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may be expressed as <math>b_k = \left ( \frac{1}{T} \right ) \int \limits_T x \left ( t - t_0 \right ) e^{-j k w_0 t}\, dt</math>. | may be expressed as <math>b_k = \left ( \frac{1}{T} \right ) \int \limits_T x \left ( t - t_0 \right ) e^{-j k w_0 t}\, dt</math>. | ||
| + | |||
| + | Letting <math>\tao</math> = t - <math>t_0</math> in the new integral and noting that the new variable <math>\tao</math> will | ||
| + | |||
| + | also range over an interval of duration T, we obtain: | ||
| + | |||
| + | <math>\qquad \left ( \frac{1}{T} \right ) \int \limits_T x \left ( \tao \right ) e^{-j k w_0 \left ( \tao + t_0 \right )}\, d\tao</math> | ||
Revision as of 02:45, 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 $ \tao $ = t - $ t_0 $ in the new integral and noting that the new variable $ \tao $ will
also range over an interval of duration T, we obtain:
$ \qquad \left ( \frac{1}{T} \right ) \int \limits_T x \left ( \tao \right ) e^{-j k w_0 \left ( \tao + t_0 \right )}\, d\tao $
