| Line 2: | Line 2: | ||
When a time shift is applied to a periodic signal x(t), the period T of the signal is preserved.<br> | When a time shift is applied to a periodic signal x(t), the period T of the signal is preserved.<br> | ||
The Fourier series coefficients <math>b_{k}</math> of the resulting signal y(t)=x(t-<math>t_{0}</math>) may be expressed as<br> | The Fourier series coefficients <math>b_{k}</math> of the resulting signal y(t)=x(t-<math>t_{0}</math>) may be expressed as<br> | ||
| − | <math>b_{k}=\frac{1}{T}\int_T x(t-t_{0})e^{-jk w_{0} t}dt</math><br> | + | <math>b_{k}=\frac{1}{T}\int_T x(t-t_{0})e^{-jk w_{0} t}dt</math>.<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> | ||
| + | <math>\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau+\tau_{0})}d\tau</math><br> | ||
Revision as of 03:31, 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 $
