Fourier transform of cos(wt) and cos(t).

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SUMMARY

The Fourier transform of cos(w0t) results in π(dirac(w + w0) + dirac(w - w0)), confirming that the frequency component w0 is essential for the transformation. In contrast, the Fourier transform of cos(t) yields π(dirac(w + 1) + dirac(w - 1)), indicating that the frequency component is 1. The discussion clarifies that the scaling factor is not present in the transformation of cos(t) when using the exponential form of the cosine function.

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thomas49th
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Is there a difference? My notes are inconsistent and very poor. Google search doesn't seem to be having much use.

Which one transforms into pi(dirac(w+w0) + dirac(w-w0))?

Thanks
Thomas
 
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Neither since ω0 doesn't appear in either function.
 
Okay Fourier transform of cos(w0t) and cos(t).

What do they transform into? I presume cos(w0t) is pi(dirac(w+w0) + dirac(w-w0))?

but what about cos(t). I'm guessing somewhere like pi(dirac(w+t) + dirac(w-t)) but is there some scaling factor?
 
Use cos(t)=(e^(it)+e^(-it))/2. You know the Fourier transform of e^(it) is a delta function, right?
 

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