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Sce 504: Digital Signal Processing Question Paper
Sce 504: Digital Signal Processing
Course:Bachelor Of Science In Computer Engineering
Institution: Kenyatta University question papers
Exam Year:2011
DATE: Wednesday, 14th December, 2011
TIME: 11.00 a.m. – 1.00 p.m.
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INSTRUCTIONS:
(i)
This paper contains FIVE (8) questions.
(ii)
Answer all question in Section A and any TWO questions from each Sections B
and C.
SECTION A:
Question 1:
a) Explain the Gibbs’s phenomenon in FIR filters and show how it
can be reduced
(6 marks)
b) Briefly describe and compare FIR and IIR filters, giving the main
advantages and disadvantages of each
(6 marks)
c)Explain two basic uses of digital filters
(2 marks)
d)An input sequence corresponding to a sampled data signal is given
as {1, 0.5, 0, -0.8, -3}. What is the z-transform of the input signal? (4 marks)
e)Does the filter with transfer function H(z)=1-z-1 have a linear
phase? Why?
(3 marks)
Page 1 of 4
f)When designing digital filters from existing analog filters, what
properties should any mapping from the s-plane to the z-plane have? (4 marks)
g) Given the following structure below
T
1
1
+
y(n)
T
2
x(n)
1
0.5
1
i) Give the expression for the output y(n)?
(3 marks)
ii)State wether the filter represented is recussive or non-
recursive?
(1 mark)
iii)Give reason for your answer in (ii) above?
(2 marks)
SECTION B
Question 2:
Given
1
H (z) ?
,
1
?
2
(1?1.6z ? 0.6z? )
a) Write the difference equation for the output of H(z) and
calculate the first three samples of the step response of the
filter (6 marks)
b) Is H(z) stable ?
(2 marks)
c) Give reason for your answer in (b) above
(2 marks)
Question 3:
Determine the ideal impulse response coefficient of a lowpass filter of
order 21 to satisfy the following specification;
(10 marks)
w
? 2? ?3000rads / sec , w
? 2? ?4000rads / sec and f ? 20KHz
pass
stop
s
Page 2 of 4
Question 4:
N 1
?
2?
The DFT X(n) of the sequence x(k) is given by
? j
kn
2
X (n) ? ? x(k)e
.
k ?0
i)X(n) is periodic in n. What is the period?
(1 mark)
ii) For what range of n values is X(n) normally calculated?
(1 mark)
iii) How many real multiplications and additions are needed to
directly calculate X(n) for this range?
(2 marks)
iv)If the decimation in time FFT algorithm is used to calculate
X(n) where N is an integer power of 2, how many real
multiplications and additions are needed?
(2 marks)
v) Calculate the DFT coefficients X(0) and X(1) for the
sequence x(k):
x(k)=0,-0.5,+0.5, 0
(4 marks)
SECTION C
Question 5:
Given the following simple transfer function,
z ?1
H (z) ?
1
3
(z ? )(z ? )
2
4
i) Find the poles and zeros of the system.
(3 marks)
ii)Sketch pole-zero diagram of this transfer function and in your
diagram show posible region of convergence(ROC).
(4 marks)
iii)Explain whether this system is stable or causal?
(3 marks)
Question 6:
Given the transfer function of a simple analog filter as
1
H (s) ?
,
(1? s)
Page 3 of 4
i)Use this transfer function and the bilinear transform method
to design the corresponding digital lowpass filter whose
bandwidth is 1000Hz and the sampling frequency is 8000Hz (5 marks)
ii) Explain how frequency warping occurs when using bilinear
transform
(3marks)
iii)How can frequency warping be compensated?
(2marks)
Question 7:
Consider the signal
x(t)
e-at
( )
?at
x t ? e u(t) , a>0 where u(t) is the unit step function defined by
u(t) =
0
t
1 t > 0
0
t < 0
i)Find the fourier transform of x(t)
(6 marks)
ii)Sketch its magnitude and phase spectra?
(4 marks)
Page 4 of 4
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