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| + | [[Category:problem solving]] | ||
| + | [[Category:ECE301]] | ||
| + | [[Category:ECE]] | ||
| + | [[Category:Fourier transform]] | ||
| + | [[Category:inverse Fourier transform]] | ||
| + | [[Category:signals and systems]] | ||
| + | == Example of Computation of inverse Fourier transform (CT signals) == | ||
| + | A [[CT_Fourier_transform_practice_problems_list|practice problem on CT Fourier transform]] | ||
| + | ---- | ||
| + | |||
== Specify a Fourier transform <math>X(w)</math> == | == Specify a Fourier transform <math>X(w)</math> == | ||
| − | :<math> X(w)=\frac{ | + | :<math> X(w)=2\pi \delta\left ( w- \frac{\pi}{4}\right )+2\pi \delta\left ( w+ \frac{\pi}{4}\right ) </math> |
== Inverse Fourier transform of <math>X(w)</math>== | == Inverse Fourier transform of <math>X(w)</math>== | ||
:<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}\left (\frac{1}{4+w}\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 | ||
| + | |||
| + | \\& =e^{-j\frac{\pi}{4}t}+e^{j\frac{\pi}{4}t} | ||
| + | \\& =2cos\left (\frac{\pi}{4}\right ) t | ||
| + | \end{align} | ||
| + | </math> | ||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:44, 16 September 2013
Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
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} $
Back to Practice Problems on CT Fourier transform
\end{align}</math>
