology Conference, 2008. ECTC 2008. 58th 27-30 May 2008 pp. + 1231 - 1235. + </ol> +</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="sigproc_example_matrix_linfitsvd_ssm.html"><img src= + "b_prev.gif" border="0" align="bottom" alt= + "Iterative linear parameter estimation for multichannel systems - ssm system model in time domain"></a> </td> + + <td align="left">Iterative linear parameter estimation for multichannel systems - ssm system model in time domain</td> + + <td> </td> + + <td align="right">S-Domain Fit</td> + + <td align="right" width="20"><a href= + "sdomainfit.html"><img src="b_next.gif" border="0" align= + "bottom" alt="S-Domain Fit"></a></td> + </tr> + </table><br> + + <p class="copy">©LTP Team</p> +</body> +</html>