Answer to Question #200643 in Electrical Engineering for Alock kumar

Question

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)

Answer 1

a) The system is linear because if

$y_1(t) = x_1(t − 2) + x_1(2 − t)$

$y_2(t) = x_2(t − 2) + x_2(2 − t)$

$x(t) = αx_1(t) + βx_2(t)$

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

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

$= (αx1 + βx_2)(t - 2) + (αx_1 + βx_2)(2 - t)$

$= αx_1(t - 2) + βx_2(t - 2) + αx_1(2 - t) + βx_2(2 - t)$

$= α (x_1(t - 2) + x_1(2 - t)) + β (x_2(t - 2) + x_2(2 - t))$

$= αy_1(t) + βy_2(t)$

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

$y(t) = (αx_1 + βx_2)(t)$

$= α(t^2 x_1(t-1))(t) + β(t^2 x_2(t-1))(t)$