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+ − <!-- $Id: zdomainfit_content.html,v 1.6 2009/08/27 11:38:58 luigi Exp $ -->
+ −
+ − <!-- ================================================== -->
+ − <!-- 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>