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| author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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| date | Wed, 23 Nov 2011 19:22:13 +0100 |
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd"> <html lang="en"> <head> <meta name="generator" content= "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org"> <meta http-equiv="Content-Type" content= "text/html; charset=us-ascii"> <title>Discrete Derivative (LTPDA Toolbox)</title> <link rel="stylesheet" href="docstyle.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.52.2"> <meta name="description" content= "Presents an overview of the features, system requirements, and starting the toolbox."> </head> <body> <a name="top_of_page" id="top_of_page"></a> <p style="font-size:1px;"> </p> <table class="nav" summary="Navigation aid" border="0" width= "100%" cellpadding="0" cellspacing="0"> <tr> <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td> <td valign="baseline" align="right"><a href= "sigproc_applyfilt.html"><img src="b_prev.gif" border="0" align= "bottom" alt="Applying digital filters to data"></a> <a href= "sigproc_spec.html"><img src="b_next.gif" border="0" align= "bottom" alt="Spectral Estimation"></a></td> </tr> </table> <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Discrete Derivative</h1> <hr> <p> <!-- ================================================== --> <!-- BEGIN CONTENT FILE --> <!-- ================================================== --> <!-- ===== link box: Begin ===== --> <p> <table border="1" width="80%"> <tr> <td> <table border="0" cellpadding="5" class="categorylist" width="100%"> <colgroup> <col width="37%"/> <col width="63%"/> </colgroup> <tbody> <tr valign="top"> <td> <a href="#description">Description</a> </td> <td>Discrete derivative estimation in LTPDA.</td> </tr> <tr valign="top"> <td> <a href="#algorithm">Algorithm</a> </td> <td>Derivatives Algorithms.</td> </tr> <tr valign="top"> <td> <a href="#examples">Examples</a> </td> <td>Usage examples of discrete derivative estimation tools.</td> </tr> <tr valign="top"> <td> <a href="#references">References</a> </td> <td>Bibliographic references.</td> </tr> </tbody> </table> </td> </tr> </table> </p> <!-- ===== link box: End ====== --> <p> </p> <h2><a name="description">Derivative calculation for dicrete data series</a></h2> <p> Derivative estimation on discrete data series is implemented by the function <a href="matlab:doc('ao/diff')">ao/diff</a>. This function embeds several algorithms for the calculation of zero, first and second order derivative. Where with zero order derivative we intend a particular category of data smoothers [1]. </p> <h2><a name="algorithm">Algorithm</a></h2> <p> <table cellspacing="0" class="body" cellpadding="2" border="0" width="80%"> <colgroup> <col width="15%"/> <col width="85%"/> </colgroup> <thead> <tr valign="top"> <th class="categorylist">Method</th> <th class="categorylist">Description</th> </tr> </thead> <tbody> <tr valign="top"> <td bgcolor="#f3f4f5"> <p><span class="string">'2POINT'</span></p> </td> <td bgcolor="#f3f4f5"> <p> Compute first derivative with two point equation according to: <div align="center"> <IMG src="images/sigproc_diff_algo01.gif" align="center" border="0"> </div> </p> </td> </tr> <tr valign="top"> <td bgcolor="#f3f4f5"> <p><span class="string">'3POINT'</span></p> </td> <td bgcolor="#f3f4f5"> <p> Compute first derivative with three point equation according to: <div align="center"> <IMG src="images/sigproc_diff_algo02.gif" align="center" border="0"> </div> </p> </td> </tr> <tr valign="top"> <td bgcolor="#f3f4f5"> <p><span class="string">'5POINT'</span></p> </td> <td bgcolor="#f3f4f5"> <p> Compute first derivative with five point equation according to: <div align="center"> <IMG src="images/sigproc_diff_algo03.gif" align="center" border="0"> </div> </p> </td> </tr> <tr valign="top"> <td bgcolor="#f3f4f5"> <p><span class="string">'FPS'</span></p> </td> <td bgcolor="#f3f4f5"> <p> Five Point Stencil is a generalized method to calculate zero, first and second order discrete derivative of a given time series. Derivative approximation, at a given time <i>t = kT</i> (<i>k</i> being an integer and <i>T</i> being the sampling time), is calculated by means of finite differences between the element at <i>t</i> with its four neighbors: <div align="center"> <IMG src="images/sigproc_diff_algo04.gif" align="center" border="0"> </div> </p> <p> It can be demonstrated that the coefficients of the expansion can be expressed as a function of one of them [1]. This allows the construction of a family of discrete derivative estimators characterized by a good low frequency accuracy and a smoothing behavior at high frequencies (near the nyquist frequency). <br/> Non-trivial values for the <span class="string">'COEFF'</span> parameter are: <ul> <li> Parabolic fit approximation <br/> These coefficients can be obtained by a parabolic fit procedure on a generic set of data [1]. <ul> <li> Zeroth order -3/35 <li> First order -1/5 <li> Second order 2/7 </ul> <li> Taylor series expansion <br/> These coefficients can be obtained by a series expansion of a generic set of data [1 - 3]. <ul> <li> First order 1/12 <li> Second order -1/12 </ul> <li> PI <br/> This coefficient allows to define a second derivative estimator with a notch feature at the nyquist frequency [1]. <ul> <li> Second order 1/4 </ul> </ul> </p> </td> </tr> </tbody> </table> </p> <h2><a name="examples"></a>Examples</h2> Consider <tt>a</tt> as a time series analysis object. First and second derivative of <tt>a</tt> can be easily obtained with a call to <a href="matlab:doc('ao/diff')">diff</a>. Please refer to <a href="matlab:doc('ao/diff')">ao/diff</a> documantation page for the meaning of any parameter. <p> Frequency response of first and second order estimators is reported in figures 1 and 2 respectively. </p> <h3>First derivative</h3> <div class="fragment"><pre> pl = plist(... <span class="string">'method'</span>, <span class="string">'2POINT'</span>); b = diff(a, pl); pl = plist(... <span class="string">'method'</span>, <span class="string">'ORDER2SMOOTH'</span>); c = diff(a, pl); pl = plist(... <span class="string">'method'</span>, <span class="string">'3POINT'</span>); d = diff(a, pl); pl = plist(... <span class="string">'method'</span>, <span class="string">'5POINT'</span>); e = diff(a, pl); pl = plist(... <span class="string">'method'</span>, <span class="string">'FPS'</span>, ... <span class="string">'ORDER'</span>, <span class="string">'FIRST'</span>, ... <span class="string">'COEFF'</span>, -1/5); f = diff(a, pl); </pre></div> <h3>Second derivative</h3> <div class="fragment"><pre> pl = plist(... <span class="string">'method'</span>, <span class="string">'FPS'</span>, ... <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ... <span class="string">'COEFF'</span>, 2/7); b = diff(a, pl); pl = plist(... <span class="string">'method'</span>, <span class="string">'FPS'</span>, ... <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ... <span class="string">'COEFF'</span>, -1/12); c = diff(a, pl); pl = plist(... <span class="string">'method'</span>, <span class="string">'FPS'</span>, ... <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ... <span class="string">'COEFF'</span>, 1/4); d = diff(a, pl); </pre></div> <div align="center"> <p> <IMG src="images/sigproc_diff_algo05.png" align="center" border="0"> </p> <p> <b> Figure 1:</b> Frequency response of first derivative estimators. </p> </div> <div align="center"> <p> </p> <IMG src="images/sigproc_diff_algo06.png" align="center" border="0"> <p> <b> Figure 2:</b> Frequency response of second derivative estimators. </p> </div> <h2><a name="references">References</a></h2> <ol> <li> L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative estimation in LISA Pathfinder data reduction, <a href="matlab:web('http://www.iop.org/EJ/abstract/0264-9381/26/9/094013/','-browser')">Class. Quantum Grav. 26 (2009) 094013.</a>. <br/> L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative estimation in LISA Pathfinder data reduction <a href="matlab:web('http://arxiv.org/abs/0903.0324v1','-browser')">arXiv:0903.0324v1</a> <li> Steven E. Koonin and Dawn C. Meredith, Computational Physics, Westview Press (1990). <li> John H. Mathews, Computer derivations of numerical differentiation formulae, <i>Int. J. Math. Educ. Sci. Technol.<i>, 34:2, 280 - 287. </ol> </p> <br> <br> <table class="nav" summary="Navigation aid" border="0" width= "100%" cellpadding="0" cellspacing="0"> <tr valign="top"> <td align="left" width="20"><a href="sigproc_applyfilt.html"><img src= "b_prev.gif" border="0" align="bottom" alt= "Applying digital filters to data"></a> </td> <td align="left">Applying digital filters to data</td> <td> </td> <td align="right">Spectral Estimation</td> <td align="right" width="20"><a href= "sigproc_spec.html"><img src="b_next.gif" border="0" align= "bottom" alt="Spectral Estimation"></a></td> </tr> </table><br> <p class="copy">©LTP Team</p> </body> </html>