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I am studying Signals and Systems. Anyone, please help me to answer the problems.

  1. $y(t)=dx(t)/dt$ is time invariant: True or False ?

  2. $h(t)$ is the impulse response of an LTI system. If $h(t)$ is periodic and non-zero, the system is unstable: True or False?

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For question 1: apply the definition of time invariant: find the output as normal; find the output with the same input but delayed by $T$ $$ y_1(t) = \frac{dx(t)}{dt}\\ y_2(t) = \frac{dx(t-T)}{dt}\\ $$ Does $y_2(t) = y_1(t-T)$ ?

For question 2: Find a definition of stability and apply it. For example, for a system to be BIBO stable it needs to have

$$ \int_{-\infty}^{+\infty} \left|h(t)\right| dt < \infty $$

If $h(t)$ is periodic and non-trivial (zero), can that be true?

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  • $\begingroup$ Peter, for stability we require the $L_1$ norm to be finite, not the $L_2$ norm. E.g., an ideal low pass filter satisfies $\int_{-\infty}^{\infty}|h(t)|^2dt<\infty$, but it's not stable. $\endgroup$
    – Matt L.
    Commented Oct 26, 2015 at 16:16
  • $\begingroup$ @MattL. D'oh! You are, as usual, correct! Thanks! Corrected. $\endgroup$
    – Peter K. ♦
    Commented Oct 26, 2015 at 16:22
  • $\begingroup$ 1) False, 2) True.. is that corect? $\endgroup$
    – Aadnan Farooq A
    Commented Oct 26, 2015 at 16:47
  • $\begingroup$ The Q1 statement being false means that if I take a derivative of a signal this morning it will give me one answer and then I take the derivative of a signal this afternoon it'll give me a different answer. $\endgroup$
    – Peter K. ♦
    Commented Oct 26, 2015 at 16:57
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    $\begingroup$ @AadnanFarooqA: Correct! $\endgroup$
    – Peter K. ♦
    Commented Oct 26, 2015 at 18:05

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