As far as I understand Fourier Transforms are for non-periodic signals and Fourier Series for periodic signals.
So why is it we can take the Fourier Transform of a cosine when it is a periodic function, assuming the above paragraph is correct?
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Sign up to join this communityAs far as I understand Fourier Transforms are for non-periodic signals and Fourier Series for periodic signals.
So why is it we can take the Fourier Transform of a cosine when it is a periodic function, assuming the above paragraph is correct?
Indeed there are two things you have to know.
First, it can be shown that the continuous-time Fourier transform can be obtained from the continuous-time Fourier series by letting the period $T$ go to infinity.
Second, formally speaking the Fourier transform integral for periodic signals do not converge, hence do not exist. The solution is a generalisation of the Fourier transform by the use of Dirac impulse functions.
The result is an interpretation that the Fourier transform of periodic functions is a sum of scaled Dirac impulses at the Fourier harmonic frequencies, and the scale being the corresponding Fourier series coefficients.