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<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> | <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> | ||
| + | |||
| + | <math>= e^{-j k w_0 t_0} a_k = e^{-j k \left ( 2\pi / T \right ) t_0} a_k</math>, | ||
| + | |||
| + | where <math>a_k</math> is the <math>k^{th}</math> Fourier series coefficient of x(t). That is, if | ||
| + | |||
| + | <math>x \left ( t \right ) \Longleftrightarrow^{\mathit{FS}} a_k</math>, then | ||
| + | |||
| + | <math>x \left ( t - t_0 \right ) \Longleftrightarrow^{\mathit{FS}} e^{-j k w_0 t_0} a_k = e^{-j k \left ( 2\pi / T \right ) t_0} a_k</math>. | ||
| + | |||
| + | One consequence of this property is that, when a periodic signal is shifted in time, the | ||
| + | |||
| + | <math>\mathit{magnitudes}</math> of its Fourier series coefficients remain unaltered. | ||
| + | |||
| + | That is, <math> | b_k | = | a_k |</math>. | ||
Latest revision as of 03:10, 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 $
$ = e^{-j k w_0 t_0} a_k = e^{-j k \left ( 2\pi / T \right ) t_0} a_k $,
where $ a_k $ is the $ k^{th} $ Fourier series coefficient of x(t). That is, if
$ x \left ( t \right ) \Longleftrightarrow^{\mathit{FS}} a_k $, then
$ x \left ( t - t_0 \right ) \Longleftrightarrow^{\mathit{FS}} e^{-j k w_0 t_0} a_k = e^{-j k \left ( 2\pi / T \right ) t_0} a_k $.
One consequence of this property is that, when a periodic signal is shifted in time, the
$ \mathit{magnitudes} $ of its Fourier series coefficients remain unaltered.
That is, $ | b_k | = | a_k | $.
