How to derive the Fourier transform of a comb function

In summary, the conversation is about the discrete Fourier transform and how to derive the equation for I(ν). The equation for I(ν) involves a comb function, and the transformation from kT/N to lN/T is important in understanding the derivation. The equation involves a summation and integration, and the conversation discusses the conditions under which the value of I(ν) will be zero.
  • #1
arcTomato
105
27
TL;DR 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.
 
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  • #2
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
Thanks for reply @mitochan.
I cannnot understand what is going on this part

mitochan said:
[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
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
ok thank you.
I think I got it ;>
 

Related to How to derive the Fourier transform of a comb function

1. What is a comb function?

A comb function is a mathematical function that consists of a series of equally spaced spikes or impulses. It is often used to model periodic phenomena in signal processing and other areas of science and engineering.

2. What is the Fourier transform of a comb function?

The Fourier transform of a comb function is a series of equally spaced impulses in the frequency domain. This means that the comb function has a periodic spectrum with peaks at the same intervals as the spikes in the time domain.

3. How do you derive the Fourier transform of a comb function?

To derive the Fourier transform of a comb function, you can use the Fourier transform formula, which is an integral that transforms a function from the time domain to the frequency domain. For a comb function, this integral simplifies to a series of delta functions, resulting in the periodic spectrum mentioned earlier.

4. What is the significance of the Fourier transform of a comb function?

The Fourier transform of a comb function is significant because it allows us to analyze and understand the frequency components of a periodic signal. It also has many practical applications, such as in filtering, signal processing, and communication systems.

5. Are there any variations of the Fourier transform for comb functions?

Yes, there are variations of the Fourier transform for comb functions, such as the discrete Fourier transform (DFT) and the fast Fourier transform (FFT). These variations are more efficient and commonly used in digital signal processing and other applications where the signal is discrete and finite.

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