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Determine the transfer function of a causal discrete-time LTI system described by the following equation: yn+3 yn−1+ 2 yn−2=xn−1 Exercise 8.. Determine the homogeneous solution for the s

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VIETNAM NATIONAL UNIVERSITY, HANOI

UNIVERSITY OF ENGINEERING AND TECHNOLOGY

- Course title: Signals and Systems

- Course ID: ELT2035

Faculty of Electronics and Telecommunications

University of Engineering and Technology

Vietnam National University, Hanoi

Phone number: (04) 3754 9271 Email: halv@vnu.edu.vn

3 List of Questions and Exercises

3.1 Multiple-choice questions

Question 1 Which one of the following statements is INCORRECT?

A An energy signal can not be periodic

B A power signal can not have finite energy

C A sinusoidal is a power signal

D A finite-length signal can be a power signal

A The impulse response of a causal LTI system is a causal signal

B The impulse response of a stable LTI system is an energy signal

C The impulse response of a stable LTI system is a finite-length signal

D The frequency response of a discrete-time stable LTI system is a frequency function

continuous-E The frequency response of a discrete-time LTI system is periodic

Question 3 Given a system described by the equation y(n)y(n1) = x(n), which one of

the following statements is INCORRECT about this system?

A The system is linear

B The system is time-invariant

C The system is stable

D The system is causal

Types of credit hours: Lecture hours; Tutorial/Lab hours; Self preparatory hours.

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A All poles of a stable continuous-time causal LTI system must be in the right half

of the s-plane

B All poles of a stable discrete-time causal LTI system must be inside the unit circle

in the z-plane

C The region of convergence (ROC) of the transfer function of a stable

continuous-time LTI system must contain the j axis of the s-plane

D The region of convergence (ROC) of the transfer function of a stable time LTI system must contain the unit circle in the z-plane

discrete-Question 5 Which one of the following signals is NOT periodic:

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Question 14 What is the appropriate Fourier representation of the following signal:

x (t)=e−t cos(2 π t)u (t)

A The continuous-time Fourier transform (FT)

B The discrete-time Fourier transform (DTFT)

C The continuous-time Fourier series (FS)

D The discrete-time Fourier series (DTFS)

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x (n)={cos(π n /10)+ jsin (π n/10) (∣n∣<10)

x (t)=e 1+t u (2−t)

x (t)=∣sin(2 π t)∣

Question 18 What is the appropriate description of the system described by the following impulse response:

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Question 25 What is the final value of the signal x (t) , given its Laplace transform as follows:

A x (n)=δ(n−1)

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B x (n)=δ(n+1)

C x (n)=(2/3)∣n∣

D x (n)=(1/ 4) n u(−n)

Question 30 Which one of the systems described by the following transfer functions can

be both causal and stable?

x(n)=(−1) n

Exercise 4 Find the fundamental period of the following periodic signal:

x( n)=cos(2 π n)

Exercise 5 Determine the impulse response of a continuous-time LTI system described

by the following equation:

Exercise 7 Determine the transfer function of a causal discrete-time LTI system described by the following equation:

y(n)+3 y(n−1)+ 2 y(n−2)=x(n−1)

Exercise 8 Determine the step response of a causal discrete-time LTI system described

by the following equation:

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Exercise 12 The negative feedback control system shown in figure bellow has a plant P

and a feedback coefficient of K, in which the plant P is described by the equation

dy(t )

dt −2 y(t )=x(t )

and K is a real value.

Compute the transfer function of the feedback control system

Exercise 13 The negative feedback control system shown in figure bellow has a plant P

and a feedback coefficient of K, in which the plant P is described by the equation

dy(t )

dt −2 y(t )=x(t )

and K is a real value.

Determine K so that the system is causal and stable.

Exercise 14 Determine the frequency response of an LTI system having the following impulse response:

h(t)=sin (t)[u(t )−u(t−1)]

Exercise 15 Determine the response of an LTI system having the following impulse response:

h(t)=sin (t)[u(t )−u(t−1)]

to the input signal:

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Exercise 16 Determine the step response of the system having the following impulse response:

Exercise 24 Determine the homogeneous solution for the system described by the following differential equation:

d2y(t)

dt2 +4 y(t)=3

dx(t ) dt

Exercise 25 Determine the homogeneous solution for the system described by the following differential equation:

Exercise 27 Determine the homogeneous solution for the system described by the following difference equation:

y(n)−ay(n−1)=2 x(n)

y(n)−(1/ 4) y(n−1)−(1/8) y (n−2)=x(n)+ x(n−1)

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given the input:

x(t )=cos(t )+sin(t )

Exercise 37 Determine the particular solution for the system described by the following differential equation:

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given the input:

given the input x(t )=2 e−t

.Exercise 39 Determine the particular solution for the system described by the following differential equation:

given the input:

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y(n)+ y(n−1)+(1/ 2) y(n−2)=x(n)+ 2x(n−1)

given the input:

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dt2 +y(t)=3

dx(t ) dt

given the input:

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the initial conditions:

y(−1)=1 and y(−2)=0

Exercise 59 Determine the output of the system described by the following difference equation:

y( n)+(1/ 4) y(n−1)−(1/8) y( n−2)=x(n)+ x(n−1)

given the input

x(n)=(−1) n u(n)

and the initial conditions:

y(−1)=4 and y(−2)=−2

Exercise 60 Determine the output of the system described by the following difference equation:

y( n)−(3/ 4) y(n−1)+(1/8) y(n−2)=2 x(n)

given the input:

x(n)=2 u(n)

y(−1)=1 and y(−2)=−1

Exercise 61 Determine the natural and forced responses of the system described by the following difference equation:

y(−1)=1 and y(−2)=0

y( n)+(1/ 4) y(n−1)−(1/8) y( n−2)=x(n)+ x(n−1)

given the input:

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x( n)=(−1) n u(n)

y(−1)=4 and y(−2)=−2

y(n)−(3/ 4) y(n−1)+(1/8) y(n−2)=2 x(n)

given the input:

x( n)=2 u(n)

y(−1)=1 and y(−2)=−1

Exercise 65 Find the difference equation describing the discrete-time system represented

by the following block diagram:

Exercise 66 Find the difference equation describing the discrete-time system represented

by the following block diagram:

Exercise 67 Draw the block-diagram representation of a discrete-time LTI system described by the following state-variable representation matrices:

A=[ 1 −1/21/ 3 0 ] , B=[1

2] , C =[1 1] , and D=[0]

Exercise 68 Draw the block-diagram representation of a discrete-time LTI system described by the following state-variable representation matrices:

A=[ 1 −1/ 21/3 0 ] , B=[1

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Exercise 84 Determine the time-domain signal represented by the following Fourier series coefficients:

X (ω)=e−2 ωu (ω)

Exercise 99 Determine the time-domain signal corresponding to the following Fourier transform:

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and the fundamental period of the signal T =1

Exercise 105 Determine the time-domain signal corresponding to the following frequency-domain representation:

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Exercise 115 Determine the time-domain signal corresponding to the following Fourier transform:

X (ω)=4 sin(2 ω−4)

4sin (2 ω+4)

2 ω+4Exercise 116 Determine the time-domain signal corresponding to the following Fourier transform:

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Evaluate the Fourier transform of the following signal:

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Without determining x (t) , find the Fourier series representation of the following signal:

x (t)=e−t u(t) and y (t)=e−2 tu(t)+e−3 tu(t )

Exercise 141 Determine the frequency response and the impulse response of a system, given the following pair of input and output signals for the system:

x (t)=e−3 tu(t) and y (t)=e−3(t−2 )u (t−2)

Exercise 142 Determine the frequency response and the impulse response of a system, given the following pair of input and output signals for the system:

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Exercise 146 Determine the frequency response and the impulse response of the system described by the following differential equation:

d2y (t )

dt2 +5dy (t)

dt +6 y(t)=

−dx (t) dt

Exercise 147 Determine the frequency response and the impulse response of the system described by the following difference equation:

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Exercise 157 Given the Laplace transform of the signal x (t) as follows:

Exercise 160 Determine the forced and natural responses for the LTI system described

by the following differential equation with the specified initial and input conditions:

dy (t)

dt +10 y (t)=10 x (t) , y (0− )=1 , and x (t)=u (t)

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u(t) and y (t)=e−2 tcos(t)u(t )

Exercise 174 Determine the transfer function and the impulse response of a stable system, given a pair of its input and output signals as follows:

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Exercise 177 Determine the transfer function and the impulse response of a causal system described by the following differential equation:

d2y (t )

dt2 −dy (t)

dt −2 y (t)=−4 x (t)+5

dx (t) dt

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