Find the response of the system to the following inputs

Question

Regular Problems

Problem 1. Skitch the following signals.

a. x_a(t)= sin(πt/2) [u(t)-u(t-8)]

Problem 2. A linear time-invariant system is found to have an output y(t) when the input is x(t) where x(t) and y(t) are shown in the figure below.

Find the response of the system to the following inputs signals:

a.

b. x₂(t)=u(t)

Problem 3. For each of the systems defined below. determine whether or not they are (a) mentoryless. (b) causal, (c) stable. (d) linear, (e) time-invariant. Show your work on how you determined whether or not the system has or does not have each property.

(a) y(t) = x(t - 1)u(t - 1)

(b) y(t) = 2x(t)+cos (πt)

(c) y(t) = x(t/2)

Problem 4. It is known that if x(t) = e^-t u(t- 1) is the input to a system then the response is y(t) tu(t). What properties. such as memoryless, causal, stabl, linear, and time-invariant (if any) can be established for this system? Explain. (Think carefully about this one. All you are given is the response to only one input).

Problem 5. Evaluate the integral y(t) = _-∞∫^t P(T)dT, where x(t) is a unit pulse centered at the origin of duration T = 1,

and make a carefully labeled plot of y(t) for all t.

Problem 6. Evaluate the integral y(t) = ₀∫¹ e^-(t-T)dT