Question

For the system, y (t) = x (\(\frac{t}{2}\)), which of the following holds true?

(a) System is Linear, time-invariant, causal and stable

(b) System is Linear and time-invariant

(c) System is Linear and causal

(d) System is Linear and stable

I have been asked this question in class test.

The origin of the question is Properties of Systems in section Signals and Systems Basics of Signals and Systems

Answer 1

The correct option is (d) System is Linear and stable

For explanation: y1 (t) = v (\(\frac{t}{2}\))

And y2 (t) = k v (\(\frac{t}{2}\)) = k y1 (t)

If x3 = v (t) + w (t),

Then, y3 (t) = v (\(\frac{t}{2}\)) + w (\(\frac{t}{2}\))

= y1 (t) + y2 (t)

Since, it satisfies both law of homogeneity and additivity, it is also linear.

Again, y1 (t) = v (\(\frac{t}{2}\)), y2 (\(\frac{t}{2}\)-t0) ≠ y (t-t0) = v (\(\frac{t-t_0}{2})\)

∴ The system is time variant.

If x (t) is bounded then y (t) is bounded, so a stable system.

At time t=-2, y (-2) = x (-1), therefore, the response at time t=-2 depends on the excitation at a later time, t=-1, so a Non-causal system.