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ECE 300
Signals and Systems
Exam 2
30 April, 2009
NAME ________________________________________
This exam is closed-book in nature. You may use a calculator for simple
calculations during the exam, but not for integration. Do not write on the back of
any page, use the extra pages at the end of the exam.
Problem 1 ________ / 20
Problem 2 ________ / 25
Problem 3 ________ / 15
Problem 4 ________ / 20
Problem 5 ________ / 20
Exam 2 Total Score: _______ / 100
1) (20 points) The spectrum of periodic signal x ( ) t is shown below. The period of this
signal is T 0 (^) = 3 seconds and all angles are multiples of 45 degrees.
-4 -3 -2 -1 0 1 2 3 4
0
1
2
Magnitude
Harmonic
-4 -3 -2 -1 0 1 2 3 4
0
100
200
Phase (degrees)
Harmonic
a) Determine a closed-form expression for x ( ) t in terms of cosines.
b) Sketch the single-sided power spectrum for this signal as power versus harmonic. Be
sure to label all significant points (values) on your graph.
c) Compute the average power of this signal.
d) Compute the average value of this signal.
3) (15 points) Assume the periodic signal x ( ) t with the Fourier series representation
x jk (^) ot k k
x t ce
ω
is the input to an LTI system described by the differential equation
y t ( ) + ay t ( ) = dx t ( − b )
Since the system is LTI the output will be periodic with Fourier series representation
jk (^) ot k
y
k
y t ce
ω
a) Determine an algebraic relationship between
x c k and
y c k
b) Determine the (continuous frequency) transfer function H ( j ω )relating the input and
output.
4) (20 points) The periodic signal x ( ) t has the Fourier series representation
j t
k
k
k x t e kj
=
=−
∞
∞
x ( ) t is the input to an LTI system (a band reject or notch filter) with the transfer function
3 2 | 0 ( 0 3.
j e and H j
ω
− ⎧ (^) ≤ ≤ ≤ = <
The steady state output of the system can be written as y t ( ) = ax t ( − b ) + d cos( et + f )
, e
Determine numerical values for the parameters (^) a b d , , and f
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
