Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Summer 3rd Sem 3131705 Dynamics Of Linear Systems Previous Question Paper

Enrolment No.___________

**GUJARAT TECHNOLOGICAL UNIVERSITY**

**BE - SEMESTER? III EXAMINATION ? SUMMER 2020**

**Subject Code: 3131705**

**Date:27/10/2020**

**Subject Name: Dynamics of Linear Systems**

**Time: 02:30 PM TO 05:00 PM**

**Total Marks: 70**

**Instructions:**

**1. Attempt all questions.**

**2. Make suitable assumptions wherever necessary.**

**3. Figures to the right indicate full marks.**

**Marks**

**Q.1 (a)**Explain continuous-time and discrete-time signals with

**03**

suitable example.

**(b)**For each of the following input-output relationships,

**04**

determine whether the corresponding system is linear, time

invariant or both.

1. y(t) = t2 x(t- 1)

2. y[n] = x[n + 1] - x[n- 1]

**(c)**What is time-variant and time-invariant system? Determine

**07**

causality and stability of the following discrete-time systems

with justification. Consider y[n] is the system output and x[n]

is the system input.

1. y[n] = x[- n]

2. y[n] = x[n - 2] - 2x[n - 8]

**Q.2 (a)**Explain LTI systems with and without memory.

**03**

**(b)**A linear time-invariant system is characterized by its impulse

**04**

1

*n*

response

*h*[

*n*] =

*u*(

*n*).

2

Determine energy density spectrum of the output signal when the

1

*n*

[

*x n*]

*u*(

*n*).

system is excited by the signal

4

**(c)**Compute and plot the convolution

*y[ n ] = x[ n ] * h[ n ],*where

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*n*

1

*x*[

*n*] = and

*h[ n ] = u[ n - 1].*

3

**OR**

**(c)**Explain commutative and distributive property of a LTI

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system.

**Q.3 (a)**

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For

*x*(

*t*) 1 sin

*w t*2cos

*w t*co

*w t*

0

0

s 2

0

4 .

Determine Fou rier series coefficient using complex exponential

representation.

**(b)**Discuss applications of frequency-selective filters.

**04**

**(c)**Discuss the properties of continuous-time Fourier series.

**07**

1

**OR**

**Q.3 (a)**Each of the two sequences

*x*[ ]

*n*and

*x*[ ]

*n*has a period N =

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1

2

4, and the corresponding Fourier series coefficients are

specified as

*x*[

*n*]

*a*and

*x*[

*n*]

*b*

1

*k*

2

*k*

Where,

1

1

*a*=

*a*=

*a*=

*a*=1 and

*b*=

*b*=

*b*=

*b*=1. Using the

0

3

1

2

2

2

0

1

2

3

multiplication property, determine the Fourier series coefficients

*c*for the signal

*g*[

*n*] =

*x*[ ]

*n x*[ ]

*n*.

*k*

1

2

**(b)**Discuss applications of frequency-shaping filters.

**04**

Determine whether each of the following statements is true or

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**(c)**false. Justify your answers.

1. An odd and imaginary signal always has an odd and

imaginary Fourier transform.

2. The convolution of an odd Fourier transform with an

even Fourier transform is always odd.

**Q.4 (a)**Explain time reversal and linearity property for the discrete

**03**

time Fourier transforms.

**(b)**Determine the Fourier transform for

*w*< for the

**04**

periodic signal

*x*(

*n*) sin

*n*

3

4 .

**(c)**Consider a discrete-time LTI system with impulse response

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1

*n*

*h*(

*n*)

*u*(

*n*).

2

Use Fourier transforms to determine the response for the input

3

*n*

*x*(

*n*)

*u*(

*n*).

4

**OR**

**Q.4 (a)**Explain differentiation and integration property for the

**03**

continuous time Fourier transforms.

**(b)**Determine the Fourier transform of periodic signal

**04**

*x*(

*t*) 1 co

s 6

*t*

8 .

**(c)**Compute the Fourier transform of each of the following

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signals:

1. [

*x n*] [

*u n*2] [

*u n*6]

1

*n*

2. [

*x n*]

[

*u*

*n*]

1

2

**Q.5 (a)**Determine the Laplace transform and the associated region of

**03**

convergence for (

*x t*)

2

*e*

*tu*(

*t*)

3

*e*

*tu*(

*t*).

**(b)**Determine the function of time x(t), for the following Laplace

**04**

transforms and their associated regions of convergence:

1

,

{

*e*}

*s*>0.

2

*s*9

**(c)**Explain properties of the Z-transform.

**07**

2

**OR**

**Q.5 (a)**Find Z-transform and region of convergence of

**03**

1

*n*

1

*n*

*x*(

*n*) 7

*u*(

*n*) 6

*u*(

*n*).

3

2

**(b)**Find inverse Z-transform for

*X*(

*z*) log 1

(

1

*az*) ,

*z*>

*a*.

**04**

**(c)**Consider the system function corresponding to causal LTI

**07**

1

systems: H(z)=

.

1

1 2

2 1 1

1

(

*z*

*z*

1

)(

2

*z*

*z*)

4

3

9

1. Draw a direct-form block diagram.

2. Draw a block diagram that corresponds to the cascade

connection of two second-order block diagrams. Each

second-order block diagram should be in direct form.

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This post was last modified on 04 March 2021