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<math>a_k=\frac{1}{T}\int_T x(t)e^{-jkw_0t}dt</math> | <math>a_k=\frac{1}{T}\int_T x(t)e^{-jkw_0t}dt</math> | ||
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| + | The coefficients of the transformed function are then | ||
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| + | <math>\frac{1}{T}\int_T x(t)e^{-jkw_0(t-t_0)}dt</math> | ||
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| + | <math>=\frac{1}{T}\int_T x(t)e^{-jkw_0t}e^{jkw_0t_0}dt</math> | ||
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| + | <math></math> | ||
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| + | <math></math> | ||
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| + | <math></math> | ||
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| + | <math></math> | ||
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Revision as of 16:45, 8 July 2009
Time Shifting Property
The time shifting property states that if the periodic signal $ x(t) $ is shifted by $ t_0 $ to created the shifted signal $ x(t-t_0) $, the Fourier series coefficients of the shifted will be $ a_k e^{-jkw_0t_0} $, where $ a_k $ are the coefficients of $ x(t) $.
Proof
Let $ a_k $ be the Fourier series coefficients of $ x(t) $, so
$ a_k=\frac{1}{T}\int_T x(t)e^{-jkw_0t}dt $
The coefficients of the transformed function are then
$ \frac{1}{T}\int_T x(t)e^{-jkw_0(t-t_0)}dt $
$ =\frac{1}{T}\int_T x(t)e^{-jkw_0t}e^{jkw_0t_0}dt $
