MAE143 A  Signals and Systems  Winter 11
Final
Instructions
(i) This exam is open book. You may use whatever written materials you choose, including
your class notes and textbook. You may use a hand calculator with no communication
capabilities
(ii) You have 3 hours
(iii) Do not forget to write your name, student number, and instructor
1.
Filtering and sampling.
The continuoustime signal
x
(
t
) = 100 sinc
2
(100
t
)
goes through an
ADC block that samples signals at a frequency
f
s
= 190
Hz. Answer the following questions
(a) (1 point) Compute the Fourier transform of
x
(
t
)
. What is its bandwidth?
(b) (1 point) Would you be able to reconstruct the original signal out of the samples taken by
the ADC block? Why?
(c) (2 points) Consider a system whose impulse response is given by
h
(
t
) = 150 sinc(150
t
)
What kind of filter is this? What is its cutoff frequency in Hz? Plot the magnitude and
phase of the transfer function.
(d) Suppose the signal
x
(
t
)
passes first through the system in (c) to produce the signal
y
(
t
)
and then goes through the ADC block to produce the samples, see Figure 1.
h(t)
ADC
x(t)
samples
y(t)
Figure 1: Block diagram for question 3, part (d).
i. (1 point) What is the bandwidth of
y
(
t
)
? Would you be able to reconstruct the signal
y
(
t
)
out of the samples taken by the ADC block? Justify your answer.
ii. (1 point) Why would one call the system in (c) an antialiasing filter?
(e) (1 point (bonus)) With knowledge of
y
(
t
)
, would you be able to recover the original sig
nal
x
(
t
)
? Why?
Solution:
(a) Using the table of basic transforms and properties (time scaling), the Fourier
transform of
x
(
t
)
is
X
(
f
) = tri
f
100
(
+ .5 point
)
The bandwidth of this signal is therefore
f
B
= 100
Hz.
(+ .5 point)
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(b) No, because the sampling frequency is below the Nyquist rate,
f
s
= 190
<
2
f
B
= 200
.
(+ 1 point)
(c) We compute the transfer function of the system as
H
(
f
) = rect
f
150
(
+ 1 point
)
Therefore, this is an ideal lowpass filter with cutoff frequency
f
c
= 75
Hz.
(+ .5 point)
The phase plot of the filter is trivial (identically zero). The magnitude plot looks like
100
50
50
100
0.2
0.4
0.6
0.8
1
(+ .5 point)
(d.i) After the signal
x
(
t
)
goes through the ideal filter, all frequencies above the cutoff
frequency (
75
Hz) get cut. Therefore, the bandwidth of
y
(
t
)
is
75
Hz.
(+ .5 point)
Since the sampling frequency of the ADC block is
190
>
150 = 2
*
75
, we will be able to
reconstruct the signal
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 Summer '08
 MIROSLAVKRSTIC
 Fourier Series, Dirac delta function

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