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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?

  • $\begingroup$ 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 $\endgroup$ Commented Dec 8, 2022 at 22:25
  • $\begingroup$ 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. $\endgroup$
    – JellyTree
    Commented Dec 8, 2022 at 23:42
  • $\begingroup$ 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 $t_0$. $\endgroup$
    – TimWescott
    Commented Dec 9, 2022 at 0:41
  • $\begingroup$ (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.) $\endgroup$
    – TimWescott
    Commented Dec 9, 2022 at 0:42
  • $\begingroup$ 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". $\endgroup$
    – JellyTree
    Commented Dec 9, 2022 at 1:42


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