Description
1. (20 points) Linear systems Determine whether each of the following systems is linear or
not. Explain your answer.
(a) y(t) = |x(t)| + x(t)
(b) y(t) = 1 + x(t) cos(ωt)
(c) y(t) = cos(ωt + x(t))
(d) y(t) = (x(t) + x(−t)) u(t)
2. (13 points) LTI systems
(a) (7 points) Consider an LTI (linear time-invariant) system whose response to x1(t) is
y1(t), where x1(t) and y1(t) are illustrated as follows:
Sketch the response of the system to the input x2(t).
(b) (6 points) Assume we have a linear system with the following input-output pairs:
• the output is y1(t) = e
−tu(t) when the input is x1(t) = u(t);
• the output is y2(t) = e
−t
(u(t) − u(t − 1)) when the input is x2(t) = rect(t −
1
2
).
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Is the system time-invariant?
3. (42 points) Convolution
(a) (10 points) For each pair of the signals given below, compute their convolution using
the flip-and-drag technique. Please provide a piecewise formula for y(t).
i. f(t) = 2 rect(t −
3
2
), g(t) = 2 r(t − 1)rect(t −
3
2
)
ii. f(t) = u(−t − 1), g(t) = e
−tu(t)
(b) (12 points) For each of the following, find a function h(t) such that y(t) = x(t) ∗ h(t).
i. y(t) = R t
t−T
x(τ )dτ
ii. y(t) = x(t − 1)
iii. y(t) = P∞
n=−∞ x(t − nTs)
Note: this last operation creates a periodic extension of x(t) where the period is Ts.
(c) (12 points) Use the properties of convolution to simplify the following expressions:
i. hR t
−∞ u(−τ + 3)δ(τ − 1)dτi
∗ (δ(t − 2) + δ(2t − 8))
ii. [δ(t − 3) + δ(t + 2)] ∗
e
3tu(−t) + δ(t + 2) + 2
iii. d
dt [(u(t) − u(t − 1)) ∗ u(t − 2)], Hint: Show first that u(t) ∗ u(t) = r(t) where r(t)
is the ramp function.
(d) (8 points) Explain whether each of the following statements is true or false.
i. If x(t) and h(t) are both odd functions, and y(t) = x(t) ∗ h(t), then y(t) is an even
function.
ii. If y(t) = x(t) ∗ h(t), then y(2t) = h(2t) ∗ x(2t).
4. (9 points) Impulse response and LTI systems
Consider the following three LTI systems:
• The first system S1 is given by its input-output relationship: y(t) = R t+t0
−∞ x(τ )dτ ;
• The second system S2 is given by its impulse response: h2(t) = u(t − 2);
• The third system S3 is given by its impulse response: h3(t) = δ(t − 3).
(a) (3 points) Compute the impulse responses h1(t) of system S1.
(b) (3 points) The three systems are interconnected as shown below.
Determine the impulse response heq(t) of the equivalent system.
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