(→Inverse Fourier transform of X(w)) |
(→Inverse Fourier transform of X(w)) |
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:<math>\begin{align} x(t)&=\frac{1}{2\pi}\int_{-\infty}^{\infty}X( \omega)e^{j\omega t}d\omega | :<math>\begin{align} x(t)&=\frac{1}{2\pi}\int_{-\infty}^{\infty}X( \omega)e^{j\omega t}d\omega | ||
\\& =\frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi \delta\left ( w- \frac{\pi}{4}\right )e^{j\omega t}d\omega+ \frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi \delta\left ( w+ \frac{\pi}{4}\right )e^{j\omega t}d\omega | \\& =\frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi \delta\left ( w- \frac{\pi}{4}\right )e^{j\omega t}d\omega+ \frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi \delta\left ( w+ \frac{\pi}{4}\right )e^{j\omega t}d\omega | ||
| + | |||
\\& =\int_{-\infty}^{\infty}\delta\left ( w- \frac{\pi}{4}\right )e^{j\omega t}d\omega+ \int_{-\infty}^{\infty}\delta\left ( w+ \frac{\pi}{4}\right )e^{j\omega t}d\omega | \\& =\int_{-\infty}^{\infty}\delta\left ( w- \frac{\pi}{4}\right )e^{j\omega t}d\omega+ \int_{-\infty}^{\infty}\delta\left ( w+ \frac{\pi}{4}\right )e^{j\omega t}d\omega | ||
| + | |||
| + | \\& =e^{-j\frac{\pi}{4}t}+e^{j\frac{\pi}{4}t} | ||
| + | |||
| + | \\& =2cos\left (\frac{\pi}{4}\right ) t | ||
Revision as of 18:17, 8 October 2008
Specify a Fourier transform $ X(w) $
- $ X(w)=2\pi \delta\left ( w- \frac{\pi}{4}\right )+2\pi \delta\left ( w+ \frac{\pi}{4}\right ) $
Inverse Fourier transform of $ X(w) $
- $ \begin{align} x(t)&=\frac{1}{2\pi}\int_{-\infty}^{\infty}X( \omega)e^{j\omega t}d\omega \\& =\frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi \delta\left ( w- \frac{\pi}{4}\right )e^{j\omega t}d\omega+ \frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi \delta\left ( w+ \frac{\pi}{4}\right )e^{j\omega t}d\omega \\& =\int_{-\infty}^{\infty}\delta\left ( w- \frac{\pi}{4}\right )e^{j\omega t}d\omega+ \int_{-\infty}^{\infty}\delta\left ( w+ \frac{\pi}{4}\right )e^{j\omega t}d\omega \\& =e^{-j\frac{\pi}{4}t}+e^{j\frac{\pi}{4}t} \\& =2cos\left (\frac{\pi}{4}\right ) t \end{align} $
