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P4-Exam.-{)ct-09-317 ~ [; . c ~ 0CYK\J~ CR)Con. 5914-09. (REVISEDCOURSE)

--- ~ ';.. 6 . (3 H >u s-hI \)./ (Cn~S ~ u~S~SN.S.: 1) Question No.1 is com~UISOry,(2) Answer five question in all.(3) All parts of the same questions must be written in continuation.

(4) Assume suitable data, if required and state them clearly.

1. Answer any four :-

(a) A discrete time periodic sequence is given by xp(n) = A cos [n;](i) Determine the period of the sequence(ii) Sketchthe sequencex(n) for the variablen for one period.

D

(b) S howthat flx(t)1 dt


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Con. 5914-SP-8606-Q9. 2 -4. (a) Find the Fourier transform of the unit impulse train function T(t), where T(t)is 10

defined by00

T(t) = + o(t + 2T) + o(t + T) + o(t - T) + o(t - 2T) + } = o(t - nT)n=-D(b) Evaluate the followingconvolutions -

(i) o(t) * o(t)

(ii) x(t) * o(t - to)

10

5. (a) Find the z transform of -

(i) x(n) = (-1)n cos 1tn u(n)3(ii) x(n) = (n + 1)an.

10

(b) Dete rmine a cascade realization for the followingtransfer function 10

0 .7 (z2 - o. 36)H(z)= 2Z + 0 .1z - 0.72

6. (a) Find the Fourier series for the function x(t) defined by 10

0x(t) =

ASinwot

-T

2


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7. (a) Show that if xe(n) is an even signal and xo(n) is an odd signal, then xe(n) .xo(n) is an odd signal.

(b) Find theJ f:lp.I.~9.~ransform of the signal shown below

. ~(-t).- - -- -

Z..

(c) Find the state transition matrix for

A -[

0 1

]6 -5

(d) Obta in the inverse z transform of a rational function

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-~ t;-~-t e c ~""rt:>n t'C:~--'~~ ~e \ e.. CO W\ vv.-~~T; t-1 ~~8; 1st ialf. 1O-AM(I) -S-t W\ v' / Ie.-' .Con. 3724-10. (REVISED COURSE) AN-4288

,. ~j ~ SlJSf-e~'5'J ~ (3 Hours) [Total Marks: 100N.B.: (1) Question No.1 is compulsory.

(2) Attempt five questions in all.(3) All parts of the same questions must be written in continuation.(4) Assume suitable data, if required and state them clearly.

1. Answer any four of the following :- 20(a) Find the ROC of the given signal

x(t) = 3e-21 u(t) - 2e-1 u(t).

(b) Determine the direct form - I realisation for the following transfer functionH(z) = 1 - 0.7 Z-1 + 0.4 Z-2

(c) A linear-time invariant (LTI) system is characterized by the following difference. equation:

y(n) = a y(n - 1) + b x(n) for 0 < a < 1Find the magnitude and phase of the frequency response H( eiW) f the system.

~, (d) Determine he signal energy and signal power for the following signals:(i) x(t) = e-(ii) x(t) = e-$1

(e) State and explain =convolution property of Z.transform.r, ,

2. (a) Con sider the analog signal xa(t) = 5 sin 2007tt 10(i) Determine the minimum required sampling rate to avoid sam pling.

(ii) Suppose that the signal is sampled at the rate F s = 100 H z. What is'the discrete time signal obtained after sampling?

(iii) Suppose that the signal is sampled at the rate F = 300 Hz, what isthe discrete time signal obtained after sampling.

""""'"\ (b) Impulse response of a discrete-time LTI system is expressed as under: 10

;:~' h(n) = 1, 2, 3 },,::;" Find the i/p sequence x(n) for output response which is given by -y(n) = { 1, 1, 2, -1, 3}.

..

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.

111111111.

.I.I..~3. (a) Compute he response f the system y(n) = 0.7 y(n - 1) - 0.12 y(n - 2) + x(n -1) + x(n - 2) 10

to input x(n) = nu(n).Is the system stable?

Obtain the Fourier transform of a rectangular pulse of duration 2 seconds and 10

having a magnitude of 10 volts.

4. (a) Find the Fourier series for the function x(t) defined by 10

{ 0 =f


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Hdl1-fnr .,'k"I-upq-"'Cl' h!f~' I " ~ 7

N.S. (1) Question NO.1 is compulsory.

(2) Attempt any four from remaining six questions.(3) Assume any suitable data if necessary and Make it clearly.

1. (a) Determine whether following signals are energy signals or power 4signals? Calculate their energy or power.

(i) x (t) = A cos (27t fa t + 0)

(ii) x [nJ := [.2.T u [nJ4J

(b) Determine whether following signals are periodic or non periodic? If periodic, 4find f undamental period.

(i) x (t) = 5 cos (47tt) + 3 sin (8 7tt)

(ii) x [nJ = sin [6;'1 n + 1](c) Check whether following systems are linear or nonlinear, Time-Inv ariant or 4

Time variant, causal or non causal.

(i) Y (t) = x (t). cos 100 7t t

(ii) Y (n) =: x (n) + n . x [n + 1](d) Find the initial value X (0) and final value X (00) of following: 4

1x (z) = -1 -2

1+ 2z --3z

(e) Find the fourier Transform of double sided exponential signal. 4

2. (a) Determine the exponential form of fourier series representation of signal shown 10

in fig 2'1 Hence determine the trignometric. form of fourier series.

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Con. 6618-MP-3895-11. 2

(b) (i) Plot the signal with respect to time. 10

x (t) = u (t) - r (t - 1) + 2r (t - 2) - r (t - 3) + u (t - 4) - 2u (t - 5)(ii) Find the even and odd parts of this signal.

3. (a) Obtain the fourier Transform of periodic gate function of amplitude A, period 10To and width "C as shown in fig 3'1. Plot. the magnitude spectrum.

(b) State and prove following properties of Fourier Transform.(i) Convolution in Time domain

(ii) Differentiation in Time domain.(c) Explain Gibb's phenomenon.

An analog signal.

xa (t) = sin [480 7t t] + 3 sin (720 7t t) is sampled 600 times per second.(i) Determine the Nyquist sampling rate for x a (t).

(ii) Determine the folding frequency.(iii) What are the frequencies, in radians, in resulting discrete time

signal x (n).(iv) If x (n) is passed through an ideal D/A converter, what is the reconstructed

signal Y a (t) . ?www.studentacademy.in
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Con.6Ei18-MP-3895-11. 3

(b) Determine the laplace transform of the signals shown in Fig. 41 and Fig. 42. 6

(c) Determine the laplace transform of following using properties of laplace 4

transform.x (t) = {t2 - 2t} u (t - 1).

5. (a) Determine the z - transform and sketch R.O.C. 8

(i) x1 [nJ=[;r ;n~O(ii) x2 [n] = x1 [n + 4]



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Con. 6618-MP-3895-11. 4

(b) Determine the convolution of following pairs of signals by means of 8z - transform.

X1 [nJ = [ : r u [n -1J

6. (a) The difference equation of system is given by-y (n) = 3y [n - 2] + 4y [n - 1] + x [n]If x [n] = [0'5Jn u [n] andy [- 1] =1,y [-2] =0

Find (i) Zero Input Response(ii) Zero state Response

(iii) Total Response.(b) A syste m is described by following difference equation.

1 1Y[n] = 2 Y (n-1) + 4 Y (n-2) + x(n) + x (n-1)

Obtain(i) Direct form I Realization

(ii) Direct form II Realization(iii) Cascade Realization.

A [-12 2]= -36 !1

(c) Find relationship between Discrete Time fourier Transform and z - Transform. 4

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SAE- IMP Questions SECOND YEAR (ALL BRANCHES) MATHS M3 & M4(www.studentacademy.in ) 8082540401

1 Expected questions for SECOND YEAR (ALL BRANCHES) MATHS M3 & M4-2013| Student Academy ofEngineering

The following category of sums normally appear in thepaper

VECTORS irrotational/ conservative field/ solenoidal greens theorem stokes theorem gauss divergence theorem work done

MATRICES eigen values and eigen vectors diagonalizable, diagonal form and tranmsforming matrix

derogatory or non derogatory and its minimal polynomial orthogonal matrix and find unknown linearly dependent or independent fine non singular matrix P & Q normal form and rank of matrix inverse of matrix and adjoint of A canonical form orthogonal and find unknown verify caley hamilton matrix

PROBABILITY problem based on bayes theorem mean and variance of given function sum on normal distribution sum on binomial distribution mean and std deviation of normal distribution mean and variance of binomial distribution

LAPLACE basic laplace transform

evaluate laplace transform differential laplace transform inverse laplace transform using formulae inverse laplace transform using convolution inverse laplace transform of log function laplace transform of heaviside function


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SAE- IMP Questions SECOND YEAR (ALL BRANCHES) MATHS M3 & M4(www.studentacademy.in ) 8082540401

2 Expected questions for SECOND YEAR (ALL BRANCHES) MATHS M3 & M4-2013| Student Academy of

FOURIER half range cosine or sine series

fourier series even and odd function fourier integral fourier series (0,2) / ( - ,) / (0,2l) / ( -l,l)

COMPLEX orthogonal trajectory harmonic, find conjugate and analytical function laurents series expansion bilinear transformation residue theorem (2 sums)

construct analytical function

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