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EECS 150: Continuous-time Signals and Systems
Final Exam, Winter 2021
115 minutes

1. Given that the highest frequencies contained in x(t) and h(t) are 40 Hz and 50
Hz respectively, find the highest frequency contained in the following signals.
a) m(t) = 2x(t)h(t)
b) n(t) = 2x(t) +h(t)
c) o(t) = x(t)*h2(t)
d) p(t) = h(3(t-2))
e) () =
2()
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f) What is the Nyquist rate for case (a), m(t) = 2x(t)h(t)?


2. The input-output relationship of an LTI system is y(t) = 4x(t)cos(5t). If the
input is
x(t) = (8) ,
a) what is Y()? Sketch it.
b) what is ∫ ()
+∞
−∞
?
c) compute the Energy of y(t), Ey = ∫ |()|2
+∞
−∞
.
d) What is the impulse response of this LTI system?


3. The signal x(t) is the input of an LTI system with transfer function () =

2
−0.4
.
y(t) is the output of this system.
a) What is the Fourier transform of the impulse response of this system?
b) What is the DC gain of this system?
c) What is the impulse response of this system, if the Fourier transform of the
impulse response exists?
d) What is Y(s) if the input is e3tu(-t+2)?
e) What is y(t) if the input is ete-jt?
f) Find the differential equation showing the relationship between x(t) and y(t)


4. Signal x(t) = sin(20t) + cos(30t) is sampled at the rate of 50 rad/s and named
xs(t). Sketch the Fourier Transform of xs(t). The actual magnitudes and
frequencies should be shown.



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5. The periodic signal x(t) is shown in the following figure.

a) Write the mathematical representation of x(t)
b) What is the Fourier Transform of x(t)?
c) Sketch the Fourier Transform of x(t), showing the important parameters.
Make sure you show the magnitude of X() at 
d) What are the Fourier Series coefficients of x(t)? Let’s call them Ck.
e) Sketch Ck vs k.

6. H(s) is the Laplace Transform of an LTI system. The figure shows the zero-
pole plot of H(s). (No explanation is necessary for parts a to e)

a) What is the region of convergence of H(s) if the impulse response is left-
sided?
b) What is the region of convergence of H(s) if the impulse response is finite
in time?
c) What is the region of convergence of H(s) if the system is stable?
d) What is the region of convergence of H(s) if the system is causal?
e) What is the region of convergence of H(s) if h(t)e2t is absolutely integrable?
f) Let’s assume we cascade the system H(s) with an LTI system H2(s) and we
call the combined system Hc(s). Design the system H2(s) (find its Laplace
transform) so that Hc(s) could be both stable and causal at the same time.
Explain what you are trying to do.



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Parseval’s Theorem:
∫ |()|2
+∞
−∞
=
1
2
∫ |()|2
+∞
−∞


cos() =
+ −
2

sin() =
− −
2



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