CT LTI sytem
An example system would be:
y(t) = 2*x(t)
Part A: The unit impulse response and system function H(s)
The unit impulse response:
$ x(t) \to \delta(t) * h(t) = 2*\delta(t) $
The system function, H(s) derivation:
$ y(t) = \int_{-\infty}^{\infty} x(\tau) * h(\tau) *d\tau $
$ y(t) = \int_{-\infty}^{\infty} e^{-j*w(t-k)} * 2\delta(\tau) *d\tau $
$ y(t) = e^{j*w*t} \int_{-\infty}^{\infty} e^{-j*w*k} * 2\delta(\tau) * d\tau $
$ H(s) = \int_{-\infty}^{\infty} e^{-j*w*k} * 2\delta(\tau) d\tau $
$ 2*1 = 2 $
$ H(s) = 2 $
