Question

The input x(t) and the corresponding output y(t) of a linear time-invariant (LTI) system are x(t) = u(t) – uſt – 1), y(t) = r(t) – 2r(t – 1) + r(t – 2) Determine the outputs yi(t), i : 1; 2; 3 corresponding to the following inputs

x(t) = u(t - 2) + u(t - 3);

Answer 1

To find the outputs yi(t) corresponding to the given input x(t) = u(t - 2) + u(t - 3), we substitute the input into the output equation y(t) = r(t) - 2r(t - 1) + r(t - 2). Simplifying the equation gives us the corresponding outputs y1(t), y2(t), and y3(t).

Step-by-step explanation:

To determine the outputs y1(t), y2(t), and y3(t) corresponding to the input x(t) = u(t - 2) + u(t - 3), we substitute the given input into the output equation y(t) = r(t) - 2r(t - 1) + r(t - 2).

By substituting x(t), we get:

y(t) = r(t) - 2r(t - 1) + r(t - 2) = [u(t - 2) + u(t - 3)] - 2[u(t - 3) + u(t - 4)] + [u(t -4) + u(t - 5)].

Simplifying further:

y(t) = u(t - 2) -u(t - 3) - [2u(t - 3) - 2u(t - 4)] + u(t - 4) + u(t - 5) = u(t - 2) - u(t - 3) + 2u(t - 4) - u(t - 5).

So, the corresponding outputs yi(t) are:

• y1(t) = u(t - 2) - u(t - 3) + 2u(t - 4) - u(t - 5)
• y2(t) = u(t - 3) - u(t - 4) + 2u(t - 5) - u(t - 6)
• y3(t) = u(t - 4) - u(t - 5) + 2u(t - 6) - u(t - 7)