L-3/T -2/EEE Date: 01108/2016
BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY, DHAKA
L-3/T-2 B. Sc. Engineering Examinations 2014-2015
Sub: EEE 311 (Digital Signal Processing I)
Full Marks: 210 Time: 3 Hours
The figures in the margin indicate full marks.
USE SEP ARA TE SCRIPTS FOR EACH SECTION
SECTION -A
There are FOUR questions in this Section. Answer any THREE questions.
1. (a) A causa11inear time-invariant system is described by the difference equation (15)
y[n]-5y[n -1 ]+6y[n - 2]= 2x[n -1]
(i) Determine the homogenous response of the system.
(ii) Determine the impulse response of the system.
(iii) Determine the step response of the system.
(b) For the system in Figure Q. No. 1(b), determine the outputy(n] when the input x(n] is
jOJ 1 IlUl < ~
bIn] and H(dj is an idea110w pass filter, H (e ) = 0 (20)
{ ~ < IlUl ~ Jr
7' .... .. )~ ., (~l~U.l
X(h3-J:i.J~ EI)" '" :JG>1-
~ __ .. =,~)==I-~~~~~~-I.',-"
~ __~. .~i_-~-HC..~J_' ..{i~~~W-.
--=r I
-2/r
IJ- -X
cI:J=_~.~..
-1Y2,. 7'2.- K
1
2/t
'. :tV--
'"----- --_._------------~---------- --~---------
2. (a) A linear time-invariant (LTI) system is described by the input-output relation
y[n]= x[n]+ 2x[n -1 ]+x[n - 2]' (18)
I
I '
\ .
(i) Is this a stable system?
(ii) Determine Frequency response of the system, H(dj.
(iii) Plot the magnitude and phase of the frequency response.
(iv) Consider a new system, HI (ejOJ )= H(ej(OJ+7r)). Determine h[n].
(b) Consider the cascade ofLTI discrete - time system shown in Figure Q. No. 2(b). (17)
_~.a:a-L.T-~JI:t-~-r
'-'~'-'--C-, -.- -'----~-. ,.-~. -----~---~_.-
kf..~!lLTttt~ f--~
~cr.J- F~{b)
" H J (te."~) ,
The first system, HI elOJ = ( . ) {1o IlUl < 0.5Jr .
0.5Jr ~ IlUl < Jr
The second system; y[n] = w[n]- w[n -1]
The input to the system is x[n] = cos(0.6Jrn)+35[n-5]+2
Determine the output y[n] and Y(e jOJ
).
Contd P/2
1.
. "r,
"
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3. (a) The input to an LTI system is x(n]= (~r u(n]+ 2nu(-n -1].
(20)
(i) Find the system function H(z) of the system. Plot the poles and zeros of H(z) and
indicate the Region of convergence.
(ii) Find the impulse response h(n] of the system.
(iii) Write the difference equation that characterizes the system.
(iv) Is the system stable? Is it causal?
(b) let x(n] be the sequence with the pole-zero plot shown in Figure Q. No. 3(b). Sketch
the pole-zero plot for, y(n] = (~) n x(n]. Specify the region of convergence of Y(z). (15)
-----
~
_._' _'.~"'ft~
i '-Pc-fi~C2-t*3q,-)-
4. (a)
--1 ~ ..._--.-.-.---
~r".J ~H ~[".1 £H.0'1-j
For the system in Figure Q. No. 4(a), given that Hd(z) is causal, stable and non- minimum
phase, given by
Choose Hc(z) so that it is stable, causal and the magnitude of overall effective frequency
response is unity. (15)
(b) The system for changing the sampling rate by a non-integer factor is given by Figure
L = up sampling factor = 2. M = down sampling factor = 3
Contd P/3
1
('..
•
_ )
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Contd ... Q. No. 4(b)
Sketch
(i) IX(e j
()) ~
(ii) IXe (ej(V ~
(iii) IX (e
i
j
()) ~
(iv) IX d (e j
()) ~
Where, x[n] is obtained from xc(t), xc(jQ) is given in Figure Q. No. 4(b)-2.
SECTION-B
There are FOUR questions in this Section. Answer any THREE questions.
Symbols have their usual meanings.
5. (a) What are the two main problems in designing ideal low pass filter? Briefly describe
the principle of Windowing method of FIR filter design. (10)
(b) What is the limitation of the impulse invariance method of IIR filter design? How the
principle used in the bilinear transfer method can overcome the limitation? (10)
(c) Fig for Q. No. 5(c) shows the signal flow graph for a causal discrete time LTI system (15)
(i) Find H(z) = Y(z) .
X(z)
(ii) Determine h[l], the impulse response at n = 1.
.- -- - ~. ----- -----:-,
,
,
i 1-
X[!)) ~[nJ-.
-I
~c; -1:
-I 2
,_._------
-I
~
4
_____ F_~31a.No. ;(e)
6. (a) Consider a finite-length sequence x[n] oflength N. Two finite-length sequences xl[n]
and x2[n] oflength 2N are constructed from x[n]. In Fig. for Q. No. 6(a), x[n]' xl[n] and
x2[n] are shown (solid line is used to suggest the envelope of the sequence values. The
N-point DFT of x[n] is denote by X[k] and the 2N point DFTs of xl[n] and x2[n] are
denoted by X1[k] and X2[k], respectively. (15)
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Contd ... Q. No. 6(a)
(i) Specify how X2[k] can be obtained if X[k] is given.
(ii) Show that X[k] can be obtained by decimating XI[k] by 2.
--' ---------'--------------------------'-------~
(b) Given that xI[n] = [ 1 0 1 2] (20)
t
x2[n] = [ 1 0 1]
t
Yl[n] =xd«n-2))4], Y2[n] =xd«n+l))4],
(i) Find YI[k] and Y2[k]
(ii) IfYc[n] = YI[n] @ Y2[n], find Yc[k] and using Yc[k] find yc[n] .
7. (a) For the system shown in Fig. for Q. No. 7(a), for the basic filter block H(ejOl) it is
given that
h[n]={ h[-n] , -Lsns.L
o ,otherwise
H( dOl) satisfies the following approximation error specification: (15)
(1- ° 1) s IH(e)l1J ~ ~ (1+ °1), 0S OJ S OJ p
- 02 S IH(e)l1J ~ s ° 2, OJs S OJ S 7r
(i) Determine whether the overall impulse response g[n] is FIR and symmetric.
(ii) Find the approximation error specification for G( dOl).
~[nJ t.rnJ
Heeiw)
. .
.... 'Wlt\} 14 ce?"')
2.
Contd P/5
\,
J
\
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Contd ... Q. No.7
(b) By using Kaiser Window method design Type-I FIR filter with generalized linear
phase that meets the following specifications: (20)
jaJ
0.9 < JH(e ~ < 1.1, 0::;;levi::;; 0.27l"
- 0.06 < IH(e jaJ ~ < 0.06, 0.37l"::;; levi::;; 0.4757l"
1.9 < IH(ejaJ ~ < 2.1, 0.5257l" ::;; levi::;; 7l"
(i) Determine appropriate values of ripple 8, ~ L1co and filter length.
(ii) Find ~[n] and h[n].
8. (a) The system function of a discrete-time system is
Assume that this discrete-time filter was designed by the impulse-invariance method with
h[n] = 3hc(3n), where l\;(t) is real. Find the system function Hc(s) of a continuous time
filter that could have been the basis for the design. Is your answer unique? (10)
. (b) Consider designing a discrete-time filter with system function H(z) from a continuous
time filter with rational system function Hc(s) by the transformation
(i) Show that the j n axis of the s-plane is mapped to the unit circle of z-plane.
(ii) Find the mapping between n and co.
(iii) Consider the following specifications for the desired discrete-time filter
0.8::;; IH(ejaJ ~ ::;;1, 0::;;levi::;; 0.257l"
IH(ejaJ~::;;0.15, 0.47l"17l"1::;;7l"
using the transformation specified at the beginning find H(z). (25)
