LTI System: $ y(t) = Kx(t)\, $ where K is a constant
Unit Impulse Response: $ h(t) = K \delta(t) $
Frequency Response:
$ x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\omega_0 t}\, $
then $ y(t)=\sum^{\infty}_{k = -\infty}a_k*[h(t)*e^{j\omega_0 t}] $
$ H(s) = \int^{\infty}_{-\infty} h(t)e^{-j\omega_0 t} dt $ by definition
$ H(s) = \int^{\infty}_{-\infty} K \delta(t) e^{-j\omega_0 t} dt $
$ H(s) = K e^{-jw0} $
$ H(s) = K $
Response of the CT LTI system in 4.1:
$ x(t) = 1+\sin \omega_0 t + \cos(2\omega_0 t+ \frac{\pi}{4}) $
$ x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\omega_0 t} $
$ y(t) = \sum^{\infty}_{k = -\infty} a_k H(s) e^{jk\omega_0 t} $
$ y(t) = \sum^{\infty}_{k = -\infty} a_k (10) e^{jk\omega_0 t} $
$ y(t) = K\sum^{\infty}_{k = -\infty} a_k e^{jk\omega_0 t} $
$ y(t) = K+K\sin \omega_0 t + K\cos(2\omega_0 t+ \frac{\pi}{4}) $
