JNTUH USED 08-11-2020 AM

Code No: 133BQ

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

B.Tech II Year I Semester Examinations, October - 2020

SIGNALS AND STOCHASTIC PROCESS (Common to ECE, ETM)

Time: 2 hours Max. Marks: 75

Answer any five questions

All questions carry equal marks

- - -

1.a) Prove that the complex exponential functions are orthogonal functions.

b) Write short notes on:

i) Impulse, step and signum functions ii) Sinusoidal signals. [7+8]

2.a) Explain causality and physical relization of a system and hence give poly-wiener

criterion.

b) What is ideal filter? Find impulse response of a ideal Low Pass Filter. [8+7]

3.a) Derive the relation between Trigonometric and exponential Fourier series Co-efficient.

b) Find the Fourier transform periodic impulse train. [7+8]

4.a) State and prove time convolution and time differentiation properties of Fourier transform.

b) What is sampling? Explain the need for sampling and clearly discuss the process of

sampling low pass signals and derive conditions for optimum reconstruction of signal.

[7+8]

5.a) Verify time shifting and time scaling property of Laplace transform.

b) Find the inverse Laplace transform of X s =2+2se−2s +4e−4s

s2+4s+3, Re s > −1. [7+8]

6.a) Distinguish Laplace, Fourier and z transforms in detail.

b) Find the inverse Z transform of X z =1

1−az−1 2 ; ROC z > a . [7+8]

7.a) Differentiate between stationary and ergodic random processes.

b) Explain the following:

i) Time average function

ii) Time auto correlation function. [7+8]

8. Derive the relation between the power spectrum and auto correlation function. [15]

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JNTUH USED 08-11-2020 AM

Code No: 133BQ


B.Tech II Year I Semester Examinations, May/June - 2019


Time: 3 Hours Max. Marks: 75

Note: This question paper contains two parts A and B.

Part A is compulsory which carries 25 marks. Answer all questions in Part A.

Part B consists of 5 Units. Answer any one full question from each unit.

Each question carries 10 marks and may have a, b, c as sub questions.

PART- A

(25 Marks)

1.a) Give the condition for the physical reliability of a system. [2]

b) What are the properties of convolution? [3]

c) State any two properties of Fourier series. [2]

d) Find the Fourier transform of the signal x(t) = 20 sinc (20t). [3]

e) Explain the concept of region of convergence for Laplace transforms. [2]

f) Write the differentiation in time property of Laplace transform. [3]

g) Define random process. [2]

h) Give the relation between correlation and Convolution. [3]

i) Verify that the cross spectral density of two uncorrelated stationary random

processes is an impulse function. [2]

j) Define cross –spectral density and its examples. [3]

PART-B

(50 Marks)

2. Graphically convolve the signals

1

1;( )

0;

for T t TX t

elsewhere

and 2

1; 2 2( )

0;

for T t TX t

elsewhere

[10]

OR

3.a) What is an LTI system? Explain the properties of it.

b) Find whether x (t) = A e-α (t)

u(t) , α > 0 is an energy signal or not. [5+5]

4.a) Obtain the Fourier series coefficients for x(t) = A Sin ω0t.

b) What is the Significance of Hilbert Transform? Explain. [5+5]

OR

5. Define Fourier transform. Explain the properties of Fourier transform. [10]

6.a) Find the Laplace transform of x(t) = -t

2e

-at u(-t) and indicate its ROC.

b) Find the inverse Laplace transform of

x(s) = 5(s+5)/ s(s+3) (s+7); Re(s) > -3. [5+5]

OR

7.a) Find the inverse Z- transform of 1

1 2

1 3( )

1 3 2

zX z

z z

for different possible ROCs.

b) Give the relationship between z-transform and Laplace Transform. [7+3]

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8.a) A Random Process X(t)= A Cos (2πfct) , where A is a Gaussian Random Variable with

zero mean and unity variance, is applied to an ideal integrator, that integrates with

respect to ‘t’, over (0,t). Check the output of integrator for stationarity.

b) A random Process is defined as X(t)=3 Cos(2πt+Y), where Y is a random Variable with

p(Y=0)=p(Y=π)=1/2. Find the mean and Variance of the Random Variable X(2). [5+5]

OR

9.a) State and prove properties of cross correlation function.

b) If the PSD of X(t) is Sxx(ω ). Find the PSD of dx(t)/dt. [5+5]

10.a) Find and plot the Autocorrelation function of

(i) Wide band White noise (ii) Band Pass White noise.

b) Derive the expression for the Cross Spectral Density of the input Process X(t) and the

output process Y(t) of an LTI system in terms of its Transfer function. [5+5]

OR

11. The auto correlation function of a random process X(t) is RXX(τ) = 3+2 exp (−4τ2)

a) Evaluate the power spectrum and average power of X(t).

b) Calculate the power in the frequency band −1/√2 < ω < 1/√2. [5+5]

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JNTUH USED 03-06-2019AM

Code No: 133BQ


B.Tech II Year I Semester Examinations, November/December - 2018







PART- A

(25 Marks)

1.a) Is the system described by the equation 2y t x t time invariant or not? Why? [2]

b) Give the relation between bandwidth and Rise time of a signal. [3]

c) What are the effects of aliasing and how can you minimize the aliasing error? [2]

d) Distinguish between series and transform in the Fourier Representation of a signal.[3]

e) Let x(s) = L{x(t)}, determine the initial value, x(0) and the final value x , for the

following signal using initial value and final value theorems. [2]

7 6

( )(3 5)

sx s

s s

f) How the stability of a system can be found in Z-Transform and what is the condition for

causality in terms of Z-Transform. [3]

g) Prove that ( ) ( )xy yxR R . [2]

h) If the customers arrive at a bank according to a Poisson process with mean rate 2 per

minute, find the probability that during a 1-minute interval no customer arrives. [3]

i) Prove that the power spectral density of a real random process is an even function. [2]

j) Find the auto correlation function, whose spectral density is: [3]

, 1

0,s

otherwise

PART- B

(50 Marks)

2.a) Prove that the set 0sin mw t and 0sin nw t are orthogonal for m n , where

m = 0,1,2……. and n = 0,1,2…….. , over to, 0

0

2.t

b) Explain the concepts of unit step function and Signum function. [5+5]

OR 3.a) Explain causality and physical reliability of a system and explain Paley-wiener criterion.

b) Consider a stable LTI system characterized by the differential equation:

( )

2 ( ) ( ).dy t

y t x tdt

Find its impulse response. [5+5]

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4.a) Find the Fourier Transform of the signal 2 .atx t e u t

b) Define sampling theorem for time limited signal and find the Nyquist rate for the

following signals.

i) 300rect t ii) 10 300cos t [4+6]

OR 5.a) Derive the expression for trigonometric Fourier series coefficients.

b) Determine the exponential form of the Fourier series representation of the signal shown

in figure 1. [4+6]

Figure 1

6.a) By using the power series expression technique, find the inverse Z-Transform of the

following X(z).

2

1;

2 3 1 2

zX z z

z z

.

b) Distinguish between the Laplace, Fourier and Z-Transforms. [7+3]

OR 7.a) Find the Laplace Transform of the periodic, rectangular wave shown in figure 2.

Figure 2

b) Find the Laplace Transform of following functions:

i) Exponential function

ii) Unit step function. [6+4]


8.a) Explain the characteristics of a first order and strict sense stationary process using

relevant expressions.

b) State and prove the properties of auto correlation of a random process. [5+5]

OR 9.a) Find the mean, variance and Root Mean Square value of the process, whose auto

correlation function is 2

2

25 36

6.25 4xxR

.

b) Consider two random processes 3x t cos t and 2y t cos t , where

2

and is uniformly distributed over (0,2π), verify (0) (0)xy xx yyR R R .

[5+5]

10.a) Derive the relation between input and output power spectral densities of a linear system.

b) The cross power spectrum of real random process x(t) and y(t) is given by:

, 1

0 ,xy

a ib ifS

elsewhere

Find the cross correlation function. [5+5]

OR

11.a) Consider a random process 0 0 ,X t A cos t where 0A and 0 are constants and

is a uniform random variable in the interval 0, , find whether X(t) is WSS process.

b) Show that 2

( ) .yy xxS H S Where xxS and yyS are the power spectral

density functions of the input x(t) and the output y(t) respectively and H is the

system transfer function. [5+5]

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Code No: 133BQ


B.Tech II Year I Semester Examinations, April/May - 2018

SIGNALS AND STOCHASTIC PROCESS (Electronics and Communication Engineering)






PART- A

(25 Marks)

1.a) What is meant by Total response? [2]

b) Define Unit step function and Signum function. [3]

c) State “time shift” property of Fourier transform. [2]

d) Define aliasing effect? How can you overcome? [3]

e) What is the time shifting property of Z transform? [2]

f) Define inverse Laplace transform. State the linearity property for Laplace transforms.

[3]

g) Give an example of evolutionary random process. [2]

h) List the properties of Cross correlation function. [3]

i) Define wiener khinchine relations. [2]

j) State any two properties of cross-power density spectrum. [3]

PART-B

(50 Marks)

2. Define the error function while approximating signals and hence derive the expression for

condition for orthogonality between two waveforms f1(t) and f2(t). [10]

OR

3. Obtain the impulse response of an LTI system defined by dy(t)/dt + 2y(t) = x(t). Also

obtain the response of this system when excited by e-2t

u(t). [10]

4. State and prove sampling theorem for band limited signals. [10]

OR

5.a) State and prove Differentiation and integration properties of Fourier Transform.

b) Obtain the expressions to represent trigonometric Fourier coefficients in terms of

exponential Fourier coefficients. [5+5]

6. Determine the inverse Laplace of the following functions.

a) 1/s(s+1) (s+3) b) 3s2+8s+6 / (s+8) (s

2+6s+1). [5+5]

OR

7.a) Find the Inverse Z transform of

2

2( )

4 2 3

zX z

z z

;

3

4z

b) Find the Z transform of x [n] = an+1

u [n+1]. [5+5]

R16 JNTUH USED 18-05-2018 PM

8.a) X(t) is a random process with mean =3 and Autocorrelation function

Rxx(τ) =10[exp(- 0.3|τ|)+2]. Find the second central Moment of the random variable

Y=X(3)-X(5).

b) X(t)=2ACos(Wct+2θ) is a random Process, where „θ‟ is a uniform random variable, over

(0,2π). Check the process for mean ergodicity. [5+5]

OR

9.a) A random process is defined as X(t) = A Cos(ωot +Θ), where Θ is a uniformly distributed

random variable in the interval (0,π/2). Check for its wide sense stationarity? A and ωo

are constants.

b) Given the auto correlation function for a stationary ergodic process with no periodic

components is RXX(τ) = 25+4/(1+6τ2). Find mean and variance of process X(t). [5+5]

10.a) Compare and contrast Auto and cross correlations.

b) If Y(t) = A Cos(w0t+θ)+N(t), where „θ‟ is a uniform random variable over (-π,π), and

N(t) is a band limited Gaussian white noise process with PSD=K/2. If „θ‟ and N(t) are

independent, find the PSD of Y(t). [5+5]

OR

11. Given RXX (τ) = Ae-α and h(t) = e

-βt u(t) where u(t) =

1; 0

0;

t

otherwise

. Find the spectral

density of the output Y (t). [10]

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JNTUH USED 18-05-2018 PM

Code No: 133BQ


B.Tech II Year I Semester Examinations, November/December - 2017

SIGNALS AND STOCHASTIC PROCESS (Electronics and Communication Engineering)






PART- A

(25 Marks)

1.a) Write about unit step function and unit impulse function. [2]

b) Define signal bandwidth and system bandwidth. [3]

c) Determine the complex exponential Fourier series representation for

x t = cos 2t +π

4 . [2]

d) Find the Fourier transform of x t = ejω0t . [3]

e) Find the Laplace transform of x t = −eat u −t . [2]

f) Write the differences between the continuous-time signal ejω0t and the discrete-time

signal ejω0n . [3]

g) Explain about second order stationary process. [2]

h) Explain about Cross- Covariance function. [3]

i) Define Cross-Power Spectrum function. [2]

j) Find auto correlation function for SXX ω =8

9+ω2 2. [3]

PART-B

(50 Marks)

2.a) Define orthogonal signal space and orthogonal vector space. Bring out clearly its

applications in representing a signal and vector respectively.

b) Explain how functions can be approximated using orthogonal functions. [5+5]

OR

3.a) Derive the relationship between rise time and bandwidth.

b) State and Prove the Convolution property of Fourier transform. [5+5]

4.a) Expand following function f(t) by trigonometric Fourier series over the interval (0,1).

In this interval f(t) is expressed as f(t) = At

b) State and prove multiplication property of continuous time Fourier series. [5+5]

OR

5.a) Find the Fourier transform of symmetrical gate pulse and sketch the spectrum.

b) State and prove sampling theorem for band limited signals using analytical approach.

[5+5]

6.a) State and prove the properties of ROC of Laplace transform.

b) Find the inverse Laplace transform of X s =5s+13

s(s2+4s+13), Re s > 0. [5+5]

OR

7.a) Find X(z) and sketch the zero-pole plot and the ROC for a < 1 and a > 1 for the

signal x n = a n .

b) Determine the inverse Z transform of X z = log 1

1−az−1 ; ROC z > a . [5+5]

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JNTUH USED28-11-2017AM

8.a) Explain briefly about Gaussian and Poisson Random Process.

b) Show that the random process X t = Acos (ω0t + θ) is wide-sense stationary if it is

assumed that 𝐴 and 𝜔0 are constants and 𝜃 is a uniformly density random variable

over the interval (0,2𝜋). [5+5]

OR

9.a) Explain about Auto-correlation function with their properties.

b) Show that mean square value of output response is independent of time t. [5+5]

10. Explain about cross power spectrum density and its properties with proofs. [10]

OR

11. Derive the relationship between cross-power spectrum and cross correlation function.

[10]

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JNTUH USED28-11-2017AM

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