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| − | <math>X(w) = \pi \delta (w - 2 \pi)(3j - 7) + \pi \delta (w + 2 \pi) ( | + | [[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]] | ||
| + | ---- | ||
| + | <math>X(w) = \pi \delta (w - 2 \pi)(3j - 7) + \pi \delta (w + 2 \pi) (3j + 7) </math> | ||
<math>x(t) = \frac{1}{2 \pi} \int^{\infty}_{- \infty} X(w) e^{jwt} dw</math> | <math>x(t) = \frac{1}{2 \pi} \int^{\infty}_{- \infty} X(w) e^{jwt} dw</math> | ||
| − | <math>\frac{1}{2 \pi} \int^{\infty}_{- \infty} [ \pi \delta (w - 2 \pi)(3j - 7) + \pi \delta (w + 2 \pi) ( | + | <math> = \frac{1}{2 \pi} \int^{\infty}_{- \infty} [ \pi \delta (w - 2 \pi)(3j - 7) + \pi \delta (w + 2 \pi) (3j + 7)] e^{jwt} dw</math> |
| − | <math>\frac{3j - 7}{2} \int^{\infty}_{- \infty} \delta (w -2 \pi) e^{jwt} dw + \frac | + | <math> = \frac{3j - 7}{2}\int^{\infty}_{- \infty}\delta (w -2\pi) e^{jwt} dw + \frac {3j + 7}{2}\int^{\infty}_{- \infty}\delta (w + 2\pi) e^{jwt} dw</math> |
| + | |||
| + | <math> = \frac{3j - 7}{2} e^{j2\pi t} + \frac{3j + 7}{2} e^{-j2\pi t}</math> | ||
| + | |||
| + | <math> = \frac{3j}{2} e^{j 2\pi t} - \frac{7}{2} e^{j 2\pi t} + \frac{3j}{2} e^{-j 2\pi t} + \frac{7}{2} e^{-j 2\pi t}</math> | ||
| + | |||
| + | <math> = \frac{3j(e^{j 2\pi t} + e^{-j 2\pi t})}{2} + \frac{7(- e^{j 2\pi t} + e^{-j 2\pi t})}{2}</math> | ||
| + | |||
| + | <math> = 3sin(2\pi) + 7cos(2\pi)</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
$ X(w) = \pi \delta (w - 2 \pi)(3j - 7) + \pi \delta (w + 2 \pi) (3j + 7) $
$ x(t) = \frac{1}{2 \pi} \int^{\infty}_{- \infty} X(w) e^{jwt} dw $
$ = \frac{1}{2 \pi} \int^{\infty}_{- \infty} [ \pi \delta (w - 2 \pi)(3j - 7) + \pi \delta (w + 2 \pi) (3j + 7)] e^{jwt} dw $
$ = \frac{3j - 7}{2}\int^{\infty}_{- \infty}\delta (w -2\pi) e^{jwt} dw + \frac {3j + 7}{2}\int^{\infty}_{- \infty}\delta (w + 2\pi) e^{jwt} dw $
$ = \frac{3j - 7}{2} e^{j2\pi t} + \frac{3j + 7}{2} e^{-j2\pi t} $
$ = \frac{3j}{2} e^{j 2\pi t} - \frac{7}{2} e^{j 2\pi t} + \frac{3j}{2} e^{-j 2\pi t} + \frac{7}{2} e^{-j 2\pi t} $
$ = \frac{3j(e^{j 2\pi t} + e^{-j 2\pi t})}{2} + \frac{7(- e^{j 2\pi t} + e^{-j 2\pi t})}{2} $
$ = 3sin(2\pi) + 7cos(2\pi) $
