Complex Fourier transform (represented by Σ)

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  • Thread starter arcTomato
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  • #1
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Summary:
I can't understand how to derive fourier transform(described by Σ).
Dear all.

I can't understand how to derive Eq.(2.3a).

Fourier coefficients, ##A_j## and ##B_j## are described by summation in this paper as (2.2). I think this is weird.
Because this paper said "In this section 2.1 ,the Fourier transform is introduced in very general terms".
and I understand Fourier coefficients as integral.( I think integral version is more general than ##Σ## version.) so I'm confused.

I would like to ask you how to derive (2.3a) in this paper( in ##Σ## version).
(this is the paper I readFourier_techniques.pdf)

スクリーンショット 2019-10-21 9.40.02.png
スクリーンショット 2019-10-21 9.41.40.png
 

Answers and Replies

  • #2
tnich
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Assume that you are integrating from ##-T## to ##+T##, and consider the evenness or oddness of ##\cos^2\omega_jt##, ##\sin^2\omega_jt##, and ##\sin\omega_jt\cos\omega_jt##. Then take the small step from integration to discrete summation.
 
  • #3
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I appreciate for you, @tnich.

Can I use ##t_k=kT/N,ω_j=2πj/T##in the process to derive (2.2)??
This paper does not say anything about ##ω_j## in section 2.1.
So you say like this??Can I approximate to Σ from integral?
1571622923271.png
 
  • #4
tnich
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Yes, I am saying that you can approximate the integrals with discrete summations. Try looking at the definition of an integral as a limit of discrete summations to see how to do the approximation.
I am having trouble reading your derivation, so I don't know if you got it right. I see that you kept ##k## and ##t_k## in the integral. I think you need to integrate over ##t## from ##-T## to ##+T##.
But, that said, on page 2 the author says that this section is a general overview and that he will go into the details later. If you understand the general idea, I would be tempted to just read on.
 
  • #5
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Ok, I think I got it😊
Thanks for your kindness,@tnich!!
 

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