Message Boards Message Boards

3 Replies
2 Total Likes
View groups...
Share this post:

Question about Fourier Coefficents

Posted 11 years ago
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!
3 Replies
 (* 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
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
Posted 11 years 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
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
or Discard

Group Abstract Group Abstract