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Up until now, I have dealt with finding Fourier Coefficients for functions: $f(t) > 0$

Which made it convenient calculating the Fourier Analysis Integral. However, I am now presented with functions: $f(t) < 0$

My question is, would I need 3 different bounds when performing integration for this particular function?

i.e. For the function above, if the lower bound was limited to 0, I would only need to define the integration from -1 to 1 to fully represent the Fouier Analysis Integral: $ \int_{-1}^{1} 1$$ (e)^{-jwkt} $$ \,dt$

However, since the lower bound reaches to -1, would I need 3 different bounds of integration to fully represent and calculate the Fourier Analysis Integral / Coefficients?

Are there any easier "tricks" for finding the Fourier coefficients?

JellyTree
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  • The sign of the values of $f(t)$ don't play a role at all in how many integrals you need - the Fourier transform here is really just a one-dimensional integral transform, so you would need a single integral, if that exists. It's really not clear what you mean with the integrals! Please expand your question by editing to illustrate what you mean – Marcus Müller Dec 08 '22 at 22:25
  • I have edited the post! Thanks for clarifying my concept. I think you might have answered my question indirectly if what I asked above is correct. – JellyTree Dec 08 '22 at 23:42
  • You talk about the "Fourier Analysis Integral". I haven't run across that name before. Are you attempting to find the Fourier Series of this waveform, or are you attempting to find the Fourier Transform of it? At any rate, that's a periodic function with a period of 4. If you don't take any of the symmetry shortcuts, that means you need to integrate over the interval $t \in [t_0,\ t_0+4]$ for just about any $t_0$. – TimWescott Dec 09 '22 at 00:41
  • (Anyone who's about to critique me for using a doubly-closed interval -- yes, you're right. I had a choice between exactly right and simple; I chose simple.) – TimWescott Dec 09 '22 at 00:42
  • I'm attempting to utilize Fourier Series to find the Fourier Coefficients. You mentioned that the function has a period of 4, and that I should integrate over any interval t, t+4 - which I agree. I guess my true question would be, how would you describe the function? I've sort of answered my own question thanks to your first comment, but I was asking if its valid to think that: since integration is a linear operation, I am able to "create" a piecewise function that describes the "above" function, and take respective integration over those piecewise "bounds". – JellyTree Dec 09 '22 at 01:42
  • @JellyTree: Your last comment shows that you've understood what to do. Now try to realize that it's totally irrelevant whether $f(t)>0$ or not. That's why your question was a bit confusing. It's all about how to integrate a piecewise defined function. – Matt L. Dec 09 '22 at 07:34
  • @JellyTree I think I agree with Matt: The question you ask here in the comments – can I, and if so, how, piece together the Fourier transform from piecewise integrals – is pretty different from what you wrote in the question. I think it'd be best if you edited this question, and introduced a canonical notation for what you want to do. Your question (I hope I understand it correctly, I'm not 100% sure) seems to be a good one, and you might be pretty close to what the DFT actually is, theoretically. (To foreshadow: the FT of a single rectangle is a sinc. A rectangle wave has a line spectrum!) – Marcus Müller Dec 09 '22 at 09:54

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