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The time shifting property states that if the periodic signal <math>x(t)</math> is shifted by <math>t_0</math> to created the shifted signal <math>x(t-t_0)</math>, the Fourier series coefficients of the shifted will be <math>a_k e^{-jkw_0t_0}</math>, where <math>a_k</math> are the coefficients of <math>x(t)</math>. | The time shifting property states that if the periodic signal <math>x(t)</math> is shifted by <math>t_0</math> to created the shifted signal <math>x(t-t_0)</math>, the Fourier series coefficients of the shifted will be <math>a_k e^{-jkw_0t_0}</math>, where <math>a_k</math> are the coefficients of <math>x(t)</math>. | ||
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| + | ===Proof=== | ||
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| + | Let <math>a_k</math> be the Fourier series coefficients of <math>x(t)</math>, so | ||
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| + | <math>a_k=\frac{1}{T}\int_T x(t)e^{-jkw_0t}dt</math> | ||
Revision as of 16:42, 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 $
