- #1
billiards
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Homework Statement
Show that for a fixed value of [itex]\omega[/itex] that [itex]G(\omega)e^{-i\omega t}[/itex] is the response of the system to the input signal [itex]e^{-i\omega t}[/itex].
(From Roel Snieder's book 'A Guided Tour of Mathematical Methods for the Physical Sciences', pg 233 (Section 15.7, Problem e))
2. Homework Equations (I think)
The Fourier transform pair:
[itex]f(t)=\int^{\infty}_{-\infty}F(\omega)e^{-i\omega t}d\omega[/itex]
[itex]F(\omega)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(t)e^{i\omega t}dt[/itex]
Convolution relations:
[itex]o(t)=\int^{\infty}_{-\infty}g(t-\tau)i(\tau)d\tau[/itex]
[itex]O(\omega)=2\pi G(\omega)I(\omega)[/itex]
The Attempt at a Solution
I'm not really sure I understand the question. I am assuming that the 'response of the system' is [itex]g(t)[/itex] and that 'input signal' is [itex]i(t)[/itex]. I could well be wrong on that.
So I am reading the question as saying: show that for the case [itex]i(t)=e^{-i\omega t}[/itex] ...
[itex]g(t)=G(\omega)e^{-i\omega t}[/itex].
I have tried playing around with the symbols but I have not managed to get thing to click, which makes me suspect that I am misunderstanding the question. I have been sitting on this problem for weeks and have managed to progress through the book no problem, but not getting this question is bugging me.
Any help would be greatly appreciated.