Time Shifting Property
The time shifting property states that if the periodic signal $ x(t) $ is shifted by $ t_0 $ to created the shifted signal $ x(t-t_0) $, the Fourier series coefficients of the shifted will be $ a_k e^{-jkw_0t_0} $, where $ a_k $ are the coefficients of $ x(t) $.
Proof
Let $ a_k $ be the Fourier series coefficients of $ x(t) $, so
$ a_k=\frac{1}{T}\int_T x(t)e^{-jkw_0t}dt $
The coefficients of the transformed function are then
$ \frac{1}{T}\int_T x(t-t_0)e^{-jkw_0t}dt $
Substituting $ \tau = t - t_0 $ into the equation results in
$ =\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau+t_0}d\tau $
$ =\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau}e^{-jkw_0(t_0}d\tau $
Because $ e^{-jkw_0t_0} $ is constant over $ \tau $ it can be factored out of the integral
$ =(e^{-jkw_0t_0})\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau}d\tau $
By substitution of $ a_k=\frac{1}{T}\int_T x(t)e^{-jkw_0t}dt $ he coefficients of the transformed function can then be set to equal
$ a_k e^{-jkw_0t_0} $
Thereby proving the time shifting property
Adam Siembida
--Asiembid 20:57, 8 July 2009 (UTC)
