(New page: ===System=== y(t) = 5x(t) ===Unit Impulse Response=== <math>x(t) = \delta(t)</math> <math>h(t) = 5\delta(t)</math> ===System Function=== :<math>y(t) = \int^{\infty}_{-\infty} h(\tau) * ...) |
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:<math>H(s) = 5 e^{-jw0}</math> | :<math>H(s) = 5 e^{-jw0}</math> | ||
:<math>H(s) = 5\,</math> | :<math>H(s) = 5\,</math> | ||
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| + | ===Response to a signal=== | ||
| + | |||
| + | :<math> x(t) = 2sin(2\pi t) + cos(\pi t). </math> | ||
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| + | :<math> x(t) = \frac{1}{j}(e^{2 \pi jt} + e^{-2 \pi jt}) + \frac{1}{2}(e^{\pi jt}+e^{-\pi jt}) </math> | ||
| + | |||
| + | :<math> y(t) = H(s)e^{-st}x(t) </math> | ||
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| + | :<math> y(t) = 5e^{-st}(\frac{1}{j}(e^{2 \pi jt} + e^{-2 \pi jt}) + \frac{1}{2}(e^{\pi jt}+e^{-\pi jt})) </math> | ||
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| + | :<math> y(t) = \frac{5e^{-st}}{j}(e^{2 \pi jt} + e^{-2 \pi jt}) + \frac{5e^{-st}}{2}(e^{\pi jt}+e^{-\pi jt}) </math> | ||
Latest revision as of 07:43, 25 September 2008
System
y(t) = 5x(t)
Unit Impulse Response
$ x(t) = \delta(t) $ $ h(t) = 5\delta(t) $
System Function
- $ y(t) = \int^{\infty}_{-\infty} h(\tau) * x(\tau) d\tau, $
- $ y(t) = \int^{\infty}_{-\infty} 5\delta(\tau) * e^{-jw(t -\tau)} d \tau, $
- $ y(t) = e^{jwt} \int^{\infty}_{-\infty} 5 \delta(\tau) e^{-jw\tau} d\tau $
- $ H(s) = \int^{\infty}_{-\infty} 5 \delta(\tau) e^{-jw\tau} d\tau $
- $ H(s) = 5 e^{-jw0} $
- $ H(s) = 5\, $
Response to a signal
- $ x(t) = 2sin(2\pi t) + cos(\pi t). $
- $ x(t) = \frac{1}{j}(e^{2 \pi jt} + e^{-2 \pi jt}) + \frac{1}{2}(e^{\pi jt}+e^{-\pi jt}) $
- $ y(t) = H(s)e^{-st}x(t) $
- $ y(t) = 5e^{-st}(\frac{1}{j}(e^{2 \pi jt} + e^{-2 \pi jt}) + \frac{1}{2}(e^{\pi jt}+e^{-\pi jt})) $
- $ y(t) = \frac{5e^{-st}}{j}(e^{2 \pi jt} + e^{-2 \pi jt}) + \frac{5e^{-st}}{2}(e^{\pi jt}+e^{-\pi jt}) $
