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  <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>









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