Question

Determine the nature of the system: y(t)= t^2 x(t-1)

(a) Linear, time invariant

(b) Linear, time variant

(c) Non-linear, time invariant

(d) Non-linear, time variant

I had been asked this question by my school teacher while I was bunking the class.

My question comes from Properties of Systems in division Signals and Systems Basics of Signals and Systems

Answer 1

The correct option is (b) Linear, time variant

The best explanation: For linearity:

For input x1(t): y1 (t)= t^2 x1 (t-1)

For input x2(t): y2 (t)= t^2 x2 (t-1)

⇒ ay1 (t)+by2 (t)= t^2 [x1 (t-1)+ bx2 (t-1)]  Equation 1

For input x3(t): y3 (t)= t^2 x3 (t-1) = t^2 [ax1 (t-1)+ bx2 (t-1)] Equation 2

∴ The system is linear.

For time invariancy: Shift in input:

⇒y(t,T)= t^2 x(t-1-T)

Shift in output:  y(t- T)= (t-T)^2 x(t-1-T)

∵ The shift in output is not equal to the shift in input, therefore, the system is time variant.