Difference between revisions of "User:Tohline/Appendix/Ramblings/FourierSeries"
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To declare that the bounded periodic function of period <math>~2L</math>, <math>~f(x)</math>, may be represented in the form of a ''Fourier series'', means that, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~f(x)</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
\frac{a_0}{2} + \sum_{n=1}^{\infty} \biggl[ a_n\cos \biggl(\frac{n\pi x}{L}\biggr) + b_n\sin \biggl(\frac{n\pi x}{L}\biggr) \biggr] \, , | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
where, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~a_n</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
<math>~b_n</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
==One-Dimensional Aperture== | ==One-Dimensional Aperture== |
Revision as of 17:57, 10 November 2017
Fourier Series
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To declare that the bounded periodic function of period <math>~2L</math>, <math>~f(x)</math>, may be represented in the form of a Fourier series, means that,
<math>~f(x)</math> |
<math>~=</math> |
<math>~ \frac{a_0}{2} + \sum_{n=1}^{\infty} \biggl[ a_n\cos \biggl(\frac{n\pi x}{L}\biggr) + b_n\sin \biggl(\frac{n\pi x}{L}\biggr) \biggr] \, , </math> |
where,
<math>~a_n</math> |
<math>~=</math> |
<math>~</math> |
<math>~b_n</math> |
<math>~=</math> |
<math>~</math> |
One-Dimensional Aperture
General Concept
Hence, we have,
<math>~A(y_1)</math> |
<math>~=</math> |
<math>~A_0 \sum_j a_j e^{-i[2\pi y_1 Y_j/(\lambda L)]} \, , </math> |
|
<math>~=</math> |
<math>~A_0 \sum_j a_j \biggl[ \cos\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) - i \sin\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \biggr] \, , </math> |
where, now, <math>~A_0 = e^{i2\pi L/\lambda}</math>. When written in this form, it should immediately be apparent why discrete Fourier transform techniques (specifically FFT techniques) are useful tools for evaluation of the complex amplitude, <math>~A</math>.
See Also
- Tohline, J. E., (2008) Computing in Science & Engineering, vol. 10, no. 4, pp. 84-85 — Where is My Digital Holographic Display? [ PDF ]
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