LTI System
$ y(t)=2x(t)+x(t+2) $
Unit Impulse and System Function
The unit impulse is the systems response to an input of the function $ \delta(t) $.
$ x(t)=\delta(t) $
$ h(t)=2\delta(t)+\delta(t+2) $ is the Unit Unit Impulse Response.
$ H(s)=\int_{-\infty}^\infty h(\tau)e^{-j\omega\tau}d\tau $ is the equation used to find the system response.
$ H(s)=\int_{-\infty}^\infty (2\delta(t)+\delta(t+2))e^{-j\omega\tau}d\tau $
$ H(s)=2+e^{2j\omega} $
Response to a Signal
My signal in Part 1 was: $ x(t)=sin(\pi t) + cos(2\pi t) $
$ x(t)=sin(\pi t) + cos(2\pi t) = \frac{1}{j}e^{-j}+e^{j}+ \frac{1}{2}e^{2j}+ \frac{1}{2}e^{-2j} $
