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!

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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

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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

.

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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

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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

]

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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 )

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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

]|

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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

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←− please encode your student number, andwrite your first and last name below.

First and last name:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

How to select answer B and C ::

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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

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