Why is the Fourier transform of 1 the 2pi*dirac(w) function

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Homework Help Overview

The discussion revolves around the Fourier transform of the constant function 1, specifically questioning why it results in the expression 2πδ(w). Participants explore the implications of a constant function in the frequency domain and its relationship to oscillation and frequency content.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants question the nature of the Fourier transform of a constant function, with some suggesting that it should represent infinite frequencies. Others discuss the idea that a constant function has only zero frequency and relate this to the properties of the Dirac delta function.

Discussion Status

The discussion is active, with participants sharing insights and asking clarifying questions about the normalization factor of 2π in the Fourier transform. There is an exploration of different conventions and the implications of these choices on the interpretation of the transform.

Contextual Notes

Some participants note the potential confusion surrounding the definition of frequency in the context of a constant function and the normalization conventions used in Fourier transforms.

thomas49th
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Homework Statement


Hi, I was wondering why the Fourier transform of 1 is

2\pi\delta(w)

I would of though that one would be of infinite frequencies (like a square wave).

Further more if g(t) = 1, for all t, g(t) = 1. Why does the Fourier transform have the argument of g(t) = 1 have an argument (w). g(t) has no frequency!

Thanks
Thomas
 
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Wouldn't g(t)=1 have only zero frequency? It never oscillates at all. And isn't delta(w) the function in the frequency domain that only has zero frequency? It's only 'nonzero' at w=0. Make perfect sense to me.
 
Last edited:
good intuition. May I ask where the 2pi comes from again :)
 
thomas49th said:
good intuition. May I ask where the 2pi comes from again :)

It's a normalization of the Fourier transform. There's more than one convention for doing that. What's yours? I thought this question was more about intuition. The 2*pi has nothing to do with that. Take the inverse Fourier transform of 2*pi*delta(w). Do you get 1?
 

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