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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
parents f0afece42f48
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Daniele Nicolodi <nicolodi@science.unitn.it>
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1 <!-- $Id: sigproc_diff_content.html,v 1.10 2011/04/05 08:12:13 hewitson Exp $ -->
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2
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3 <!-- ================================================== -->
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4 <!-- BEGIN CONTENT FILE -->
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5 <!-- ================================================== -->
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6 <!-- ===== link box: Begin ===== -->
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7 <p>
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8 <table border="1" width="80%">
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9 <tr>
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10 <td>
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11 <table border="0" cellpadding="5" class="categorylist" width="100%">
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12 <colgroup>
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13 <col width="37%"/>
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14 <col width="63%"/>
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15 </colgroup>
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16 <tbody>
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17 <tr valign="top">
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18 <td>
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19 <a href="#description">Description</a>
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20 </td>
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21 <td>Discrete derivative estimation in LTPDA.</td>
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22 </tr>
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23 <tr valign="top">
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24 <td>
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25 <a href="#algorithm">Algorithm</a>
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26 </td>
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27 <td>Derivatives Algorithms.</td>
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28 </tr>
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29 <tr valign="top">
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30 <td>
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31 <a href="#examples">Examples</a>
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32 </td>
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33 <td>Usage examples of discrete derivative estimation tools.</td>
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34 </tr>
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35 <tr valign="top">
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36 <td>
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37 <a href="#references">References</a>
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38 </td>
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39 <td>Bibliographic references.</td>
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40 </tr>
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41 </tbody>
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42 </table>
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43 </td>
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44 </tr>
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45 </table>
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46 </p>
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47 <!-- ===== link box: End ====== -->
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48
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49
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50
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51 <p>
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52
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53 </p>
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54
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55
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56 <h2><a name="description">Derivative calculation for dicrete data series</a></h2>
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57
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58 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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59 Derivative estimation on discrete data series is implemented by the function
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Daniele Nicolodi <nicolodi@science.unitn.it>
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60 <a href="matlab:doc('ao/diff')">ao/diff</a>.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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61 This function embeds several algorithms for the calculation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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62 of zero, first and second order derivative. Where with zero order derivative we intend
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Daniele Nicolodi <nicolodi@science.unitn.it>
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63 a particular category of data smoothers [1].
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64 </p>
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65
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66
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67
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68 <h2><a name="algorithm">Algorithm</a></h2>
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69
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70 <p>
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71 <table cellspacing="0" class="body" cellpadding="2" border="0" width="80%">
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72 <colgroup>
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73 <col width="15%"/>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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74 <col width="85%"/>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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75 </colgroup>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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76 <thead>
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77 <tr valign="top">
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78 <th class="categorylist">Method</th>
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79 <th class="categorylist">Description</th>
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80 </tr>
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81 </thead>
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82 <tbody>
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83 <tr valign="top">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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84 <td bgcolor="#f3f4f5">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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85 <p><span class="string">'2POINT'</span></p>
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86 </td>
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87 <td bgcolor="#f3f4f5">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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88 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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89 Compute first derivative with two point equation according to:
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90 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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91 <IMG src="images/sigproc_diff_algo01.gif" align="center" border="0">
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92 </div>
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93 </p>
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94 </td>
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95 </tr>
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96 <tr valign="top">
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97 <td bgcolor="#f3f4f5">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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98 <p><span class="string">'3POINT'</span></p>
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99 </td>
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100 <td bgcolor="#f3f4f5">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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101 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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102 Compute first derivative with three point equation according to:
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103 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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104 <IMG src="images/sigproc_diff_algo02.gif" align="center" border="0">
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105 </div>
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106 </p>
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107 </td>
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108 </tr>
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109 <tr valign="top">
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110 <td bgcolor="#f3f4f5">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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111 <p><span class="string">'5POINT'</span></p>
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112 </td>
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113 <td bgcolor="#f3f4f5">
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114 <p>
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115 Compute first derivative with five point equation according to:
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116 <div align="center">
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117 <IMG src="images/sigproc_diff_algo03.gif" align="center" border="0">
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118 </div>
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119 </p>
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120 </td>
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121 </tr>
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122 <tr valign="top">
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123 <td bgcolor="#f3f4f5">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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124 <p><span class="string">'FPS'</span></p>
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125 </td>
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126 <td bgcolor="#f3f4f5">
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127 <p>
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128 Five Point Stencil is a generalized method to calculate zero, first and second
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129 order discrete derivative of a given time series. Derivative approximation,
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130 at a given time <i>t = kT</i> (<i>k</i> being an integer and <i>T</i>
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131 being the sampling time), is calculated by means of finite differences
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132 between the element at <i>t</i> with its four neighbors:
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133 <div align="center">
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134 <IMG src="images/sigproc_diff_algo04.gif" align="center" border="0">
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135 </div>
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136 </p>
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137 <p>
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138 It can be demonstrated that the coefficients of the expansion can be
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139 expressed as a function of one of them [1]. This allows the construction
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Daniele Nicolodi <nicolodi@science.unitn.it>
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140 of a family of discrete derivative estimators characterized by a
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Daniele Nicolodi <nicolodi@science.unitn.it>
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141 good low frequency accuracy and a smoothing behavior at high frequencies
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142 (near the nyquist frequency). <br/>
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143 Non-trivial values for the <span class="string">'COEFF'</span> parameter are:
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144 <ul>
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145 <li> Parabolic fit approximation <br/>
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146 These coefficients can be obtained by a parabolic fit procedure on
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Daniele Nicolodi <nicolodi@science.unitn.it>
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147 a generic set of data [1].
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148 <ul>
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149 <li> Zeroth order -3/35
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150 <li> First order -1/5
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151 <li> Second order 2/7
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152 </ul>
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153 <li> Taylor series expansion <br/>
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154 These coefficients can be obtained by a series expansion of a generic set of data [1 - 3].
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155 <ul>
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156 <li> First order 1/12
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157 <li> Second order -1/12
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158 </ul>
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159 <li> PI <br/>
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160 This coefficient allows to define a second derivative estimator with
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Daniele Nicolodi <nicolodi@science.unitn.it>
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161 a notch feature at the nyquist frequency [1].
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162 <ul>
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163 <li> Second order 1/4
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164 </ul>
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165 </ul>
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166 </p>
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167 </td>
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168 </tr>
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169 </tbody>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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170 </table>
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171 </p>
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172
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173 <h2><a name="examples"></a>Examples</h2>
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174
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175 Consider <tt>a</tt> as a time series analysis object. First and second
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176 derivative of <tt>a</tt> can be easily obtained with a call to
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177 <a href="matlab:doc('ao/diff')">diff</a>. Please refer to
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178 <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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179 meaning of any parameter.
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180 <p>
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181 Frequency response of first and second order estimators is reported in
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182 figures 1 and 2 respectively.
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183 </p>
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184
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185 <h3>First derivative</h3>
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186
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187 <div class="fragment"><pre>
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188
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189 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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190 <span class="string">'method'</span>, <span class="string">'2POINT'</span>);
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191 b = diff(a, pl);
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192
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193 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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194 <span class="string">'method'</span>, <span class="string">'ORDER2SMOOTH'</span>);
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195 c = diff(a, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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196
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Daniele Nicolodi <nicolodi@science.unitn.it>
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197 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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198 <span class="string">'method'</span>, <span class="string">'3POINT'</span>);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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199 d = diff(a, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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200
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Daniele Nicolodi <nicolodi@science.unitn.it>
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201 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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202 <span class="string">'method'</span>, <span class="string">'5POINT'</span>);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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203 e = diff(a, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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204
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Daniele Nicolodi <nicolodi@science.unitn.it>
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205 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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206 <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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207 <span class="string">'ORDER'</span>, <span class="string">'FIRST'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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208 <span class="string">'COEFF'</span>, -1/5);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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209 f = diff(a, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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210
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Daniele Nicolodi <nicolodi@science.unitn.it>
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211 </pre></div>
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212
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213 <h3>Second derivative</h3>
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214
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Daniele Nicolodi <nicolodi@science.unitn.it>
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215 <div class="fragment"><pre>
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216
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Daniele Nicolodi <nicolodi@science.unitn.it>
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217 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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218 <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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219 <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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220 <span class="string">'COEFF'</span>, 2/7);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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221 b = diff(a, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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222
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223 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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224 <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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225 <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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226 <span class="string">'COEFF'</span>, -1/12);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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227 c = diff(a, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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228
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Daniele Nicolodi <nicolodi@science.unitn.it>
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229 pl = plist(...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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230 <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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231 <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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232 <span class="string">'COEFF'</span>, 1/4);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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233 d = diff(a, pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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234
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Daniele Nicolodi <nicolodi@science.unitn.it>
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235 </pre></div>
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236
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237
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Daniele Nicolodi <nicolodi@science.unitn.it>
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238 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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239 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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240 <IMG src="images/sigproc_diff_algo05.png" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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241 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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242 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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243 <b> Figure 1:</b> Frequency response of first derivative estimators.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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244 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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245
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Daniele Nicolodi <nicolodi@science.unitn.it>
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246 </div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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247 <div align="center">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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248 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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249 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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250 <IMG src="images/sigproc_diff_algo06.png" align="center" border="0">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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251 <p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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252 <b> Figure 2:</b> Frequency response of second derivative estimators.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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253 </p>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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254 </div>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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255
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Daniele Nicolodi <nicolodi@science.unitn.it>
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256
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Daniele Nicolodi <nicolodi@science.unitn.it>
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257 <h2><a name="references">References</a></h2>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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258
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Daniele Nicolodi <nicolodi@science.unitn.it>
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259 <ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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260 <li> L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative
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Daniele Nicolodi <nicolodi@science.unitn.it>
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261 estimation in LISA Pathfinder data reduction,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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262 <a href="matlab:web('http://www.iop.org/EJ/abstract/0264-9381/26/9/094013/','-browser')">Class. Quantum Grav. 26 (2009) 094013.</a>. <br/>
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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263 L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative
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Daniele Nicolodi <nicolodi@science.unitn.it>
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264 estimation in LISA Pathfinder data reduction
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Daniele Nicolodi <nicolodi@science.unitn.it>
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265 <a href="matlab:web('http://arxiv.org/abs/0903.0324v1','-browser')">arXiv:0903.0324v1</a>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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266 <li> Steven E. Koonin and Dawn C. Meredith, Computational Physics, Westview Press (1990).
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Daniele Nicolodi <nicolodi@science.unitn.it>
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267 <li> John H. Mathews, Computer derivations of numerical differentiation formulae,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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268 <i>Int. J. Math. Educ. Sci. Technol.<i>, 34:2, 280 - 287.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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269 </ol>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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270
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Daniele Nicolodi <nicolodi@science.unitn.it>
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271
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Daniele Nicolodi <nicolodi@science.unitn.it>
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272
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Daniele Nicolodi <nicolodi@science.unitn.it>
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273
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Daniele Nicolodi <nicolodi@science.unitn.it>
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274
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Daniele Nicolodi <nicolodi@science.unitn.it>
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275
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Daniele Nicolodi <nicolodi@science.unitn.it>
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276
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277