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, I\. Con. 3176-07. ' (REVISED COURSE)

(3 Hours)

ttlb(01-.

ND-1969

[TotalMarks: 100

N.B.: (1) Questibn No.1 is compulsory.(2) Attempt any four out of remaining six questions.(3). Figures to the right indicate full marks.(4) Assume suitable data if necessary.

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1. (a) State whether each of the following statement is true or false. Justify your 12answer (any three) :

(i) A stable filter is always causal(ii) LTI system is stable if its response is absolutely summable;

t (iii) Hi1(z)and H2(z) both have zeros at (+0.5) an,d (-0.2). However H1(z) hasboth the poles at origin, whereas ~2(Z) has only one pole and it is-situatedat origin. Both systems are causal FIR syster'!1s.

(iv)-ROC'oftheltransferfunction of stable filter must include z = 0.(b) Test the ~ollowing systems for Linearity, Stability, Time Invariance, Causality:

(i) y~n) = x(n) cos (won) , .

(ii) y(n) = Trunc {x(n)} Le. Truncation of x(n).

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2. When the input to LTI system is ~

x(n)~ G)" u(n)+ 2nu(-n-1) the outputisy(n)~ 6Gf u(n)- 6GF u(n)(a) Find system function H(z) of the system

(b) Show OF - I, OF ~ II realization of the system(c) Obtain impulse response of the system(d) Obtain expression for Magnitude Response and Phase Response and Sketch it(e) Identify the system type based on its passband and based on its phase response." ' ""

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3. (a) x1(n) = {1, 1,2, 2} ; x2(n) = {2, 3, 4, 2}. Find X1(k) and X2(k) of the above seque~cesby computing OFT only once.

(b) (i)Using £DIT- FFT, find X(k) of the following sequence: .xGn) = 1, ° 5 n 5 3

. =0, 45n57 -(ii)Using above OFT, compute OFT of following sequence:

x(n) = 1, n = °=0,15n54=1,55n57

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"4. (a) Using overlap ana ~aVe fechhique perform convolution ofi ..'

x(n) = {1,j2, 3, 4,5,6,7, 8} with h(n) = {1, 1/2}'(b) Using OFT / 10FT method, find response of the system with impulse response

h(n) = 2 ~(n) + 5 8(n) if the input to the system is -x(n) = 2 8(n) + 3 8(n - 1) + 5 8(n - 2).

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[TURN OVER

ws May 07-238

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Con. 3176-ND-1969-07. 2

5. (a) The difference equation of the system is given by -: ' .. 1

y(n) =3 y(n - 2) + 2 y(n - 1) + x(n) - 2 x(n - 1)

to the system ilp x(n) = (~J u(n) y(-1) =1, y(-2) = 0

Find (i) - Zero ilp response(ii) Zero state response

(iii) Total response of the system.(b) Tile system function of the LTIsystem is given as -

3 - 4 Z-1H(z) - ,

- 1 - 3.5 Z-1+ 1.5 Z-2

Specify ROC of H(z)and determine unit sample response h(n) for followingconditions:(i) Stable system(ii) Causal system

(iii) Anticausal system.

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6. (a) Consider a causal LTIsystem whose system function is :

1 1 -1. +-zH(z) = 5

(11 -1 1 -2

) (11 ..

..

-1

)--z +-z +-~

2 3 . 2

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Implement the system in each of the following forms:(i) Direct form I(ii) Direct form II

(iii) Cascadeform(iv) Parallel form. /

(b) Determine energy of the signal:

x(n) = (~J for n ~ 0

= 3n fqr n < 0

. (c) Obtain inverse z-transform of -X(z) = In(1 + az-1), Izl > lal.

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7. (a) With the help of neat block diagram explain anyone DSP processor in detail.(b) Explain relationship between OFT, DTFT and' z-transforms.(c) Write short note on system classification. . .

(d) Compare, IIR and FIR systems.

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