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}) $
