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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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2 "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd">
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3
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4 <html lang="en">
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5 <head>
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6 <meta name="generator" content=
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7 "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org">
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8 <meta http-equiv="Content-Type" content=
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9 "text/html; charset=us-ascii">
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10
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11 <title>Discrete Derivative (LTPDA Toolbox)</title>
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12 <link rel="stylesheet" href="docstyle.css" type="text/css">
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13 <meta name="generator" content="DocBook XSL Stylesheets V1.52.2">
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14 <meta name="description" content=
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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16 </head>
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17
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18 <body>
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19 <a name="top_of_page" id="top_of_page"></a>
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20
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21 <p style="font-size:1px;">&nbsp;</p>
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22
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23 <table class="nav" summary="Navigation aid" border="0" width=
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24 "100%" cellpadding="0" cellspacing="0">
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25 <tr>
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26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
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27
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28 <td valign="baseline" align="right"><a href=
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29 "sigproc_applyfilt.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Applying digital filters to data"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "sigproc_spec.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Spectral Estimation"></a></td>
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33 </tr>
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34 </table>
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35
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36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Discrete Derivative</h1>
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37 <hr>
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38
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39 <p>
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40
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41 <!-- ================================================== -->
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42 <!-- BEGIN CONTENT FILE -->
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43 <!-- ================================================== -->
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44 <!-- ===== link box: Begin ===== -->
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45 <p>
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46 <table border="1" width="80%">
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47 <tr>
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48 <td>
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49 <table border="0" cellpadding="5" class="categorylist" width="100%">
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50 <colgroup>
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51 <col width="37%"/>
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52 <col width="63%"/>
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53 </colgroup>
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54 <tbody>
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55 <tr valign="top">
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56 <td>
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57 <a href="#description">Description</a>
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58 </td>
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59 <td>Discrete derivative estimation in LTPDA.</td>
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60 </tr>
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61 <tr valign="top">
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62 <td>
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63 <a href="#algorithm">Algorithm</a>
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64 </td>
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65 <td>Derivatives Algorithms.</td>
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66 </tr>
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67 <tr valign="top">
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68 <td>
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69 <a href="#examples">Examples</a>
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70 </td>
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71 <td>Usage examples of discrete derivative estimation tools.</td>
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72 </tr>
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73 <tr valign="top">
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74 <td>
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75 <a href="#references">References</a>
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76 </td>
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77 <td>Bibliographic references.</td>
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78 </tr>
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79 </tbody>
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80 </table>
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81 </td>
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82 </tr>
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83 </table>
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84 </p>
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85 <!-- ===== link box: End ====== -->
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86
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88
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89 <p>
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90
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91 </p>
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92
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93
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94 <h2><a name="description">Derivative calculation for dicrete data series</a></h2>
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95
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96 <p>
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97 Derivative estimation on discrete data series is implemented by the function
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98 <a href="matlab:doc('ao/diff')">ao/diff</a>.
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99 This function embeds several algorithms for the calculation
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100 of zero, first and second order derivative. Where with zero order derivative we intend
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101 a particular category of data smoothers [1].
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102 </p>
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103
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104
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105
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106 <h2><a name="algorithm">Algorithm</a></h2>
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107
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108 <p>
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109 <table cellspacing="0" class="body" cellpadding="2" border="0" width="80%">
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110 <colgroup>
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111 <col width="15%"/>
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112 <col width="85%"/>
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113 </colgroup>
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114 <thead>
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115 <tr valign="top">
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116 <th class="categorylist">Method</th>
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117 <th class="categorylist">Description</th>
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118 </tr>
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119 </thead>
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120 <tbody>
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121 <tr valign="top">
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122 <td bgcolor="#f3f4f5">
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123 <p><span class="string">'2POINT'</span></p>
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124 </td>
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125 <td bgcolor="#f3f4f5">
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126 <p>
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127 Compute first derivative with two point equation according to:
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128 <div align="center">
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129 <IMG src="images/sigproc_diff_algo01.gif" align="center" border="0">
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130 </div>
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131 </p>
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132 </td>
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133 </tr>
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134 <tr valign="top">
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135 <td bgcolor="#f3f4f5">
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136 <p><span class="string">'3POINT'</span></p>
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137 </td>
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138 <td bgcolor="#f3f4f5">
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139 <p>
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140 Compute first derivative with three point equation according to:
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141 <div align="center">
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142 <IMG src="images/sigproc_diff_algo02.gif" align="center" border="0">
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143 </div>
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144 </p>
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145 </td>
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146 </tr>
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147 <tr valign="top">
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148 <td bgcolor="#f3f4f5">
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149 <p><span class="string">'5POINT'</span></p>
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150 </td>
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151 <td bgcolor="#f3f4f5">
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152 <p>
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153 Compute first derivative with five point equation according to:
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154 <div align="center">
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155 <IMG src="images/sigproc_diff_algo03.gif" align="center" border="0">
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156 </div>
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157 </p>
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158 </td>
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159 </tr>
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160 <tr valign="top">
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161 <td bgcolor="#f3f4f5">
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162 <p><span class="string">'FPS'</span></p>
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163 </td>
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164 <td bgcolor="#f3f4f5">
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165 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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166 Five Point Stencil is a generalized method to calculate zero, first and second
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
167 order discrete derivative of a given time series. Derivative approximation,
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
168 at a given time <i>t = kT</i> (<i>k</i> being an integer and <i>T</i>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
169 being the sampling time), is calculated by means of finite differences
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
170 between the element at <i>t</i> with its four neighbors:
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
171 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
172 <IMG src="images/sigproc_diff_algo04.gif" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
173 </div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
174 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
175 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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176 It can be demonstrated that the coefficients of the expansion can be
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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177 expressed as a function of one of them [1]. This allows the construction
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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178 of a family of discrete derivative estimators characterized by a
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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179 good low frequency accuracy and a smoothing behavior at high frequencies
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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180 (near the nyquist frequency). <br/>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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181 Non-trivial values for the <span class="string">'COEFF'</span> parameter are:
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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182 <ul>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
183 <li> Parabolic fit approximation <br/>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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184 These coefficients can be obtained by a parabolic fit procedure on
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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185 a generic set of data [1].
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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186 <ul>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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187 <li> Zeroth order -3/35
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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188 <li> First order -1/5
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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189 <li> Second order 2/7
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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190 </ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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191 <li> Taylor series expansion <br/>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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192 These coefficients can be obtained by a series expansion of a generic set of data [1 - 3].
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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193 <ul>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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194 <li> First order 1/12
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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195 <li> Second order -1/12
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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196 </ul>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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197 <li> PI <br/>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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198 This coefficient allows to define a second derivative estimator with
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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199 a notch feature at the nyquist frequency [1].
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Daniele Nicolodi <nicolodi@science.unitn.it>
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200 <ul>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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201 <li> Second order 1/4
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
202 </ul>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
203 </ul>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
204 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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205 </td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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206 </tr>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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207 </tbody>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
208 </table>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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209 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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210
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Daniele Nicolodi <nicolodi@science.unitn.it>
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211 <h2><a name="examples"></a>Examples</h2>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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212
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Daniele Nicolodi <nicolodi@science.unitn.it>
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213 Consider <tt>a</tt> as a time series analysis object. First and second
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Daniele Nicolodi <nicolodi@science.unitn.it>
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214 derivative of <tt>a</tt> can be easily obtained with a call to
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Daniele Nicolodi <nicolodi@science.unitn.it>
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215 <a href="matlab:doc('ao/diff')">diff</a>. Please refer to
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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216 <a href="matlab:doc('ao/diff')">ao/diff</a> documantation page for the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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217 meaning of any parameter.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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218 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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219 Frequency response of first and second order estimators is reported in
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Daniele Nicolodi <nicolodi@science.unitn.it>
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220 figures 1 and 2 respectively.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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221 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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222
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
223 <h3>First derivative</h3>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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224
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
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225 <div class="fragment"><pre>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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226
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Daniele Nicolodi <nicolodi@science.unitn.it>
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diff changeset
227 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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228 <span class="string">'method'</span>, <span class="string">'2POINT'</span>);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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229 b = diff(a, pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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230
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
231 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
232 <span class="string">'method'</span>, <span class="string">'ORDER2SMOOTH'</span>);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
233 c = diff(a, pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
234
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
235 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
236 <span class="string">'method'</span>, <span class="string">'3POINT'</span>);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
237 d = diff(a, pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
238
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
239 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
240 <span class="string">'method'</span>, <span class="string">'5POINT'</span>);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
241 e = diff(a, pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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242
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
243 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
244 <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
245 <span class="string">'ORDER'</span>, <span class="string">'FIRST'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
246 <span class="string">'COEFF'</span>, -1/5);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
247 f = diff(a, pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
248
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
249 </pre></div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
250
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
251 <h3>Second derivative</h3>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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252
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
253 <div class="fragment"><pre>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
254
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
255 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
256 <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
257 <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
258 <span class="string">'COEFF'</span>, 2/7);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
259 b = diff(a, pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
260
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
261 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
262 <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
263 <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
264 <span class="string">'COEFF'</span>, -1/12);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
265 c = diff(a, pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
266
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
267 pl = plist(...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
268 <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
269 <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
270 <span class="string">'COEFF'</span>, 1/4);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
271 d = diff(a, pl);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
272
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
273 </pre></div>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
274
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
275
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
276 <div align="center">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
277 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
278 <IMG src="images/sigproc_diff_algo05.png" align="center" border="0">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
279 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
280 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
281 <b> Figure 1:</b> Frequency response of first derivative estimators.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
282 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
283
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
284 </div>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
285 <div align="center">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
286 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
287 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
288 <IMG src="images/sigproc_diff_algo06.png" align="center" border="0">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
289 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
290 <b> Figure 2:</b> Frequency response of second derivative estimators.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
291 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
292 </div>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
293
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
294
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
295 <h2><a name="references">References</a></h2>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
296
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
297 <ol>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
298 <li> L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
299 estimation in LISA Pathfinder data reduction,
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
300 <a href="matlab:web('http://www.iop.org/EJ/abstract/0264-9381/26/9/094013/','-browser')">Class. Quantum Grav. 26 (2009) 094013.</a>. <br/>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
301 L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
302 estimation in LISA Pathfinder data reduction
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
303 <a href="matlab:web('http://arxiv.org/abs/0903.0324v1','-browser')">arXiv:0903.0324v1</a>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
304 <li> Steven E. Koonin and Dawn C. Meredith, Computational Physics, Westview Press (1990).
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
305 <li> John H. Mathews, Computer derivations of numerical differentiation formulae,
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
306 <i>Int. J. Math. Educ. Sci. Technol.<i>, 34:2, 280 - 287.
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
307 </ol>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
308
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
309
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
310
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
311
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
312
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
313
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
314
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
315
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
316
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
317 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
318
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
319 <br>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
320 <br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
321 <table class="nav" summary="Navigation aid" border="0" width=
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
322 "100%" cellpadding="0" cellspacing="0">
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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323 <tr valign="top">
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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324 <td align="left" width="20"><a href="sigproc_applyfilt.html"><img src=
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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325 "b_prev.gif" border="0" align="bottom" alt=
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
326 "Applying digital filters to data"></a>&nbsp;</td>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
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327
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328 <td align="left">Applying digital filters to data</td>
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329
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330 <td>&nbsp;</td>
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331
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332 <td align="right">Spectral Estimation</td>
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333
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334 <td align="right" width="20"><a href=
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335 "sigproc_spec.html"><img src="b_next.gif" border="0" align=
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336 "bottom" alt="Spectral Estimation"></a></td>
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337 </tr>
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338 </table><br>
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339
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340 <p class="copy">&copy;LTP Team</p>
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341 </body>
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342 </html>