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== Unit Impulse == | == Unit Impulse == | ||
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<math> y(t) = H(s)x(t)\,</math><br> | <math> y(t) = H(s)x(t)\,</math><br> | ||
| − | <math> y(t) = \frac{1}{2}e^{\frac{2j2\pi t}{6}} + \frac{1}{2}e^{\frac{-2j2\pi t}{6}} -2je^{\frac{2j5\pi t}{6}} + 2je^{\frac{-2j5\pi t}{6}}</math> | + | <math> y(t) = \frac{1}{j2}\frac{1}{2}e^{\frac{2j2\pi t}{6}} + \frac{1}{-j2}\frac{1}{2}e^{\frac{-2j2\pi t}{6}} \frac{1}{j5}-2je^{\frac{2j5\pi t}{6}} + \frac{1}{-j5}2je^{\frac{-2j5\pi t}{6}}</math> |
Revision as of 17:45, 25 September 2008
Unit Impulse
$ h(t) = u(t-1) \, $
$ H(s) = \int^{\infty}_{-\infty} u(t-1)e^{-jw_0 t} dt\, $
$ H(s) = \int^{\infty}_{1}e^{-jw_0 t} dt\, $
$ H(s) = \frac{1}{jw_0} $
Repsonse of the CT system
$ x(t) = cos({\frac{2\pi t}{3}})+ 4sin({\frac{5\pi t}{3}})\, $
$ y(t) = H(s)x(t)\, $
$ y(t) = \frac{1}{j2}\frac{1}{2}e^{\frac{2j2\pi t}{6}} + \frac{1}{-j2}\frac{1}{2}e^{\frac{-2j2\pi t}{6}} \frac{1}{j5}-2je^{\frac{2j5\pi t}{6}} + \frac{1}{-j5}2je^{\frac{-2j5\pi t}{6}} $
