Problem
Given the system $ y(t) = 5x(t-1)\, $, where $ y(t)\, $ is the output and $ x(t)\, $ is the input, find the unit impulse response $ h(t)\, $ and the system function $ H(s)\, $.
Then find the response to $ x(t) = 5cos(3\pi t) + sin(\pi t)\, $
Analysis
$ h(t) = 5\delta (t-1)\, $
$ \,H(s)=\int_{-\infty}^{\infty}h(t)e^{-st}dt = \int_{-\infty}^{\infty}5\delta (t-1)e^{-st}dt = 5e^{-s}\, $
From HW4.1 William Schmidt_ECE301Fall2008mboutin the input $ x(t)\, $ is equivalent to:
- $ x(t) = \frac{5}{2} e^{3\pi jt} + \frac{5}{2} e^{-3\pi jt} + \frac{1}{2j} e^{\pi jt} - \frac{1}{2j} e^{-\pi jt} $
We know because the system is LTI the response is:
- $ \,y(t)=\sum_{k=-\infty}^{\infty}a_kH(jkw_o)e^{jkw_ot}\, $, with $ w_o = \pi\, $
The total response is:
- $ y(t) = 5e^{-3\pi}\frac{5}{2} e^{3\pi jt} + 5e^{3\pi}\frac{5}{2} e^{-3\pi jt} + 5e^{-\pi}\frac{1}{2j} e^{\pi jt} - 5e^{3\pi}\frac{1}{2j} e^{-\pi jt} $
