(New page: Adam Frey Proof of ''Parseval's Relation'' in Continuous-Time Transform 4.3.7 If x(t) and X(jt) are a Fourier transform pair, then : <math> \int_{-\infty}^\infty |x(t)|^2\,dt = \fr...) |
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<math> \int_{-\infty}^\infty |x(t)|^2\,dt = \frac{1}{2\pi} \int_{-\infty}^\infty | X(jw) |^2 dw </math> | <math> \int_{-\infty}^\infty |x(t)|^2\,dt = \frac{1}{2\pi} \int_{-\infty}^\infty | X(jw) |^2 dw </math> | ||
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Latest revision as of 11:13, 8 July 2009
Adam Frey
Proof of Parseval's Relation in Continuous-Time Transform 4.3.7
If x(t) and X(jt) are a Fourier transform pair, then :
$ \int_{-\infty}^\infty |x(t)|^2\,dt = \frac{1}{2\pi} \int_{-\infty}^\infty |X(jw)|^2\,dw $
This is known as Parseval's Relation and results from direct application of the Fourier transform :
$ \int_{-\infty}^\infty |x(t)|^2\,dt = \int_{-\infty}^\infty x(t)x^*(t)\,dt $
$ = \int_{-\infty}^\infty x(t)[ \frac{1}{2\pi} \int_{-\infty}^\infty X^* (jw)e^{(-jwt)} dw ]dt $
Reversing the order of integration results in:
$ \int_{-\infty}^\infty |x(t)|^2\,dt = \frac{1}{2\pi} \int_{-\infty}^\infty X^*(jw) [ \int_{-\infty}^\infty x(t) e^{(-jwt)} dt ]dw $
The bracketed term is simply the Fourier transform of x(t); therefore,
$ \int_{-\infty}^\infty |x(t)|^2\,dt = \frac{1}{2\pi} \int_{-\infty}^\infty | X(jw) |^2 dw $
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