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Question about Fourier Coefficents

GROUPS:
Hello Everyone,

i have recently a Problem about the Fourier Coefficents, which will be descreibed as follow:
so i have got a Equation of light intensity in a Optic System. Each term on the right-hand side of Equation below has a different frequency dependence.


This Equation is measured for different retardation plate rotation angles beta_i, the  retardation plate angles beta_i can be controlled by a stepping motor drive so that a set of data will consist of N data points obtained at the angular position beta_i(i=1 to N), with the setp size "delta_beta=beta_i+1 - beta_i" generally being a small fraction of 360 Degree.

And this equation can be noted as a Fourier Series:

The Equation above can ben exactly inverted in a finite and discrete Fourier transform.
The Question is, how can i get the Fourier series coefficients blow for an even number of data points N=2L? If there is a way that these Fourier series coefficents can be calculated by the Wolfram.


Thank you all!
POSTED BY: Ke Zheng
Answer
1 year ago
Having given 
Subscript[I, T] (\[Beta])
 as Fourier series, its Fourier coefficients Subscript[C, i] and Subscript[S, i] are computed as written in ref/FourierSeries. Converting these integrals into a Riemann sum approximation should result in the formulae you cited - no need to use "the Wolfram". "The Wolfram" tries to solve the integrals, not to approximate them by sums.
POSTED BY: Udo Krause
Answer
1 year ago
Thank you Udo,

i have tried the Riemann-Sum, but i still cant understand  the Delta Function there. And by the way i have tried to calculate the Fourier coefficients via Mathematic for the situation, which the parameter alpha=0, delta=Pi/2, and beta is variable.  Then i got the same coefficients except the first term. So i have used another way to compute coefficients. Anyway, thank you for ur Tipps. emoticon
POSTED BY: Ke Zheng
Answer
1 year ago
 (* zero-th cosine fourier series coefficient *)
 Subscript[c, 0] = (1/\[Pi]) Integrate[f[\[Beta]], {\[Beta], 0, \[Pi]}]
 (* other cosine fourier series coefficients *)
 Subscript[c, k] = (2/\[Pi]) Integrate[f[\[Beta]] Cos[k \[Beta]] , {\[Beta], 0, \[Pi]}]
 (* l.h.s. approximation, very crude *)
 Subscript[c, k] = (2/\[Pi]) Sum [f[Subscript[\[Beta], i]] Cos[
      k Subscript[\[Beta], i]] \[CapitalDelta]\[Beta], {i, 1, N}]
 \[CapitalDelta]\[Beta] = \[Pi]/N
 Subscript[c, k] = (2/N) Sum[
   f[Subscript[\[Beta], i]] Cos[k (i - 1) \[CapitalDelta]\[Beta]], {i,
     1, N}]
Subscript[c, k] = (2/N) Sum[
   f[Subscript[\[Beta], i]] Cos[\[Pi] k (i - 1)/N], {i, 1, N}]
(* k = L = 2 N *)
Subscript[c, L] = (1/N) Sum[f[Subscript[\[Beta], i]] Cos[2 \[Pi] (i - 1)], {i, 1, N}]
(* Cos[2 \[Pi] (i - 1)] === 1, so Subscript[c, L] has to have the same norm as \
Subscript[c, 0] which explains
the delta functions Subscript[\[Delta], k0] and the Subscript[\[Delta], kL] *)
I could not reproduce the
Subscript[\[Omega], k]
as seen in your inlet.
POSTED BY: Udo Krause
Answer
1 year ago