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Letting <math>\tau</math>=t-<math>t_{0}</math> in the integral, and noticing that the new variable <math>\tau</math> will also range over<br> | Letting <math>\tau</math>=t-<math>t_{0}</math> in the integral, and noticing that the new variable <math>\tau</math> will also range over<br> | ||
an interval of duration T, we obtain<br> | an interval of duration T, we obtain<br> | ||
| − | <math>\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau+\tau_{0})}d\tau</math><br> | + | <math>\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau+\tau_{0})}d\tau=e^{-jkw_{0}t_{0}}\frac{1}{T}\int_T x(\tau)e^{-jkw_0\tau}d\tau</math><br> |
Revision as of 03:34, 9 July 2009
== Time Shifting Property of Continuous-Time Fourier Series ==
When a time shift is applied to a periodic signal x(t), the period T of the signal is preserved.
The Fourier series coefficients $ b_{k} $ of the resulting signal y(t)=x(t-$ t_{0} $) may be expressed as
$ b_{k}=\frac{1}{T}\int_T x(t-t_{0})e^{-jk w_{0} t}dt $.
Letting $ \tau $=t-$ t_{0} $ in the integral, and noticing that the new variable $ \tau $ will also range over
an interval of duration T, we obtain
$ \frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau+\tau_{0})}d\tau=e^{-jkw_{0}t_{0}}\frac{1}{T}\int_T x(\tau)e^{-jkw_0\tau}d\tau $
