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ECE 300
Signals and Systems
Exam 2
18 October 2007
NAME ________________________________________
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
- Short Answer Questions (5 points each):
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 ( )?
1 ( )^ ( )
t h t e u t
−
= h 2^ ( ) t^^ =^2 δ^ (^ t −1)
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.
- Assume periodic signal x ( ) t has Fourier series representation
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 ( )
- Assume x ( ) t is a periodic signal with period T 0 (^) = 3. x ( ) t is defined over one period as
t x t t
⎧^ −^ <^ ≤
⎩ <^ ≤
a) Determine the fundamental frequency ω 0.
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
⎜ ⎟ =^ ⎜ −^ +^ ⎟ −^ ⎜ −^ −
⎝ ⎠ ⎝ ⎠ ⎝^2
