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Sample Final Exam Covering Chapters 1017 (finals03)
1
Sample Final Exam (finals03)
Covering Chapters 1017 of
Fundamentals of Signals & Systems
Problem 1 (25 marks)
Consider the discretetime system shown below, where
N
↓
represents decimation by
N
. This
system transmits a signal
[]
x n
coming in at 1000 samples/s over two lowbitrate channels.
Numerical values:
lowpass filters' cutoff frequencies
12
2
clp
clp
π
ωω
==
,
highpass filter's cutoff frequency
2
chp
ω
=
,
signal's spectrum over
[,
]
ππ
−
:
43
1,
34
()
3
0,
4
j
Xe
ωπ
−≤
=
<<
.
(a) [7 marks] Sketch the spectra
j
X e
,
1
j
X
e
,
2
j
X
e
,
2
j
We
, indicating the important
frequencies and magnitudes.
j
hp
H
e
j
lp
H
e
1
N
↓
2
N
↓
Receiver
x n
1
x n
2
x n
1
vn
2
yn
channels
ω
−π
1
0.75π
π
−0.75π
j
X e
ω
−π
1
0.5π
π
−0.5π
1/3
1
j
X
e
cos(
)
2
n
×
2
wn
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View Full Document Sample Final Exam Covering Chapters 1017 (finals03)
2
(b) [10 marks] Let the decimation factors be
1
2
N
=
and
2
4
N
=
. Sketch the corresponding spectra
1
()
j
Ve
ω
,
2
j
X
e
,
2
j
, indicating the important frequencies and magnitudes. Assuming for
the moment that
12
[]
, []
vn vn
are quantized using 16bit quantizers, find the bit rate of each
channel, and the total bit rate. How would this compare to the bit rate for of a direct transmission
of
x n
using a 16bit quantizer?
Answer:
After sampling in decimation operation:
With
1
2
N
=
, the first channel transmits at a bit rate of:
1000
samples/s 16bits/sample = 8000 bits/s
2
×
And with
2
4
N
=
, the second channel transmits at a bit rate of:
ω
−π
1
0.5π
π
0.75π
−0.5π
−0.75π
1/3
2
j
X
e
ω
−π
1/2
0.5π
π
−0.5π
1/6
1
j
ω
−π
π
1/12
0.5π
−0.5π
2
j
2
j
p
We
ω
−π
0.25π
π
0.75π
−0.25π
−0.75π
1/6
2
j
ω
−π
π
−0.25π
−0.75π
1/12
Sample Final Exam Covering Chapters 1017 (finals03)
3
1000
samples/s 16bits/sample = 4000 bits/s
4
×
Thus, the total bit rate is
12000 bits/s
.
A direct transmission of the signal would require a bit rate of
1000samples/s 16bits/sample = 16000 bits/s
×
(c) [8 marks] Design the receiver system (draw its block diagram) such that
[]
yn
xn
=
(assume
that there is no quantization of the signals.) You can use upsamplers (symbol
{ }
lp
N
↑
, with
embedded ideal lowpass filters of cutoff frequency
N
π
and gain
N
), synchronous demodulators,
ideal filters and summing junctions. Sketch the spectra of all signals in your receiver.
Answer:
+
{ }
1
lp
N
↑
{ }
2
lp
N
↑
2
kn
2
x n
+
1
vn
2
ω
−π
2
0.5π
π
−0.5π
0.75π
−0.75π
1
x n
cos(
)
2
n
×
2
()
j
K
e
ω
ω
−π
0.25π
π
0.75π
−0.25π
−0.75π
1/3
ω
−π
1
0.5π
π
−0.5π
1/3
1
j
X
e
2
j
Qe
ω
−π
π
−0.25π
−0.75π
1/6
2
qn
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View Full DocumentSample Final Exam Covering Chapters 1017 (finals03)
4
Problem 2 (20 marks)
You are the engineer in charge of the design of a rocket's guidance control system so that the rocket
can track a desired pitch angle trajectory
α
des
t
()
in a vertical plane during the takeoff phase. The
transfer function from the rocket's thrust vector angle command with respect to its longitudinal axis,
call it
θ
t
, to the angle between the rocket's pitch angle
t
(angle between the longitudinal axis
and the inertial vertical axis), is given by
2
ˆ
1
:
ˆ
12
(1
)
93
s
Gs
s
ss
s
==
++
.
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This note was uploaded on 09/16/2011 for the course ECSE 361 taught by Professor Franciscodgaliana during the Winter '09 term at McGill.
 Winter '09
 FRANCISCODGALIANA

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