# properties of System

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?

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?

• 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. – Matt L. Oct 26 '15 at 16:16
• @MattL. D'oh! You are, as usual, correct! Thanks! Corrected. – Peter K. Oct 26 '15 at 16:22
• 1) False, 2) True.. is that corect? – Aadnan Farooq A Oct 26 '15 at 16:47
• 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. – Peter K. Oct 26 '15 at 16:57
• @AadnanFarooqA: Correct! – Peter K. Oct 26 '15 at 18:05