Digital Signals and Systems (April – 2017) [Question Paper | CBSGS: 75:25 Pattern]
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The document is a question paper for the Digital Signals and Systems course at Mumbai University, part of the B.Sc.IT program, from April 2017, following a 75:25 pattern. It contains multiple questions divided into different sections, covering topics such as properties of functions, Laplace and Z-transforms, digital filters, and convolution. Students are instructed to attempt various questions and provide detailed answers and diagrams where necessary.
![Question Paper
[CBSGS – 75:25 PATTERN]
DIGITAL
SIGNALS AND SYSTEMS](https://image.slidesharecdn.com/cbsgs-dss-apr-2017-180708053839/85/Digital-Signals-and-Systems-April-2017-Question-Paper-CBSGS-75-25-Pattern-1-320.jpg)
![MUMBAI UNIVERSITY
QUESTION PAPER
DIGITAL SIGNALS AND SYSTEMS
(APRIL – 2017 | 75:25 PATTERN)
B.SC.IT
(SEMESTER – VI)
KjT MUMBAI B.SC.IT STUDY
FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB
PAGE
1
Time: 2 ½ Hours Total Marks: 75
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.
(6) Use of Non – Programmable Calculator is allowed.
Q.1 ATTEMPT ANY TWO QUESTIONS: (10 MARKS)
(A) State and explain the properties of Unit Impulse Function (𝑡). (5)
(B) How are Continuous and Discrete Time Systems classified? Explain. (5)
(C) What are Energy and Power Signals?
Determine if the following signals are energy signals or power signals or neither:
(i) 𝑥(𝑡) = 𝑡𝑢(𝑡)
(ii) 𝑥(𝑛) = (−0.5) 𝑛
𝑢(𝑛)
(5)
(D) Obtain the Trigonometric Fourier series for the half wave rectified sine wave shown below: (5)
Q.2 ATTEMPT ANY TWO QUESTIONS: (10 MARKS)
(A) Find the Laplace Transform of the following functions:
(i) 𝑓(𝑡) =
1−𝑒 𝑡
𝑡
(ii) 𝑓(𝑡) = 𝑐𝑜𝑠3
3𝑡
(5)
(B) Find the Laplace transform of the full wave rectified output as shown below: (5)
(C) Find the inverse Laplace transform of {
𝑆2−𝑆−3
(𝑆+5)(𝑆+4)2} (5)
(D) The unit step of a network is (1 – 𝑒−𝑎𝑡
). Determine the impulse response ℎ(𝑡) of the network. (5)
Q.3 ATTEMPT ANY TWO QUESTIONS: (10 MARKS)
(A) A system has an impulse response ℎ(𝑛) = {1,2,3} and output response
𝑦(𝑛) = {1,1,2, −1,3}. Determine the input sequence 𝑥(𝑛).
(5)
(B) Determine the z-transform for the analog input signal 𝑥(𝑡) = 𝑒−𝑎𝑡
applied to a Digital Filter. (5)
(C) How is z-transform obtained from Laplace Transform? Derive the z-transform of 𝑓(𝑛𝑇) = 𝑐𝑜𝑠 𝜔𝑛𝑇 (5)
(D) Define one-sided z-Transform, Two-sided z-Transform and Inverse z-Transform. (5)
[TURN OVER]](https://image.slidesharecdn.com/cbsgs-dss-apr-2017-180708053839/85/Digital-Signals-and-Systems-April-2017-Question-Paper-CBSGS-75-25-Pattern-2-320.jpg)
![MUMBAI UNIVERSITY
QUESTION PAPER
DIGITAL SIGNALS AND SYSTEMS
(APRIL – 2017 | 75:25 PATTERN)
B.SC.IT
(SEMESTER – VI)
KjT MUMBAI B.SC.IT STUDY
FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB
PAGE
2
Q.4 ATTEMPT ANY TWO QUESTIONS: (10 MARKS)
(A) Explain the Paley – Wiener criteria. (5)
(B) Consider a causal and stable LTI system whose input 𝑥(𝑛) and output 𝑦(𝑛) are related through the
second order difference equation. 𝑦(𝑛) −
1
12
𝑦(𝑛 − 1) −
1
12
𝑦(𝑛 − 2) = 𝑥(𝑛)
Determine the step response for the system.
(5)
(C) Find the response of the following difference equation
𝑦(𝑛) − 5𝑦(𝑛 − 1) + 6𝑦(𝑛 − 2) = 𝑥(𝑛) 𝑓𝑜𝑟 𝑥(𝑛) = 𝑢(𝑛)
(5)
(D) A second order discrete time system is characterised by the difference equation
𝑦(𝑛) − 0.1𝑦(𝑛 − 1) − 0.02𝑦(𝑛 − 2) = 2𝑥(𝑛) − 𝑥(𝑛 − 1)
Determine y(n) for 𝑛 ≥ 0 when 𝑥(𝑛) = 𝑢(𝑛) and the initial conditions are
𝑦(– 1) =– 10 𝑎𝑛𝑑 𝑦(– 2 ) = 5
(5)
Q.5 ATTEMPT ANY TWO QUESTIONS: (10 MARKS)
(A) Find the 4-point DFT of the sequence 𝑥(𝑛) = 𝑐𝑜𝑠
𝑛𝜋
4
. (5)
(B) Compute the circular periodic convolution graphically of the two sequences:
𝑥(𝑛) = 𝛿(𝑛) + 𝛿(𝑛 − 1) − 𝛿(𝑛 − 2) − 𝛿(𝑛 − 3) 𝑎𝑛𝑑
ℎ(𝑛) = 𝛿(𝑛) − 𝛿(𝑛 − 2) + 𝛿(𝑛 − 4)
(5)
(C) Determine the cross-correlation values of the two sequences 𝑥(𝑛) = {1,0,0,1} and ℎ(𝑛) =
{4,3,2,1}.
(5)
(D) Distinguish between linear convolution and circular convolution. (5)
Q.6 ATTEMPT ANY TWO QUESTIONS: (10 MARKS)
(A) Design a digital Chebyshev filter to satisfy the constrains
0.707 ≤ │𝐻(𝑒 𝑗𝜔
)│ ≤ 1, 0 ≤ 𝜔 ≤ 0.2𝜋
│𝐻(𝑒) 𝑗𝜔
│ ≤ 0.1, 0.5𝜋 ≤ 𝜔 ≤ 𝜋
Using bilinear transformation and assuming 𝑇 = 1𝑠.
(5)
(B) 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)
(C) An analog filter has the following system function. Convert this filter into a digital filter using backward
difference for the derivative.
𝐻(𝑠) =
1
(𝑠 + 0.1)2 + 9
(5)
(D) Design a digital Chebyshev filter to satisfy the constrains
0.707 ≤ │𝐻(𝑒 𝑗𝜔
)│ ≤ 1, 0 ≤ 𝜔 ≤ 0.2𝜋
│𝐻(𝑒) 𝑗𝜔
│ ≤ 0.1, 0.5𝜋 ≤ 𝜔 ≤ 𝜋
Using bilinear transformation and assuming 𝑇 = 1𝑠.
(5)
[TURN OVER]](https://image.slidesharecdn.com/cbsgs-dss-apr-2017-180708053839/85/Digital-Signals-and-Systems-April-2017-Question-Paper-CBSGS-75-25-Pattern-3-320.jpg)
Digital Signals and Systems (April – 2017) [Question Paper | CBSGS: 75:25 Pattern]
- 1. Question Paper [CBSGS – 75:25 PATTERN] DIGITAL SIGNALS AND SYSTEMS
- 2. MUMBAI UNIVERSITY QUESTION PAPER DIGITAL SIGNALS AND SYSTEMS (APRIL – 2017 | 75:25 PATTERN) B.SC.IT (SEMESTER – VI) KjT MUMBAI B.SC.IT STUDY FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB PAGE 1 Time: 2 ½ Hours Total Marks: 75 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. (6) Use of Non – Programmable Calculator is allowed. Q.1 ATTEMPT ANY TWO QUESTIONS: (10 MARKS) (A) State and explain the properties of Unit Impulse Function (𝑡). (5) (B) How are Continuous and Discrete Time Systems classified? Explain. (5) (C) What are Energy and Power Signals? Determine if the following signals are energy signals or power signals or neither: (i) 𝑥(𝑡) = 𝑡𝑢(𝑡) (ii) 𝑥(𝑛) = (−0.5) 𝑛 𝑢(𝑛) (5) (D) Obtain the Trigonometric Fourier series for the half wave rectified sine wave shown below: (5) Q.2 ATTEMPT ANY TWO QUESTIONS: (10 MARKS) (A) Find the Laplace Transform of the following functions: (i) 𝑓(𝑡) = 1−𝑒 𝑡 𝑡 (ii) 𝑓(𝑡) = 𝑐𝑜𝑠3 3𝑡 (5) (B) Find the Laplace transform of the full wave rectified output as shown below: (5) (C) Find the inverse Laplace transform of { 𝑆2−𝑆−3 (𝑆+5)(𝑆+4)2} (5) (D) The unit step of a network is (1 – 𝑒−𝑎𝑡 ). Determine the impulse response ℎ(𝑡) of the network. (5) Q.3 ATTEMPT ANY TWO QUESTIONS: (10 MARKS) (A) A system has an impulse response ℎ(𝑛) = {1,2,3} and output response 𝑦(𝑛) = {1,1,2, −1,3}. Determine the input sequence 𝑥(𝑛). (5) (B) Determine the z-transform for the analog input signal 𝑥(𝑡) = 𝑒−𝑎𝑡 applied to a Digital Filter. (5) (C) How is z-transform obtained from Laplace Transform? Derive the z-transform of 𝑓(𝑛𝑇) = 𝑐𝑜𝑠 𝜔𝑛𝑇 (5) (D) Define one-sided z-Transform, Two-sided z-Transform and Inverse z-Transform. (5) [TURN OVER]
- 3. MUMBAI UNIVERSITY QUESTION PAPER DIGITAL SIGNALS AND SYSTEMS (APRIL – 2017 | 75:25 PATTERN) B.SC.IT (SEMESTER – VI) KjT MUMBAI B.SC.IT STUDY FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB PAGE 2 Q.4 ATTEMPT ANY TWO QUESTIONS: (10 MARKS) (A) Explain the Paley – Wiener criteria. (5) (B) Consider a causal and stable LTI system whose input 𝑥(𝑛) and output 𝑦(𝑛) are related through the second order difference equation. 𝑦(𝑛) − 1 12 𝑦(𝑛 − 1) − 1 12 𝑦(𝑛 − 2) = 𝑥(𝑛) Determine the step response for the system. (5) (C) Find the response of the following difference equation 𝑦(𝑛) − 5𝑦(𝑛 − 1) + 6𝑦(𝑛 − 2) = 𝑥(𝑛) 𝑓𝑜𝑟 𝑥(𝑛) = 𝑢(𝑛) (5) (D) A second order discrete time system is characterised by the difference equation 𝑦(𝑛) − 0.1𝑦(𝑛 − 1) − 0.02𝑦(𝑛 − 2) = 2𝑥(𝑛) − 𝑥(𝑛 − 1) Determine y(n) for 𝑛 ≥ 0 when 𝑥(𝑛) = 𝑢(𝑛) and the initial conditions are 𝑦(– 1) =– 10 𝑎𝑛𝑑 𝑦(– 2 ) = 5 (5) Q.5 ATTEMPT ANY TWO QUESTIONS: (10 MARKS) (A) Find the 4-point DFT of the sequence 𝑥(𝑛) = 𝑐𝑜𝑠 𝑛𝜋 4 . (5) (B) Compute the circular periodic convolution graphically of the two sequences: 𝑥(𝑛) = 𝛿(𝑛) + 𝛿(𝑛 − 1) − 𝛿(𝑛 − 2) − 𝛿(𝑛 − 3) 𝑎𝑛𝑑 ℎ(𝑛) = 𝛿(𝑛) − 𝛿(𝑛 − 2) + 𝛿(𝑛 − 4) (5) (C) Determine the cross-correlation values of the two sequences 𝑥(𝑛) = {1,0,0,1} and ℎ(𝑛) = {4,3,2,1}. (5) (D) Distinguish between linear convolution and circular convolution. (5) Q.6 ATTEMPT ANY TWO QUESTIONS: (10 MARKS) (A) Design a digital Chebyshev filter to satisfy the constrains 0.707 ≤ │𝐻(𝑒 𝑗𝜔 )│ ≤ 1, 0 ≤ 𝜔 ≤ 0.2𝜋 │𝐻(𝑒) 𝑗𝜔 │ ≤ 0.1, 0.5𝜋 ≤ 𝜔 ≤ 𝜋 Using bilinear transformation and assuming 𝑇 = 1𝑠. (5) (B) 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) (C) An analog filter has the following system function. Convert this filter into a digital filter using backward difference for the derivative. 𝐻(𝑠) = 1 (𝑠 + 0.1)2 + 9 (5) (D) Design a digital Chebyshev filter to satisfy the constrains 0.707 ≤ │𝐻(𝑒 𝑗𝜔 )│ ≤ 1, 0 ≤ 𝜔 ≤ 0.2𝜋 │𝐻(𝑒) 𝑗𝜔 │ ≤ 0.1, 0.5𝜋 ≤ 𝜔 ≤ 𝜋 Using bilinear transformation and assuming 𝑇 = 1𝑠. (5) [TURN OVER]
- 4. MUMBAI UNIVERSITY QUESTION PAPER DIGITAL SIGNALS AND SYSTEMS (APRIL – 2017 | 75:25 PATTERN) B.SC.IT (SEMESTER – VI) KjT MUMBAI B.SC.IT STUDY FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB PAGE 3 Q.7 ATTEMPT ANY THREE QUESTIONS: (15 MARKS) (A) Write a short note on Dirichlet’s conditions. (5) (B) In the parallel RLC circuit. I0 = 5 A, L = 0.2 H, C = 2 F And 𝑅 = 0.5 𝛺. Switch S is opened at time 𝑡 = 0. Obtain the complete particular solution for the voltage 𝑣(𝑡) across the parallel network. Assume zero current through inductor L and zero voltage across capacitor C before switching. (5) (C) Convolve the sequences 𝑥(𝑛) and ℎ(𝑛) where 𝑥(𝑛) = 0, 𝑛 < 0 = 𝑎 𝑛 , 𝑛 ≥ 0 ℎ(𝑛) = 0, 𝑛 < 0 = 𝑏 𝑛 , 𝑛 ≥ 0 Specify the answers if (i) 𝑎 = 𝑏 and (ii) 𝑎 ≠ 𝑏 (5) (D) Find the convolution of the two signals 𝑥(𝑛) = 𝑢(𝑛) 𝑎𝑛𝑑 ℎ(𝑛) = 𝑎 𝑛 𝑢(𝑛), ROC: |𝑎| < 1; 𝑛 ≥ 0 (5) (E) 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) (F) Design an analog BPF to satisfy the following specifications: (i) 3 dB upper and lower cut-off frequencies are 100 Hz and 3.8 kHz (ii) Stop band attenuation of 20 dB at 20 Hz and 8 kHz. (iii) No ripple with both passband and stopband. (5)