Digital Signals and System (October – 2016) [Revised Syllabus | Question Paper]
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The document is an examination question paper for a course in digital signal processing with a total of 100 marks and a duration of 3 hours. It consists of various questions divided into multiple sections, covering topics such as Laplace transforms, Fourier series, z-transforms, stability of systems, and filter design. Students are instructed to make assumptions where necessary, answer questions in groups, and include diagrams when appropriate.
![Q u e s t i o n P a p e r ( O c t o b e r – 2 0 1 6 ) [ R e v i s e d S y l l a b u s ]
1 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Time: 3 Hours Total Marks: 100
N.B.: (1) All Question are Compulsory.
(2) Make Suitable Assumptions Wherever Necessary And State The Assumptions Made.
(3) Answer To The Same Question Must Be Written Together.
(4) Number To The Right Indicates Marks.
(5) Draw Neat Labeled Diagrams Wherever Necessary.
Q.1 Attempt Any Two Question: (10 Marks)
(A) Find the Laplace Transform for the following signal: (5)
(B) By first differentiating 𝑥(𝑧) and then using appropriate properties, determine 𝑥(𝑛) for
𝑥(𝑧) = 𝑙𝑜𝑔(1 − 2𝑧) |𝑧| <
1
2
(5)
(C) Determine the exponential Fourier Series for the waveform show below: (5)
(D) What are the applications of Digital Signal Processing? (5)
Q.2 Attempt Any Three Question: (15 Marks)
(A) Explain the following properties of Fourier Transform:
(i) Linearity (ii) Symmetry (iii) Scaling
(5)
(B) Show that the product of two even signals or two odd signals is an even signal and that
the product of an event and an odd signals is an odd signal.
(5)
(C) What is even signal and odd signal? Determine the even and odd components of
𝑥(𝑡) = cos 𝑡 + sin 𝑡
(5)
(D) Draw and explain the block diagram of an Analog – To – Digital Converter. (5)
(E) What is meant by sampling? State Sampling Theorem. (5)
(F) Write a note on Dirichlet’s Conditions. (5)](https://image.slidesharecdn.com/dss-rs-qp-october-2016-180428083748/85/Digital-Signals-and-System-October-2016-Revised-Syllabus-Question-Paper-1-320.jpg)
![Q u e s t i o n P a p e r ( O c t o b e r – 2 0 1 6 ) [ R e v i s e d S y l l a b u s ]
2 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Q.3 Attempt Any Three Question: (15 Marks)
(A) Determine poles, zeroes of 𝐹(𝑠). Obtain 𝐹(𝑠) = 4.
(𝑆+1)(𝑆+3)
(𝑆+2)(𝑆+4)
(5)
(B) Obtain Laplace Transform for step and Impulse Response of Series R-C Circuit. (5)
(C) Find Inverse Laplace Transform of 𝐹1(𝑆) =
𝑆2+5
𝑆3+𝑍𝑆2+4𝑆
(5)
(D) Discuss initial Value Theorem in Laplace Transform Domain. (5)
(E) Define Laplace Transform and Inverse Laplace Transform. What is region of
convergence?
(5)
(F) Find the Laplace Transform of:
(i) 𝑒−𝑡
sin 4𝑡
(ii) 𝑒2𝑡
+ 2𝑡𝑒−2𝑡
− 𝑡2
(5)
Q.4 Attempt Any Three Question: (15 Marks)
(A) Compare the properties of two-sided z-transform with those of one-sided z-
Transform.
(5)
(B) Find the z-transform of the following:
(i) 𝑛𝑢(𝑛)
(ii) 𝑛𝑑 𝑛
𝑢(𝑛)
(5)
(C) Determine the convolution of the two sequences 𝑥(𝑛) = {2,1,1,0,5} and 𝑥(𝑛) =
{1,2,5,4,1,1}.
(5)
(D) What is Z-transform? Explain the use of Z-transform. Determine the z-transform
of 𝑥(𝑛) = {1,2,5,4,1,1}.
(5)
(E) Let 𝑥(𝑛) be the sequence with z-transform 𝑋(𝑧). Determine in terms of 𝑋(𝑧), the z-
transforms of the following signals:
(i) 𝑥1(𝑛) = 𝑥 (
𝑛
2
) If n is even, 0 otherwise.
(ii) 𝑥2(𝑛) = 𝑥(2𝑛)
(5)
(F) Determine the causal signal having z-transform 𝑋(𝑧) =
𝑧2+𝑧
(𝑧−
1
2
)
3
(𝑧−
1
4
)
for the region of
coverage |𝑧| >
1
2
(5)
CONTINUE NEXT ->](https://image.slidesharecdn.com/dss-rs-qp-october-2016-180428083748/85/Digital-Signals-and-System-October-2016-Revised-Syllabus-Question-Paper-2-320.jpg)
![Q u e s t i o n P a p e r ( O c t o b e r – 2 0 1 6 ) [ R e v i s e d S y l l a b u s ]
3 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Q.5 Attempt Any Three Question: (15 Marks)
(A) Explain the Paley-Wiener Criterion. (5)
(B) Explain stability in Linear Time Invariant System. What is the condition for a system o
be BIBO Stable?
(5)
(C) Determine whether the system described by 𝐹[𝑥(𝑛)] = 𝑎[𝑥(𝑛)]2
+ 𝑏 𝑥(𝑛) is linear
and time invariant.
(5)
(D) Obtain Frequency Response for 𝑦(𝑛) = 𝑥(𝑛) + 10𝑦(𝑛 − 1) with initial
condition 𝑦(−1) = 0.
(5)
(E) Determine the impulse response and unit step response of the system described by
the difference equations
(i) 𝑦(𝑛) = 0.6𝑦(𝑛 − 1) − 0.08𝑦(𝑛 − 2) + 𝑥(𝑛)
(ii) 𝑦(𝑛) = 0.7𝑦(𝑛 − 2) − 0.1𝑦(𝑛) + 2𝑥(𝑛) − 𝑥(𝑛 − 2)
(5)
(F) Find the impulse response, frequency response, magnitude response and phase
response of the second order system characterised by the difference equation:
𝑦(𝑛) − 𝑦(𝑛 − 1) +
3
16
𝑦(𝑛 − 2) = 𝑥(𝑛) −
1
2
𝑥(𝑛 − 1)
(5)
Q.6 Attempt Any Three Question: (15 Marks)
(A) Consider two periodic sequences 𝑥(𝑛) and 𝑦(𝑛) with period M and N respectively.
The sequence 𝑤(𝑛) is defined as 𝑤(𝑛) = 𝑥(𝑛) + 𝑦(𝑛). Show that 𝑤(𝑛) is periodic
with period MN.
(5)
(B) Determine the circular correlation values of the two sequences 𝑥(𝑛) = {1,2,0,1}
and ℎ(𝑛) = {3,3,2,1}.
(5)
(C) What are the methods used to perform Fast Convolution. Explain any one method
giving all the steps involved to perform Fast Convolution.
(5)
(D) Define Discrete Fourier Transform (DFT) and Inverse Discrete Fourier Transform
(IDFT). Also state the Complex Conjugate property and Circular Convolution property
of Discrete Fourier Transform (DFT).
(5)
(E) Compute 8-point DFT of the sequence 𝑥(𝑛) = {1,1,1,1,1,1,1,1} by using DIF FFT
Algorithm.
(5)
(F) Find the circular periodic convolution using DFT and IDFT of the two sequences:
𝑥(𝑛) = {1,1,2,2} and ℎ(𝑛) = {1,2,3,4}
(5)
CONTINUE NEXT ->](https://image.slidesharecdn.com/dss-rs-qp-october-2016-180428083748/85/Digital-Signals-and-System-October-2016-Revised-Syllabus-Question-Paper-3-320.jpg)
![Q u e s t i o n P a p e r ( O c t o b e r – 2 0 1 6 ) [ R e v i s e d S y l l a b u s ]
4 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Q.7 Attempt Any Three Question: (15 Marks)
(A) Explain the effects of windowing. Define Rectangular and Hamming Window
functions.
(5)
(B) What are the advantages of FIR Filter over IIR Filters? (5)
(C) Describe Elliptical Filters in detail. (5)
(D) What is an IIR Filter? Compare its characteristics with an FIR Filter. (5)
(E) Design a Finite Impulse Response low pass filter with a cut-off frequency of 1kHz and
sampling rate of 4kHz with eleven samples using Fourier series.
(5)
(F) Design a bandpass filter to pass frequencies in the range 1 – 2 rad/sec using Hanning
window N = 5.
(5)](https://image.slidesharecdn.com/dss-rs-qp-october-2016-180428083748/85/Digital-Signals-and-System-October-2016-Revised-Syllabus-Question-Paper-4-320.jpg)
Digital Signals and System (October – 2016) [Revised Syllabus | Question Paper]
- 1. Q u e s t i o n P a p e r ( O c t o b e r – 2 0 1 6 ) [ R e v i s e d S y l l a b u s ] 1 | Page M u m b a i B . S c . I T S t u d y F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e – Kamal T. Time: 3 Hours Total Marks: 100 N.B.: (1) All Question are Compulsory. (2) Make Suitable Assumptions Wherever Necessary And State The Assumptions Made. (3) Answer To The Same Question Must Be Written Together. (4) Number To The Right Indicates Marks. (5) Draw Neat Labeled Diagrams Wherever Necessary. Q.1 Attempt Any Two Question: (10 Marks) (A) Find the Laplace Transform for the following signal: (5) (B) By first differentiating 𝑥(𝑧) and then using appropriate properties, determine 𝑥(𝑛) for 𝑥(𝑧) = 𝑙𝑜𝑔(1 − 2𝑧) |𝑧| < 1 2 (5) (C) Determine the exponential Fourier Series for the waveform show below: (5) (D) What are the applications of Digital Signal Processing? (5) Q.2 Attempt Any Three Question: (15 Marks) (A) Explain the following properties of Fourier Transform: (i) Linearity (ii) Symmetry (iii) Scaling (5) (B) Show that the product of two even signals or two odd signals is an even signal and that the product of an event and an odd signals is an odd signal. (5) (C) What is even signal and odd signal? Determine the even and odd components of 𝑥(𝑡) = cos 𝑡 + sin 𝑡 (5) (D) Draw and explain the block diagram of an Analog – To – Digital Converter. (5) (E) What is meant by sampling? State Sampling Theorem. (5) (F) Write a note on Dirichlet’s Conditions. (5)
- 2. Q u e s t i o n P a p e r ( O c t o b e r – 2 0 1 6 ) [ R e v i s e d S y l l a b u s ] 2 | Page M u m b a i B . S c . I T S t u d y F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e – Kamal T. Q.3 Attempt Any Three Question: (15 Marks) (A) Determine poles, zeroes of 𝐹(𝑠). Obtain 𝐹(𝑠) = 4. (𝑆+1)(𝑆+3) (𝑆+2)(𝑆+4) (5) (B) Obtain Laplace Transform for step and Impulse Response of Series R-C Circuit. (5) (C) Find Inverse Laplace Transform of 𝐹1(𝑆) = 𝑆2+5 𝑆3+𝑍𝑆2+4𝑆 (5) (D) Discuss initial Value Theorem in Laplace Transform Domain. (5) (E) Define Laplace Transform and Inverse Laplace Transform. What is region of convergence? (5) (F) Find the Laplace Transform of: (i) 𝑒−𝑡 sin 4𝑡 (ii) 𝑒2𝑡 + 2𝑡𝑒−2𝑡 − 𝑡2 (5) Q.4 Attempt Any Three Question: (15 Marks) (A) Compare the properties of two-sided z-transform with those of one-sided z- Transform. (5) (B) Find the z-transform of the following: (i) 𝑛𝑢(𝑛) (ii) 𝑛𝑑 𝑛 𝑢(𝑛) (5) (C) Determine the convolution of the two sequences 𝑥(𝑛) = {2,1,1,0,5} and 𝑥(𝑛) = {1,2,5,4,1,1}. (5) (D) What is Z-transform? Explain the use of Z-transform. Determine the z-transform of 𝑥(𝑛) = {1,2,5,4,1,1}. (5) (E) Let 𝑥(𝑛) be the sequence with z-transform 𝑋(𝑧). Determine in terms of 𝑋(𝑧), the z- transforms of the following signals: (i) 𝑥1(𝑛) = 𝑥 ( 𝑛 2 ) If n is even, 0 otherwise. (ii) 𝑥2(𝑛) = 𝑥(2𝑛) (5) (F) Determine the causal signal having z-transform 𝑋(𝑧) = 𝑧2+𝑧 (𝑧− 1 2 ) 3 (𝑧− 1 4 ) for the region of coverage |𝑧| > 1 2 (5) CONTINUE NEXT ->
- 3. Q u e s t i o n P a p e r ( O c t o b e r – 2 0 1 6 ) [ R e v i s e d S y l l a b u s ] 3 | Page M u m b a i B . S c . I T S t u d y F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e – Kamal T. Q.5 Attempt Any Three Question: (15 Marks) (A) Explain the Paley-Wiener Criterion. (5) (B) Explain stability in Linear Time Invariant System. What is the condition for a system o be BIBO Stable? (5) (C) Determine whether the system described by 𝐹[𝑥(𝑛)] = 𝑎[𝑥(𝑛)]2 + 𝑏 𝑥(𝑛) is linear and time invariant. (5) (D) Obtain Frequency Response for 𝑦(𝑛) = 𝑥(𝑛) + 10𝑦(𝑛 − 1) with initial condition 𝑦(−1) = 0. (5) (E) Determine the impulse response and unit step response of the system described by the difference equations (i) 𝑦(𝑛) = 0.6𝑦(𝑛 − 1) − 0.08𝑦(𝑛 − 2) + 𝑥(𝑛) (ii) 𝑦(𝑛) = 0.7𝑦(𝑛 − 2) − 0.1𝑦(𝑛) + 2𝑥(𝑛) − 𝑥(𝑛 − 2) (5) (F) Find the impulse response, frequency response, magnitude response and phase response of the second order system characterised by the difference equation: 𝑦(𝑛) − 𝑦(𝑛 − 1) + 3 16 𝑦(𝑛 − 2) = 𝑥(𝑛) − 1 2 𝑥(𝑛 − 1) (5) Q.6 Attempt Any Three Question: (15 Marks) (A) Consider two periodic sequences 𝑥(𝑛) and 𝑦(𝑛) with period M and N respectively. The sequence 𝑤(𝑛) is defined as 𝑤(𝑛) = 𝑥(𝑛) + 𝑦(𝑛). Show that 𝑤(𝑛) is periodic with period MN. (5) (B) Determine the circular correlation values of the two sequences 𝑥(𝑛) = {1,2,0,1} and ℎ(𝑛) = {3,3,2,1}. (5) (C) What are the methods used to perform Fast Convolution. Explain any one method giving all the steps involved to perform Fast Convolution. (5) (D) Define Discrete Fourier Transform (DFT) and Inverse Discrete Fourier Transform (IDFT). Also state the Complex Conjugate property and Circular Convolution property of Discrete Fourier Transform (DFT). (5) (E) Compute 8-point DFT of the sequence 𝑥(𝑛) = {1,1,1,1,1,1,1,1} by using DIF FFT Algorithm. (5) (F) Find the circular periodic convolution using DFT and IDFT of the two sequences: 𝑥(𝑛) = {1,1,2,2} and ℎ(𝑛) = {1,2,3,4} (5) CONTINUE NEXT ->
- 4. Q u e s t i o n P a p e r ( O c t o b e r – 2 0 1 6 ) [ R e v i s e d S y l l a b u s ] 4 | Page M u m b a i B . S c . I T S t u d y F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e – Kamal T. Q.7 Attempt Any Three Question: (15 Marks) (A) Explain the effects of windowing. Define Rectangular and Hamming Window functions. (5) (B) What are the advantages of FIR Filter over IIR Filters? (5) (C) Describe Elliptical Filters in detail. (5) (D) What is an IIR Filter? Compare its characteristics with an FIR Filter. (5) (E) Design a Finite Impulse Response low pass filter with a cut-off frequency of 1kHz and sampling rate of 4kHz with eleven samples using Fourier series. (5) (F) Design a bandpass filter to pass frequencies in the range 1 – 2 rad/sec using Hanning window N = 5. (5)