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<HTML>
<HEAD>
<TITLE>Discrete Fourier series</TITLE>
</HEAD>
<BODY bgcolor="#FFFFFF" fgcolor="#000000">
<p> <font size="+1" color="green"><B>Discrete Fourier series</B></font></P>
<p>
Given <code>2N</code> samples of real data <img src="fft4.png">
(where <i>j</i> <code>= 0;1;2;...;2N-1</code>) taken at equally spaced intervals
<img src="fft5.png">, where <i>T</i> is the period, the corresponding
Fourier series is:</p>
<p>
<center><img align="top" src="fft6.png"></center></p>
<p>
where <img src="fft7.png"> and
<img src="fft1.png"> is the mean value of
<img src="fft8.png">.</p>
<p>
From the original <code>2N</code> data points, we have exactly <code>2N</code>
calculated coefficients, that is, we have <code>N+1</code>
<img src="H.png">'s and <code>N-1</code>
<img src="G.png">'s with "real" information.</p>
<p>
If <i>t</i><code>=0</code> at <img src="fft9.png">, then for each
<img src="fft10.png"> we have <img src="fft11.png">,
for <i>j</i> <code>= 0;1;...;2N-1</code>, and so</p>
<p>
<center><img align="top" src="fft12.png"></center></p>
<p>
The amplitude, <i>A</i>, and the phase, <i>P</i>, are calculated as follows:</p>
<p>
<center><img align="top" src="fft13.png"></center></p>
<p>
<center><img align="top" src="fft14.png"></center></p>
<p>
<center><img align="top" src="fft15.png"></center></p>
<p>
<center><img align="top" src="fft16.png"></center></p>
<P>
<a href="coefficients.htm"><img align="top" border="0" src="../../../shadow_left.gif">
<font size="+1" color="olive">Fourier coefficients</font></a><br />
<a href="restrictions.htm"><img align="top" border="0" src="../../../shadow_right.gif">
<font size="+1" color="olive">Restrictions</font></a></P>
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