ASN4 - Fourier Transform property: Parseval's Theorem: Difference between revisions
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== Parseval's Theorem == |
== Parseval's Theorem == |
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<math>\int_{- \infty}^{\infty} (|s(t)|)^2 dt</math> in time transforms to <math>\int_{- \infty}^{\infty} (|S(f)|)^2 df</math> in frequency |
<math>\int_{- \infty}^{\infty} (|s(t)|)^2 dt</math> in time transforms to <math>\int_{- \infty}^{\infty} (|S(f)|)^2 df</math> in frequency |
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The magnitude of <math>s(t)</math> is also the Inverse Fourier Transform of <math>S(f)</math>. |
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<math> |s(t)|= F ^{-1}[S(f)]=|\int_{- \infty}^{\infty}S(f)e^{j 2 \pi f t} df | </math> |
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<math> |s(t)|= |
The above equation of <math>|s(t)|</math> simplifies to then <math>|s(t)|= \int_{- \infty}^{\infty}S(f) df= |S(f)|</math> |
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Therefore,squaring the function and intergrating it in the time domain <math>\int_{- \infty}^{\infty} (|s(t)|)^2 dt</math> is to do the same in the frequency domain <math>\int_{- \infty}^{\infty} (|S(f)|)^2 df</math> |
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Latest revision as of 08:17, 3 December 2009
Parseval's Theorem
in time transforms to in frequency
The magnitude of is also the Inverse Fourier Transform of .
Note that
The above equation of simplifies to then
Therefore,squaring the function and intergrating it in the time domain is to do the same in the frequency domain