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Signals and Systems - Engineering

Q1:

Which one is a linear system?

A y[n] = x[n] x x[n - 1]

B y[n] = x[n] + x[n - 10]

C y[n] = x2[n]

D (a) and (c)

ANS:B - y[n] = x[n] + x[n - 10]

For linearity y1[n] = x1[n] + x2[n - 10] ...(1) y2 = x2[n] + x2[n - 10] ...(2) y1[n] = x1[n] + x2[n] + x2[n - 10] + x2[n - 10] ...(3) Now find y1[n] + y2[n] Corresponding to x1[n] + x2[n] It is same as equation (3) hence linear. But in part (c) y[n] = x2[n] ⇒ y1[n] = x21[n], y2[n] = x22[n]⇒ y1[n] + y2[n] = x22[n]...3 But y1[n] + y2[n] Corresponing x1[n] + x2[n] is y1[n] + y2[n] = {x1[n] + x2[n]}2 = x12[n] + x22[n] + 2x1[n] x2[n]....4 Equations (3) and (4) are not same hence not linear.