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\pi t}\, $
then $ y(t)=\sum^{\infty}_{k = -\infty}a_k*(h(t)*e^{jj\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(r) e^{-jwr} dr $
$ H(s) = K e^{-jw0} $
$ H(s) = K $
