User:Tohline/Appendix/Ramblings/FourierSeries
Fourier Series
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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,
| Standard Fourier Series Expression | |||
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where,
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<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> | |
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<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
Alternatively, if we set <math>~a_n = c_n \cos\phi_n</math> and <math>~b_n = - c_n \sin\phi_n</math>, then,
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<math>~a_n\cos \biggl(\frac{n\pi x}{L}\biggr) + b_n\sin \biggl(\frac{n\pi x}{L}\biggr) </math> |
<math>~=</math> |
<math>~ c_n \cos\phi_n \cos \biggl(\frac{n\pi x}{L}\biggr) - c_n \sin\phi_n \sin \biggl(\frac{n\pi x}{L}\biggr) </math> |
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<math>~=</math> |
<math>~ c_n \cos \biggl(\frac{n\pi x}{L} + \phi_n \biggr) \, , </math> |
in which case we may rewrite the Fourier series expression in the form,
| Alternate Fourier Series Expression | |||
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where,
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<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> |
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> <math>~\Rightarrow</math> <math>~e^{-i\alpha} = \cos\alpha - i \sin\alpha \, ,</math>
in which case we may write,
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<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,
| Complex Fourier Series Expression | |||
|---|---|---|---|
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where, for <math>~n = 0, \pm 1, \pm 2, \pm 3, \dots~</math>,
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<math>~\omega_n</math> |
<math>~=</math> |
<math>~ \frac{n\pi }{L} \, , </math> |
and the complex coefficients,
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<math>~d_n = a_{n} -i b_{n} </math> |
<math>~=</math> |
<math>~ \frac{1}{L} \int_{-L}^{L} f(x) \cos\biggl( \frac{n\pi x}{L} \biggr) dx - i\frac{1}{L} \int_{-L}^{L} f(x) \sin\biggl( \frac{n\pi x}{L} \biggr) dx </math> |
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<math>~=</math> |
<math>~ \frac{1}{L} \int_{-L}^{L} f(x) \biggl[ \cos\biggl( \frac{n\pi x}{L} \biggr) - i\sin\biggl( \frac{n\pi x}{L} \biggr)\biggr] dx </math> |
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<math>~=</math> |
<math>~ \frac{1}{L} \int_{-L}^{L} f(x)e^{-i\omega_n x} dx \, . </math> |
Let's demonstrate that this rewritten (complex) expression for <math>~f(x)</math> matches the standard Fourier series expression. First, we will refer to the above standard definitions of <math>~a_n</math> and <math>~b_n</math> as, respectively, <math>~a_{|n|}</math> and <math>~b_{|n|}</math>, and recognize that, as the summation is extended to negative numbers, the following mapping is appropriate:
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<math>~a_n ~ \rightarrow ~ a_{|n|}</math> |
and |
<math>~b_n ~ \rightarrow ~ b_{|n|} \, ,</math> |
for | <math>~n > 0 \, ;</math> |
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<math>~a_n ~ \rightarrow ~ a_{|n|}</math> |
and |
<math>~b_n ~ \rightarrow ~ - b_{|n|} \, ,</math> |
for | <math>~n < 0 \, .</math> |
Hence, we have,
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<math>~2f(x)</math> |
<math>~=</math> |
<math>~ \sum_{n = -\infty}^{n = + \infty} (a_n - ib_n)e^{i\omega_n x} </math> |
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<math>~=</math> |
<math>~ \sum_{n = 1}^{n = + \infty} (a_{|n|} - ib_{|n|})e^{i\omega_{|n|} x} + a_0 + \sum_{n = \infty}^{n = 1} (a_{|n|} + ib_{|n|})e^{- i\omega_{|n|} x} </math> |
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<math>~=</math> |
<math>~ \sum_{n = 1}^{n = + \infty} (a_{|n|} - ib_{|n|}) [ \cos (\omega_{|n|} x) + i\sin (\omega_{|n|} x)] + a_0 + \sum_{n = 1}^{n = \infty} (a_{|n|} + ib_{|n|}) [ \cos (\omega_{|n|} x) - i\sin (\omega_{|n|} x)] </math> |
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<math>~=</math> |
<math>~ a_0 + \sum_{n = 1}^{n = + \infty}\biggl\{ (a_{|n|} - ib_{|n|}) [ \cos (\omega_{|n|} x) + i\sin (\omega_{|n|} x)] + (a_{|n|} + ib_{|n|}) [ \cos (\omega_{|n|} x) - i\sin (\omega_{|n|} x)] \biggr\} </math> |
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<math>~=</math> |
<math>~ a_0 + \sum_{n = 1}^{n = + \infty}\biggl\{2a_{|n|} \cos (\omega_{|n|} x) + 2b_{|n|} \sin (\omega_{|n|} x)] \biggr\} </math> |
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<math>~\Rightarrow ~~~ f(x)</math> |
<math>~=</math> |
<math>~ \frac{a_0}{2} + \sum_{n = 1}^{n = + \infty}\biggl\{a_{|n|} \cos (\omega_{|n|} x) + b_{|n|} \sin (\omega_{|n|} x)] \biggr\} \, . </math> |
Q.E.D.
From Williams & Tohline (1987)
Here, we replicate the discussion of a Fourier series analysis that was presented by H. A. Williams & J. E. Tohline (1987, ApJ, 315, 594) — see especially their §III — in the context of their discussion of nonlinear dynamic instabilities in rotating polytropes.
A useful way of analyzing the growth and pattern speed of nonaxisymmetric structures is to Fourier transform the (discrete) density distribution, <math>~\rho(\theta_L)</math>, in angle space, <math>~\theta_L = L \delta\theta</math>, where, <math>~\delta\theta \equiv 2\pi/L_\mathrm{max}</math>. On the discrete angular grid, the Fourier transform equations are
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<math>~a_m</math> |
<math>~=</math> |
<math>~ \frac{2}{L_\mathrm{max}} \cdot \sum_{L=1}^{L_\mathrm{max}} \rho(\theta_L) \cos(m\theta_L) \, , </math> |
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<math>~b_m</math> |
<math>~=</math> |
<math>~ \frac{2}{L_\mathrm{max}} \cdot \sum_{L=1}^{L_\mathrm{max}} \rho(\theta_L) \sin(m\theta_L) \, . </math> |
Notice that, <math>~a_0 = 2\bar\rho</math>, where <math>~\bar\rho</math> is the average density. The density function can be reconstructed via the expression,
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<math>~\rho(\theta_L)</math> |
<math>~=</math> |
<math>~ \frac{a_0}{2} + \sum_{m=1}^{L_\mathrm{max}/2} \biggl[ a_m \cos(m\theta_L) + b_m\sin(m\theta_L) \biggr] \, . </math> |
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Suppose the density distribution is given by the expression,
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Example #1: Suppose <math>~\bar\rho = 3</math>; <math>~(j, k, L_\mathrm{max}) = (1, 2, 8)</math>; and,
In this case, the specified density distribution is given by the expression, <math>~\rho_i = 3 + \tfrac{3}{4}\sin(\theta + 0.9\pi) + \tfrac{2}{5}\cos(2\theta) \, . </math>
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Alternatively, we can switch from the Fourier series coefficients, <math>~a_m</math> and <math>~b_m</math>, to the coefficient/phase definitions, <math>~c_m</math> and <math>~\phi_m</math>, such that,
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<math>~c_m</math> |
<math>~=</math> |
<math>~ [a_m^2 + b_m^2]^{1 / 2} \, , </math> |
and, if <math>~a_m</math> is positive,
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<math>~\phi_m</math> |
<math>~=</math> |
<math>~ \tan^{-1}\biggl( \frac{-b_m}{a_m} \biggr) \, , </math> |
otherwise, given that <math>~a_m</math> is negative,
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<math>~\phi_m</math> |
<math>~=</math> |
<math>~ \tan^{-1}\biggl( \frac{-b_m}{a_m} \biggr) +\pi \, . </math> |
Notice that, when <math>~a_m = 0</math>, <math>~\tan^{-1}(-b_m/a_m) = \pi/2</math>. Using <math>~c_m</math> and <math>~\phi_m</math>, the discrete density distribution can be exactly reconstructed via the Fourier series,
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<math>~\rho(\theta)</math> |
<math>~=</math> |
<math>~ \frac{c_0}{2} + \sum_1^{L_\mathrm{max}} c_m \cos\biggl[m\theta_L + \phi_m\biggr] \, . </math> |
One-Dimensional Aperture
General Concept
Hence, we have,
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<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> |
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<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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© 2014 - 2021 by Joel E. Tohline |