Define a CT LTI System
Let $ y(t) = 3x(t) + x(t-3) $
Let $ x(t) = \delta (t) $ so that $ h(t)= 3 \delta (t) + \delta (t-3) $
Response to My Function From Part 1
The function I chose for part 1 was
$ x(t) = 4sin(3t) + 8cos(7t) = \frac{2e^{j3t} }{j}- \frac{2e^{j3t} }{j} + 4e^{j7t} + 4e^{-j7t} $
Now to compute H(jw)
$ H(j\omega)=\int_{-\infty}^\infty h(\tau)e^{-j\omega\tau}d\tau $
$ H(j\omega)=\int_{-\infty}^\infty (3\delta(\tau) + \delta(\tau-3))e^{-j\omega\tau}d\tau $
Using the sifting property we get
$ H(j\omega)=(3 + e^{3 j\omega}) $
So since y(t) = x(t)*H(t)
$ y(t) = \frac{2e^{j3t}(3 + e^{9 j\omega})} {j}- \frac{2e^{j3t}(3 + e^{9 j\omega})} {j} + 4e^{j7t}(3 + e^{21 j\omega}) + 4e^{-j7t}(3 + e^{-21 j\omega}) $
