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The exam questions for the signals and systems course (ece 300) at a university. The exam covers topics such as stability of systems, impulse responses, periodic signals, fourier series, and lti systems. Students are required to solve problems related to these topics and provide calculations and expressions in the exam. The document also includes some potentially useful relationships for reference.
Typology: Exams
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This exam is closed-book in nature. You may use a calculator for simple
calculations, but not for things like integrals. Credit will not be given if your work
is not shown!
Problem 1 ________ / 25
Problem 2 ________ / 25
Problem 3 ________ / 25
Problem 4 ________ / 25
Exam 2 Total Score: _______ / 100
a) Is the system with impulse response ( ) ( )BIBO stable? Why or why not?
t h t = e u t
b) Is the system
( ) cos ( )
y t x t
⎟ BIBO stable? Why or why not?
c) What is the impulse response for the system
1
( ) ( 2)
t t y t e e x d
λ
− −
−∞
? Be sure to
include appropriate unit step functions.
d) Consider the two LTI systems shown below, with impulse responses shown. What is
the impulse response between x ( ) t and y t ( )?
t h t e u t
−
x ( ) t v t ( ) y t ( )
e) Is the function ( ) cos(4 ) sin(6 )
2
x t t t
π = π + + π periodic? If yes, determine the
fundamental period.
k jkt
k
jk x t e jk
=∞
=−∞
x ( ) t is the input to an LTI system with transfer function given by
H j j
otherwise
Determine the steady state output of the system. For full credit your answer
must be written in terms of cosines (and/or sines). Clearly indicate whether you are
writing your phase in degrees or in radians.
, y t ( )
t x t t
b) Determine the average value of x ( ) t.
c) Determine the average power in the DC component of x ( ) t.
d) Determine an expression for the expansion coefficients,. X (^) k , where ( )
jk (^) ot x t X ek
ω
must write your expression in terms of the sinc function, and possibly a leading
nential term.
You
expo
Some Potentially Useful Relationships
T 2 2
T T
E lim x t dt x t dt
∞
∞ →∞ − −∞
T 2
T T
P lim x t d 2T
∞ (^) →∞ −
t
jx e = cos x + jsin x j = − 1
(^1) jx jx cos x e e 2
− = ⎡ + ⎤ ⎣ ⎦
(^1) jx jx sin x e e 2 j
− = ⎡ − ⎤ ⎣ ⎦
cos x cos 2x 2 2
sin x cos 2x 2 2
0 0 0
t t T T rect u t t u t t T 2