Question

Is the system y[n]=2x[n]+2 linear?

(a) YES

(b) NO

I had been asked this question during an online exam.

This intriguing question comes from Properties of Systems topic in portion Signals and Systems Basics of Signals and Systems

Answer 1

The correct answer is (b) NO

The best explanation: The system needs to satisfy superposition principle for linearity:

For input x1[n], y1 [n] = 2x1 [n] + 2

For input x2[n], y2 [n] = 2x2 [n] + 2

⇒ ay1 [n]+ by2 [n] = 2(ax1 [n]+ bx2 [n]) + 2(a+b)  Equation 1

For, x3[n], y3 [n]=2x3 [n]+2 = 2(ax1 [n]+ bx2 [n]) + 2 Equation 2

Clearly, Equation 1 is not equal to equation

∴ The system does not satisfy superposition principle ⇒ The system is not linear.