# WBUT Question Papers EC

## Digital Signal Processing Semester B Tech Fifth Sem June 2008

Time : 3 Hours J

I Full Marks : 70

GROUP -A

(Multiple Choice Tjrpe Questions}                                                                                                                                                                                 Choose the correct alternatives for any ten of the following :     10 x 1 = 10

, i) The Z-transform of u(n- 1) is

a) 1/(1-Z-1)               b) ZK1-Z-1)

c)                                                                                  1/lZQ-Z-1)]   d) (1+Z-1).

ii) Considering x as the sampling period, transfer function of a discrete-time integrator employing trapezoidal integration is xl

a)                 2

Hi) The transfer function of a system with Impulse response h (n)» u (n)- u (n- 1) is,

a) 2                       » TT1 ,

c —I                      • di i.                 CZ!

1 (Z-1)(Z*1)

iv)       A DTLTI system with impulse response g(n) is BIBO stable if +» .

. a) ^\9{n)\<oo                                 b) ^g{n)<»

+00 +»

VI-267211 (1-A)
C) i?ig,(rc)|<0°                                          d) ^|g(n)|<l.

Ct/B.TSC8 (BCSl/aSH-e/ICMWl/M                     4

v)         If xtn) Is a flnlte-duration, two-sided sequence. ROC of its Z-transform is entire Z-plane except ,

a) Z « 0 b> Z – 1 ,       c) Z =»                   d) both Z = 0 and Z = °°.                                                _

vi)       If x(n)* |2, 4, 6, 1 j then x (n – 2) is  *

a) |2, 4, 6, l}           b) |2, 4, 6, lj

c) |2, 4, 6, 1, oj        d) jo, 2, 4, 6, lj.       1  1

vii)      If x(n) is a sequence of L samples and h(n) of M samples, the convolution of x(n) and h(n) contains

a) Max (L, M) samples     b) L + M – 1 samples

c) L + M – 2 samples      d) L + M samples.          1 I

vlii) The inverse Z-transform of l/|l-Z-1J is

a) u(n) as well as -u (- n – 1) b) u(n) but not u (- n – 1)

c) u(n) as well as u(-n)  d) u(-n) but not u(n).

Ix) the Fourier transform of an aperiodic discrete-time sequence is

a)          discrete & periodic function of frequency

b)          discrete & aperiodic function of frequency

c)          continuous & periodic function of frequency

d) continuous & aperiodic function of frequency. 1 1

x)         Z-transform of a causal sequence x(n) is 2/|l-~Z_l j. Then x(0) is equal to

a)                    b) 2

c) 1                      d) 4.                        I I

vi-a^ran trar

xi)        For a rectangular window of Af samples, width of the main lobe Is

a) 2n/m            b)         n/m

c) 6*/m            d)         4*/m. xll) Ifx(n)» jl, 0, 0, lj, the DFT valueX(0) is

a) 2               b)         1+j

cj 0               d)         1-j.

xlli) Two non-lnteractlng DTLTI systems in cascade have impulse responses gin) and h(n). The impulse response of the combination is

a)   g(n) h(n) b) g(n)+h(n)

c) g(n)*h(n)              d) [g(n) h(n)]^.

GROUP-B

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

1. 2.         Find the Inverse Z-transform of X(Z) =  ROC : \Z \ >2.

Z2-3Z+2

1. 3.         Find the DFT of a sequence xn -{1, 1, 0, 0}.
2. 4.         A DTLTI system with impulse response Hp) «• jl, 1, l| is excited by a sequence x(n) – j#, 3, 2, lj. Determine the output y(n) of the systcan.
3. 5.         The output y(n) and the Input x(n) of a discrete-time system are related by the equation y(n)- ex<n). Determine whether the system is linear, time-invariant and stable.
4. 6.         A signal x(t)s3cos200nt + 2cos500nt is uniformly sampled at a rate 150 samples/second. Determine the frequency information carried by the sampled version of x(t).

rTCMWii (l-ATl

GROUP -C ( Long Answer Type Questions )

Answer any three of the following questions.

The output and the Input of a recursive DTLTI system are related by the equation y(n) – -0 ■ li/(n -1)+0 • 2y(n – 2)+3x(n) + 3 • 6x(n -1)+0 • 6x(n – 2). Derive and draw the direct form-II structure for realising the system.  5

Derive the sketch the cascade and parallel structures for the system with transfer

2(Z+2)______                                                      10

function H(Z)~

(Z-0-l)(Z + 0-5XZ + 0-4)

Determine the Impulse response of the system with x(n) as Input and y(n) as output shown In figure below. Impulse responses of the subsystems are h, (n) – (1 / 3)” u(n), hj (n) – (1 / 2)” un & h3 (n) – (1 / 4)” u„. Also determine expression for

frequency response of the system.

1. 9.         a) Find the IDFT of the sequence X(K)-{6, -2+j2, -2, -2 – j2}.

b)          State the “Sampling Theorem”.

c)          Point out the properties of ROC of Z-transform.

1. 10.      a) Design a digital Butterworth filter to satisfy the following constraints

0-9*|//(e*’)|*l; Osws^

|tf(e*”)|s0-2;

Use bilinear transformation. Consider a sampling period of 1 second.

ivi-aayai

 b) An analog filter has transfer function

G(S)——

• ‘ (S+1XS+2)

Discretize the filter to obtain the transfer function of an equivalent discrete time filter by impulse-invariant technique. Consider a sampling frequency of 2 Hz. 5

1.                                                                                                                                                                                   Write short notes on any two of the following:                          2×7^
2. i) Mapping of S-plane into Z-plane,                    • .

11) DIF algorithm

ill)     Design of linear phase FIR filter

iv) Architecture of digital signal processors.

END