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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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11 <title>Z-Domain Fit (LTPDA Toolbox)</title>
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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18 <body>
19 <a name="top_of_page" id="top_of_page"></a>
20
21 <p style="font-size:1px;">&nbsp;</p>
22
23 <table class="nav" summary="Navigation aid" border="0" width=
24 "100%" cellpadding="0" cellspacing="0">
25 <tr>
26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
27
28 <td valign="baseline" align="right"><a href=
29 "sigproc_example_matrix_linfitsvd_ssm.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Iterative linear parameter estimation for multichannel systems - ssm system model in time domain"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "sdomainfit.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="S-Domain Fit"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Z-Domain Fit</h1>
37 <hr>
38
39 <p>
40
41 <!-- ================================================== -->
42 <!-- BEGIN CONTENT FILE -->
43 <!-- ================================================== -->
44 <!-- ===== link box: Begin ===== -->
45 <p>
46 <table border="1" width="80%">
47 <tr>
48 <td>
49 <table border="0" cellpadding="5" class="categorylist" width="100%">
50 <colgroup>
51 <col width="37%"/>
52 <col width="63%"/>
53 </colgroup>
54 <tbody>
55 <tr valign="top">
56 <td>
57 <a href="#description">Description</a>
58 </td>
59 <td>Z-domain system identification in LTPDA.</td>
60 </tr>
61 <tr valign="top">
62 <td>
63 <a href="#algorithm">Algorithm</a>
64 </td>
65 <td>Fit Algorithm.</td>
66 </tr>
67 <tr valign="top">
68 <td>
69 <a href="#examples">Examples</a>
70 </td>
71 <td>Usage example of z-domain system identification tool.</td>
72 </tr>
73 <tr valign="top">
74 <td>
75 <a href="#references">References</a>
76 </td>
77 <td>Bibliographic references.</td>
78 </tr>
79 </tbody>
80 </table>
81 </td>
82 </tr>
83 </table>
84 </p>
85 <!-- ===== link box: End ====== -->
86
87
88
89 <h2><a name="description">Z-domain system identification in LTPDA</a></h2>
90 <p>
91 System identification in z-domain is performed with the function
92 <a href="matlab:doc('ao/zDomainFit')">zDomainFit</a>.
93 It is based on a modeified version of the vector fitting algorithm that was
94 adapted to fit in z-domain. Details on the core agorithm can be found in [1 - 3].
95 </p>
96 <p>
97 If you provide more than one AO as input, they will be fitted
98 together with a common set of poles.
99 Only frequency domain (<a href="matlab:doc('fsdata')">fsdata</a>) data can be
100 fitted. Each non fsdata object is ignored. Input
101 objects must have the same number of elements.
102 </p>
103
104
105 <h2><a name="algorithm">Fit algorithm</a></h2>
106
107 <p>
108 The function performs a fitting loop to automatically identify model
109 order and parameters in z-domain. Output is a z-domain model expanded
110 in partial fractions:
111 </p>
112 <p>
113 <div>
114 <IMG src="images/zdomainfit_1.gif" border="0">
115 </div>
116 </p>
117 <p>
118 Each element of the partial fraction expansion can be seen as a
119 <a href="sigproc_iir.html">miir</a> filter. Therefore the complete expansion
120 is simply a parallel <a href="sigproc_filterbanks.html">filterbank</a> of
121 <a href="sigproc_iir.html">miir</a> filters.
122 Since the function can fit more than one input analysis object at a time
123 with a common set of poles, output filterbank are embedded in a
124 <a href="class_desc_matrix.html">matrix</a> (note that this characteristic
125 will be probably changed becausse of the introduction of the
126 <a href="class_desc_collection.html">collection</a> class).
127 </p>
128 <p>
129 Identification loop stops when the stop condition is reached.
130 Stop criterion is based on three different approaches:
131 <ol>
132 <li> Mean Squared Error and variation <br>
133 Check if the normalized mean squared error is lower than the value specified in
134 <tt>FITTOL</tt> and if the relative variation of the mean squared error is lower
135 than the value specified in <tt>MSEVARTOL</tt>.
136 E.g. <tt>FITTOL = 1e-3</tt>, <tt>MSEVARTOL = 1e-2</tt> search for a fit with
137 normalized meam square error lower than <tt>1e-3</tt> and <tt>MSE</tt> relative
138 variation lower than <tt>1e-2</tt>.
139 </li>
140 <li> Log residuals difference and root mean squared error
141 <ul>
142 <li> Log Residuals difference <br>
143 Check if the minimum of the logarithmic difference between data and
144 residuals is larger than a specified value. ie. if the conditioning
145 value is <tt>2</tt>, the function ensures that the difference between data and
146 residuals is at lest two order of magnitude lower than data itsleves.
147 <li> Root Mean Squared Error <br>
148 Check that the variation of the root mean squared error is lower than
149 <tt>10^(-1*value)</tt>.
150 </ul>
151 </li>
152 <li> Residuals spectral flatness and root mean squared error
153 <ul>
154 <li> Residuals Spectral Flatness <br>
155 In case of a fit on noisy data, the residuals from a good fit are
156 expected to be as much as possible similar to a white noise. This
157 property can be used to test the accuracy of a fit procedure. In
158 particular it can be tested that the spectral flatness coefficient of
159 the residuals is larger than a certain qiantity sf such that <tt>0 < sf < 1</tt>.
160 <li> Root Mean Squared Error <br>
161 Check that the variation of the root mean squared error is lower than
162 <tt>10^(-1*value)</tt>.
163 </ul>
164 </li>
165 </ol>
166 Fitting loop stops when the two stopping conditions are satisfied, in both cases.
167 </p>
168 <p>
169 The function can also perform a single loop without taking care of
170 the stop conditions. This happens when <span class="string">'AUTOSEARCH'</span> parameter is
171 set to <span class="string">'OFF'</span>.
172 </p>
173
174
175
176 <h2><a name="examples">Usage example of z-domain system identification tool</a></h2>
177 <p>
178 In this example we fit a given frequency response to get a stable <tt>miir</tt> filter.
179 For the meaning of any parameter please refer to
180 <a href="matlab:doc('ao')">ao</a> and
181 <a href="matlab:doc('ao/zDomainFit')">zDomainFit</a>
182 documentation pages.
183 </p>
184
185 <div class="fragment"><pre>
186 pl = plist(...
187 <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>,...
188 <span class="string">'f1'</span>, 1e-6,...
189 <span class="string">'f2'</span>, 5,...
190 <span class="string">'nf'</span>, 100);
191
192 a = ao(pl);
193 a.setName;
194
195 <span class="comment">% Fit parameter list</span>
196 pl_fit = plist(<span class="string">'FS'</span>,10,...
197 <span class="string">'AutoSearch'</span>,<span class="string">'on'</span>,...
198 <span class="string">'StartPolesOpt'</span>,<span class="string">'clog'</span>,...
199 <span class="string">'maxiter'</span>,50,...
200 <span class="string">'minorder'</span>,15,...
201 <span class="string">'maxorder'</span>,30,...
202 <span class="string">'weightparam'</span>,<span class="string">'abs'</span>,...
203 <span class="string">'CONDTYPE'</span>,<span class="string">'MSE'</span>,...
204 <span class="string">'FITTOL'</span>,1e-2,...
205 <span class="string">'MSEVARTOL'</span>,1e-1,...
206 <span class="string">'Plot'</span>,<span class="string">'on'</span>,...
207 <span class="string">'ForceStability'</span>,<span class="string">'on'</span>);
208
209 <span class="comment">% Do fit</span>
210 mod = zDomainFit(a, pl_fit);
211 </pre></div>
212
213 <p>
214 <tt>mod</tt> is a <tt>matrix</tt> object containing a <tt>filterbank</tt> object.
215 </p>
216
217 <div class="fragment"><pre>
218 >> mod
219 ---- matrix 1 ----
220 name: fit(a)
221 size: 1x1
222 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]))
223 description:
224 UUID: 9274455a-68e8-4bf1-b1ad-db81551f3cd6
225 ------------------
226 </pre></div>
227
228 <p>
229 The <tt>filterbank</tt> object contains a parallel bank of 18 filters.
230 </p>
231
232 <div class="fragment"><pre>
233 >> mod.objs
234 ---- filterbank 1 ----
235 name: fit(a)
236 type: parallel
237 01: fit(a)(fs=10.00, ntaps=2.00, a=[-1.19e+005 0], b=[1 0.0223])
238 02: fit(a)(fs=10.00, ntaps=2.00, a=[1.67e+005 0], b=[1 0.137])
239 03: fit(a)(fs=10.00, ntaps=2.00, a=[-5.41e+004 0], b=[1 0.348])
240 04: fit(a)(fs=10.00, ntaps=2.00, a=[1.15e+004 0], b=[1 0.603])
241 05: fit(a)(fs=10.00, ntaps=2.00, a=[-1.69e+005 0], b=[1 0.639])
242 06: fit(a)(fs=10.00, ntaps=2.00, a=[1.6e+005 0], b=[1 0.64])
243 07: fit(a)(fs=10.00, ntaps=2.00, a=[9.99e-009 0], b=[1 -1])
244 08: fit(a)(fs=10.00, ntaps=2.00, a=[-4.95e-010 0], b=[1 1])
245 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])
246 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])
247 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])
248 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])
249 13: fit(a)(fs=10.00, ntaps=2.00, a=[-1.67e+004+i*432 0], b=[1 0.171-i*0.14])
250 14: fit(a)(fs=10.00, ntaps=2.00, a=[-1.67e+004-i*432 0], b=[1 0.171+i*0.14])
251 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])
252 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])
253 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])
254 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])
255 description:
256 UUID: 21af6960-61a8-4351-b504-e6f2b5e55b06
257 ----------------------
258 </pre></div>
259
260 <p>
261 Each object of the <tt>filterbank</tt> is a <tt>miir</tt> filter.
262 </p>
263
264 <div class="fragment"><pre>
265 filt = mod.objs.filters.index(3)
266 ------ miir/1 -------
267 b: [1 0.348484501572296]
268 histin: 0
269 version: $Id: zdomainfit_content.html,v 1.6 2009/08/27 11:38:58 luigi Exp $
270 ntaps: 2
271 fs: 10
272 infile:
273 a: [-54055.7700068032 0]
274 histout: 0
275 iunits: [] [1x1 unit]
276 ounits: [] [1x1 unit]
277 hist: miir.hist [1x1 history]
278 procinfo: (empty-plist) [1x1 plist]
279 plotinfo: (empty-plist) [1x1 plist]
280 name: (fit(a)(3,1))(3)
281 description:
282 mdlfile:
283 UUID: 6e2a1cd8-f17d-4c9d-aea9-4d9a96e41e68
284 ---------------------
285 </pre></div>
286
287
288 <h2><a name="references">References</a></h2>
289 <p>
290 <ol>
291 <li> B. Gustavsen and A. Semlyen, "Rational approximation of frequency
292 domain responses by Vector Fitting", IEEE Trans. Power Delivery
293 vol. 14, no. 3, pp. 1052-1061, July 1999.
294 <li> B. Gustavsen, "Improving the Pole Relocating Properties of Vector
295 Fitting", IEEE Trans. Power Delivery vol. 21, no. 3, pp.
296 1587-1592, July 2006.
297 <li> Y. S. Mekonnen and J. E. Schutt-Aine, "Fast broadband
298 macromodeling technique of sampled time/frequency data using
299 z-domain vector-fitting method", Electronic Components and
300 Technology Conference, 2008. ECTC 2008. 58th 27-30 May 2008 pp.
301 1231 - 1235.
302 </ol>
303 </p>
304 </p>
305
306 <br>
307 <br>
308 <table class="nav" summary="Navigation aid" border="0" width=
309 "100%" cellpadding="0" cellspacing="0">
310 <tr valign="top">
311 <td align="left" width="20"><a href="sigproc_example_matrix_linfitsvd_ssm.html"><img src=
312 "b_prev.gif" border="0" align="bottom" alt=
313 "Iterative linear parameter estimation for multichannel systems - ssm system model in time domain"></a>&nbsp;</td>
314
315 <td align="left">Iterative linear parameter estimation for multichannel systems - ssm system model in time domain</td>
316
317 <td>&nbsp;</td>
318
319 <td align="right">S-Domain Fit</td>
320
321 <td align="right" width="20"><a href=
322 "sdomainfit.html"><img src="b_next.gif" border="0" align=
323 "bottom" alt="S-Domain Fit"></a></td>
324 </tr>
325 </table><br>
326
327 <p class="copy">&copy;LTP Team</p>
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329 </html>