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| author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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| date | Wed, 07 Dec 2011 17:29:47 +0100 |
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<!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>