Description

(12 pts) Analyze whether the following systems have these properties: memory, stability, causality, linearity, invertibility, timeinvariance. Provide your answer in detail.
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P
(a) (6 pts) y[n] = x[n k]
k=1


(6 pts) y(t) = tx(2t + 3)


(13 pts) Consider an LTI system given by the following block diagram:

x(t)
+
Z
y(t)
5


(3 pts) Find the di erential equation which represents this system.



(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.


(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].


(5 pts) Given x(t) = u(t 1) + u(t + 1) and h(t) = e ^{t} sin(t)u(t), calculate y(t) = ^{dx}_{dt}^{(t)} h(t).


(20 pts) Evaluate the following convolutions.


(10 pts) Given h(t) = e ^{2t}u(t) and x(t) = e ^{t}u(t), nd y(t) = x(t) h(t).

(b) (10 pts) Given h(t) = e^{3t}u(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) 3y^{00}(t) + 4y^{0}(t) 2y(t) = 0, y^{00}(0) = 2, y^{0}(0) = 1 and y(0) = 3.
6. (20 pts) Consider the following discrete time LTI system which is initially at rest:
x[n] 
h_{0}[n] 
w[n] 
h_{0}[n] 
y[n] 

1 

where w[n] 
w[n 1] = x[n]. 

2 


(10 pts) Find h_{0}[n].

(5 pts) Find the overall impulse response, h[n], of this system.

(5 pts) Find the di erence equation which represents the relationship between the input x[n] and the output y[n].
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