Question

The impulse response function of four linear systems $S_1,S_2,S_3,S_4$ are given respectively by $$h_1\left(t\right)=1h_2\left(t\right)=u\left(t\right)$$$$h_3\left(t\right)=\frac{u\left(t\right)}{t+1}{h}_4\left(t\right)=e^{-3t}u\left(t\right)$$Where u(t) is the unit step function. Which of these systems is time invariant, casual, and stable?

Answer 1

$h_1\left(t\right)\ne 0fort<0$
$\therefore s_1$ $\text{is non causal}$

$h_2\left(t\right)=u\left(t\right)=0fort<0$$\underset{-\infty }{\overset{\infty }{\int }}{h}_2\left(t\right)dt=\underset{-\infty }{\overset{\infty }{\int }}u\left(t\right)dt=\underset{0}{\overset{\infty }{\int }}dt=\infty$
$\therefore s_2$ $\text{is unstable and causal}$

$h_3\left(t\right)=\frac{u\left(t\right)}{t+1}$$att=-1$$h_3\left(t\right)=\infty ,h_3\left(t\right)=0fort<0$
$\therefore s_3$$\text{is unstable and causal}$

$h_4\left(t\right)=e^{-3t}u\left(t\right)$
$s_4$ $\text{is time invariant, causal and stable.}$