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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>Whitening noise (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 "ltpda_training_topic_2_5.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Remove trends from a time-series AO"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "ltpda_training_topic_2_7.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Select and find data from an AO"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Whitening noise</h1>
37 <hr>
38
39 <p>
40 <p>
41 The LTPDA toolbox offers various ways in which you could whiten data. Perhaps you know the whitening
42 filter you want to use, in which case you can build the filter and filter the data. Alternatively, you
43 may have a model for the spectral content of the data, in which case you can use the method <tt>ao/whiten1D</tt>
44 if you are dealing with single, uncorrelated data streams, or <tt>ao/whiten2D</tt> if you have a pair of
45 correlated data streams. You can also use <tt>ao/whiten1D</tt> in the case where you don't have a model for
46 the spectral content of the data. In this case, the method
47 calculates the spectrum of the data, re-bins the spectrum so to
48 reduce the individual points fluctuations, and fits a model
49 of the spectrum as a series of partial fractions z-domain filters.
50 </p>
51 <p>
52 The whitening algorithms are highly configurable and accept a large number of parameters. The main ones that
53 we will change from the defaults in the following examples are
54 <table cellspacing="0" class="body" cellpadding="2" border="0" width="80%">
55 <colgroup>
56 <col width="25%"/>
57 <col width="75%"/>
58 </colgroup>
59 <thead>
60 <tr valign="top">
61 <th class="categorylist">Key</th>
62 <th class="categorylist">Description</th>
63 </tr>
64 </thead>
65 <tbody>
66 <!-- Key 'Plot' -->
67 <tr valign="top">
68 <td bgcolor="#f3f4f5">
69 <p><tt>PLOT</tt></p>
70 </td>
71 <td bgcolor="#f3f4f5">
72 <p>Plot the result of the fitting as it proceeds.</p>
73 </td>
74 </tr>
75 <!-- Key 'MAXORDER' -->
76 <tr valign="top">
77 <td bgcolor="#f3f4f5">
78 <p><tt>MAXORDER</tt></p>
79 </td>
80 <td bgcolor="#f3f4f5">
81 <p>Specify the maximum allowed model order that can be fit.</p>
82 </td>
83 </tr>
84 <!-- Key 'weights' -->
85 <tr valign="top">
86 <td bgcolor="#f3f4f5">
87 <p><tt>WEIGHTS</tt></p>
88 </td>
89 <td bgcolor="#f3f4f5">
90 <p>Choose the way the data is weighted in the fitting procedure.</p>
91 </td>
92 </tr>
93 <!-- Key 'RMSVAR' -->
94 <tr valign="top">
95 <td bgcolor="#f3f4f5">
96 <p><tt>RMSVAR</tt></p>
97 </td>
98 <td bgcolor="#f3f4f5">
99 <p>Check if the variation of the RMS error is smaller than 10^(-b),
100 where b is the value given in the plist.</p>
101 </td>
102 </tr>
103 </tbody>
104 </table>
105 </p>
106 <p>
107 We will start by whitening some data using this last method, i.e., allowing <tt>whiten1D</tt> to determine
108 the whitening filter from the data itself.
109 </p>
110 <p>
111 The data we will whiten can be found in your data packet in the 'topic2' sub-directory.
112 </p>
113 <p>
114 We start by loading the mat file:
115 </p>
116 <div class="fragment"><pre>
117 a = ao(<span class="string">'topic2/whiten.mat'</span>);
118 </pre>
119 </div>
120 <p>
121 The AO stored in the variable <tt>a</tt> is a coloured noise time-series.
122 Let's have a look at this times series using <tt>iplot</tt>.
123 </p>
124 <div class="fragment"><pre>
125 >> iplot(a);
126 </pre></div>
127 <p>The result should be similar to: </p>
128 <img src="images/ltpda_training_1/topic2/coloured.png" alt="coloured" border="1">
129 <p>
130 Before we can whiten the data, we have to define the parameter list for the whitening tool:
131 </p>
132 <div class="fragment"><pre>
133 pl = plist(...
134 <span class="string">'Plot'</span>, true, ...
135 <span class="string">'MaxOrder'</span>, 9, ...
136 <span class="string">'Weights'</span>, 2);</pre>
137 </div>
138 <p>
139 Now we can call the whitening function <tt>whiten1D</tt> with our input AO, <tt>a</tt> and the
140 parameter list <tt>pl</tt>:
141 </p>
142 <div class="fragment"><pre> >> aw = whiten1D(a,pl); </pre></div>
143 <p>
144 To compare the whitened data with the coloured noise we compute the power spectrum (for details see <a
145 href="ltpda_training_topic_3_2.html"><tt>Power spectral density estimation</tt></a>):
146 </p>
147 <div class="fragment"><pre>
148 awxx = aw.lpsd;
149 axx = a.lpsd;
150 </pre></div>
151 <p>
152 and finally plot our result in the frequency domain; in particular we plot the whitened data (<tt>awxx</tt>) compared to the coloured noise that was our input (<tt>axx</tt>).
153 </p>
154 </pre> </div>
155 <div class="fragment"><pre>
156 iplot(axx, awxx);
157 </pre>
158 </div>
159 <img src="images/ltpda_training_1/topic2/whiten.png" alt="white" border="1">
160
161 </p>
162
163 <br>
164 <br>
165 <table class="nav" summary="Navigation aid" border="0" width=
166 "100%" cellpadding="0" cellspacing="0">
167 <tr valign="top">
168 <td align="left" width="20"><a href="ltpda_training_topic_2_5.html"><img src=
169 "b_prev.gif" border="0" align="bottom" alt=
170 "Remove trends from a time-series AO"></a>&nbsp;</td>
171
172 <td align="left">Remove trends from a time-series AO</td>
173
174 <td>&nbsp;</td>
175
176 <td align="right">Select and find data from an AO</td>
177
178 <td align="right" width="20"><a href=
179 "ltpda_training_topic_2_7.html"><img src="b_next.gif" border="0" align=
180 "bottom" alt="Select and find data from an AO"></a></td>
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182 </table><br>
183
184 <p class="copy">&copy;LTP Team</p>
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