Question

For the system, y (t) = x (t-5) – x (3-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, time-invariant and stable

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

I got this question in a dream while sleeping

Asked question is from Properties of Systems in portion Signals and Systems Basics of Signals and Systems

Answer 1

Correct choice is (c) System is Linear, time-invariant and stable

For explanation I would say: y1 (t) = v (t-5) – v (3-t)

y2 (t) = k v (t-5) – k v (3-t) = ky1 (t)

Let x1 (t) = v (t), then y1 (t) = v (t-5) – v (3-t)

Let x2 (t) = 2w (t), then y2 (t) = w (t-5)-w (3-t)

Let x3 (t) = x (t) + w (t)

Then, y3 (t) = y1 (t) + y2 (t)

Hence it is linear.

Again, y1 (t) = v (t-5) – v (3-t)

∴ y2 (t) = y1 (t-t0)

Hence, system is time-invariant

If x (t) is bounded, then, x (t-5) and x (3-t) are also bounded, so stable system.

At t=0, y (0) = x (-5)-x (3), therefore, the response at t=0 depends on the excitation at a later time t=3.

Therefore Non-Causal.