JNTUWORLD

Code No-: R21044

II B. Tech I Semester, Supplementary Examinations, May - 2012

SIGNALS & SYSTEMS (Com. to ECE, EIE, ECC, BME)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry Equal Marks

~~~~~~~~~~~~~~~~~~~~~~~~

1. a) Discuss orthogonal vector space and orthogonal signal space and its importance in signal

analysis.

b) Find the expression for mean square error using the expression of a function using

orthogonal signal space.

2. a) Write Drichlet’s conditions to obtain Fourier series representation of any signal.

b) Find the Trigonometric Fourier series for the periodic ware from shown below and draw its

magnitude spectra.

3. a) State and prove differentiation property of Fourier transforms.

b) Find the fourier transform of following waveform using the property of Fourier transform.

1 of 2

SET - 1 R10

1

0.5

1 3 2 4 5

t

f(t)

-2T -T/2 T

V

2T

Time

-T T/2

-V

f(t)

www.jntuworld.com

www.jntuworld.com

www.jwjobs.net

JNTUWORLD

Code No-: R21044

4. a) Explain the properties of a Linear Time Invariant (LT I) systems.

b) Clearly discuss the filter characteristics of linear systems and Mention conditions for

distortion less transmission of a signal through the system.

5. a) Explain Energy Density Spectrum of a signal and also explain about energy densities of the

input and the response of a system.

b) Obtain the relationship between auto correlation and Energy Density Spectrum of a signal.

6. a) State and Prove sampling theorem for low pass signals. Also, explain the recovery of

original signal from its sampled signal. Draw neat diagrams wherever necessary.

b) Obtain the Nyquist rate of the signal,

.5000cos2010000sin102000cos)( ttttx πππ ++=

7.

a) Find the Laplace Transform of the signal .)()(.)( 32tuetuetx

tt+=

− Also draw its region of

convergence.

b) Find the inverse Laplace transform of the following, )52()1(

1)(

2+++

−=

sss

ssX

8. a) Explain briefly ROC and its important properties with regards to Z-Transform.

b) Find Z-transforms of the following sequences and give their region of convergence.

( ) )()( nuin

−21 ( ) ( ){ } )()( nuii

nn−+ 4

121

2 of 2

SET - 1 R10

www.jntuworld.com

www.jntuworld.com

www.jwjobs.net

JNTUWORLD

Code No-: R21044

II B. Tech I Semester, Supplementary Examinations, May - 2012

SIGNALS & SYSTEMS (Com. to ECE, EIE, ECC, BME)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry Equal Marks

~~~~~~~~~~~~~~~~~~~~~~~~

1. a) Define orthogonal signal space and approximate a function by a set of mutually orthogonal

functions.

b) Derive expression for mean square error using the expression obtained above.

2. a) Explain the symmetry of waveform and comment on Fourier co-efficient

b) Obtain the trigonometric Fourier series for the waveform shown below.

3. a) State and prove time convolution property of Fourier transforms

b) Find the Fourier Transform of the function below.

4. a) Explain filter characteristics of linear systems and what are the conditions to obtain

distortion less transmission through the linear systems.

b) Obtain relationship between rise time and band width of a low pass filter when unit step

signal is applied.

1 of 2

SET - 2 R10

- π

x(t)

A

t -2π π 2π 3 π

2

π−

2

π

A

t

f(t)

A cos t

www.jntuworld.com

www.jntuworld.com

www.jwjobs.net

JNTUWORLD

Code No-: R21044

5. a) A two port network has an impulse response h(t) given by h(t)= e-t u(t). The input to the

network is a pulse of unit amplitude extending from t=0 to t=2 sec. Using convolution,

determine the output.

b) State and Prove Parseval’s theorem for energy signals.

6. a) State and Prove sampling theorem. Also, show using neat diagrams the effect of Aliasing

due to under sampling.

b) Calculate the Nyquist rate of the function.

ttttx πππ 5000cos2010000sin102000cos)( ++=

7. a) Find the Laplace transform of the following, ).1(2cos...3)( 4−=

−ttetf

t

b) Obtain inverse Laplace Transform, .)52()23(

27223)(

22

2

++++

++=

ssss

sssX

8. a) Find Z- Transforms of the following sequences and give their Region of convergences.

i) )(2

1nu

n

−

ii) )(4

1

2

1nu

nn

+

b) Find Inverse Z- transform for the following transform2

1;

2

11

1)(

1

<

+

=−

z

z

zX .

2 of 2

SET - 2 R10

www.jntuworld.com

www.jntuworld.com

www.jwjobs.net

JNTUWORLD

Code No-: R21044

II B. Tech I Semester, Supplementary Examinations, May - 2012

SIGNALS & SYSTEMS

(Com. to ECE, EIE, ECC, BME)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry Equal Marks

~~~~~~~~~~~~~~~~~~~~~~~~

1. a) Explain how vector analogy with signal is useful for spectral or signal analysis.

b) Explain how a function can be represented by a closed or a complete set of mutually

orthogonal functions.

2. a) State strong and weak Drichlet’s conditions.

b) Find the Fourier series expansion of the half wave rectified sine wave shown below.

3. a) State and prove Frequency translation property of Fourier Transform

b) Using convolution theorem find the Fourier transform F(ω) of the following function

)(.sin)( tutwetf o

at−=

4. a) Obtain the condition for distortionless transmission of signal through linear system.

b) Define causality and physical realisability of a system. Also, explain about Paley-Wiener

criterion for physical realisalbltity of system.

5. a) Define correlation. Explain auto correlation, cross-correlation and give their properties.

b) Prove that the power spectral density and the correlation function of a periodic waveform

Forms a Fourier transform pair.

1 of 2

SET - 3 R10

-π

x(t)

t

π 2π 3π

A

0 -2π

www.jntuworld.com

www.jntuworld.com

www.jwjobs.net

JNTUWORLD

Code No-: R21044

6. a) State uniform sampling theorem and prove both graphically and analytically. Also, mention

all the results used.

b) Explain reconstruction of signal from its sampled signal.

7. a) Find the Laplace transform of the following function

<<

=

−

otherwise

Ttetx

at

0

0)(

Plot its region of convergence

b) Find inverse Laplace Transform of

+

+=

bs

assX ln)(

8. a) Find z-transform of [ ] )1(.1−=

−nuanx

n

b) Obtain Inverse z- transform of X(z) using the properties.

.2;2

3)( >

−= z

zzX

2 of 2

SET - 3 R10

www.jntuworld.com

www.jntuworld.com

www.jwjobs.net

JNTUWORLD

Code No-: R21044

II B. Tech I Semester, Supplementary Examinations, May - 2012

SIGNALS & SYSTEMS

(Com. to ECE, EIE, ECC, BME)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry Equal Marks

~~~~~~~~~~~~~~~~~~~~~~~~

1. a) Derive the condition for two signals f1(t) and f2(2) to be orthogonal to each other.

b) Define impulse function and explain clearly how impulse function can be used for analyzing

any system.

2. a) Obtain the relations between the coefficients of trigonometric Fourier series and Exponential

Fourier series.

b) Find the exponential Fourier series for the periodic waveform shown below and draw its

spectrum.

3. a) State and prove time scaling property of Fourier transforms.

b) Using the properties of Fourier transform, find the Fourier transform of the following.

)2()( 3−=

−tuetx

t atetg

−=)(

4. a) Derive the transfer function of a LTI system and hence explain filter characteristics of linear

system.

b) When a unit step signal is applied to a low pass filter, obtain the relationship between the

Rise time and band width of the filter.

5. a) State and prove frequency convolution theorem.

b) Determine the auto correlation Rg(τ) for )(.)( tuetgat−

= and hence prove that Rg(τ) exhibits

conjugate property what is the significance of Rg(0).

1 of 2

SET - 4 R10

-8

t

-4 -2 2 4 8

1

f(t)

www.jntuworld.com

www.jntuworld.com

www.jwjobs.net

JNTUWORLD

Code No-: R21044

6. a) State uniform sampling theorem and prove both graphically and analytically. Mention all

the results used.

b) Explain reconstruction of the signal from its sampled signal.

7. a) State and prove Differentiation Theorem for Laplace Transforms.

b) Find Laplace Transform of x(t) & Sketch Pole-Zero plot with ROC

)()(.)( 23tuetuetx

tt−+=

−

8. a) Find Z-transform of x[n] & Sketch Pole-Zero plot with ROC

)1(2

1)(

3

1][ −−

+

= nunuxx

nn

b) Find the inverse Z-Transform of ;43

)(21

1

−−

−

+−=

ZZ

ZzX R O C 1>Z .

2 of 2

SET - 4 R10

www.jntuworld.com

www.jntuworld.com

www.jwjobs.net