User:Tohline/Appendix/Ramblings/FourierSeries
Fourier Series
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
Representations
The following Fourier series representations have been drawn primarily from pp. 458 - 460 of the 1971 (19th) edition of the CRC's Standard Mathematical Tables, published by the Chemical Rubber Co., Cleveland, Ohio, U.S.A.
Standard
If <math>~f(x)</math> is a bounded periodic function of period <math>~2L</math>, it may be represented by the Fourier series,
|
<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>~ \frac{1}{L} \int_{-L}^{L} f(x) \cos\biggl( \frac{n\pi x}{L} \biggr) dx </math> |
for | <math>~n = 0, 1, 2, 3, \dots \, ;</math> | |
|
<math>~b_n</math> |
<math>~=</math> |
<math>~ \frac{1}{L} \int_{-L}^{L} f(x) \sin\biggl( \frac{n\pi x}{L} \biggr) dx </math> |
for | <math>~n = 1, 2, 3, \dots </math> |
Alternate
Equivalently, we may write the Fourier series expression in the form,
|
<math>~f(x)</math> |
<math>~=</math> |
<math>~ \frac{a_0}{2} + \sum_{n=1}^{\infty} c_n\cos \biggl(\frac{n\pi x}{L} + \phi_n\biggr) \, , </math> |
where,
|
<math>~c_n = \sqrt{a_n^2 + b_n^2}</math> |
and |
<math>~\phi_n = \tan^{-1}\biggl(\frac{-b_n}{a_n}\biggr)</math> |
and in which case, <math>~a_n = c_n\sin\phi_n</math> and <math>~b_n = c_n\cos\phi_n</math> .
Complex
Here we make use of the exponential/complex relation — also referred to as Euler's equation,
<math>~e^{i\alpha} = \cos\alpha + i \sin\alpha \, ,</math>
in which case we may write,
|
<math>~\cos\alpha = \frac{1}{2} \biggl[ e^{i\alpha} + e^{-i\alpha}\biggr] \, ,</math> |
and |
<math>~\sin\alpha = \frac{1}{2i} \biggl[ e^{i\alpha} - e^{-i\alpha}\biggr]\, .</math> |
Employing these definitions of the trigonometric relations <math>~\cos\alpha</math> and <math>~\sin\alpha</math>, the standard representation of the Fourier series may be rewritten as,
|
<math>~f(x)</math> |
<math>~=</math> |
<math>~ \frac{1}{2}\sum_{n = -\infty}^{n = + \infty} c_n e^{i\omega_n x} \, , </math> |
where,
|
<math>~\omega_n</math> |
<math>~=</math> |
<math>~ \frac{n\pi }{L} </math> |
for | <math>~n = 0, \pm 1, \pm 2, \dots \, ;</math> |
and the, now complex, coefficients,
|
<math>~c_n</math> |
<math>~=</math> |
<math>~ \frac{1}{L} \int_{-L}^{L} f(x) e^{-i\omega_n x} dx </math> |
for | <math>~n = 0, \pm 1, \pm 2, \pm 3, \dots \, .</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 |