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June 8th, 2016, 02:39 PM   #1
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Question Interpreting the Fourier transform of cosine

I wasn’t quite sure where to post this thread, so I apologize if it’s in the wrong place. Anyway, I thought that I would spend some of my free time this summer learning about the Fourier transform. So far, things have been going very well, but I do have a question about the Fourier transform of cos(2piat). I can compute the transform fine, but I’m unsure about how to interpret it. Taking the transform I get 1/2(delta(s + a) + delta(s - a)). Now it makes sense that the spectrum would only contain +/- a because that’s the only contributing frequency, but how do I interpret the delta? The way I’ve viewed the graphs of Fourier transforms before was seeing the magnitude for each frequency as corresponding to the amplitude of a sinusoid of that frequency, and adding all the contributing frequencies together (with their amplitudes and phase shifts) would recover the signal. Is this correct, and if so what about the amplitude in this case? It seems like the amplitude should be finite (1/2 each contributed from positive and negative a), but doesn’t delta(0) = infinity? Or am I misunderstanding the notation? The graphs all seem to contain vertical arrows at a and -a which seem to denote that the height is infinite, but that doesn’t really seem to make sense based on the interpretation that I just explained.

If anybody could offer some clarity, it would be greatly appreciated. I feel like the issue most likely lies in the notation.

Last edited by skipjack; June 9th, 2016 at 12:20 AM.
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June 8th, 2016, 11:23 PM   #2
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I think I have my answer, although I’m not sure about all the details. I started by questioning my assumption about how the magnitude for a given frequency directly corresponds to the amplitude of the sinusoid of that frequency contributing to the signal, and to do this I thought that I’d start with comparing the magnitude of the Fourier coefficient against the amplitude of the corresponding sinusoid for the Fourier series of a given function (I chose a square wave.). I’d then extrapolate this to the Fourier transform, which is just a limiting case of the series. Well the Fourier coefficient checked out like it was supposed to, but it didn’t work for the Fourier transform. I then realized that the Fourier coefficient was scaled by 1/T and that there’s no such scaling factor for the Fourier transform. Anyway, it looks like the Fourier transforms of functions don’t directly correspond to amplitudes. They have to be scaled first. This also explains my confusion regarding the delta function.

Still, if anybody has more information on scaling the results of a Fourier transform, that would be appreciated.

Last edited by Mihaila; June 8th, 2016 at 11:25 PM. Reason: typo
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