(New page: Let <font size = '4'><math>x(t) = e^{-at} u(t)</math></font> <math>\chi(w) = \mathcal{F} (x(t)) = \int^{\infty}_{-\infty} e^{-at} u(t) e^{-jwt} dt</math> <math>= \int^{\infty}_{0} e^{-at...) |
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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]] | ||
| + | ---- | ||
| − | <math> | + | Let <font size = '4'><math>x(t) = e^{-a(t+1)} u(t + 1)</math></font> |
| − | <math>= \int^{\infty}_{ | + | <math>\chi(w) = \mathcal{F} (x(t)) = \int^{\infty}_{-\infty} e^{-at}e^{-a} u(t + 1) e^{-jwt} dt</math> |
| − | <math>= \int^{\infty}_{ | + | <math>= e^{-a} \int^{\infty}_{-1} e^{-at}.e^{-jwt} dt</math> |
| − | <math>= -\ | + | <math>= e^{-a} \int^{\infty}_{-1}e^{-(a+jw)t} dt</math> |
| − | <math>= -\frac{ | + | <math>= -\frac{e^{-a}}{a+jw} [e^{-(a+jw)t}]^{\infty}_{1} </math> |
| − | <math>=\frac{ | + | <math>= -\frac{e^{-a}}{a+jw} [-e^{-(a+jw)}]</math> |
| + | |||
| + | <math>=\frac{e^{-(2a+jw)}}{a+jw}</math> | ||
| + | ---- | ||
| + | [[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]] | ||
Latest revision as of 11:36, 16 September 2013
Example of Computation of Fourier transform of a CT SIGNAL
A practice problem on CT Fourier transform
Let $ x(t) = e^{-a(t+1)} u(t + 1) $
$ \chi(w) = \mathcal{F} (x(t)) = \int^{\infty}_{-\infty} e^{-at}e^{-a} u(t + 1) e^{-jwt} dt $
$ = e^{-a} \int^{\infty}_{-1} e^{-at}.e^{-jwt} dt $
$ = e^{-a} \int^{\infty}_{-1}e^{-(a+jw)t} dt $
$ = -\frac{e^{-a}}{a+jw} [e^{-(a+jw)t}]^{\infty}_{1} $
$ = -\frac{e^{-a}}{a+jw} [-e^{-(a+jw)}] $
$ =\frac{e^{-(2a+jw)}}{a+jw} $
