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<h1 class="title"><a name="f3-12899" id="f3-12899"></a>Z-Domain Fit</h1>
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<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>
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<td>
<a href="#examples">Examples</a>
</td>
<td>Usage example of z-domain system identification tool.</td>
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<a href="#references">References</a>
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<td>Bibliographic references.</td>
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<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>
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