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Which one is a linear system?

Correct Answer: y[n] = x[n] + x[n - 10]

Explanation:

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.


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