(New page: {| | align="left" style="padding-left: 0em;" | Parseval's property |} <math> \mathcal{X}(\omega)=\mathcal{X}(2\pi f) \ </math> <div align="left" style="padding-left: 0em;"> <math> \begin{...) |
(No difference)
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Revision as of 20:30, 9 September 2010
| Parseval's property |
$ \mathcal{X}(\omega)=\mathcal{X}(2\pi f) \ $
$ \begin{align} \int_{-\infty}^{\infty} |x(t)|^2 dt &= \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(2\pi f)|^2 d2\pi f \\ &= \int_{-\infty}^{\infty} |\mathcal{X}(2\pi f)|^2 df \\ &= \int_{-\infty}^{\infty} |X(f)|^2 df \end{align} $
$ Since\ X(\alpha)=\mathcal{X}(2\pi \alpha) $
