1. (12 pts) Analyze whether the following systems have these properties: memory, stability, causality, linearity, invertibility, time-invariance. Provide your answer in detail.
1
P
(a) (6 pts) y[n] = x[n k]
k=1
(b) (6 pts) y(t) = tx(2t + 3)
2. (13 pts) Consider an LTI system given by the following block diagram:
x(t)
+
Z
y(t)
5
(a) (3 pts) Find the di erential equation which represents this system.
(b) (10 pts) Find the output y(t), when the input x(t) = (e t + e 3t)u(t). Assume that the system is initially at rest.
3. (15 pts) Evaluate the following convolutions.
(a) (10 pts) Given x[n] = 2 [n] + [n + 1] and h[n] = [n 1] + 2 [n + 1], compute and draw y[n] = x[n] h[n].
(b) (5 pts) Given x(t) = u(t 1) + u(t + 1) and h(t) = e t sin(t)u(t), calculate y(t) = dxdt(t) h(t).
4. (20 pts) Evaluate the following convolutions.
(a) (10 pts) Given h(t) = e 2tu(t) and x(t) = e tu(t), nd y(t) = x(t) h(t).
(b) (10 pts) Given h(t) = e3tu(t) and x(t) = u(t) u(t 1), nd y(t) = x(t) h(t).
5. (20 pts) Solve the following homogeneous di erence and di erential equations with the speci ed initial conditions.
(a) (10 pts) 2y[n + 2] 3y[n + 1] + y[n] = 0, y[0] = 1 and y[1] = 0.
(b) (10 pts) y(3)(t) 3y00(t) + 4y0(t) 2y(t) = 0, y00(0) = 2, y0(0) = 1 and y(0) = 3.
6. (20 pts) Consider the following discrete time LTI system which is initially at rest:
x[n]
h0[n]
w[n]
h0[n]
y[n]
1
where w[n]
w[n 1] = x[n].
2
(a) (10 pts) Find h0[n].
(b) (5 pts) Find the overall impulse response, h[n], of this system.
(c) (5 pts) Find the di erence equation which represents the relationship between the input x[n] and the output y[n].
1
