quiz - eth z...y +1/6/55+ y 3 frequency domain box 4: question 12 let fu[n]gbe an input sequence...
Post on 21-Jan-2020
0 Views
Preview:
TRANSCRIPT
Quiz November 8th, 2018
Signals & Systems (151-0575-01) Dr. A. Carron, Dr. G. Ducard
Quiz
Exam Duration: 40 Min
Number of Problems: 20
Number of Points: 20
Permitted aids: None
Important: Questions must be answered on the provided answer sheet;answers given in the booklet will not be considered.
There exist multiple versions of the exam, where the orderof the answers has been permuted randomly.
Every question has a unique correct answer.
Every question is worth one point for a correct answer, andzero points otherwise.
Giving multiple answers to a question will invalidate theanswer.
No negative points will be given for incorrect answers.
Partial points (Teilpunkte) will not be awarded.
You do not need to justify your answers; your calculationswill not be graded.
Use only the provided paper for your calculations; addi-tional paper is available from the supervisors.
Good luck!
y +1/2/59+ y
y y
y +1/3/58+ y1 Discretization
Question 1 Consider the CT signal x(t) = 1 + ej400π
7 t − ej200π
5 t. The signal is uniformlysampled with sampling frequency fs = 100Hz. What is the fundamental period N of the obtainedDT signal?
N = 70
N = 48
N = 72
N = 100
N = 35
N = 5
Question 2 Consider a causal stable continuous-time autonomous system q(t) = Acq(t), i.e.,for all eigenvalues λ of Ac, Re(λ) < 0. The continuous-time system is discretized with both theEuler discretization and the exact discretization resulting in the discrete-time systems q(k + 1) =AdEulerq(k) and q(k + 1) = Adexactq(k), respectively. Which statement is true?
q(k + 1) = Adexactq(k) is stable for all sampling times Ts
The eigenvalues of AdEuler are independent of the sampling time Ts
The eigenvalues of Adexact are independent of the sampling time Ts
q(k + 1) = AdEulerq(k) is stable for all sampling times Ts
Question 3 A continuous-time LTI system Gc is discretized using the exact discretizationmethod for a zero-order hold input and a uniformly sampled output, such that Gd = SGcH, withsampling time Ts = 0.1s. The system Gc can be described in state-space form as
q(t) = 2u(t), y(t) = q(t) + u(t).
Its discrete-time counterpart Gd = SGcH can be described as
q[n+ 1] = adq[n] + bdu[n], y[n] = cdq[n] + ddu[n].
Calculate bd
bd = 0.5
bd = 1
bd = 0.1
bd = 0.2
bd = 0
bd = 2
y y
y +1/4/57+ y2 Systems Properties
Box 1: Questions 4, 5, 6, 7
A system has the following input-output relationship
y[n] =
k=∞∑k=−∞
u[k]g[n− 2k],
where u[n] is the input, y[n] the output, g[n] = s[n− s[n− 4], and s[n] the unit step. Letalso δ[n] be the unit impulse. Answer the following questions with respect to the above systemdescription.
Question 4 If u[n] = δ[n− 2] then y[n] = s[n− 4]− s[n− 8] True False
Question 5 The system is unstable True False
Question 6 The system is LTI True False
Question 7 The system is causal True False
.
y y
y +1/5/56+ yBox 2: Questions 8, 9
Which of the following systems, defined by difference equations with input u[n] and output y[n],are linear and/or time-invariant?
Question 8 y[n] =(12
)nu[n]
Linear Time-Invariant
Linear Time-Variant
Nonlinear Time-Invariant
Nonlinear Time-Variant
Question 9 y[n] = eu[n]
Linear Time-Invariant
Linear Time-Variant
Nonlinear Time-Invariant
Nonlinear Time-Variant
Box 3: Questions 10, 11
For which of the following systems, defined by difference equations with input u[n] and outputy[n], the system description provides sufficient evidence that the system must be stable and/orcausal?
Question 10 y[n] =(12
)nu[n] + 2nu[1− n]
Stable and Causal
Stable and Non-causal
Unstable and Causal
Unstable and Non-causal
None
Question 11 y[n] = sin(π2u[n])
Stable and Causal
Stable and Non-causal
Unstable and Causal
Unstable and Non-causal
None
y y
y +1/6/55+ y3 Frequency Domain
Box 4: Question 12
Let u[n] be an input sequence applied to the causal LTI system G. The following plots showthe input sequence, and the poles and zero of the system’s transfer function, respectively.
Recall that: zp is a pole of H(z) (× on a pole-zero plot), if H(zp) =∞zz is a zero of H(z) ( on a pole-zero plot), if H(zz) = 0.
−3 −2 −1 0 1 2 3 4 5
−1
0
1
n
u[n
]
1
G
Re
Im
Question 12 Which one of the following plots corresponds to the output of the LTI system G?
−1 0 1 2 3 4 5 6 7 8 9−2
−1
0
1
2
n
y 1[n
]
−1 0 1 2 3 4 5 6 7 8 9−2
−1
0
1
2
n
y 3[n
]
−1 0 1 2 3 4 5 6 7 8 9−2
−1
0
1
2
n
y 5[n
]
−1 0 1 2 3 4 5 6 7 8 9−2
−1
0
1
2
n
y 2[n
]
−1 0 1 2 3 4 5 6 7 8 9−2
−1
0
1
2
n
y 4[n
]
−1 0 1 2 3 4 5 6 7 8 9−2
−1
0
1
2
n
y 6[n
]
y y
y +1/7/54+ yBox 5: Questions 13, 14, 15
Consider the magnitude and phase responses of four causal LTI systems GA,GB ,GC , and GD.
0 π4
π3
π2
2π3
3π4
π
1√2
5
10
|HA
(Ω)|
0 π4
π3
π2
2π3
3π4
π
−π4−π2− 3π
4
−π
0
∠HA
(Ω)
0 π4
π3
π2
2π3
3π4
π0
0.5
11√2
|HB
(Ω)|
0 π4
π3
π2
2π3
3π4
π
−π4−π2− 3π
4
−π
0
∠HB
(Ω)
0 π4
π3
π2
2π3
3π4
π0
0.5
11√2
|HC
(Ω)|
0 π4
π3
π2
2π3
3π4
π
−π4−π2− 3π
4
−π
0∠HC
(Ω)
0 π4
π3
π2
2π3
3π4
π0
0.5
11√2
Ω
|HD
(Ω)|
0 π4
π3
π2
2π3
3π4
π
−π4−π2− 3π
4
−π
0
Ω
∠HA
(Ω)
Question 13 Which one of the following systems could have Fourier TransformH(Ω) = (ejΩ+1)2
4ej2Ω ?
GA GB GC GD
Question 14 Which one of the four LTI systems has an output sequence y[n] = − 12ejπn
given the input sequence u[n] = ejπn?
GA GB GC GD None
Question 15 Consider system GA, what is the output sequence y[n] given the input se-quence u[n] = cos( 3π
4 n)?
y[n] = 10 cos( 3π4 n+ 7π
8 )
y[n] = 1√2
cos( 3π4 n−
7π8 )
y[n] = 1√2
cos( 3π4 n+ π
8 )
y[n] = 1√2
cos(π4n+ π8 )
y y
y +1/8/53+ yQuestion 16 Let the sequence x[n] be periodic with period N = 4, and have DFS coeffi-cients X[k]. The following plot shows the magnitude and phase of the DFS coefficients X[k]:
0 1 2 30
2
4
k
|X[k
]|
0 1 2 3
π
π2
0
−π2−π
k
∠X
[k]
Which one of the following sequences has DFS coefficients X[k], as shown in the above plot?
x[n] = sin(πn)
x[n] = e2πn
x[n] = sin(π2n) + ejπn
x[n] = 2 + eπ2 n
x[n] = sin(πn) + cos(π2n)
None of the above
Question 17 Consider the LTI system G, with transfer function H(z) = 1 − z−1. Let thesequence u[n] = cos(π2n) be the input to G for all times n. If y[n] = Gu[n] and if Y [k] arethe DFS coefficients of y[n], which one of the following plots shows to the magnitude of Y [k]?
0 1 2 30
1
2
3
k
|Y1[k
]|
0 1 2 30
1
2
3
k
|Y3[k
]|
0 1 2 30
1
2
3
k
|Y2[k
]|
0 1 2 30
1
2
3
k
|Y4[k
]|
y y
y +1/9/52+ yBox 6: Questions 18, 19, 20
The following plots show the magnitude of the DFT coefficients of six different DT signals,computed using a DFT length of 18.
0 5 10 150
3
6
9
12
|XA
[k]|
Plot A
0 5 10 150
3
6
9
12
|XB
[k]|
Plot B
0 5 10 150
3
6
9
12
|XC
[k]|
Plot C
0 5 10 150
3
6
9
12
|XD
[k]|
Plot D
0 5 10 150
3
6
9
12
k
|XE
[k]|
Plot E
0 5 10 150
3
6
9
12
k
|XF
[k]|
Plot F
Answer the following questions with respect to the above plots.
Question 18 Let x1[n] = cos(π2n). Which of the above plots shows the magnitudeof X1[k]?
Plot A Plot B Plot C Plot D Plot E Plot F
Question 19 Let x2[n] = cos( 5π9 n) + 1
2 sin( 2π3 n)− ej 5π
9 n. Which of the above plots showsthe magnitude of X2[k]?
Plot A Plot B Plot C Plot D Plot E Plot F
Question 20 Let x3[n] = 12ej π3 n− 1
2ej 2π
3 n. Which of the above plots shows the magnitudeof X3[k]?
Plot A Plot B Plot C Plot D Plot E Plot F
y y
y +1/10/51+ y
y y
y +1/11/50+ y
y y
y +1/12/49+ y
y y
y +1/13/48+ y
y y
y +1/14/47+ y
y y
y +1/15/46+ y
y y
y +1/16/45+ y
y y
y +1/17/44+ y
y y
y +1/18/43+ y
y y
y +1/19/42+ yAnswer sheet:
student number
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
←− please encode your student number, andwrite your first and last name below.
First and last name:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How to select answer B and C ::
3
3 (Desired answers clear)
3 (Desired answers clear)
7 (Desired answers unclear)
Answers must be given exclusively on this sheet;answers provided on any other sheet will not be counted.
1 Discretization
Q1: A B C D E F
Q2: A B C D
Q3: A B C D E F
2 Systems Properties
Q4: A B
Q5: A B
Q6: A B
Q7: A B
Q8: A B C D
Q9: A B C D
Q10: A B C D E
Q11: A B C D E
3 Frequency Domain
Q12: A B C D E F
Q13: A B C D
Q14: A B C D E
Q15: A B C D
Q16: A B C D E F
Q17: A B C D
Q18: A B C D E F
Q19: A B C D E F
Q20: A B C D E F
y y
y +1/20/41+ y
y y
top related