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Date
Feb 20, 2024
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ECE 442/542 Exam 1 (White) February 19, 2020 Procedure: On-Campus Students 5:30pm-6:45pm (75 minutes), Room Gittings 201. 1.
One 8.5" x 11" sheet of notes (front and back). 2.
Table of z-Transforms (Appendix B from Schaum’s Outline) or a One-Page z-Transform Table (Student’s Choice). 3.
Calculator. (Allowed calculators are up to TI-89, TI-Nspire, & HP-51.) 4.
Writing Instrument. 5.
Closed-book. 6.
Please indicate your answers to each question on the Answer Sheet
. You should show your work on the exam itself. Your work will verify your understanding of the material. I have completed this exam without receiving any unauthorized help and will not discuss the exam with anyone until 10pm (Tucson, AZ time) on Sunday, 2/23/2020. Name: Signed: Date:
ECE 442/542 Exam 1 (White) February 19, 2020 Name: Problem 1. Given 𝐺𝐺
[
𝑧𝑧
] =
𝑧𝑧−0
.
75
𝑧𝑧
2
−0
.
75𝑧𝑧+0
.
125
, find the 1% settling time expressed in terms of the number of sample periods. 1.
N = 3 2.
N = 4 3.
N = 6 4.
N = 7 5.
None of the above. Problem 2. Given 𝐺𝐺
[
𝑧𝑧
] =
𝑧𝑧−0
.
75
𝑧𝑧
2
−0
.
75𝑧𝑧+0
.
125
, find the structure of the natural response. You may assume the response is causal, i.e., you may assume 𝑛𝑛 ≥
0 . 1.
𝑎𝑎
1
(
−
0.25)
𝑛𝑛
+
𝑎𝑎
2
(
−
0.5)
𝑛𝑛
2.
𝑎𝑎
1
(
−
0.75)
𝑛𝑛
+
𝑎𝑎
2
(0.125)
𝑛𝑛
3.
𝑎𝑎
1
(0.25)
𝑛𝑛
+
𝑎𝑎
2
(0.5)
𝑛𝑛
4.
𝑎𝑎
1
(0.75)
𝑛𝑛
+
𝑎𝑎
2
(
−
0.125)
𝑛𝑛
5.
None of the above.
Problem 3. Given 𝐺𝐺
[
𝑧𝑧
] =
𝑧𝑧−0
.
75
𝑧𝑧
2
−0
.
75𝑧𝑧+0
.
125
, is the system BIBO stable? 1.
The system is BIBO stable. 2.
The system is unstable. 3.
Not enough information is provided. 4.
None of the above. Problem 4. Given 𝐺𝐺
[
𝑧𝑧
] =
𝑧𝑧−0
.
75
𝑧𝑧
2
−0
.
75𝑧𝑧+0
.
125
and an input equal to 𝑢𝑢
(
𝑛𝑛
) = 2 ,
𝑛𝑛 ≥
0 , what is the structure of the total response. 1.
𝐴𝐴
1
(
−
0.25)
𝑛𝑛
+
𝐵𝐵
1
(
−
0.5)
𝑛𝑛
+
𝐶𝐶
1
2.
𝐴𝐴
2
(
−
0.75)
𝑛𝑛
+
𝐵𝐵
2
(0.125)
𝑛𝑛
+
𝐶𝐶
2
3.
𝐴𝐴
3
(0.25)
𝑛𝑛
+
𝐵𝐵
3
(0.5)
𝑛𝑛
+
𝐶𝐶
3
4.
𝐴𝐴
4
(0.75)
𝑛𝑛
+
𝐵𝐵
4
(
−
0.125)
𝑛𝑛
+
𝐶𝐶
4
5.
None of the above.
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Use the following all-delay block diagram for Problems 5, 6, and 7. Problem 5. What is a possible system matrix, 𝛷𝛷
? 1.
𝛷𝛷
=
�
0.7
5
0.1
1
�
2.
𝛷𝛷
=
�
0.7
5
0.6
1
�
3.
𝛷𝛷
=
�
0.7
5
0.1
0.2
�
4.
𝛷𝛷
=
�
0.7
1
0.6
0.2
�
5.
None of the above. Problem 6. For the states consistent with Problem 5, what is the input matrix, 𝛤𝛤
? 1.
𝛤𝛤
=
�
6
2
�
2.
𝛤𝛤
=
�
2
4
�
3.
𝛤𝛤
=
�
2
3
�
4.
𝛤𝛤
=
�
3
2
�
5.
None of the above. Problem 7. For the states consistent with Problem 5, what is the output matrix, 𝐻𝐻
? 1.
𝐻𝐻
= [
6
5
] 2.
𝐻𝐻
= [
3
4
] 3.
𝐻𝐻
= [
6
12
] 4.
𝐻𝐻
= [
6
4
] 5.
None of the above.
Use the following all-delay block diagram for Problems 8 and 9. Problem 8. What is a possible difference equation for the above system ? 1.
𝑦𝑦
(
𝑛𝑛
)
−
1.5
𝑦𝑦
(
𝑛𝑛 −
1) + 0.5
𝑦𝑦
(
𝑛𝑛 −
2) + 0.5
𝑢𝑢
(
𝑛𝑛 −
2) = 0
2.
𝑦𝑦
(
𝑛𝑛
)
−
0.5
𝑦𝑦
(
𝑛𝑛 −
1)
−
1.5
𝑦𝑦
(
𝑛𝑛 −
2)
−
1.5
𝑢𝑢
(
𝑛𝑛 −
2) = 0
3.
𝑦𝑦
(
𝑛𝑛
) + 0.5
𝑦𝑦
(
𝑛𝑛 −
1) + 1.5
𝑦𝑦
(
𝑛𝑛 −
2) + 1.5
𝑢𝑢
(
𝑛𝑛 −
2) = 0
4.
𝑦𝑦
(
𝑛𝑛
) + 1.5
𝑦𝑦
(
𝑛𝑛 −
1)
−
0.5
𝑦𝑦
(
𝑛𝑛 −
2)
−
0.5
𝑢𝑢
(
𝑛𝑛 −
2) = 0
5.
None of the above. Problem 9. Find the poles of the above system ? 1.
−
1.0,
−
0.5 2.
−
1.5,
−
0.5 3.
1.0, 0.5
4.
1.5, 0.5
5.
None of the above.
Problem 10. If a continuous-time system’s transfer function is given by 𝐺𝐺
[
𝑠𝑠
] =
(
𝑠𝑠+3
)
(
𝑠𝑠
2
+5
.
5𝑠𝑠+2
.
5
)(
𝑠𝑠+2
)
and one wants to control the system with a discrete-time controller without changing the system’s bandwidth, what is a reasonable sample period? 1.
𝑇𝑇
= 0.001 seconds 2.
𝑇𝑇
= 0.1 seconds 3.
𝑇𝑇
= 0.4 seconds
4.
𝑇𝑇
= 0.04 seconds
5.
None of the above. Problem 11. If a continuous-time system’s transfer function is given by 𝐺𝐺
[
𝑠𝑠
] =
(
𝑠𝑠+3
)
(
𝑠𝑠
2
+5
.
5𝑠𝑠+2
.
5
)(
𝑠𝑠+2
)
what is the system’s longest time constant? 1.
𝜏𝜏
= 0.2 seconds 2.
𝜏𝜏
= 0.5 seconds 3.
𝜏𝜏
= 1.0 seconds 4.
𝜏𝜏
= 2.0 seconds 5.
None of the above. Problem 12. Find 𝑋𝑋
[
𝑧𝑧
] , given 𝑥𝑥
(
𝑛𝑛
) = (2)
𝑛𝑛
,
𝑛𝑛 ≥
−
1 . 1.
𝑋𝑋
[
𝑧𝑧
] =
0
.
5𝑧𝑧
𝑧𝑧−2
2.
𝑋𝑋
[
𝑧𝑧
] =
0
.
5𝑧𝑧
2
𝑧𝑧−2
3.
𝑋𝑋
[
𝑧𝑧
] =
2𝑧𝑧
𝑧𝑧−2
4.
𝑋𝑋
[
𝑧𝑧
] =
2𝑧𝑧
2
𝑧𝑧−2
5.
None of the above.
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Problem 13. For the 𝑋𝑋
[
𝑧𝑧
] in Problem 12, what is the Region of Convergence, ROC? 1.
2 < |
𝑧𝑧
|
2.
|
𝑧𝑧
| < 2
3.
0.5 < |
𝑧𝑧
|
4.
|
𝑧𝑧
| < 0.5
5.
None of the above. Problem 14. For the given 𝑥𝑥
(
𝑛𝑛
) = (2)
𝑛𝑛
,
𝑛𝑛 ≥
−
1 in Problem 12, describe its time response behavior. 1.
Exponentially decays. 2.
Exponentially grows. 3.
Remains constant. 4.
Not enough information is provided. 5.
None of the above. Problem 15. Find 𝑌𝑌
[
𝑧𝑧
] , given 𝑦𝑦
(
𝑛𝑛
) = (0.5)
𝑛𝑛
+ (1.5)
𝑛𝑛
,
𝑛𝑛 ≥
1 . 1.
𝑌𝑌
[
𝑧𝑧
] =
𝑧𝑧
𝑧𝑧−0
.
5
+
𝑧𝑧
𝑧𝑧−1
.
5
2.
𝑌𝑌
[
𝑧𝑧
] =
0
.
5𝑧𝑧
𝑧𝑧−0
.
5
+
1
.
5𝑧𝑧
𝑧𝑧−1
.
5
3.
𝑌𝑌
[
𝑧𝑧
] =
0
.
5𝑧𝑧
−1
𝑧𝑧−0
.
5
+
1
.
5𝑧𝑧
−1
𝑧𝑧−1
.
5
4.
𝑌𝑌
[
𝑧𝑧
] =
0
.
5
𝑧𝑧−0
.
5
+
1
.
5
𝑧𝑧−1
.
5
5.
None of the above.
Problem 16. For the 𝑌𝑌
[
𝑧𝑧
] in Problem 15, what is the Region of Convergence, ROC? 1.
0.5 < |
𝑧𝑧
| < 1.5
2.
0.5 < |
𝑧𝑧
|
3.
1.5 < |
𝑧𝑧
|
4.
|
𝑧𝑧
| < 0.5
5.
None of the above. Problem 17. For the given 𝑦𝑦
(
𝑛𝑛
) = (0.5)
𝑛𝑛
+ (1.5)
𝑛𝑛
,
𝑛𝑛 ≥
1 in Problem 15, describe its time response behavior. 1.
Exponentially decays. 2.
Exponentially grows. 3.
Remains constant. 4.
Not enough information is provided. 5.
None of the above. Problem 18. Find the poles of the system given by 𝑥𝑥
(
𝑛𝑛
+ 1) = 𝛷𝛷𝑥𝑥
(
𝑛𝑛
) +
𝛤𝛤𝑢𝑢
(
𝑛𝑛
)
𝑦𝑦
(
𝑛𝑛
) = 𝐻𝐻𝑥𝑥
(
𝑛𝑛
) +
𝐽𝐽𝑢𝑢
(
𝑛𝑛
)
when 𝛷𝛷
=
�
1
0.25
−
1
0
�
, 𝛤𝛤
=
�
1
−
1
�
, 𝐻𝐻
= [
1
0
]
, 𝐽𝐽
= 0 . 1.
−
0.5,
−
0.5 2.
−
1, 0.25 3.
0.5, 0.5
4.
1,
−
0.25
5.
None of the above.
Problem 19. Find the transfer function associated with the given difference equation, where 𝑦𝑦
(
𝑛𝑛
) is the output and 𝑢𝑢
(
𝑛𝑛
) is the input. 𝑦𝑦
(
𝑛𝑛
+ 2) = 2.5
𝑦𝑦
(
𝑛𝑛
+ 1)
− 𝑦𝑦
(
𝑛𝑛
) +
𝑢𝑢
(
𝑛𝑛
+ 1)
−
0.25
𝑢𝑢
(
𝑛𝑛
)
1.
𝑧𝑧−0
.
25
𝑧𝑧
2
−2
.
5𝑧𝑧+1
2.
𝑧𝑧−0
.
25
𝑧𝑧
2
+2
.
5𝑧𝑧−1
3.
−𝑧𝑧+0
.
25
𝑧𝑧
2
−2
.
5𝑧𝑧+1
4.
−𝑧𝑧+0
.
25
𝑧𝑧
2
+2
.
5𝑧𝑧−1
5.
None of the above. Problem 20. For the given circuit, what is a valid differential equation? 1.
−𝑣𝑣
1
(
𝑡𝑡
) +
𝑥𝑥
1
(
𝑡𝑡
) +
𝑅𝑅
1
𝐶𝐶𝑥𝑥
1
′
(
𝑡𝑡
)
− 𝑅𝑅
1
𝑥𝑥
2
(
𝑡𝑡
) = 0
2.
−𝑣𝑣
1
(
𝑡𝑡
) +
𝑥𝑥
1
(
𝑡𝑡
)
− 𝑅𝑅
1
𝐶𝐶𝑥𝑥
1
′
(
𝑡𝑡
)
− 𝑅𝑅
1
𝑥𝑥
2
(
𝑡𝑡
) = 0
3.
−𝑣𝑣
1
(
𝑡𝑡
) +
𝑥𝑥
1
(
𝑡𝑡
)
− 𝑅𝑅
1
𝐶𝐶𝑥𝑥
1
′
(
𝑡𝑡
) +
𝑅𝑅
1
𝑥𝑥
2
(
𝑡𝑡
) = 0
4.
−𝑣𝑣
1
(
𝑡𝑡
) +
𝑥𝑥
1
(
𝑡𝑡
) +
𝑅𝑅
1
𝐶𝐶𝑥𝑥
1
′
(
𝑡𝑡
)
∓ 𝑥𝑥
2
(
𝑡𝑡
) = 0
5.
None of the above.
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