# WBUT Exam Papers EE Digital Signal Processing B Tech 6th Sem 2011

# WBUT Exam Papers EE

## Digital Signal Processing B Tech 6^{th} Sem 2011

Time Allotted *: 3 Hours *

Full Marks *:* *70*

The figures in the margin indicate full marks.

Candidates are required to give their answers in their own words

as far as practicable.

GROUP – A ( Multiple Choice Type Questions )

1. Choose the correct alternatives for any ten of the

following :

i) The system described by y(n) = x(ri) + 2x(n – 2) + 3x(n – 3) is

a) causal and stable

b) causal and unstable

c) non causal and stable

d) noncausal and unstable.

ii) If x(n) – { 2, 1, 3, 0, 1, 2, 4}, then x ( ~ n + 2 } is

given by

a) {2, L 3, 0, 1, 2, 4}

f ■

b) {2, 1, 3, 0, 1, 2, 4}

T

c) {4, 2, 1, 0, 3, 1, 2 }

t

d) {4, 2, 1, 0, 3, 1, 2}.

T

^{lfAr(}“^{,and/,(n,a}>-e two finite ienrth

their convolu tion has length

^{se}<l«ences, then

**a) **8

**b) **10

cJ li

d) 9.

‘^{Vi The} °^{Verali} tapulse response of

two systems with impulse responseTh M °^{f}

) _{b} ^{(,,)and/,2(n)is} _{c)} . , . ‘ ^{h}^2(n)

^{c) h}i(n)*h_{2}(n)

* *^{a) h}> (^{n})~h_{2}(n).

^{A dl}—te-time LTI system is causal if ^{3)} ‘^{mPUlSe res}P°nse h(n)> _{0}, _{n > 0} ^ ^{taPUlSe res}P°^{ns}e h In )<_{0}._{n>} o

^{C) impu,se res}P°^{n}se n ( „ ) . o, „ > o ‘

^{d) }^{impU,SC res}P°nse h ( „ , . o. „ < o.

– The ROC of an infinite causal sequence is the ‘ interior of a circle

^{hl}^{ ex}terior of a circle

^{C) entU e Z}-Plane except * = _{0}

^{d>}^{ entire z}‘P’ane except z = ».

^{W}° ^{The Z}‘^{transf}°rm of u [n – i] _{ls}

a) l/fl-z-i) _{?}

^{1} ‘ . ^{b}> z/(i-z-‘)

^{c)} i/[z(i-_{z}-‘)l „ ,

t ‘ /] d) (l + 2-ij viii) If x ( K ) represents the 8 point DFT of a: ( n ) = { 1, l, l,

- 1, 1,0,0} then x ( 0 ) is

a) 3 b) 6

^{c) 1} d) 0. ‘

tx) The mapping from analog to digital domain in impulse invariant method is

a) one to many b) many to one

c) one to one d) none of these.

x) Overlap save method is used to find

**a) **circular convolution b) linear convolution c) DFT d) Z-transform.

xi) Number of multiplications is FFT algorithm is

**a) **n log (n ) b) (n/2)*log(n)

c) ( n/2 ) * log ( n/2 ) d) n log ( n/2 ).

xii) FIR filter is

a) recursive and linear

b) non-recursive linear

c) recursive and non-linear

d) none of these.

3 [ Turn over

GROUP -B ( Short Answer Type Questions )

Answer any three of the following. 3×5=15

^{2}‘ ^{ThC imPUlSe reSp}°^{nse} — W system is * ^ _{= {]},_{2}. _ _{J}}

Determine the response of the system to the , *

y teni to the input signal

*(” = {1,2, 3. 1}.

If a discrete-time I ti •

Lii system is BIBO u

stable, show that the

ROC of its system function H ( _{z} ) m ,

^{z} ) must contain the unit

circle, i.e., |z|=l.

**4- **Explain the relationship between _{S}–_{plane and} ,._{plane}

**5- **a, _{Flnd thf DTFr of the sequence} ,_{{n)} , ^

**b) **Find the IDTFT of x(e“°) = _{e}W^{1} ^ ^{1} \ .

‘ ^{1} ‘ (2 2^{COS<u}l’ 2 + 3

6- Determine the convolution of the two fr.ll

^{lne two} following sequences

using overlap add method :

^{X}^^{n>}^{ =} (^{3}’ ^{2}‘ ^{2}) %) = { 1, 2, 1, 1}.

GROUP -C ( Long Answer Type Questions )

Answer any three of the following. 3×15 = 45

7. a) Justify whether the system is LTI or not.

* „ /

2

*y(n)* = *y(n* – 1) + ^x(n-k).

k-0r-

**b) **Compute the circular convolution of the two sequences given below.

x(n)-{2 -1 0 1 -2 3 0 1}. h(n)-{l 2-1. 1}.

**c) **Determine the linear convolution of the above sequences using over-lap save method.

- a) What is ROC ? State its properties.

b) Find the system function & impulse response of the system described by _{y}(_{n}) – _{x}(n) + 2x(n – l) – 4x(n – 2) + x(n – 3)

c) Find the Inverse Z-transform of ‘ X(Z) = z[z^{2} -4Z + 5)/(Z -3)(Z- 2)(Z – 1)

d) Prove that an LTI system is BIBO stable if the ROC system function includes the unit circle. 2 + 5 + 5 + 3

- a) Find the 8-point DFT using decimation in time FFT algorithm for a sequence x(n) – {1, 3, 5, 7, 2, 4, 6, 8}.

b) What do you mean by zero padding ?

c) Using linear convolution find y(n) = x(n) * h(n) for the sequence x(n) = {l_{f} 2, -1, – 2, 0, 1, 3, -1}. Compare the result by solving the problem using

i) overlap save method

ii) overlap add method. 5 + 2 + 8

- Following specifications are given for a filter function :

“pass – 4 dB, a_{stop} = 48 dB, /_{stop} = 7 kHz, = 2 kHz, /_{8ampltIlg} . 20 kHz

Determine an IIR filter using Butterworth approximation and impulse invariant method.

- a) Design a digital Butteworth IIR filter for the given

frequency response :

si, for 0 s (o s 0-2 n

H^e^{jw} j | <s 0 • 02, for 0-45 n <, go s n

Use impulse invariant method.

b) Convert the analog filter with system function

S + 0 * 1

~ (s + 0 – I)^{2} + 16 ^{mt}° ^{3}^{ digital fllter usin}g bilinear

transformation. The digital filter should have a resonant

frequency of = ~ radian. _{8 + 7}

12. Write short notes on any three of the following : 3 _{x} 5

a) Causal and non-causal system

b) Circular convolution and linear convolution

c) DIT-FFT algorithm

d) Difference between DTFT and DFT

e) Bilinear transformation.

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