Question

For the system, y (t) = cos 2πt x (t), which of the following holds true?

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

(b) System is time-invariant, causal and stable

(c) System is Linear, causal and stable

(d) System is Linear, time-invariant and stable

I have been asked this question in final exam.

This key question is from Properties of Systems in section Signals and Systems Basics of Signals and Systems

Answer 1

The correct choice is (c) System is Linear, causal and stable

Easiest explanation: y2 (t) k cos 2πt v (t) = k y1 (t)

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

So, y3 (t) = cos 2πt [v (t) + w (t)]

= y1 (t) + y2 (t)

Since, the system is both homogeneous and additive, therefore it is linear.

Again, y1 (t) = cos 2πt v (t)

And y2 (t) = cos 2πt (t-t0) ≠ y (t-t0)

= cos [2π (t-t0)] v (t-t0)

∴ The system is time variant.

The response at any time t=t0, depends only on the excitation at that time and not on the excitation at any later time, so Causal system.

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