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also range over an interval of duration T, we obtain: | also range over an interval of duration T, we obtain: | ||
| − | <math>\qquad \left ( \frac{1}{T} \right ) \int \limits_T x \left ( \tau \right ) e^{-j k w_0 \left ( \tau + t_0 \right )}\, d\tau</math> | + | <math>\qquad \left ( \frac{1}{T} \right ) \int \limits_T x \left ( \tau \right ) e^{-j k w_0 \left ( \tau + t_0 \right )}\, d\tau = e^{-j k w_0 t_0} \left ( \frac{1}{T} \right ) \int \limits_T x \left ( \tau \right ) e^{-j k w_0 \tau}\, d\tau</math> |
Revision as of 02:50, 23 July 2009
CTFS Time Shifting Property
If x(t) has CTFS coefficients $ a_k $ and y(t) has CTFS coefficients $ b_k $,
then the Fourier series coefficients $ b_k $ of the resulting signal y(t) = x(t - $ t_0 $)
may be expressed as $ b_k = \left ( \frac{1}{T} \right ) \int \limits_T x \left ( t - t_0 \right ) e^{-j k w_0 t}\, dt $.
Letting $ \tau $ = t - $ t_0 $ in the new integral and noting that the new variable $ \tau $ will
also range over an interval of duration T, we obtain:
$ \qquad \left ( \frac{1}{T} \right ) \int \limits_T x \left ( \tau \right ) e^{-j k w_0 \left ( \tau + t_0 \right )}\, d\tau = e^{-j k w_0 t_0} \left ( \frac{1}{T} \right ) \int \limits_T x \left ( \tau \right ) e^{-j k w_0 \tau}\, d\tau $
