Continous - Time Fourier Series: Time Reversal
The period T of a periodic signal x(t) remains unchanged when it goes through time reversal
$ x(-t) = \sum_{k=-\infty}^\infty a_k e^{-jk2\pi t/T} $
Substitute k = -m
y(t) = $ x(-t) = \sum_{m=-\infty}^\infty a_{-m} e^{-jm2\pi t/T} $
Right-hand side of the equation has the form of a Fourier series synthesis equation for x(-t)
$ b_k = a_{-k} $
$ x(t)\mathcal F\Longleftrightarrow a_k $
$ x(-t)\mathcal F\Longleftrightarrow a_{-k} $
