(New page: == Specify a signal x(t) == <math>x(t)=cos(8 \pi x)e^{-x^{2}}</math>) |
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| + | [[Category:problem solving]] | ||
| + | [[Category:ECE301]] | ||
| + | [[Category:ECE]] | ||
| + | [[Category:Fourier transform]] | ||
| + | [[Category:signals and systems]] | ||
| + | == Example of Computation of Fourier transform of a CT SIGNAL == | ||
| + | A [[CT_Fourier_transform_practice_problems_list|practice problem on CT Fourier transform]] | ||
| + | ---- | ||
| + | |||
== Specify a signal x(t) == | == Specify a signal x(t) == | ||
| − | <math>x(t)=cos(8 \pi | + | <math>x(t)=cos(8 \pi t)e^{-t^{2}}</math> |
| + | |||
| + | == Fourier Transform of x(t) == | ||
| + | :<math>\begin{align} X(\omega) &=\int_{-\infty}^{\infty} x(t) e^{-j\omega t}dt | ||
| + | \\&= \int_{-\infty}^{\infty} cos(8 \pi t)e^{-t^{2}}e^{-j\omega t}dt | ||
| + | \\&= \int_{-\infty}^{\infty}\frac{e^{j8\pi t}-e^{-j8\pi t}}{2}e^{-t^{2}}e^{-j\omega t}dt | ||
| + | \\&= \int_{-\infty}^{\infty}\frac{e^{j8\pi t-t^2}-e^{-j8\pi t-t^2}}{2}e^{-j\omega t}dt | ||
| + | \\&= \int_{-\infty}^{\infty}\frac{e^{t(j8\pi -t)}-e^{-t(j8\pi +t)}}{2}e^{-j\omega t}dt | ||
| + | \\&= \int_{-\infty}^{\infty}\frac{e^{(t-4\pi j)^2+16\pi ^2}-e^{-(t-4\pi j)^2-16\pi ^2}}{2}e^{-j\omega t}dt | ||
| + | \\&= \int_{-\infty}^{\infty}\frac{e^{(t-4\pi j)^2+16\pi ^2}}{2}e^{-j\omega t}dt -\int_{-\infty}^{\infty}\frac{e^{-(t-4\pi j)^2-16\pi ^2}}{2}e^{-j\omega t}dt | ||
| + | |||
| + | \end{align}</math> | ||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 12:25, 16 September 2013
Example of Computation of Fourier transform of a CT SIGNAL
A practice problem on CT Fourier transform
Specify a signal x(t)
$ x(t)=cos(8 \pi t)e^{-t^{2}} $
Fourier Transform of x(t)
- $ \begin{align} X(\omega) &=\int_{-\infty}^{\infty} x(t) e^{-j\omega t}dt \\&= \int_{-\infty}^{\infty} cos(8 \pi t)e^{-t^{2}}e^{-j\omega t}dt \\&= \int_{-\infty}^{\infty}\frac{e^{j8\pi t}-e^{-j8\pi t}}{2}e^{-t^{2}}e^{-j\omega t}dt \\&= \int_{-\infty}^{\infty}\frac{e^{j8\pi t-t^2}-e^{-j8\pi t-t^2}}{2}e^{-j\omega t}dt \\&= \int_{-\infty}^{\infty}\frac{e^{t(j8\pi -t)}-e^{-t(j8\pi +t)}}{2}e^{-j\omega t}dt \\&= \int_{-\infty}^{\infty}\frac{e^{(t-4\pi j)^2+16\pi ^2}-e^{-(t-4\pi j)^2-16\pi ^2}}{2}e^{-j\omega t}dt \\&= \int_{-\infty}^{\infty}\frac{e^{(t-4\pi j)^2+16\pi ^2}}{2}e^{-j\omega t}dt -\int_{-\infty}^{\infty}\frac{e^{-(t-4\pi j)^2-16\pi ^2}}{2}e^{-j\omega t}dt \end{align} $
