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−<!-- $Id: sigproc_diff_content.html,v 1.10 2011/04/05 08:12:13 hewitson Exp $ -->
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−<!-- ===== link box: Begin ===== -->
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−<p>
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−  <table border="1"  width="80%">
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−    <tr>
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−      <td>
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−        <table border="0" cellpadding="5" class="categorylist" width="100%">
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−          <colgroup>
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−            <col width="37%"/>
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−            <col width="63%"/>
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−          </colgroup>
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−          <tbody>
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−            <tr valign="top">
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−              <td>
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−                <a href="#description">Description</a>
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−              </td>
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−              <td>Discrete derivative estimation in LTPDA.</td>
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−            </tr>
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−            <tr valign="top">
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−              <td>
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−                <a href="#algorithm">Algorithm</a>
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−              </td>
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−              <td>Derivatives Algorithms.</td>
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−            </tr>
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−            <tr valign="top">
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−              <td>
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−                <a href="#examples">Examples</a>
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−              </td>
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−              <td>Usage examples of discrete derivative estimation tools.</td>
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−            </tr>
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−            <tr valign="top">
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−              <td>
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−                <a href="#references">References</a>
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−              </td>
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−              <td>Bibliographic references.</td>
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−            </tr>
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−          </tbody>
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−        </table>
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−      </td>
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−    </tr>
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−  </table>
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−</p>
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−<!-- ===== link box: End ====== -->
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−
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−
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−
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−<p>
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−  
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−</p>
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−
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−
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−<h2><a name="description">Derivative calculation for dicrete data series</a></h2>
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−
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−<p>
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−  Derivative estimation on discrete data series is implemented by the function
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−  <a href="matlab:doc('ao/diff')">ao/diff</a>.
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−  This function embeds several algorithms for the calculation
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−  of zero, first and second order derivative. Where with zero order derivative we intend
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−  a particular category of data smoothers [1].
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−</p>
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−
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−
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−
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−<h2><a name="algorithm">Algorithm</a></h2>
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−
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−<p>
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−  <table cellspacing="0" class="body" cellpadding="2" border="0" width="80%">
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−    <colgroup>
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−      <col width="15%"/>
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−      <col width="85%"/>
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−    </colgroup>
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−    <thead>
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−      <tr valign="top">
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−        <th class="categorylist">Method</th>
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−        <th class="categorylist">Description</th>
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−      </tr>
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−    </thead>
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−    <tbody>
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−      <tr valign="top">
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−        <td bgcolor="#f3f4f5">
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−          <p><span class="string">'2POINT'</span></p>
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−        </td>
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−        <td bgcolor="#f3f4f5">
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−          <p>
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−            Compute first derivative with two point equation according to:
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−            <div align="center">
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−              <IMG src="images/sigproc_diff_algo01.gif" align="center" border="0">
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−            </div>
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−          </p>
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−        </td>
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−      </tr>
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−      <tr valign="top">
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−        <td bgcolor="#f3f4f5">
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−          <p><span class="string">'3POINT'</span></p>
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−        </td>
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−        <td bgcolor="#f3f4f5">
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−          <p>
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−            Compute first derivative with three point equation according to:
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−            <div align="center">
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−              <IMG src="images/sigproc_diff_algo02.gif" align="center" border="0">
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−            </div>
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−          </p>
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−        </td>
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−      </tr>
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−      <tr valign="top">
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−        <td bgcolor="#f3f4f5">
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−          <p><span class="string">'5POINT'</span></p>
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−        </td>
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−        <td bgcolor="#f3f4f5">
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−          <p>
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−            Compute first derivative with five point equation according to:
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−            <div align="center">
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−              <IMG src="images/sigproc_diff_algo03.gif" align="center" border="0">
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−            </div>
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−          </p>
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−        </td>
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−      </tr>
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−      <tr valign="top">
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−        <td bgcolor="#f3f4f5">
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−          <p><span class="string">'FPS'</span></p>
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−        </td>
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−        <td bgcolor="#f3f4f5">
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−          <p>
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−            Five Point Stencil is a generalized method to calculate zero, first and second
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−            order discrete derivative of a given time series. Derivative approximation,
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−            at a given time <i>t = kT</i> (<i>k</i> being an integer and <i>T</i>
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−            being the sampling time), is calculated by means of finite differences
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−            between the element at <i>t</i> with its four neighbors:
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−            <div align="center">
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−              <IMG src="images/sigproc_diff_algo04.gif" align="center" border="0">
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−            </div>
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−          </p>
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−          <p>
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−            It can be demonstrated that the coefficients of the expansion can be
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−            expressed as a function of one of them [1]. This allows the construction
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−            of a family of discrete derivative estimators characterized by a
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−            good low frequency accuracy and a smoothing behavior at high frequencies
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−            (near the nyquist frequency). <br/>
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−            Non-trivial values for the <span class="string">'COEFF'</span> parameter are:
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−            <ul>
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−              <li> Parabolic fit approximation <br/>
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−              These coefficients can be obtained by a parabolic fit procedure on
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−              a generic set of data [1].
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−              <ul>
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−                <li> Zeroth order -3/35
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−                <li> First order -1/5
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−                <li> Second order 2/7
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−              </ul>
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−              <li> Taylor series expansion <br/>
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−              These coefficients can be obtained by a series expansion of a generic set of data [1 - 3].
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−              <ul>
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−                <li> First order 1/12
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−                <li> Second order -1/12
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−              </ul>
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−              <li> PI <br/>
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−              This coefficient allows to define a second derivative estimator with
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−              a notch feature at the nyquist frequency [1].
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−              <ul>
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−                <li> Second order 1/4
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−              </ul>
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−            </ul>
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−          </p>
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−        </td>
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−      </tr>
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−    </tbody>
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−  </table>
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−</p>
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−
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−<h2><a name="examples"></a>Examples</h2>
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−
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−Consider <tt>a</tt> as a time series analysis object. First and second
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−derivative of <tt>a</tt> can be easily obtained with a call to 
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−<a href="matlab:doc('ao/diff')">diff</a>. Please refer to 
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−<a href="matlab:doc('ao/diff')">ao/diff</a> documantation page for the
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−meaning of any parameter.
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−<p>
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−  Frequency response of first and second order estimators is reported in 
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−  figures 1 and 2 respectively.
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−</p>
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−
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−<h3>First derivative</h3>
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−
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−<div class="fragment"><pre>
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−    
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−    pl = plist(...
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−      <span class="string">'method'</span>, <span class="string">'2POINT'</span>);
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−    b = diff(a, pl);
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−    
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−    pl = plist(...
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−      <span class="string">'method'</span>, <span class="string">'ORDER2SMOOTH'</span>);
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−    c = diff(a, pl);
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−    
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−    pl = plist(...
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−      <span class="string">'method'</span>, <span class="string">'3POINT'</span>);
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−    d = diff(a, pl);
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−    
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−    pl = plist(...
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−      <span class="string">'method'</span>, <span class="string">'5POINT'</span>);
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−    e = diff(a, pl);
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−    
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−    pl = plist(...
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−      <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
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−      <span class="string">'ORDER'</span>, <span class="string">'FIRST'</span>, ...
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−      <span class="string">'COEFF'</span>, -1/5);
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−    f = diff(a, pl);
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−    
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−</pre></div>
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−
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−<h3>Second derivative</h3>
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−
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−<div class="fragment"><pre>
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−    
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−    pl = plist(...
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−      <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
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−      <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
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−      <span class="string">'COEFF'</span>, 2/7);
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−    b = diff(a, pl);
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−    
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−    pl = plist(...
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−      <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
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−      <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
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−      <span class="string">'COEFF'</span>, -1/12);
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−    c = diff(a, pl);
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−    
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−    pl = plist(...
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−      <span class="string">'method'</span>, <span class="string">'FPS'</span>, ...
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−      <span class="string">'ORDER'</span>, <span class="string">'SECOND'</span>, ...
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−      <span class="string">'COEFF'</span>, 1/4);
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−    d = diff(a, pl);
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−    
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−</pre></div>
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−
+
−
+
−<div align="center">
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−  <p>
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−    <IMG src="images/sigproc_diff_algo05.png" align="center" border="0">
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−  </p>
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−  <p>
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−    <b> Figure 1:</b> Frequency response of first derivative estimators.
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−  </p>
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−  
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−</div>
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−<div align="center">
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−  <p>
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−  </p>
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−  <IMG src="images/sigproc_diff_algo06.png" align="center" border="0">
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−  <p>
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−    <b> Figure 2:</b> Frequency response of second derivative estimators.
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−  </p>
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−</div>
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−
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−
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−<h2><a name="references">References</a></h2>
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−
+
−<ol>
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−  <li> L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative
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−  estimation in LISA Pathfinder data reduction, 
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−  <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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−  L. Ferraioli, M. Hueller and S. Vitale, Discrete derivative
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−  estimation in LISA Pathfinder data reduction
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−  <a href="matlab:web('http://arxiv.org/abs/0903.0324v1','-browser')">arXiv:0903.0324v1</a>
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−  <li> Steven E. Koonin and Dawn C. Meredith, Computational Physics, Westview Press (1990).
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−  <li> John H. Mathews, Computer derivations of numerical differentiation formulae,
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−  <i>Int. J. Math. Educ. Sci. Technol.<i>, 34:2, 280 - 287.
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−</ol>
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