How to derive the Fourier transform of a comb function

  • #1
92
26
Summary:
Fourier transform
Dear all.
I'm learning about the discrete Fourier transform.

##I(\nu) \equiv \int_{-\infty}^{\infty} i(t) e^{2 \pi \nu i t} d t=\frac{N}{T} \sum_{\ell=-\infty}^{\infty} \delta\left(\nu-\ell \frac{N}{T}\right)##

this ##i(t)## is comb function
##i(t)=\sum_{k=-\infty}^{\infty} \delta\left(t-\frac{k T}{N}\right)##.

I would like to see how to derive ##I(ν)##.(Especially the part about transformation to ##lN/T from kT/N)
If you can teach me, please.
Thank you.
 
Last edited:

Answers and Replies

  • #2
283
126
Hi.
[tex]I(\nu)=\sum^\infty_{k=-\infty} \int^\infty_{-\infty}\delta(t-\frac{kT}{N})e^{2\pi\nu it}dt=\sum^\infty_{k=-\infty}e^{2\pi\nu i kT/N}=\sum^\infty_{l=-\infty}\delta(\nu T/N-l)[/tex][tex]=\frac{N}{T}\sum^\infty_{l=-\infty}\delta(\nu -\frac{lN}{T})[/tex]
 
  • #3
92
26
Thanks for reply @mitochan.
I cannnot understand what is going on this part

[tex]=\sum^\infty_{k=-\infty}e^{2\pi\nu i kT/N}=\sum^\infty_{l=-\infty}\delta(\nu T/N-l)[/tex]

Could you teach me about this detail??
 
  • #4
283
126
RHS says ##\nu T/N## must be an integer. If not LHS =0 due to summation of various phase numbers of magnitude 1. Sumamtion in RHS says any integer is OK.
 
  • #5
92
26
ok thank you.
I think I got it ;>
 

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