Code: A-23 Subject: DIGITAL SIGNAL PROCESSING
Time: 3 Hours Max. Marks: 100
NOTE: There are 11
Questions in all.
· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must
be written in the space provided for it in the answer book supplied and nowhere
else.
· Answer any THREE Questions each from Part I and Part II. Each of
these questions carries 14 marks.
· Any required data not explicitly given, may be suitably assumed and
stated.
Q.1 Choose the correct or best alternative in
the following: (2x8)
a. A signal x (n) is called an energy signal, if its energy E and power P satisfy the following:
(A) both E and P are infinite. (B) E is infinite but P is finite.
(C) E is finite, P = 0. (D) E is finite but P is infinite.
b.
If 
is the sampling
frequency, then the maximum frequency that a digital filter can process
properly is
(A) 
. (B) 
.
(C)
. (D) 0.1 fs.
c.
The convolution of a finite sequence 
with another sequence
results in a finite
sequence 
. Then 
(A) must be a finite sequence.
(B) must be an infinite sequence.
(C) may be a finite or an infinite sequence.
(D) must have finite energy.
d. The region of convergence of the z-transform of a causal signal
(A) is the interior of a circle. (B) is the exterior of a circle.
(C) is an annular region. (D) may be any of these depending on
the signal.
e. The Fourier transform of a discrete periodic sequence is
(A) continuous and periodic or non-periodic.
(B) continuous and periodic.
(C) continuous and non periodic.
(D) discrete and periodic.
f. If
the difference equation for a system is 
then the transfer
function is
(A)
. (B) 
.
(C) 
. (D) 
.
g. A
highly selective IIR band pass digital filter may become unstable due to finite
word length because of
(A)
the closeness of its poles to the
unit circle.
(B)
the closeness of its zeros to the
unit circle.
(C)
zeros lying outside the unit
circle.
(D) its poles on the real axis.
h. The
number of complex multiplications in radix-2 DIF-FFT algorithm is
(A)
N log2 N. (B) 
.
(C)
. (D) 
.
PART I
Answer any THREE Questions. Each question carries 14 marks.
Q.2 a. Define the following:
(i) Discrete periodic signal. (ii) All pass filter.
(iii)
Aliasing distortion. (6)
b. Define
the linearity propriety of a DSP system.
Test the following systems for linearity:
(i) 
(ii) 
. (8)
Q.3 a. A
digital filter is characterized by the difference equation, 
. Check the filter
for BIBO stability.
(BIBO : Bounded Input-Bounded Output) (6)
b. Show that an FIR system can be realised recursively and an IIR system can be realised non recursively. (8)
Q.4 a. What
is meant by the region of convergence of a z-transform? Give an example of a signal whose region of
convergence is an annular region in the z-plane. (6)
b. Determine the z-transform and region of convergence of the signals
(i) 
(ii) 
. (8)
Q.5 a. The system function of an FIR filter is given by,
Deduce its two different direct form realisations. (6)
b. Obtain two canonical realisations of the system function,
. (8)
Q.6 a. Write the three important steps in the design
of digital filters from analog filters.
(3)
b. What is bilinear
transformation? Write two of its
important properties. (4)
c. Given 
, find the corresponding digital transfer function 
by using bilinear
transformation with T = 2 seconds.
Sketch the magnitude characteristics of H(s) as well as 
. (7)
PART II
Answer any THREE Questions. Each question carries 14 marks.
Q.7 a. Define the N-point
DFT of a finite duration sequence x(n).
Deduce the relationship of the DFT to the z-transform. (7)
b. Using the defining
equation, compute the DFT of
(i) 
(ii) 
.
(iii) 
. (7)
Q.8 a. What
are the methods used for designing FIR filters? What do you mean by linear phase response as used in FIR filter
theory? (5)
b. Using a rectangular window, design a lowpass filter with a passband gain of unity, cutoff frequency of 1000 Hz and working at a sampling frequency of 5 KHz. Take the length of the impulse response as 7. (9)
Q.9 a. Why
is FFT needed in signal processing?
What is its main advantage? (4)
b. Given 
, find X(k) using the DIT-FFT algorithm. (10)
Q.10 a. What are the
three quantisation errors due to finite word length registers in digital
filters? Briefly explain each one of
them. (10)
b. Find x(n) using the
partial fraction expansion method for 
, if the region of convergence is 
. (4)
.
.