Question #200643

Find the following system are Linear or non-linear

1. Y(t)= X(t-2)+X(2-t)

2. Y(t)=t^2 X(t-1)

Expert's answer

a) The system is linear because if

"y_1(t) = x_1(t \u2212 2) + x_1(2 \u2212 t)"

"y_2(t) = x_2(t \u2212 2) + x_2(2 \u2212 t)"

"x(t) = \u03b1x_1(t) + \u03b2x_2(t)"

then the output y(t) corresponding to the input x(t) is

"y(t) = x(t - 2) + x(2 - t)"

"= (\u03b1x1 + \u03b2x_2)(t - 2) + (\u03b1x_1 + \u03b2x_2)(2 - t)"

"= \u03b1x_1(t - 2) + \u03b2x_2(t - 2) + \u03b1x_1(2 - t) + \u03b2x_2(2 - t)"

"= \u03b1 (x_1(t - 2) + x_1(2 - t)) + \u03b2 (x_2(t - 2) + x_2(2 - t))"

"= \u03b1y_1(t) + \u03b2y_2(t)"

b) The system is linear because if

"y_1(t)=t^2 x_1(t-1)"

"y_2(t)=t^2 x_2(t-1)"

and x(t) = αx1(t) + βx2(t), then the output y(t) corresponding to the input x(t) is

"y(t) = (\u03b1x_1 + \u03b2x_2)(t)"

"= \u03b1(t^2 x_1(t-1))(t) + \u03b2(t^2 x_2(t-1))(t)"

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