Question

For the system, \(t\frac{dy (t)}{dt}\) – 8 y (t) = x (t), which of the following holds true?

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

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

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

(d) System is Linear, causal and stable

The question was posed to me in a national level competition.

Question is taken from Properties of Systems in division Signals and Systems Basics of Signals and Systems

Answer 1

The correct choice is (c) System is time-invariant, causal and stable

Easiest explanation: All options are linear. Hence linearity is not required to be checked.

Let x1 (t) = v (t), then \(t\frac{dy_1 (t)}{dt}\) – 8 y1 (t) = v (t)

Let x2 (t) = v (t-t0)

Then, \(t\frac{dy_2 (t)}{dt}\) – 8 y2 (t) = v (t-t0)

The first equation can be written as (t-t0) \(t\frac{dy (t-t_0)}{dt}\) – 8 y (t-t0) = x (t-t0)

This equation is not satisfied if y2 (t) = y1 (t-t0). Therefore y2 (t) ≠ y1 (t-t0)

∴ 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.

The response will increase without bound as time increases, so unstable system.