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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> | ||
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| + | The coefficients of the transformed function are then | ||
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| + | <math>\frac{1}{T}\int_T x(t-t_0)e^{-jkw_0t}dt</math> | ||
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| + | Substituting <math>\tau = t - t_0</math> into the equation results in | ||
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| + | <math>=\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau+t_0}d\tau</math> | ||
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| + | <math>=\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau}e^{-jkw_0(t_0}d\tau</math> | ||
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| + | Because <math>e^{-jkw_0t_0}</math> is constant over <math>\tau</math> it can be factored out of the integral | ||
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| + | <math>=(e^{-jkw_0t_0})\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau}d\tau</math> | ||
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| + | By substitution of <math>a_k=\frac{1}{T}\int_T x(t)e^{-jkw_0t}dt</math> he coefficients of the transformed function can then be set to equal | ||
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| + | <math>a_k e^{-jkw_0t_0}</math> | ||
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| + | Thereby proving the time shifting property | ||
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| + | --[[User:Asiembid|Adam Siembida (asiembid)]] 20:57, 8 July 2009 (UTC) | ||
Latest revision as of 16:58, 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-t_0)e^{-jkw_0t}dt $
Substituting $ \tau = t - t_0 $ into the equation results in
$ =\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau+t_0}d\tau $
$ =\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau}e^{-jkw_0(t_0}d\tau $
Because $ e^{-jkw_0t_0} $ is constant over $ \tau $ it can be factored out of the integral
$ =(e^{-jkw_0t_0})\frac{1}{T}\int_T x(\tau)e^{-jkw_0(\tau}d\tau $
By substitution of $ a_k=\frac{1}{T}\int_T x(t)e^{-jkw_0t}dt $ he coefficients of the transformed function can then be set to equal
$ a_k e^{-jkw_0t_0} $
Thereby proving the time shifting property
--Adam Siembida (asiembid) 20:57, 8 July 2009 (UTC)
