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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>IFO/Temperature Example - Simulation (LTPDA Toolbox)</title>
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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 "ltpda_training_topic_4_4_2.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="By defining filter properties"></a>&nbsp;&nbsp;&nbsp;<a href=
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32 "bottom" alt="Topic 5 - Model fitting"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>IFO/Temperature Example - Simulation</h1>
37 <hr>
38
39 <p>
40 <p>
41 We now come back to the IFO/Temperature working example. Our interest here is not
42 to add more tools to the data analysis chain that you've been developing but
43 to create a simple toy model that allows you to reproduce the steps done up
44 to now but with synthetic data.
45 </p>
46 <p>
47 The problem is shown schematically in the figure below. We will generate temperature and interferometer
48 noisy data by applying a filter (<b>TMP</b> and <b>IFO</b>) to white noise data
49 (<b>WN1</b> and <b>WN2</b>). We will then add a temperature coupling (<b>K2RAD</b>) to the
50 interferometer and from the measured interferometer and temperature data we will then estimate
51 the temperature to interferometer coupling.
52 </p>
53
54 <p align="center">
55 <img src="images/ltpda_training_1/topic4/IFO2TSimulation.png" alt="temperature substraction scheme" border="1" >
56 </p>
57 <p>
58 Since we've been through the same steps that you need to apply here in the previous
59 section we will give here the step by step description of the task and let you
60 play with the models.
61 </p>
62
63 <h2> Build the models<h2>
64 <p>
65 We need three models: one to generate temperature-like data, another modelling the
66 interferometer and a third one acting as the K-to-rad transfer function.
67 </p>
68
69 <h3> STEP 1: Build a temperature noise PZMODEL with the following properties<h4>
70
71 <table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
72 <colgroup>
73 <col width="15%"/>
74 <col width="35%"/>
75 </colgroup>
76 <thead>
77 <tr valign="top">
78 <th class="categorylist">Key</th>
79 <th class="categorylist">Value</th>
80 </tr>
81 </thead>
82 <tbody>
83 <!-- Key 'filename' -->
84 <tr valign="top">
85 <td bgcolor="#f3f4f5">
86 <p><tt><span class="string">'name'</span></tt></p>
87 </td>
88 <td bgcolor="#f3f4f5">
89 <span class="string">'TMP'</span>
90 </td>
91 </tr>
92 <!-- Key 'filename' -->
93 <tr valign="top">
94 <td bgcolor="#f3f4f5">
95 <p><tt><span class="string">'ounits'</span></tt></p>
96 </td>
97 <td bgcolor="#f3f4f5">
98 <span class="string">'K'</span>
99 </td>
100 </tr>
101 <!-- Key 'filename' -->
102 <tr valign="top">
103 <td bgcolor="#f3f4f5">
104 <p><tt>GAIN</tt></p>
105 </td>
106 <td bgcolor="#f3f4f5">
107 10
108 </td>
109 </tr>
110 <!-- Key 'filename' -->
111 <tr valign="top">
112 <td bgcolor="#f3f4f5">
113 <p><tt>POLE 1</tt></p>
114 </td>
115 <td bgcolor="#f3f4f5">
116 1e-5
117 </td>
118 </tbody>
119 </table>
120 <p>
121 For example, this few lines would do the job
122 </p>
123 <div class="fragment"><pre>
124 TMP = pzmodel(10,1e-5,[]);
125 TMP.setOunits(<span class="string">'K'</span>)
126 TMP.setName(<span class="string">'TMP'</span>)
127 </pre></div>
128
129 <h3> STEP 2: Build a interferometer noise PZMODEL with the following properties<h4>
130
131 <table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
132 <colgroup>
133 <col width="15%"/>
134 <col width="35%"/>
135 </colgroup>
136 <thead>
137 <tr valign="top">
138 <th class="categorylist">Key</th>
139 <th class="categorylist">Value</th>
140 </tr>
141 </thead>
142 <tbody>
143 <!-- Key 'filename' -->
144 <tr valign="top">
145 <td bgcolor="#f3f4f5">
146 <p><tt><span class="string">'name'</span></tt></p>
147 </td>
148 <td bgcolor="#f3f4f5">
149 <span class="string">'IFO'</span>
150 </td>
151 </tr>
152 <!-- Key 'filename' -->
153 <tr valign="top">
154 <td bgcolor="#f3f4f5">
155 <p><tt><span class="string">'ounits'</span></tt></p>
156 </td>
157 <td bgcolor="#f3f4f5">
158 <span class="string">'rad'</span>
159 </td>
160 </tr>
161 <!-- Key 'filename' -->
162 <tr valign="top">
163 <td bgcolor="#f3f4f5">
164 <p><tt>GAIN</tt></p>
165 </td>
166 <td bgcolor="#f3f4f5">
167 1e-3
168 </td>
169 </tr>
170 <!-- Key 'filename' -->
171 <tr valign="top">
172 <td bgcolor="#f3f4f5">
173 <p><tt>POLE 1</tt></p>
174 </td>
175 <td bgcolor="#f3f4f5">
176 0.4
177 </td>
178 </tr>
179 </tbody>
180 </table>
181
182
183 <h3> STEP 3: Build temperature to interferometer coupling PZMODEL with the following properties<h4>
184
185 <table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
186 <colgroup>
187 <col width="15%"/>
188 <col width="35%"/>
189 </colgroup>
190 <thead>
191 <tr valign="top">
192 <th class="categorylist">Key</th>
193 <th class="categorylist">Value</th>
194 </tr>
195 </thead>
196 <tbody>
197 <!-- Key 'filename' -->
198 <tr valign="top">
199 <td bgcolor="#f3f4f5">
200 <p><tt><span class="string">'name'</span></tt></p>
201 </td>
202 <td bgcolor="#f3f4f5">
203 <span class="string">'K2RAD'</span>
204 </td>
205 </tr>
206 <!-- Key 'filename' -->
207 <tr valign="top">
208 <td bgcolor="#f3f4f5">
209 <p><tt><span class="string">'iunits'</span></tt></p>
210 </td>
211 <td bgcolor="#f3f4f5">
212 <span class="string">'K'</span>
213 </td>
214 </tr>
215 <!-- Key 'filename' -->
216 <tr valign="top">
217 <td bgcolor="#f3f4f5">
218 <p><tt><span class="string">'ounits'</span></tt></p>
219 </td>
220 <td bgcolor="#f3f4f5">
221 <span class="string">'rad'</span>
222 </td>
223 </tr>
224 <!-- Key 'filename' -->
225 <tr valign="top">
226 <td bgcolor="#f3f4f5">
227 <p><tt>GAIN</tt></p>
228 </td>
229 <td bgcolor="#f3f4f5">
230 1e-1
231 </td>
232 </tr>
233 <!-- Key 'filename' -->
234 <tr valign="top">
235 <td bgcolor="#f3f4f5">
236 <p><tt>POLE 1</tt></p>
237 </td>
238 <td bgcolor="#f3f4f5">
239 5e-4
240 </td>
241 </tr>
242 </tbody>
243 </table>
244
245 <p>
246 You can take a look at your models. Since we are interested in the projection of temperature
247 into interferometric data, we can plot the response of TMP*K2RAD against the IFO
248 </p>
249 <div class="fragment"><pre>
250 pl = plist(<span class="string">'f1'</span>,1e-5,<span class="string">'f2'</span>,0.01)
251 resp(K2RAD*TMP,IFO,pl)
252 </pre></div>
253 <p>
254 <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_1.png" alt="inteferometer data" border="1" >
255 </p>
256
257 <h2>Discretize the models<h2>
258 <p>
259 Now discretize the models at fs = 1Hz using the <tt>miir</tt> constructor.
260 After that you will obtain three digital filters
261 </p>
262 <h3> STEP 4: Discretize the three transfer (<b>TMP</b>,<b>IFO</b>,<b>K2RAD</b>) with the
263 MIIR constructor<h3>
264 <p>
265 For example, the model related to temperature noise would be discretized like this:
266 </p>
267 <div class="fragment"><pre>
268 TMPd = miir(TMP,plist(<span class="string">'fs'</span>,1));
269 </pre></div>
270 <h2> Generate white noise data<h2>
271 <p>
272 We will need two initial white noise time series, <b>WN1</b> and <b>WN2</b>, that we will use as a seed
273 to apply our filters and get noise shaped time series.
274 </p>
275 <h3> STEP 5: Generate white noise with the AO constructor<h3>
276 <p>
277 You will need the ao constructor for that. You could use the following settings
278 </p>
279 <table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
280 <colgroup>
281 <col width="15%"/>
282 <col width="35%"/>
283 </colgroup>
284 <thead>
285 <tr valign="top">
286 <th class="categorylist">Key</th>
287 <th class="categorylist">Value</th>
288 </tr>
289 </thead>
290 <tbody>
291 <!-- Key 'filename' -->
292 <tr valign="top">
293 <td bgcolor="#f3f4f5">
294 <p><tt><span class="string">'name'</span></tt></p>
295 </td>
296 <td bgcolor="#f3f4f5">
297 <span class="string">'WN1'</span>
298 </td>
299 </tr>
300 <!-- Key 'filename' -->
301 <tr valign="top">
302 <td bgcolor="#f3f4f5">
303 <p><tt><span class="string">'tsfcn'</span></tt></p>
304 </td>
305 <td bgcolor="#f3f4f5">
306 <span class="string">'randn(size(t))'</span>
307 </td>
308 </tr>
309 <!-- Key 'filename' -->
310 <tr valign="top">
311 <td bgcolor="#f3f4f5">
312 <p><tt><span class="string">'fs'</span></tt></p>
313 </td>
314 <td bgcolor="#f3f4f5">
315 1
316 </td>
317 </tr>
318 <!-- Key 'filename' -->
319 <tr valign="top">
320 <td bgcolor="#f3f4f5">
321 <p><tt><span class="string">'nsecs'</span></tt></p>
322 </td>
323 <td bgcolor="#f3f4f5">
324 250000
325 </td>
326 </tr>
327 </tbody>
328 </table>
329
330 <table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
331 <colgroup>
332 <col width="15%"/>
333 <col width="35%"/>
334 </colgroup>
335 <thead>
336 <tr valign="top">
337 <th class="categorylist">Key</th>
338 <th class="categorylist">Value</th>
339 </tr>
340 </thead>
341 <tbody>
342 <!-- Key 'filename' -->
343 <tr valign="top">
344 <td bgcolor="#f3f4f5">
345 <p><tt><span class="string">'name'</span></tt></p>
346 </td>
347 <td bgcolor="#f3f4f5">
348 <span class="string">'WN2'</span>
349 </td>
350 </tr>
351 <!-- Key 'filename' -->
352 <tr valign="top">
353 <td bgcolor="#f3f4f5">
354 <p><tt><span class="string">'tsfcn'</span></tt></p>
355 </td>
356 <td bgcolor="#f3f4f5">
357 <span class="string">'randn(size(t))'</span>
358 </td>
359 </tr>
360 <!-- Key 'filename' -->
361 <tr valign="top">
362 <td bgcolor="#f3f4f5">
363 <p><tt><span class="string">'fs'</span></tt></p>
364 </td>
365 <td bgcolor="#f3f4f5">
366 1
367 </td>
368 </tr>
369 <!-- Key 'filename' -->
370 <tr valign="top">
371 <td bgcolor="#f3f4f5">
372 <p><tt><span class="string">'nsecs'</span></tt></p>
373 </td>
374 <td bgcolor="#f3f4f5">
375 250000
376 </td>
377 </tr>
378 </tbody>
379 </table>
380
381
382 <h2>Generate the noise data streams<h2>
383 <p>
384 For each noise source you will need to apply the filter that you have designed to
385 the white noise data:
386 </p>
387
388 <h3> STEP 6: Filter white noise WN1 with the TMPd filter </h3>
389 <p>
390 For example, following our notation:
391 </p>
392 <div class="fragment"><pre>
393 T = filter(WN1,TMPd);
394 </pre></div>
395 <h3> STEP 7: Filter white noise WN2 with the IFOd filter</h3>
396 <p>
397 Temperature and interferometric noise are uncorrelated, so we need to use here the
398 second noise time series <b>WN2</b>
399 </p>
400 <p>
401 For example, following our notation:
402 </p>
403 <div class="fragment"><pre>
404 IFO = filter(WN2,IFOd);
405 </pre></div>
406 <h3> STEP 8: Filter white noise WN1 with the TMPd and the K2RADd filter</h3>
407 <p>
408 In this case you need to apply both filters in serial, you can do this in one command
409 by using the 'bank' property of the <tt>filter</tt> method.
410 </p>
411 <p>
412 <table cellspacing="0" class="note" summary="Note" cellpadding="5" border="1">
413 <tr width="90%">
414 <td>
415 Hint: you can input a vector of filters into the <tt>filter</tt> method and
416 ask it to filter the data in 'parallel' or in 'serial' (the one we are interested
417 here) by doing the following
418 <div class="fragment"><pre>
419 b = filter(<b>WN1</b>,[<b>TMPd</b> <b>K2RADd</b>],plist(<span class="string">'bank'</span>,<span class="string">'serial'</span>));
420 </pre></div>
421
422 </td>
423 </tr>
424 </table>
425 </p>
426 <h3> STEP 9: Add the IFO noise to the K2RAD noise</h3>
427 <p>
428 At this point the IFO represents the purely interferometric noise and the K2RAD the
429 contribution to interferometric noise coming from temperature. You need to add both to
430 get the final interferometric data. This only requires to add both AOs.
431 </p>
432
433 <h2>Perform the noise projection<h3>
434 <p>
435 Here we will reproduce the main steps performed in topic 3 analysis: power spectral and
436 transfer function estimation.
437 </p>
438
439 <h3> STEP 10: Split the data streams </h3>
440 <p>
441 We will do the analysis with data in the region going from 1e5 to 2e5 seconds to avoid
442 initial transients. You must then split your two data streams introducing the following
443 parameters in the <tt>split</tt> method.
444 </p>
445
446 <table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
447 <colgroup>
448 <col width="15%"/>
449 <col width="35%"/>
450 </colgroup>
451 <thead>
452 <tr valign="top">
453 <th class="categorylist">Key</th>
454 <th class="categorylist">Value</th>
455 </tr>
456 </thead>
457 <tbody>
458 <!-- Key 'filename' -->
459 <tr valign="top">
460 <td bgcolor="#f3f4f5">
461 <p><tt><span class="string">'times'</span></tt></p>
462 </td>
463 <td bgcolor="#f3f4f5">
464 [1e5 2e5]
465 </td>
466 </tr>
467 </tbody>
468 </table>
469 <p>
470 After the splitting you must have two data streams that plot together should look like the
471 ones below. The code should be similar to one in the following lines
472
473 <div class="fragment"><pre>
474 Ts = split(T,plist(<span class="string">'times'</span>,[1e5 2e5]))
475 Ts.setName(<span class="string">'Temperature'</span>)
476 IFOs = split(IFO_all,plist(<span class="string">'times'</span>,[1e5 2e5]))
477 IFOs.setName(<span class="string">'Interferometer'</span>)
478 <span class="comment">% Plot in different panels with 'subplots' options</span>
479 iplot(Ts,IFOs,plist(<span class="string">'arrangement'</span>,<span class="string">'subplots'</span>))
480 </pre></div>
481
482 <p>
483 <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_3.png" alt="inteferometer data" border="1" >
484 </p>
485
486 <h3> STEP 11: Compute power spectral estimates for the temperature and interferometric data </h3>
487 <p>
488 Here you need to apply <tt>lpsd</tt> or <tt>psd</tt> methods. For example:
489 </p>
490 <div class="fragment"><pre>
491 pl = plist(<span class="string">'order'</span>,1,<span class="string">'scale'</span>,<span class="string">'ASD'</span>)
492 psd_T = lpsd(Ts,pl)
493 psd_IFO = lpsd(IFOs,pl)
494 </pre></div>
495 <p>
496 The resulting spectrum should look like this
497 </p>
498 <p>
499 <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_4.png" alt="inteferometer data" width="800px" border="1" >
500 </p>
501
502 <h3> STEP 12: Compute transfer function estimate for the temperature and interferometric data </h3>
503 <p>
504 Here you need to apply <tt>ltfe</tt> or <tt>tfe</tt> methods. For example:
505 </p>
506 <div class="fragment"><pre>
507 T2IFO = ltfe(Ts,IFOs)
508 T2IFO.setName(<span class="string">'Transfer function'</span>)
509 </pre></div>
510 <p>
511 You can now compare the transfer function model with the estimation obtained from the data:
512 </p>
513 <div class="fragment"><pre>
514 pl = plist(<span class="string">'f1'</span>,1e-5,<span class="string">'f2'</span>,1)
515 iplot(T2IFO,resp(K2RAD,pl))
516 </pre></div>
517 <p>
518 <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_5.png" alt="inteferometer data" width="800px" border="1" >
519 </p>
520 <h3> STEP 13: Project the temperature noise </h3>
521 <p>
522 Reproducing the analysis performed in topic 3 you will be able to project the temperature noise contribution
523 into interferometric noise. The result obtained should be the one in the figure below and the code you will need
524 for that should be similar to this:
525 </p>
526
527 <div class="fragment"><pre>
528 <span class="comment">% Compute projection</span>
529 Projection = abs(T2IFO).*psd_T;
530 Projection.simplifyYunits
531 Projection.setName;
532 <span class="comment">% Plot against interferometer noise</span>
533 iplot(psd_IFO,Projection)
534 </pre></div>
535
536 <p>
537 <img src="images/ltpda_training_1/topic4/Ifo2TempEx_T4_6.png" alt="inteferometer data" width="800px" border="1" >
538 </p>
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556 </p>
557
558 <br>
559 <br>
560 <table class="nav" summary="Navigation aid" border="0" width=
561 "100%" cellpadding="0" cellspacing="0">
562 <tr valign="top">
563 <td align="left" width="20"><a href="ltpda_training_topic_4_4_2.html"><img src=
564 "b_prev.gif" border="0" align="bottom" alt=
565 "By defining filter properties"></a>&nbsp;</td>
566
567 <td align="left">By defining filter properties</td>
568
569 <td>&nbsp;</td>
570
571 <td align="right">Topic 5 - Model fitting</td>
572
573 <td align="right" width="20"><a href=
574 "ltpda_training_topic_5.html"><img src="b_next.gif" border="0" align=
575 "bottom" alt="Topic 5 - Model fitting"></a></td>
576 </tr>
577 </table><br>
578
579 <p class="copy">&copy;LTP Team</p>
580 </body>
581 </html>