User:Tohline/Appendix/Ramblings/FourierSeries
Fourier Series
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
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 ]
|
|
|---|
|
© 2014 - 2021 by Joel E. Tohline |