Sup May 2013

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Code No: R21044

II B. Tech I Semester, Supplementary Examinations, May – 2013

SIGNALS AND 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 the concepts of Impulse function and Sinc function.

b) Prove that the set tm 0sin ω and tn 0sin ω are orthogonal for nm ≠ , where

,,...2,1,0 ∞=m and ,,...2,1,0 ∞=n over 0

00

2,

ω

π+tt (7M+8M)

2. The complex exponential representation of a signal )(tf over the interval (0,T) ,

tntfn�∞

−∞=+= )2)(

4

3()( π

a) Find the numerical value of T

b) One of the components of )(tf is t3A πcos . Determine the value of A.

c) Determine the minimum no. of terms which must be maintained in representation

Of )(tf in order to include 99.9% of the energy in the interval (0, T). (5M+5M+5M)

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

b) Obtain the relationship between the bandwidth and rise time of ideal High pass filter.

(8M+7M)

4. a) Find the Fourier transform of the Rectangular Pulse and plot it's amplitude and phase

spectrum.

b) Find the Fourier transforms of the signal: )()( tuetxat

−= (8M+7M)

5. The signal tttV ππ 10cos5.05cos)( += is instantaneously sampled. The interval between

samples is ST .

a) Find the maximum allowable value for ST .

b) To reconstruct the signal )(tVS is passed through a rectangular low pass filter.

Find the minimum filter bandwidth to reconstruct the signal without distortion.

c) Explain the process of signal recovery through holding. (3M+6M+6M)

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SET - 1 R10

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Code No: R21044

6. a) Explain the difference between a time invariant system and time variant system? Write

some practical cases where you can find such systems. What do you understand by the filter

characteristics of a linear system? Explain the condition for causality of a LTI System?

b) Differentiate between linear and non-linear system. (12M+3M)

7. a) Find the Laplace transform of the function

t

etf

t−

−=

22)(

b) What is region of convergence? List the advantages of Laplace transform.

(8M+7M)

8. a) By using the Power Series expansion technique, find the inverse Z-transform of the

following X(Z):

2

1

132)(

2�

+−

= ZZZ

ZZX

b) Find the inverse Z-transform of the function

3)3

log()( �+

= ZforZ

ZZX (8M+7M)

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Code No: R21044






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1. a) Determine the period of the periodic sequence )10

sin(2)(n

nxπ

= .

b) Describe the following properties of Fourier Transform:

i) Convolution

ii) Frequency differentiation and Integration (7M+8M)

2. a) Find the Fourier Transform of the signal )()( tuetxat

−=

b) Determine the Fourier series representation of )4()32(2)( tSintSintx ππ +−=

(7M+8M)

3. a) Explain how a signal can be represented by a set of orthogonal signals.

b) Illustrate the concept of "Orthogonality” with regard to Complex Functions". (8M+7M)

4. a) Find the Fourier Transform of the Signum Function and the Fourier spectrum

b) Illustrate in detail the Complex Fourier Series (8M+7M)

5. a) Explain briefly detection of periodic signals in the presence of noise by correlation.

b) Explain briefly the process of extraction of a signal from noise by filtering. (8M+7M)

6. a) With the help of graphical example explain sampling theorem for Band limited signals.

b) Explain briefly Band pass sampling. (8M+7M)

7. a) Obtain the Laplace transform of the function t

etf

t−

−=

22)(

b) Find the Inverse Laplace transform of

)5)(1(

2)(

++

=

sssF (8M+7M)

8. a) Explain the properties of the ROC of Z- transforms.

b) Distinction between Laplace, Fourier and Z transforms. (7M+8M)

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Code No: R21044






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

1. a) Explain clearly the concept of ”orthogonal vector space”.

b) A rectangular function )(tf is defined by:

)(tf = 1 0 < t < �

= -1 � < t < 2�

Approximate the above function by a finite series of Sinusoidal functions. (7M+8M)

2. a) What is an LTI system? Explain its properties.

b) Obtain the conditions for the distortion less transmission through a system. Define the terms

Bandwidth and rise time. (7M+8M)

3. a) Explain in detail how Fourier Transform is developed from Fourier series.

b) Find the Fourier Transform of tetv

10100)( −

= . (7M+8M)

4. a) Distinguish between linear and non linear systems with suitable examples.

b) Consider a stable LTI System characterized by the differential equation

)()(2)(

txtydt

tdy=+ . Find its impulse response. (7M+8M)

5. a) A waveform m(t) has a Fourier transform M(f) whose magnitude is as shown in figure 1.

Find the normalized energy content of the waveform.

b) The signal tttv 00 3sin2cos)( ωω += is filtered by an RC low pass filter with a 3 dB

frequency 02 ff c = . Find the output power. (8M+7M)

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Figure 1

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Code No: R21044

6. a) A finite sequence x[n] is defined as }3,4,0,2,3,5{][ −−=nx . Find X [Z] and its ROC.

b) Consider the sequence x[n] = an 0 n N-1

= 0 otherwise

Find X [z]. (7M+8M)

7. a) Obtain the Laplace transform of the function t

etf

t−

−=

22)(

b) Find the Inverse Laplace transform of

)5)(1(

2)(

++

=

sssF (8M+7M)

8. a) By using the Power Series expansion technique, find the inverse Z-transform of

the following X(Z):

2

1

132)(

2�

+−

= ZZZ

ZZX

b) Determine the Laplace transform of signal shown in figure 4. (8M+7M)

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Code No: R21044






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

1. a) State the three important spectral properties of periodic power signals.

b) Assuming 20 =T , determine the Fourier series expansion of the Signal shown in figure 1.

(5M+10M)

2. a) Find the Fourier Transform of the signal )2()( tuetxat

−=

b) Determine the Fourier series representation of )4()32(2)( tSintSintx ππ +−=

(7M+8M)

3. a) Explain the following:

i) Gibbs Phenomenon ii) Convolution.

b) Determine whether each of the following sequences are periodic or not.

If periodic determine the fundamental period.

i) )7/6sin()(1 nnx π= ii) )8/sin()(2 nnx = (7M+8M)

4. a) State and prove Parsval’s theorem for power and energy type signals

b) Let )(tv be a given signal and assume that )(tv is of finite duration i.e., that 0)( =tv for

0�t , and also Tt � . Find the impulse response of an LTI system so that )( TtRxx − is

the output if )(tv is the input. (7M+8M)

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Figure 1

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Code No: R21044

5. Find the power of periodic signal g(t) shown in figure 2. Find also the powers of

a) -g(t) b) 2g(t) c) g(-t) d) g(t)/2 (15M)

6. a) Explain briefly impulse sampling.

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

signals:

i) trect 300

ii) tπ300cos10 10 (7M+8M)

7. a) Find the initial values and final values of the function )5cos3(sin)( ttetft

+=−

b) Explain the Step and Impulse responses of series R-C circuit using Laplace transforms.

(8M+7M)

8. a) Explain briefly detection of periodic signals in the presence of noise by correlation.

b) Find the Z – Transform including the region of convergence of

00)(

05)(

�=

�=

nnx

nnxn

(7M+8M)

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Figure 2

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Sup May 2013