Time-invariance, causality and stability of $h(t)$ of four given systems Question:


  The impulse response functions of four linear systems $S_1,\ S_2,\ S_3,\ S_4$ are given respectively by
  
  \begin{align}
h_1(t)&amp;=1\\
h_2(t)&amp;=u(t)\\
h_3(t)&amp;=\frac{u(t)}{(t+1)}\\
h_4(t)&amp;=\exp(-3t)*u(t)
\end{align}
  
  where $u(t)$ is the unit step function. Which of these systems is time-invariant, causal and stable?
  
  
  A) S1  
  B) S2  
  C) S3  
  D) S4
  


My Approach:

we know if $h(t)$ is not equal to $0$ for $t&lt;0$ then the system is non-causal; so option $A$ is non-causal. We also know if the impulse response function of the system is finite then the system is stable; so all the systems are stable. Now what is the condition to check Time-invariancy??
You should remember the necessary and sufficient condition for (BIBO)-stability of a system described by an impulse response $h(t)$:

$$\int_{-\infty}^{\infty}|h(t)|dt&lt;\infty\tag{1}$$

If you use condition $(1)$ you will find out that actually only one of the four systems is stable.

The question about time-invariance does not make much sense, because if we assume that the systems are characterized by their impulse response then we already imply that the systems are linear and time-invariant. For linear time-varying systems you would need a two-dimensional impulse response, and non-linear systems cannot be characterized by an impulse response.