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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)&=1\\ h_2(t)&=u(t)\\ h_3(t)&=\frac{u(t)}{(t+1)}\\ h_4(t)&=\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 $t<0$ 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??

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  • $\begingroup$ "We also know if the impulse response function of the system is finite then the system is stable; so all the systems are stable." None of them seem finite to me. $\endgroup$ – Tendero Jan 29 '18 at 19:54
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You should remember the necessary and sufficient condition for (BIBO)-stability of a system described by an impulse response $h(t)$:

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

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.

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