# Digital Signal Processing B Tech 6th Sem June- 2009

Time : 3 Hours ]

FuLL Marks : 70

GROUP – A

(Multiple Choice Type Questions)

1. Choose the correct alternatives for any ten of the following :      10×1 = 10

U Infinite memory system is also known as

a)      FIR system    b) HR system

c)     Digital system d) Analog system.

ii)              The z-transform of u {- n) is

a) m-l-1-1                 b)

(l-z-1)                 ur (1-Z)

c) I 1 _ – V               d)

( 1 – z)                   ( z – 1) •

iii)            For rectangular window used for designing FIR filters, the peak amplitude of side lobe is

a)    – 40 dB  b) – 3 dB

c)        0 dB d) – 13 dB.

iv)            The sequence x ( n = ( – 1) n is periodic with a period of

a)          6 samples b) 4 samples

c)      2 samples d) 0 sample.

£0  reduces aliasing

b)              increases time resolution

c)              increases frequency resolution

d)              has no effect.

If the Fourier transform of x ( n ) is x (co ), then the Fourier transforrm of nx ( n) is

Ivl ml

b)

„ . dx (co)

d)                 none of these.

vii)          The digital system in y ( n) = x ( n2 ) is

a)              linear and causal

c)              non-linear and causal

b)                  linear and non-causal

d)                non-linear and non-causal.

viii)         If x * ( n) is the complex conugate of x ( n) then

a)                | x( n) | 2* | x* ( n) |

b)                | x(n) | = x(n).x’|n)

c)                |x(n)|2 = x(n). x*(n)

d)              none of these.

bd If x ( k ) represents the 8-point DFT of x ( n ) = { 1, 1, 1, 1,1, 1, 0, 0 }, x ( o )x)

A discrete-tlme LTI system is known as causal system if its

a)  impulse response h ( n) is zero for n < 0

b)  impulse response h ( n) is zero for n > 0

c)   impulse response h ( n) is positive for n < 0

d)              none of these.

xi)             X ( n) is an energy signal when

a)              G = X | * ( n) | 2 is finite

b)              G= I | x(n) | 2 is infinite

 S | jc ( n) |
 ’ is finite
 P — liVVl                -a it 2iV + 1
 c)

d)              none of these.

xii)          The energy of constant amplitude complex valued exponential function x ( n) = A exp [jrw ) where A and © are constants, is given by

 b)
 a)
 A_ 2
 d)
 c)

GROUP – B ( Short Answer Type Questions )

Answer any three of the following.

1. a) State Parseval’s energy theorem.

b)               Compute the convolution of the following signals :

x ( n) = n/2 ;0<n<5         h[n) = n/2 ; – 3 < n < 5

= 0 ; otherwise              = 0 ; otherwise.

1. Prove that the energy of a real valued energy signal is equal to the stun of the energies of its even and odd components t.e. Es = Ee + E0.
2. For the analog filter having transfer function h ( s) = . * . . . Determine H ( z )

S I S t I )                                         , .

using impulse invariance method.

1. Find out the relation between Fourier transform and Laplace transform with Z-transform.
2. For a causal LT1 system, the output y ( n) = | y ( n – 1 ) + x ( n ). Calculate y ( n ) while x( rt ) = n3 u ( n + 1 ).

GROUP-C ( Long Answer Type Questions)

Answer any three of the following.   3 x 15 * 45

Find the circular convolution of two sequences Xj (n) s{ 1, 1,2, 2 } and x2 ( n ) = ( 1. 2, 3, 4, 5 }. ,     ’

b)    State and prove intial value theorem regarding z-transform.                                          3

c)    Compute DFT of the sequence x ( n) = { 1, 0, 0, 1}.                                                           5

a)   Discuss about design method of Low-pass filter.                                                              4

b)    What do you mean by Windowing ?                                                                                  2

c)  What is rectangular window ?                                                                                           2

d    How are rectangular windows used to design FIR filter ?                                                4

e)  Determine the IDFT of Y ( k) = { 1, 0, 1, 0 }.                                                                      3

a)             Find the system function and impulse response of the system described by the difference equation y ( n) =x{ n) + 2x{ n- 1)-4x( n-2 ) +x( n-3). 5

b)    Find the inverse z-transform of X ( Z) = ( z + 0.2 ) / ( z + 0.5 ) ( z – 1 ), | z | > 1.5

c)              What are the properties of Region of convergence ? Find the z-transform and ROC of the signal x ( n) = – bnu[-n- 1).

1. a) Find the order of Butterworth filter that has a-2dB passband attenuation at a frequency of 20 rad/sec and – lOdB stopband attenuation at 30 rad/sec. 5

b)               Draw the following :

i)                Direct form I

ii)              Direct form II

Parallel strutures for the system described by the difference equation

1. a) Find the convolution sum of the singnals : x ( n) = 1 for 3 < n < 6 = 0 otherwise h ( n) = 1 for – 4 < n < 3 ‘

= 0 otherwise.

b)              What is zero padding ? What are its uses ?

c)               A discrete-time system is represented by the following difference equation y(n) = 3y2(n-l)-nx(n) + 4x(n-l)-2x(n+l)

is the system                              ^

i)              linear

ii)            time-invariant ill) causal. ,

Justify.