answer all the questions in the attached file
INSTITUTE OF SPACE TECHNOLOGY
Digital Signal Processing
BS-EE
Final Exam
Fall 2018 12 January 2018
Time Allowed: 3 Hrs Max Marks: 120
Instructions:
Attempt all Questions.
You may refer to your text books during the exam.
Please show your complete reasoning to receive credit.
In case of confusion, go on to the next question until the instructor comes to you.
Answer the following Questions. Show your reasoning. (60 points)
A continuous-time signal,
is sampled at a sampling frequency 
Hz. Find the sampled discrete-time sequence 
. [1 point]A continuous-time signal is given as
. Is this signal periodic? If it is, find the fundamental period. [1 point]A discrete-time signal is given as
. Is this signal periodic? If it is, find the fundamental period. [1 point]A discrete-time signal is given as
. Is this signal periodic? If it is, find the fundamental period. [2 points]The non-zero values of a discrete-time signal are given as
and 
. Decompose into conjugate symmetric (even) and conjugate anti-symmetric (odd) signals. [3 points] The input-output relation of a discrete-time system is given as
. Is this system: [4 points]Linear?
Time-invariant?
BIBO stable?
Memoryless?
The input-output relation of a discrete-time system is given as
. Is this system: [4 points]Linear?
Time-invariant?
BIBO stable?
Memoryless?
A discrete-time system is given as
, where 
is a real scalar constant. Find:The impulse response of the system. [3 points]
The range of values of
for which the system is BIBO stable. [1 point]
Two discrete-time systems with impulse responses
and 
are combined to form a discrete-time system h
.Find the impulse response of
if the two systems are connected in parallel. [2 points]Find the impulse response of
if the two systems are connected in cascade. [3 points]
Sketch suitable pole-zero diagrams for following filters:
Stable Notch filter with notch frequencies
and 
. [2 points]Stable Digital resonator with resonant frequencies
and . [2 points] All pass filter with two zeros at
and 
. [2 points]
An unstable discrete-time system is described by the transfer function
Sketch the magnitude of the frequency response,
from 
. [3 points]If, for
, the input to the system is
find the frequencies 
and 
for which the output 
will be unbounded. [4 points]
If, for
, the input to the system is
find the frequencies and for which the output will be zero. [4 points]
The communications channel in a particular scenario can be modeled as
, where 
. Find the difference equation of the channel equalizer (the inverse system that cancels the effect of the channel). [3 points]The impulse response of an ideal low pass filter with cut-off frequency
is given as
Use this result to answer the following questions.
Find the impulse response of an ideal high pass filter with cut-off frequency
. [1 point]Find the impulse response of an ideal band pass filter with pass band frequencies from
to 
. [2 points]Find the impulse response of an ideal band stop filter with stop band frequencies from
to . [2 points]
A discrete-time signal is given as
,
where 
is the window. How would you change 
and 
so that the two frequencies in the spectrum estimate are clearly distinguishable? [3 points]
A discrete-time signal is given as
,
where is the window. How would you change and so that the two frequencies in the spectrum estimate are clearly distinguishable? [3 points]
Two periodic signals with period
are described as
Find the result of convolution of 
and 
. [3 points]
For fractional sampling rate conversion, will you use an up-sampler first or a down-sampler? [1 point]
(Sampling of Continuous-time signals, Multirate Signal Processing) (20 points)
A Continuous-time signal, 
, is defined as
.
A discrete-time signal, , is obtained by sampling at a rate 
. The sampled signal, , is then input to an upsampler followed by a downsampler. The output of the downsampler, , is then input to an ideal interpolator (Digital to Analog converter) to obtain the continuous-time signal 
, as shown in figure.
w[n]
y[n]
x[n]
D/A converter
Sampler
Analyze this system by answering the following questions.
Find an equation for the sampled signal,
and sketch the magnitude of its DTFT, from . [4 points]Find an equation for the signal,
and sketch the magnitude of its DTFT, from . [7 points]Find an equation for the signal,
and sketch the magnitude of its DTFT, from . [7 points]Find an equation for the continuous time signal
at the output of the system. [2 points]
(Hilbert Transformer) [10 points]
h1 [n]
h2 [n]
Consider the Discrete-time system shown in the diagram above. The filters, 
are 
are given as
Note that the second term in and is the Hilbert transformer. In this question, assume
Find the equation of the discrete-time signal
and plot the DTFT from . [2 points]Find the equation of the discrete-time signal
and plot the DTFT from . [2 points]Find the equation of the discrete-time signal
and plot the DTFT from . [2 points]Draw a discrete-time system that can be used to recover
, from . [4 points]
(Correlation, Discrete-Fourier Transform) (15 points)
A continuous-time signal
is sampled at a sampling rate 
. A 60 point Discrete Fourier Transform, 
, is then computed for the sampled signal. For what value of 
would you expect to observe a peak in 
?[9 points]A length 4 sequence,
, is to be correlated with a length 5 sequence 
. Draw a block diagram to show how this linear correlation can be implemented using FFT algorithm. (Note: You do not need to draw the butterfly diagram in this part. Just assume that you have a block that takes the FFT. Clearly mention the number of points in FFT block). [2 points]Derive the butterfly diagram for 4 point radix 2 FFT algorithms. [4 points]
(Design of Digital filters) (15 points)
The impulse response of an FIR low pass filter is given as
. Find the phase of this filter. [4 points]A low pass FIR filter is to be designed using windowing technique. The filter is required to have a minimum stop band attenuation of 50dB and a maximum transition bandwidth of up to
. Answer the following questions using the table given below: [5 points]Which window will you select to meet the specifications with minimum filter order?
What is the length of the filter?
| Window | Minimum Stop band attenuation | Transition Bandwidth |
| Rectangular | 20.9dB |
|
| Hann | 43.9dB |
|
| Hamming | 54.5dB |
|
| Blackman | 75.3dB |
|
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