Specimen Question Papers

Specimen Question Papers:

Important Note: The questions in these papers are meant to illustrate the general format, style, distribution, weight, degree of difficulty, time factor etc. (as there are no papers from previous years to refer to.) Students are advised to prepare for ALL topics covered in the class, and not to neglect those topics which may not have been included in the following papers.

This section carries 50 marks and is meant to be completed in 1½ hours.

Paper I

Q1 a) Discuss the classification of sequences, giving one example of each type (8) b) Comment with proof on the linearity, causality, stability and time invariance of the system y[n] = nx[n] (8) or

Q1 a) Explain the relationship between single and double pole locations in the z-plane and the impulse response. (8) b) If g[n] is an even sequence and h[n] is an odd sequence state with proof which of the following is odd and which is even 1) v[n] = h[n]h[n] 2) x[n] = g[n]h[n] (8)

Q2 a) Explain the differences between FIR and IIR Filters and mention their advantages and disadvantages. (8) b) A FIR LPF is required with specifications pass-band edge frequency = 1.5 kHz transition width =0.5 kHz, and f c is centred on the transition band. Stop-band attenuation >50 dB sampling frequency = 8 kHz Select a suitable window and find N. .If the window is w(n) = 0.54 + 0.46 Cos(2 π n/N) find filter impulse response coefficients h ( −2) and h (2) (8)

Q2 a) How will you perform the geometric evaluation of the frequency response of a LTI system? (8) b) What are window functions? How are they useful? Give the properties of the Hamming window.(8)

Q3 Any Three: (18) a) Notch and comb filters b) Sampling c) Concept of pipelining in DSP processors d) TMC320C25 processor architecture.

Paper II

Q1 a) What are the differences in the concept of frequency for continuous and discrete signal? explain. (8) b) State with proof whether y[n] = x 2[n] – x[n-1]x[n+1] is linear or not. (8)

Q1 a) Explain classification of discrete time systems with reference to their output/input relationships (8) b) A system has its impulse response h(n) defined over 0 ≤ n ≤ N-1. If N =7 and h[n] = h[N-n-1] prove that the system has a linear phase characteristic and find the phase delay. (8)

Q2 a) Write a note on the frequency sampling method of filter design (8) b) A filter is required to meet the following specifications : complete signal rejection at DC and 250 Hz, narrow pass-band centred at 125 Hz, 3 – dB bandwidth of 10 Hz. If sampling frequency is 500 Hz, obtain the transfer function and difference equation by pole-zero placement. (8)

Q2 a) Discuss the conversion of a low-pass filter H(z) and h(n) to a high-pass filter system function and impulse response. Give example/s and proof (8) b) Explain the reason for pre-warping in IIR filter design using Bilinear transformation. (8)

Q3 Answer any THREE (18) a) Types and properties of Linear phase FIR functions. b) Band transformations in IIR filter design. c) Differences between conventional and DSP processors d) The ADSP2100 processor architecture

Paper III

Q1 a) Discuss the classification of different types of signals, giving an example of each type (8) b) If y[n] = x[n +1] – 2x[n] +x[n −1], is the system linear? Time invariant? causal? (8)

Q1 a) Discuss the concepts of linearity, time invariance, stability and causality of systems (8) b) If y[n] = ay[n −1] + bx[n] n ≥ 0 and a and b are constants, find the expression for the output y[n] in terms of the input samples and initial condition y[-1]. Is the system linear if y[-1] = 1? (8)

Q2 a) Discuss the computer-aided design of FIR filter using the Remez algorithm. (8) b) A RC (analogue) filter has a normalised transfer function 1/(s+1). It is to be used to design a digital filter with a cut-off 30 Hz, and sampling frequency is 150 Hz Use the BZT to find the transfer function and difference equation of the digital filter. (8)

Q2 a) Discuss the digital resonator. (8) b) Explain the sources of errors in implementation of IIR digital filters. (8)

Q3 a) Analogue filter prototypes used for IIR digital filter design (18) b) Harvard architecture c) Simple FIR Filter design by pole-zero placement d) Architecture of DSP56000

THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV (Electronics) examination 2000 Wednesday, November 29 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) All Questions carry equal marks. iii) Suitable assumptions may be made wherever required,. Maximum marks: 100

SECTION I

Q1 a) Find the Z-transform of the following and give their ROC i) x(n) = -αn U(-n-1) ii) x(n) = cos (n π/2)U(n) b) Obtain the parallel structure for the following system 1(2 − z −1 1)( + 2z −1 + z −2 ) H z)( = 1( + 5.0 z −1 1)( − 9.0 z −1 + 81.0 z −2 )

Q1 a) Determine the causal signal x(n) having the Z-Transform

1 i) X z)( = 1( − 2z −1 1)( − z − ) 21 1− 1 z −1 ii) X z)( = 2 1− z −1 + z −2 b) Obtain the cascade structure for the system 1+ 1 z −1 H z)( = 2 ()− −1 + 1 −2 ()− −1 + 1 −2 1 z 4 z 1 z 2 z

Q2 a) Define (i) FIR, (ii) IIR and (iii) LTI Systems. The LTI system described by 1 ny )( = (ny − )2 + nx )( 4 is initially at rest. Determine its impulse response and sketch roughly its magnitude response. b) The sequence x(nT) is derived from a continuous time waveform x(t). Find the relationship between the frequency response of the discrete sequence to the Fourier transform of the continuous time waveform and explain.

Q3 a) Using the DFT, IDFT find the convolution of the following two sequences x1(n) = (2,1,2,1) x2(n) = (1,2,3,4) b) Write notes on the following (i) Periodogram (ii) Use of DFT in Power Spectral Estimation

2 OR

Q3 a) Describe fully the overlap add method. b) Discuss Decimation in Frequency algorithm to find DFT and draw the signal flow graph for N = 4 c) Write a note on Chirp-Z transform and its applications

SECTION II

jw Q4 a) If y[n] = a 1x[n+k] +a 2x[n+k −1]+ a 3x[n+k −2]+ a 2x[n+k −3]+ a 1x[n+k −4], find H(e ). For what k will the frequency response be a real function of w? b) Explain the terms Linearity, Time invariance, Causality and Stability with respect to discrete time systems. Give an example of each with your explanation.

Q4 a) Discuss the properties of the discrete time sinusoid and compare them with those of the continuous time sinusoidal function. b) A system is defined by the transfer function given below −1 2 H(z) = b 0/(1 −pz ) Is it an IIR or FIR filter? Is it High-Pass, Band-Pass, or Low-Pass? What is the condition for its stability? Assuming that it is stable, Determine b 0 and p such that H(0) = 1 and the 3-dB cut-off is at w c = π/4.

Q5 a) Discuss the Notch filter, give the pole-zero diagram and explain how the notch bandwidth may be reduced. How can you convert a notch filter into a comb filter with 4 notches? b) Discuss qualitatively the differences between the Rectangular and the Kaiser windows. (Equations are not required). c) What is the alteration theorem? How is it helpful in computer-aided FIR filter design?

Q5 a) Write a note on linear phase filters having symmetric impulse response with even length. State if the following can or cannot have a symmetric impulse response with even length: 1) a Band-pass filter and 2) High-pass filter – explain why in each case. b) Explain how you will design a band-stop IIR digital filter using a standard normalised analogue low-pass filter transfer function.

Q6 Write short notes on any three

a) Compare and contrast IIR and FIR filters, and give the advantages of each. b) Graphical method of evaluating the magnitude frequency response from the pole-zero plot in z-plane. c) Harvard architecture and pipelining. d) Pre-warping of the critical frequencies in some IIR design methods.

--END-- THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV (Electronics) examination 2001 Thursday, 19 th April 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) All Questions carry equal marks. iii) Suitable assumptions may be made wherever required,. Maximum marks: 100

SECTION I

Q1 a) Define Z-transform. Find Z transform of the following : (i) 2n δ(n-3) (ii) an cos n θ U(n) b) Using the first order sections obtain the cascade realization for the following system 1 − 1 − 1( + z 1 1)( + z 1 ) H z)( = 2 4 1 − 1 − 1 − 1( − z 1 1)( − z 1 1)( − z 1 ) 2 4 8

Q1 a) (i) Explain the term “LTI System” and “Causal System”. (ii) Find the impulse response of the system described by x[n] = y[n]-(1/4)y[n-2]. Assume the system is initially at rest b) Obtain the parallel form realization for the system − 1 − 2 + z 1 + z 2 H z)( = 4  1 −  − 1 −  1+ z 1 1+ z 1 + z 2   2  2 

Q2 a) Define DFT and IDFT. Find the DFT of the sequence x[n] = {-1} n.for N=3 using its definition b) Discuss DIF algorithm to find DFT.and draw the signal flow graph for N = 8.

Q2 a) Discuss the following (i) DFT symmetry (ii) DFT Linearity (iii) DFT Magnitudes (iv) DFT Shifting and (v) DFT leakage b) Find the linear convolution of sequences 1 0 ≤ n ≤ 2  −n ≤ ≤ = = 2 0 n 3 x1 n][  and x2 n][  0 otherwise  0 otherwise using circular convolution 2

Q3 Write notes on the following (a) Applications of DSP (b) Periodogram (c) Hilbert transform (d) Walsh functions

SECTION II

Q4 a) Give the criterion you will use to test a discrete time system for its i) Linearity ii) Time invariance and causality. Apply these criteria to the system y[n] = ax[-n] and give your conclusions. b) Explain the relation between single and double pole locations in the z-plane and the impulse response.

Q4 a) How is fundamental period defined for a periodic sequence x[n]?.State whether the following are periodic and if so find the fundamental period i) x[n] = exp[0.25 πn] and ii) g[n] = cos(0.2 πn) b) How will you perform the geometric evaluation of the frequency response from the z- plane pole zero diagram? Explain with a suitable example.

Q5 a) Discuss the comb filter. How will you convert a notch filter into a comb filter with four notches? Explain by taking a suitable transfer function for the notch filter. b) Discuss the differences between i) IIR and FIR filters ii) FIR Linear phase filters with symmetric impulse responses with ODD and EVEN lengths

Q5 a) How can you convert the low pass filter into a high pass filter? Justify. Is the filter given by the function H(z)= (1/2) (1+z –1) a high pass filter or a low pass filter? If high-pass convert it to Low-pass, and if low-pass convert it to high-pass. b) Write a note on the fixed window functions used in FIR filter design.

Q6 Write short notes on any three

a) Second order low-pass filter. b) Difference between a discrete sinusoid and a continuous sinusoid c) Special features of DSP processors. d) CAD method of IIR filter design.

END THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV (Electronics) examination 2001 Tuesday, November 27 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) All Questions carry equal marks. iii) All symbols carry usual significance. Make suitable assumptions wherever required,. Maximum marks: 100

SECTION I

Q1 a) What are discrete time systems? State whether the system given by y(n) = x(−n+2) is i) static or dynamic ii) linear or nonlinear iii)Time variant or time invariant iv)causal or noncausal and v)stable or unstable. b) Determine the impulse response for the system described by the difference equation y(n)−3y(n-1) −4y(n-2) = x(n)+2x(n-1)

Q1 a) Define z-transform, and find z-transform of i) x(n) = {sin w0n}U(n) and ii) x(n) = na nU(n) b) Determine the causal signal x(n) whose z-transform X(z) is given by

1 1( + z −1 1)( − z − ) 21

Q2 a) Obtain the Direct Form II type structure for the following system: y(n) = −0.1y(n-1) + 0.2y(n-2) + 3x(n) + 3.6x(n-1) + 0.6x(n-2) b) Define DFT and IDFT. Using definition of DFT find the DFT of the sequence x(n) = {1,1,0,0,} c) Find the IDFT of Y(k) = {1,0,1,0}

Q2 a) Obtain and draw the parallel realisation for the system

2 + z −1 + z −2 H z)( = + 1 −1 − 1 −2 + 1 −1 1( 4 z 8 z 1)( 8 z )

b) Write a note on lattice realisations c) What is sectioned convolution? When and why is it used? Describe in detail Overlap save method.

Q3 a) Describe fully Divide and Conquer approach to the computation of the DFT b) Using DIF technique, obtain 4 pt DFT of the sequence x (n) = {1,2,2,1} 2

SECTION II

Q4 a) Explain the concept of periodicity and frequency in discrete time systems. Determine the if the function x(n) = Cos (30 πn / 105) is periodic. If so, find its fundamental period N f. b) Discuss the differences between IIR and FIR systems. State the advantages and disadvantages of each.

Q4 a) Discuss with appropriate diagrams the time domain behaviour corresponding to a double real pole as a function of the pole location on the real axis. b) A system is defined by the transfer function given below 1 H z)( = − 5 −1 + 1 −2 1 6 z 6 z The input is x(n) = δ(n) – (1/3) δ(n-1). Find the output y(n)

Q5 a) Discuss the digital resonator with zeros at the origin. b) What is Gibb’s phenomena? Discuss properties of the Hamming window and Blackmann window.

Q5 a) Write a note on the Comb filter. b) Briefly describe the mapping of the s-plane into the z-plane in the different methods of IIR filter design..

Q6 Write short notes on any three

a) Linear phase filters with symmetric impulse response, with N odd. b) All-pass filters. c) IIR design using Alteration Theorem and Remez algorithm. d) Frequency sampling method for FIR filter design. e) Special features in the architecture of DSP processors f) Pole and/or zero placements for obtaining stable lowest order LowPass, HighPass, BandPass, BandStop or Band Elimination, and all-pass filters.

--END-- THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV (Electronics) examination 2002 Wednesday, April 24 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) All Questions carry equal marks. iii) Suitable assumptions may be made with due justification. Maximum marks: 100

SECTION I

Q1 a) Explain the term i) Stability of a system and ii) Causality of the system For each of the following systems determine whether or not the system is i) stable ii) causal iii) linear and iv) shift invariant, where φ is the operator and x(n) is a stable bounded sequence acting as the input i) φ [x(n)] = g(n)x(n) ii) φ [x(n)] = e x(n) b) Define z-transform. Find the z-transforms of i) x(n) = (1/2) n [U(n)-U(n-5)] ii) x(n) = [Sin w(n-2)] U(n)

Q2 a) Draw the cascaded form to implement y(n)=x(n) + 2x(n-1) + x(n-2) + 1.6y(n-1) – 0.8y(n-2) b) Obtain the Direct Form I and Direct form II structures for the following system 1[2 − z −1 1][ + 2 z −1 + z −2 ] H z)( = 1( + 5.0 z −1 1)( − 9.0 z −1 + 81.0 z −2 )

Q2 a) Obtain and draw the parallel realization with only first order sections for the system 2 + z −1 + z −2 H z)( = + 1 −1 − 1 −2 + 1 −1 1( 4 z 8 z 1)( 8 z ) b) Describe first order and second order lattice structures of the digital filters. What are its applications?

Q3 a) Define DFT and IDFT. Using DFT , IDFT find the convolution of the following sequences x 1(n) = {1, 2, 3, 4} and x 2(n) = { 2, 1, 2, 1 } b) Write a note on i) application of DSP and ii) Filtering of long data sequences using DFT and IDFT OR

Q3 a) Describe fully Radix-2 DIF FFT algorithm. Write necessary equations and draw the flow diagram for N = 8 b) Determine the response of the FIR filter with impulse response h(n) = {1,2,3} to the input sequence {1,2,2,1}

Page 1 of 2 2

SECTION II

Q4 a) Discuss the properties of discrete-time sinusoids and state how these differ from continuous time sinusoids. Give illustrative examples wherever possible b) Define periodicity of a digital sequence. Comment on the periodicity of the sequence x(n) = cos (n/8) cos ( π n/8) with proof .

Q4 a) Explain the relationship between single and double pole locations in the z-plane and the impulse response. b) Prove the equivalence of the following two systems i) y(n) = 0.2y(n-1) + x(n) - 0.3x(n-1) + 0.02x(n-2) ii) y(n) = x(n) – 0.1x(n-1)

Q5 a) Write a note on the comb filter b) A filter is required to meet the following specifications: complete signal rejection at DC and 500 Hz, narrow pass-band centred at 250 Hz, a 3 dB bandwidth of 20 Hz. If sampling frequency is 1 KHz, Obtain the transfer function and difference equation by pole-zero placement.

Q5 a) Discuss i) Kaiser window and ii) Alteration Theorem b) What is frequency warping ? Discuss the necessity for pre-warping in filter design with Bilinear Transform (BZT).

Q6 Write short notes on any three

a) Fixed window functions for FIR filters. b) Differences between IIR and FIR filters c) Harvard architecture. d) S- plane to Z-plane mapping in the Bilinear Transformation method. e) Gibb’s phenomena

--END--

Page 2 of 2 SECTION - II Q-4 [16] (a) Discuss concepts of linearity, causality and stability of discrete time systems. (b) Compute the response of the system y(n) = 0.7y(n-1) – 0.12 y(n-2) +x(n-1) + x(n-2) to the input x(n) = n u(n). 3 (c) Find inverse Z transform of X z)( = ; z > 2 z − 2 (d) Explain Overlap-save method which is used for computing convolution of long sequences.

Q-5 (a) The LTI system is characterized by the system function 3( − 4z −1 ) H z)( = 1( − 5.3 z −1 + 5.1 z −2 ) Specify the ROC of H(z) and determine h(n) for the following conditions (i) The system is stable (ii) The system is causal and (iii) The system is purely anticausal [8] (b) Determine the impulse response h[n] of the discrete time LTI system described by the difference equation: y[n] + 0.1y[n -1] – 0.06 y[n -2] = x[n] – 2x[n -1] without computing Z-transform. Let y[-1] = y[-2] = 0. [9] OR Q-5 (a) The transfer function of a causal discrete time LTI system is given by 6 − z −1 2 H z)( = + ; determine its impulse response h[n]. Also determine the response 1 + 0.5z −1 1 − 0.4 z −1 of this system for all values of n for the input given by x[n] = 1.2(-0.2) n µ[n] – 0.2(0.3) n µ[n] [9] (b) Using DFT – IDFT find the response of the system having impulse response [1, 2, 3] to the input [1, 2, 2, 1] [8]

Q-6 (a) Obtain direct form and cascade realizations for the following transfer function of an FIR filter

 1 − 3 −  1 − 1 −  H z)( = 1− z 1 + z 2 1− z 1 − z 2  [9]  4 8  8 2  (b) Perform circular convolution on the sequences: p[n] = { 2,1,2,1 } & q[n] = { 1,2,3,4 } [8] ↑ ↑ OR Q-6 (a) Explain Radix-2 DIF FFT algorithm and compute 8 –point DFT of sequence x[n] = { 0,1,2,3,4,5,6,7 } using the same algorithm. [9] (b) Develop the lattice structure to realize second order FIR filter giving all necessary details. [8]

Slip No: 02 Exam Seat No: ______

SECTION – II

Q- 4 [16] (a) (b)

Q- 5 [16] (a) (b) OR

Q- 5 (a) (b)

Q- 6 [18] (a) (b)

END

THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV(Electronics) Examination 2003 Tuesday, 8 th April 3 PM to 6 PM Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) Suitable assumptions may be made wherever required. Maximum marks: 100

SECTION I

Q1 a) Find the Z-transform of the following and give their ROC i) x(n) = -αn U(-n-1) ii) x(n) = cos (n π/2)U(n) (8) b) Determine the causal signals x(n) having the Z-Transforms 1 i) X z)( = 1( − 2z −1 1)( − z − ) 21 1− 1 z −1 ii) X z)( = 2 1− z −1 + z −2 (8) OR

Q1 a) Define (i) FIR, (ii) IIR and (iii) LTI Systems. The LTI system described by 1 y n)( = y(n − )2 + x n)( 4 is initially at rest. Determine its impulse response and sketch roughly its magnitude response. (8) b) Find circular convolution of the two sequences x 1(n) = { 2, 1, 2, 1 } and x2(n) = {1, 2, 3, 4} ↑ ↑ (8)

Q2 a) Obtain the cascade structure for the system 1+ 1 z −1 H z)( = 2 ()− −1 + 1 −2 ()− −1 + 1 −2 1 z 4 z 1 z 2 z (8) b) Write a note on lattice realisations. (8)

Q2 a) Obtain the Direct Form II type structure for the following system: y(n) = −0.1y(n-1) + 0.2y(n-2) + 3x(n) + 3.6x(n-1) + 0.6x(n-2) (8) b) Obtain and draw the parallel realisation for the system 2 + z −1 + z −2 H z)( = + 1 −1 − 1 −2 + 1 −1 1( 4 z 8 z 1)( 8 z ) (8)

Page 1 of 2 2 Q3 a) Discuss the following (i) DFT symmetry (ii) DFT Linearity (iii) DFT Magnitudes (iv) DFT Shifting and (v) DFT leakage (10) b) Using DIF technique, obtain 4 pt DFT of the sequence x (n) = {1,2,2,1} (8)

SECTION II

Q4 a) Explain the concepts of sampling and aliasing in sampled systems (8) b) A digital filter operating with sampling frequency of 250 Hz has a pole zero diagram with complex conjugate poles with r = 0.9 at 45 ° to real axis, and zeros at ±1 in the z-plane. State (i) Type of filter it represents (ii) Frequencies where its response is minimum amplitude (iii) frequencies where its amplitude response peaks (iv) its bandwidth, if any. (8)

−1 2 Q4 a) A system is defined by the transfer function H(z) = b 0/(1 −pz ) . Is it an IIR or FIR filter? Is it High-Pass, Band-Pass, or Low-Pass? What is the condition for its stability? Assuming that it is stable, Determine b 0 and p such that H(0) = 1 and the 3-dB cut-off is at w c = π/4. (8) b) Sketch the time domain behaviour of (i) a system having a double pole at +1 and (ii) a system having complex conjugate poles outside unit circle at angles ±w rad. (8)

Q5 a) Comment on frequency warping and s-plane to z plane mapping in the bilinear transform method of IIR filter design. (8) b) Discuss the CAD method of filter design. (8)

Q5 a) Discuss the Rectangular and Hamming windows. (8) b) Write a note on type I linear phase filters. (8)

Q6 Write notes on any three (18)

a) Differences between Harvard and Neumann Architecture. b) Impulse Invariance transformation. c) Frequency Sampling method. d) Differences between FIR and IIR filters. e) TMS320C20 architecture

--END--

Page 2 of 2 THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV(Electronics) Examination 2003 Wednesday, 19th November 3 PM to 6 PM Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) Suitable assumptions may be made wherever required. iii)Numbers to the right of the questions indicate maximum marks. Maximum marks: 100

SECTION I

Q1 A discrete time system can be (1) static or dynamic (2) Linear or nonlinear (3) Time variant or time invariant (4) causal or noncausal (5) Stable or unstable. Examine the following systems with respect to the properties above

i) y(n) = cos [x(n)] ii) y(n) = x(n) cos(w on) iii) y(n) = x(-n+2) n+1 iv) y(n) = ∑ x(k) (16) k ∞−=

Q2 a) Determine the Z-transform of the finite duration signal 1. x(n) = {1,2,5,7,0,1}

2. x(n) = (1/2) n n ≥ 5 = 0 n ≤ 4

b) Obtain the Direct form II and parallel structures for the system having 1(2 − z −1 1)( + 2z −1 + z −2 ) H z)( = (16) 1( + 5.0 z −1 1)( − 9.0 z −1 + 81.0 z −2 )

Q2 a) Determine the causal signal x(n) if its Z-transform X(z) is given by 1+ 3z −1 1. X z)( = 1+ 3z −1 + 2z −2 1+ 2z −2 2. X z)( = 1+ z −2 b) Obtain the cascade and the parallel form for the system 3 1 1 ny )( = (ny − )1 − (ny − )2 + nx )( + (nx − )1 4 8 3 (16)

Q3 a) Define DFT and IDFT. Describe fully divide and conquer approach to the computation of the DFT b) Determine the circular convolution of the sequences x1(n) = { 1,2,3,1} and x2(n) = {4,3,2,,2}

Page 1 of 2 2 c) Compute the 8 point DFT of the sequence x(n) ={ ½ , ½, ½ , ½ , 0, 0 ,0 0}. Use radix-2 decimation in time algorithm, give signal flow graph and keep track of all the intermediate quantities by putting them on the diagram. (18)

Q3 a) Discuss fully overlap and save method for filtering of long data sequences. b) Using DFT and IDFT determine the response of the FIR filter with impulse response h(n) = {1,2,3} to the input sequence x(n) = {1,2,2,1}.

c) Using circular convolution find the response to be determined in Q3(b) above (18)

SECTION II

Q4 a) Explain the concepts of periodicity and frequency in sampled systems, and compare these with these concepts in case of continuous time systems. (8) b) A digital filter is characterised by the following properties (1) It is high pass and has one pole and one zero (2) The pole is at r = 0.9 from the origin of z-plane (3)constant signals do not pass through the filter. Plot the pole-zero pattern and determine the system function H(z) with normalisation such that H( π)=1 and find the difference equation in the time domain (8)

Q4 a) Determine the coefficients of the filter y(n) = ax(n) + bx(n-1) + cx(n-2) such that it rejects completely a frequency component at w 0 = 2 π/3, and H(0)  =1 (8) b) Sketch the time domain behaviour of (i) a system having a single pole at -1 and (ii) a system having complex conjugate poles on unit circle at angles ±45 degrees (iii) a system that has double poles inside unit circle on negative real axis (iv) double complex conjugate poles on unit circle at ± 45 degrees (8)

Q5 a) Write a note on the stability triangle (8) b) Discuss linear phase filters of types I and III. (8)

Q5 a) Describe the Blackman and Hamming windows and their properties. (8) b) Discuss the properties of All pass filters. What are maximum phase and minimum phase filters? Show how can a mixed phase filter can be considered as a cascade of a minimum phase filter and an all pass filter. (8)

Q6 Write notes on any three (18)

a) Digital resonator. b) Schur Kohn stability test. c) Frequency Sampling method. d) Design procedure for IIR filters.

--END--

Page 2 of 2 THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV (Electronics) examination 2004 Friday, April 16 3 PM to 6 PM

Digital Signal Processing

SECTION I

Q1 a) What are discrete time systems? State whether the system given by y(n) = x(−n+2) is i) static or dynamic ii) linear or nonlinear iii)Time variant or time invariant iv)causal or non-causal and v)stable or unstable. b) Determine the causal signal x(n) whose z-transform X(z) is given by

1 1( + z −1 1)( − z − ) 21

Q1 a) Define z-transform, and find z-transform of i) x(n) = {sin w0n}U(n) and ii) x(n) = na nU(n) b) Determine the impulse response for the system described by the difference equation y(n) −3y(n-1) −4y(n-2) = x(n)+2x(n-1)

Q2 a) Obtain the Direct Form II type structure for the following system: y(n) = −0.1y(n-1) + 0.2y(n-2) + 3x(n) + 3.6x(n-1) + 0.6x(n-2) b) Write a note on lattice realisations c) Find the IDFT of Y(k) = {1,0,1,0}

Q2 a) Obtain and draw the parallel realisation for the system

2 + z −1 + z −2 H z)( = + 1 −1 − 1 −2 + 1 −1 1( 4 z 8 z 1)( 8 z )

b) Define DFT and IDFT. Using definition of DFT find the DFT of the sequence x(n) = {1,1,0,0,} c) What is sectioned convolution? When and why is it used? Describe in detail Overlap save method.

Q3 a) Discuss the following (i) DFT symmetry (ii) DFT Linearity (iii) DFT Magnitudes (iv) DFT Shifting and (v) DFT leakage b) Using DFT and IDFT determine the response of the FIR filter with impulse response h(n) = { 1, 2, 3) to the input sequence x(n) = {1, 2, 2, 1} ↑ ↑

SECTION II

Q4 a) Explain the concept of periodicity in discrete time systems. Determine the if the function x(n) = Cos (30 πn / 105) is periodic. If so, find its fundamental period N f. b) Discuss the differences between IIR and FIR systems. State the advantages and disadvantages of each.

Q4 a) Discuss with appropriate diagrams the time domain behaviour corresponding to a pair of complex conjugate poles for poles within, on and outside the unit circle. b) A system is defined by the transfer function given below 1 H z)( = − 5 −1 + 1 −2 1 6 z 6 z The input is x(n) = δ(n) – (1/3) δ(n-1). Find the output y(n)

Q5 a) How will you perform the geometric evaluation of the frequency response from the z-plane pole zero diagram? Explain with a suitable example. b) Discuss properties of the rectangular window and Hanning window.

Q5 a) Discuss the differences between FIR Linear phase filters with symmetric impulse responses with ODD and EVEN lengths. b) Write a note on the stability triangle

Q6 Write short notes on any three

a) Linear phase filters with anti symmetric impulse response, with N odd. b) The comb filter. c) Computer aided method for filter design. d) Special features in the architecture of DSP processors .

--END-- THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA First Semester of BE IV (Electronics) examination 2004 Saturday, November 27 3 PM to 6 PM

Digital Signal Processing

SECTION I

Q1 a) Define z-transform, and find z-transform of i) x(n) = {sin w0n}U(n) and ii) x(n) = na nU(n) . b) Define LTI system. Find the frequency response of the system y(n) = x(n) + 0.6 y(n-1) given y(-1) = 0. Sketch the magnitude and phase response

Q1 a) Define and explain the terms (i) cross correlation and (ii) autocorrelation of discrete sequences. Also obtain cross correlation of the following sequences x1(n) = {….,0,0,2,-1,3,7,1,2,-3,0,0,….} and x 2(n) = {….,0,0,1,-1,2,-2,4,1,-2,5,0,0,….} ↑ ↑ b) Using partial fractions method find the inverse z transform of 7z −2 + 11 z −1 + 3 zx )( = 2 − 1 −1 + −1 + −2 1( 2 z 1)( 2z 4z )

Q2 a) Obtain the Direct Form realizations of : 3 1 1 ny )( = (ny − )1 − (ny − )2 + nx )( + (nx − )1 4 8 3 What are the drawbacks of direct forms? b) Realize the transfer function given below by parallel form 1− 3 z −1 + 7 z −2 H z)( = 2 10 z 2 + 9z + 28 + 36 z −1 +16 z −2

Q3 a) Define DFT and IDFT. Find the 6 point DFT of the sequence x(n) = {3,2,1,0,1,2} b) Describe fully the DIF algorithm and draw the signal flow diagram for N = 8

Q3 a) Using DFT and IDFT determine the response of the FIR filter with impulse response h(n) = { 1, 2, 3) to the input sequence x(n) = {1, 2, 2, 1} ↑ ↑ b) Describe the different methods of filtering of long data sequences based on DFT. Also using any one of the methods you describe find output sequence if h(n) = {1,0,1} and x(n) = {1,3,2,-3,0,2,-1,0,-2,3,-2,1,…} ↑ ↑ 2

SECTION II

Q4 a) Determine if the following sequences are periodic, and if periodic find the fundamental period: (i) 3sin (0.8 πn) – 4cos (0.1 πn) (ii) cos (n/8) cos (πn/8) .. b) Show that for two sequences x(n) and y(n) with finite energy E x and E y, (i) │rxy (l) │ ≤ √(E x Ey) (ii) │rxx (l) │ ≤ rxx (0) where l is some time shift.

Q4 a) A system has zeros on unit circle at ± 60° and poles on real axis at ± 0.9. If sampling frequency is 500 s/sec, sketch the approximate frequency response of the system and indicate the frequencies (in Hz) at which the response is maximum and minimum and give its approximate bandwidth. Use a normalised y-axis. b) Describe the Schur-Cohn test for stability. . Q5 a) A FIR filter has a symmetric impulse response with (N+1) odd . Show that it has a linear phase characteristic and comment on its possible and prohibited zero positions with proof. What are the possible filter types (HP,LP,BP,BE) to which it may belong? . b) Discuss properties of the Blackman window and Kaiser window.

Q5 a) Write a note on the digital resonator. b) A filter is required to reject a signal in a narrow band centred at 250 Hz. and to pass the signal without attenuation at 0 and 500 Hz. The bandwidth of the filter is to be 20 Hz. If sampling frequency is 1000 Hz, Obtain the transfer function. Would this design change if ALL the frequency and bandwidth values given above are doubled?

Q6 Write short notes on any three

a) Frequency sampling method for FIR filter design. b) Bilinear transformation. c) Frequency band transformations. d) Block diagram of the TMS 320C20 DSP chip .

--END-- THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA First Semester of BE IV (Electronics) examination 2005 Saturday, November 26 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) All Questions unless otherwise defined carry equal marks. iii) All symbols carry usual significance. Make suitable assumptions wherever required,. Maximum marks: 100

SECTION I

Q1 a) (i) Discuss the terms “causality” and “Stability” with reference to discrete time systems. (ii) State and prove the necessary and sufficient condition for a relaxed LTI system to be BIBO stable. (iii) Determine whether the system y(n) = 3x(n) + 5 is stable? Causal? Linear? Time Variant? (9) . b) Define z-transform. (i) Find Z- transform of n(-1) n U(n). (ii) Determine the inverse Z-transform of X(z) = (1 + 2z -1 ) / (1- 2z -1 + z -2) using long division if x(n) is anti-causal. (9)

Q1 a) Using Z-transform find the step response of the system which is described by difference equation y(n) = 0.7y(n-1) – 0.1y(n-2) + 2x(n) - x(n-2). (6) b) Impulse response of the system is given by h(n) = {--- 0,0,1,2,1,-1,0,0,---}.

Determine the response of the system to the input signal x(n) = { 1,2,3,1,0,0,0,---}. (6)

c) Determine IZT of x(z) = (1 + z -1) / (1 – z -1 + 0.5z -2) having ROC : │ z │ > 1. (6)

Q2 a) Obtain the Direct Form I and Direct Form II for the following system giving their limitations if any: y(n) = 0.1y(n – 1) + 0.72y(n – 2) + 0.7x(n) – 0.252x(n – 2) (8) b) Obtain the cascade structure (of 1 st order systems) described by system function 1+ z −1 H z)( = (8) 1 − 1 − 1− z 1 − z 2 2 4

Q2 a) Realise the transfer function 7.0 z −2 − 5.1 z −1 +1 H z)( = z 2 + 9z + 28 + 36 z −1 +16 z −2 in parallel form . (8) b) What are “lattice filters”? Give their applications. Also describe fully first order and second order lattice filters. (8)

2 Q3 a) Define DFT and IDFT. Obtain DFT of the data sequence {1,0,1,0} and check the validity of your answer by calculating the IDFT. (8) b) Discuss fully “Divide and Conquer” approach for computation of DFT and write any two algorithms to compute DFT using this approach. (8)

SECTION II

Q4 a) Determine if the following sequences are periodic, and if periodic find the fundamental period: (i) sin ( π (62n /10)) (ii) 2 exp j [(n/6) – π] (8) .. b) Explain the differences between frequency and period of the analog signal and the discrete time signal giving appropriate examples. (8)

Q4 a) A system has poles at ± 60° at a distance 0.9 from the origin and zeros on real axis at ± 1. If sampling frequency is 200 samples/sec, sketch the approximate frequency response of the system and indicate the frequencies (in Hz) at which the response is maximum and minimum and give its approximate bandwidth. Use a normalised y-axis. (8) b) Explain significance of the stability triangle and its various boundaries and regions. (8) . Q5 a) A FIR filter has a symmetric impulse response with (N+1) even. Show that it has a linear phase characteristic and comment on its possible and / or prohibited zero positions with proof. What are the possible filter types (HP, LP, BP, BE) to which it may belong? (8) . b) Explain with proof and pole zero diagrams how you will convert a notch filter to a comb filter. (8)

Q5 a) Determine the coefficients of the filter y(n) = ax(n) + bx(n -1) + cx(n - 2) Such that it rejects completely a frequency component at w 0 = 2 π/3, and has H(0)  =1. (8) b) Describe the factors affecting selection of IIR or FIR filters and list the stepwise procedure for design of a filter to meet a given set of specifications. (8)

Q6 Write short notes on any three giving appropriate figures and/or examples: a) Frequency sampling method. b) Shur-Cohn test for stability. c) Time domain behaviour of signals corresponding to double real pole d) Type 3 linear phase filters (18) .

--END-- THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA First Semester of BE IV (Electronics) examination 2006 Thursday, November 16 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) All Questions unless otherwise specified carry equal marks. iii) All symbols carry usual significance. Make suitable assumptions wherever required,. Maximum marks: 100

SECTION I

Q1 a) What are LTT discrete systems? Give and explain their properties. b) “The output sequence of the discrete LTI system is the product of the input sequence and impulse response of the system.” Make your comments on the statement giving justification c) Using the input-output relation in time domain find the output of the discrete LTI system having impulse response {3 δ(n-2)- 1.5 δ(n+1)}to the input 1.5 δ(n+3) + 2δ(n-1)- 3δ(n-2)- 3δ(n+5). . d) State the necessary and sufficient condition for BIBO stability of a discrete LTI system. α A discrete LTI system is ccharacterized by the difference equation y(n) = x(n)+e y(n-1). Check this system for BIBO stability.

Q2 a) Define ‘Z’ transform. Also find the Z-transform of (n-2)sin ω(n-2).Give all the steps. z + 2 b) Find IZT of if ROC is ½

1+ 1 z −1 Q2 a) Using the properties of Z transform find IZT of 2 − 1 −1 1 2 z = 1 [(1 )n + (− 1 )n ] b) Impulse response of the system is given by h n)( 2 2 4 U n)( . Find its poles and zeros in the Z plane. How will you obtain its frequency response? Explain.  z 3( z − )4  c) The LTI system is characterised by the system function H z)( =   .  z 2 − 5.3 z + 5.1  Specify the Roc of H(z) and determine h(n) for the following conditions (i) The system is causal (ii) The system is anti-causal and (iii) The system is stable.

Q3 a) Obtain the Direct form I and Parallel structures for the system described by 1(2 − z −1 1)( + 2z −1 + z −2 ) H z)( = 1( + 5.0 z −1 1)( − 9.0 z −1 + 81.0 z −2 ) b) Using the expressions for DFT/IDFT compute (i) 4-point DFT of (1,1,2,2) and (ii) 4 point IDFT of (60,-j4,0,+j4) c) Discuss fully Overlap add method to obtain response of the filter to a very very long input sequence

Q3 a) Obtain the lattice structure for the system having input output relation given by y(n) = x(n)-0.6y(n-1)-0.3y(n-2). Derive the relation you use. b) Discuss fully Radix-2 DIF FFT algorithm. Write necessary equationsand draw the flow diagram for N = 4 c) Using the Radix-2 DIT algorithm compute 4 point DFT of (1,4,3,2) d) Obtain the cascade structure for the system described by y(n) = y(n-1)- (1/2)y(n-2)+x(n)-x(n-1)+x(n-2)

SECTION II

Q4 a) Explain the concept of periodicity in a discrete time signal. Is a discrete time sinusoid having a frequency f 0 always periodic? Explain. State whether the following are periodic, and if so find the fundamental period: (i) cos ( π (30n /105)) and (ii) cos 3n (8) .. b) Explain the difference between (i) linear and nonlinear systems (ii) Time variant and time invariant systems and (iii)Concept of frequency in analog and discrete time signals. Give suitable examples in each case. (8)

Q4 a) State and explain with reasons whether the following systems are (i) linear, (ii)time invariant (iii)causal and (iv) stable: (i) y(n) = |x(n)| and (ii) y(n) = cos [x(n)] (8) b) Explain with a suitable example the geometrical method of determining the approximate frequency response given the pole-zero diagram of the system in z-plane.(8) . Q5 a) A filter has the system function of the form H(z) = K / (1-az -1)2. If it satisfies the conditions that magnitude response at 0 frequency is 1 and 3-db cutoff is at angular digital frequency of π/4, find the values of K and a (8) . b) Describe the factors affecting selection of IIR or FIR filters. Define the parameters usually given as specification of the filter and list the stepwise procedure for design of a FIR filter to meet a given set of specifications. (8)

Q5 a) Determine the coefficients of the filter y(n) = ax(n) + bx(n -1) + cx(n - 2) Such that it rejects completely a frequency component at w 0 = 2 π/3, and has H(0)  =1. (8) b) A FIR filter has an anti-symmetric impulse response with N = 5. Show that it has a linear phase characteristic and find the group delay and write its amplitude function. Comment on prohibited zero positions if any, with proof. (8)

Q6 Write short notes on any three : a) Blackman and Kaiser windows b) Frequency Sampling method of FIR filter design c) Digital resonator d) Classification of signals (18) .

--END--

THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV (Electronics) examination 2007 Thursday, 22nd November 3 PM to 6 PM

Digital Signal Processing

Notes: - (1) Answer the two sections in separate answer books. (2) Symbols & notations carry their usual meanings unless otherwise specified. (3) You may assume additional data if required, giving proper justification. . Maximum Marks: 100

SECTION – I

Q-1 [16] (a) Comment on linearity, causality and stability of the system given as: y[n] = ax[-n] ; a is a non-zero constant. (b) Determine step response of the causal system described by the difference equation y[n]=y[n-1]+x[n]. Is this system stable? +NK 2/ K (c) Explain the term Twiddle Factor and prove that W N = - W N ; also draw DIT FFT Butterfly. (d) What is an ROC? Comment on the ROC of a two –sided sequence.

Q-2 (a) Enumerate properties of Z transform and using Differentiation property find inverse Z transform of − X(z) = log (1 + a z 1 ) ; IzI>IaI [8] (b) Determine the impulse response h[n] of the discrete time LTI system described by the difference equation: y[n] + 0.1y[n -1] – 0.06 y[n -2] = x[n] – 2x[n -1] without computing Z-transform. Let y[-1] = y[-2] = 0. [9]

OR Q-2 6 − z −1 2 (a) The transfer function of a causal discrete time LTI system is given by H z)( = + ; 1 + 0.5 z −1 1 − 0.4 z −1 determine its impulse response h[n]. Also determine the response of this system for all values of n for the input given by x[n] = 1.2(-0.2) n µ[n] – 0.2(0.3) n µ[n] [9] (b) Explain the term eigenfunction . Obtain inverse DTFT h[n] of w w − j H (e jw ) = 3( + 2cos w + 4cos 2w)cos e 2 [8] 2

Q-3 − (a) Realize the transfer function H(z) = (1 - 0.7 z 1 ) 5 in the following forms : [9] (i) Two different direct forms, (ii) cascade of one second order and one third order section (b) Perform circular convolution on the sequences: p[n] = {2, 1, 2, 1} & q[n] = { 1, 2, 3, 4 } [8] ↑ ↑

Q-3 (a) What is interlaced decomposition? Explain Radix-2 DIT FFT algorithm using necessary equations. Draw flow graph for computing 8 –point DFT using DIT FFT algorithm. [9] (b) Can the methods of filtering long data sequences be applied to IIR filters? Justify your answer and explain Overlap-save method. [8]

SECTION – II

Q4 a) Give the criterion you will use to test a discrete time system for its i) Linearity, ii)Time invariance. Apply these criteria to the system y[n] = ax[-n] and give your conclusions. [8] b) Explain the behaviour of the impulse response with respect to single and double pole locations in the z-plane. [8]

Q4 a) How would you define the fundamental period for (a) a periodic sequence x[n] and (b) a periodic function x(t)? Explain. Also discuss whether the following are periodic and if so find the fundamental period i) x[n] = exp[0.25 πn] and ii) g[n] = cos(0.2 πn) [8] b) How will you find the approximate frequency response from the z-plane pole zero diagram using geometrical methods? Explain with a suitable example for amplitude and phase plots. [8]

Q5 a) Discuss the notch filter. How will you convert a notch filter into a comb filter with four notches? Explain by taking a suitable transfer function for the notch filter. [8] b) Discuss the differences between i) IIR and FIR filters ii) FIR Linear phase filters with symmetric impulse responses with ODD and EVEN lengths. [8]

Q5 a) How can you convert the low pass filter into a high pass filter? Justify. Is the filter given by the function H(z)= (1/2) (1+z –1) a high pass filter or a low pass filter? If high-pass convert it to Low-pass, and if low-pass convert it to high-pass. [8] b) Write a note on the fixed window functions used in FIR filter design. [8]

Q6 Write short notes on any three [18]

a) Stepwise Procedures for design of filters. b) Difference between a discrete sinusoid and a continuous sinusoid c) Schur-Cohn Stability Test d) The Alteration Theorem and its use in FIR design.

END

THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA First Semester of BE IV (Electronics) examination 2008 3 PM to 6 PM

Digital Signal Processing

SECTION I

Q2 a) Define ‘Z’ transform. Also find the Z-transform of (n-2)sin ω(n-2).Give all the steps. z + 2 b) Find IZT of if ROC is ½

SECTION II

Q4 a) An analog signal is given by the expression = π + π + π xa t)( 3cos 2000 t sin5 6000 t 10 cos 12000 t This signal is sampled with a sampling frequency F s = 5000 samples/sec (i) Obtain the expression describing the resulting discrete time signal x(n) after sampling. (ii) If x(n) is passed through an ideal Low pass reconstruction filter having cut off frequency F c = 12 KHz, find the expression for the recovered analog signal y a(t). (8) b) Determine without factorising which of the following denominator polynomials have roots inside the Unit circle in z-plane. i) D z)( = 2 + 4.0 z −1 − 8.2 z −2 ii) D z)( = 1+ 814.2 z −1 + 81.0 z −2 (8) OR Q4 a) Define periodicity in discrete time signals. Determine whether or not the following signals are periodic. If so, determine the fundamental period.  n   πn  (i) nx )( = cos  cos    8   8   πn   πn   πn π  (ii) nx )( = cos   − sin   + 3cos  +  (8)  2   8   4 3  −1 = az b) The z-transform of a causal real signal is given by X z)( 2 ()1− az −1 Obtain the expression for x(n) and Sketch approximately the variations of discrete signal x(n) vs. n and pole zero plots for following conditions : (a) 0 < a < 1, (b) a = -1 and (c) a < -1 Exact calculations are not expected. (8) . K 1( − z −2 ) Q5 a) If the system transfer function of a digital filter is given by H z)( = 1+ 0.7z −2 Find: (i) value of K, (ii) the digital frequency at which amplitude frequency response is maximum, (iii) the digital frequency at which amplitude frequency response is minimum, and (iv) the amplitude frequency response at w = 4 π/9. (8) . b) Using a tabular form, compare the important parameters of (i) Rectangular, (ii) Hamming and (iii) Blackman windows. (8)

OR Q5 a) Obtain the expression for the Amplitude Frequency response and Group delay if h(n) 6 is a symmetrical impulse response, and H(z) is given by H z)( = ∑ h )( zn −n (8) n=0 b) Write a note on (i) All-Pass filter and (ii) Comb filter (8)

Q6 Write short notes on any three : a) Digital Oscillator b) Frequency Sampling method of FIR filter design c) Butterworth and Chebychev filters. d) Conversion of analog filter to digital using approximation of derivatives. (18) .

THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA First Semester of BE IV (Electronics) examination 2009 Date: 3 PM to 6 PM

Digital Signal Processing

SECTION I

Q1 a) . b c) . . d).

Q2 a). b) c).

Q2 a) b). c).

Q3 a) b) c) OR

Q3 a). b) c) d)

SECTION II

Q4 a) Explain the concept of frequency in Discrete Time Systems. Determine the if the function x(n) = Cos (30 πn / 105) is periodic. If so, find its fundamental period N f. (8) b) Discuss with appropriate diagrams the time domain behaviour corresponding to a pair of complex conjugate poles for poles within, on and outside the unit circle in z-plane. (8)

Q4 a) Determine the coefficients of the filter y(n) = ax(n) + bx(n -1) + cx(n - 2) such that it rejects completely a frequency component at w 0 = 2 π/3, and has H(0)  =1. (8) b) Discuss the differences between, and the advantages and disadvantages of IIR and FIR systems. List the steps in the design of an IIR filter to meet given specifications. (8) . Q5 a) How will you perform the geometric evaluation of the frequency response from the z-plane pole zero diagrams? Explain with a suitable example. (8) . b) Using a tabular form, compare the important parameters of (i) Hamming, and (ii) Blackman (iii) Rectangular windows. (8)

Q5 a) Obtain the expression for the Amplitude Frequency response and Group delay if h(n) is a anti-symmetrical impulse response, and N = 6 (8) b) Explain significance of the stability triangle and its various boundaries and regions. (8)

Q6 Write short notes on any three : a) Digital resonator b) Kaiser window c) Computer aided design of FIR filters. d) Linear phase filters with symmetric impulse response, with N odd. (18) .

****** THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA First Semester of BE IV (Electronics) examination 2009 3 PM to 6 PM

Digital Signal Processing

SECTION I

Determine the response of the system to the input signal x(n) = { 1,2,3,1,0,0,0,---}. (6)

c) Determine IZT of x(z) = (1 + z -1) / (1 – z -1 + 0.5z -2) having ROC : │ z │ > 1. (6)

SECTION II

Q4 a) Determine if the following sequences are periodic, and if periodic find the fundamental period: (i) sin ( π (62n /10)) (ii) 2 exp j [(n/6) – π] (8) .. b) Explain what is aliasing in the digital frequency domain, giving a suitable example. How is it avoided? (8)

Q4 a) A system has poles at ± 60° at a distance 0.9 from the origin and zeros on real axis at ± 1. If sampling frequency is 200 samples/sec, sketch the approximate frequency response of the system and indicate the frequencies (in Hz) at which the response is maximum and minimum and give its approximate bandwidth. Use a normalised y-axis. (8) b) Explain the procedure for the Schur-Cohen method for stability testing. (8) . Q5 a) A FIR filter has an anti-symmetric impulse response with (N+1) = 6. Show that it has a linear phase characteristic. Comment on its possible and / or prohibited zero positions with proof. What are the possible filter types (HP, LP, BP, BE) to which it may belong? (8) . b) Briefly describe the Bilinear Transform method of IIR filter design. (8)

Q5 a) A filter is required to meet the following specifications: complete signal rejection at DC and 500 Hz, narrow pass-band centred at 250 Hz with a 3 dB bandwidth of 20 Hz. If sampling frequency is 1 KHz, Obtain the pole-zero locations and hence the transfer function and difference equation. (8) b) Discuss i) Kaiser window and ii) Notch filter (8)

Q6 Discuss briefly any three : a) Frequency Sampling method for FIR filter design. b) Frequency band transformations. c) Time domain behaviour of signals corresponding to (i) single pole at (–1/a, 0), (ii) double pole at (+a,0), (iii) single poles at (0,+ j), (0,- j) where 0 < a < 1 d) All-Pass filter and its pole-zero locations. (18) .

--END-- THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV (Electronics) examination 1999 Friday, November 19 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) All Questions carry equal marks. iii) Suitable assumptions may be made with due justification. Maximum marks: 100

SECTION I

Q1 a) Answer the following i) What is DSP? What are its advantages and disadvantages? ii) List the application areas of DSP and give the key DSP operations. b) Describe fully the use of DSP for “spectrum analysis”

Q2 a) Define Z-transform and give its properties. Find the Z-transform of

i) x(n) = { 2, 4, 5, 7, 0, 1} ↑ ii) x(n) = [sin ωon] U(n)

b) Obtain the Direct Form I and Cascade structures for the following system 1[2 − z −1 1][ + 2 z −1 + z −2 ] H z)( = 1( + 5.0 z −1 1)( − 9.0 z −1 + 81.0 z −2 )

Q2 a) Find the causal x[n] for each of the following 1 i) X z)( = 1( + z −1 1)( − z − ) 21 1 ii) X z)( = − −1 + 1 −2 1 z 2 z b) Define LTI system The LTI system, initially at rest, is described by the difference equation 1 ny )( = (ny − )2 + nx )( 4 What is the impulse response of this system? Determine the parallel form realisation of this system.

Q3 a) Find circular convolution of the two sequences x 1(n) = { 2, 1, 2, 1 } and x2(n) = {1, 2, 3, 4} ↑ ↑ b) Using DFT and IDFT determine the response of the FIR filter with impulse response h(n) = { 1, 2, 3) to the input sequence x(n) = {1, 2, 2, 1} ↑ ↑ OR

2 Q3 a) Discuss Decimation in time algorithm to find DFT and draw the signal flow graph for N = 4. b) Compute DFT of x(n) = 1 , 0 ≤ n ≤ 3 using the signal flow graph drawn in Q3(a) above. c) What is overlap save method? Where is it used? Why? Describe.

SECTION II

Q4 a) Discuss the properties of discrete-time sinusoids and state how these differ from continuous time sinusoids. Give illustrative examples wherever possible b) A causal system is given by

y[n] = ay[n-1] + bx[n] with n ≥ 0.

Express y[n] in terms of input samples and y[ −1], the initial condition. Is the system time-invariant if y[ −1] = 1? Is it linear if y[ −1] = 0?

Q4 a) Explain the relationship between single and double pole locations in the z-plane and the impulse response. b) If g[n] is an even sequence and h[n] is an odd sequence, state with proof which of the following is odd and which is even

i) v[n] = h[n]h[n] , ii) x[n] = g[n]h[n].

Q5 a) Explain the frequency sampling method of filter design b) A filter is required to meet the following specifications: complete signal rejection at DC and 250 Hz, narrow pass-band centred at 125 Hz, a 3 dB bandwidth of 10 Hz. If sampling frequency is 500 Hz, Obtain the transfer function and difference equation by pole-zero placement.

Q5 a) Discuss the computer-aided design of FIR filters using the Remez algorithm. b) Discuss the necessity for pre-warping in IIR filter design with Bilinear Transform (BZT). In a digital BP filter the pass-band is to be from 200 to 300 Hz , The sampling frequency 2 is 2000Hz , find the pre-warped edge frequencies and the parameters W and w o required for the LP to BP band transformation

Q6 Write short notes on any three

a) Commonly used window functions for FIR filters. b) Notch and Comb filters. c) Pipelining in DSP processors. d) Harvard Architecture.

--END--

THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA

First Semester of BE IV (Electronics) examination 2000 Thursday, April 19 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) All Questions carry equal marks. iii) Suitable assumptions may be made with due justification. Maximum marks: 100

SECTION I

Q1 a) Define LTI system and describe its properties. b) For the system described by the differential equation y[n]-x[n] = 2x[n-1] + 4y[n-2] +3y[n-1] determine the response y[n]; n ≥ 0 if the input sequence x[n] is 4 n U[n]. Use y[-1] = 5, y[-2] = 0 c) Using the circular convolution determine the response of the filter having impulse response h[n] = [1,2,3] to the input sequence x[n] = [1,2,2,1]

Q2 a) Define z-transform and compare it with Laplace transform. The sequence x[n] is given by = (− 1 )n − (1 )n − − nx ][ 3 U n][ 2 U[ n ]1 Find X(z), its ROC and sketch it. b) For the causal LTI system whose system function is 1+ 1 z −1 H z)( = 5 ()− 1 −1 + 1 −2 ()+ 1 −1 1 2 z 3 z 1 4 z Obtain (i) Cascade form using first and second order direct form II sections and (ii) parallel form using first and second order direct form II sections

Q2 a) The sequence x(n) has z-transform 1+ 2z −1 + 2z −2 X z)( = − 3 −1 + 1 −2 1 2 z 2 z With ROC  z  >1. Find x[n] b) Obtain the direct form I and direct Form II structures for the system function 1+ 2z −1 + z −2 X z)( = 1− 0.75 z −1 + 0.125 z −2

Q3 a) Define and explain the term “DFT”, Also give its properties. b) By means of four point DFT and IDFT determine the response of the FIR filter having impulse response h(n) = {3,2,1} to the input sequence x(n) = {1,2,2,1}

Q3 a) Obtain DFT of the data sequence [0,1,,1,0] and check the validity of your answer by calculating its IDFT

2 b) Discuss the overlap add method and using this method convolve the two sequences h(n) = [1,0,1] and x(n) = [1,3,2,-3,0,2,-1,0,-2,3,-2,1 ….]. take the number of points required for Radix-2 FFT.

SECTION II

Q4 a) Comment with proof on the linearity, causality, stability and time invariance of the system y[n] = nx[n] b) Explain classification of discrete time systems with reference to their output/input relationships

Q4 a) How will you perform the geometric evaluation of the frequency response from the pole zero diagram? Explain with a suitable example. b) If g[n] is an even sequence and h[n] is an odd sequence, state with proof which of the following is odd and which is even

i) v[n] = g[n]g[n] , ii) x[n] = h[n]h[n].

Q5 a) Discuss the digital resonator. b) A filter is required to meet the following specifications: complete signal rejection at DC and 500 Hz, narrow pass-band centred at 250 Hz, a 3 dB bandwidth of 20 Hz. The sampling frequency is 1 kHz, Obtain the transfer function and difference equation by pole-zero placement.

Q5 a) What are window functions? How are they useful? Give the properties of the Hamming and Kaiser windows b) A RC (analogue) filter has a normalised transfer function 1/(s+1). It is to be used to design a digital filter with a cut-off at 30 Hz, and sampling frequency is 150 Hz Use the BZT to find the transfer function and difference equation of the digital filter .

Q6 Write short notes on any three

a) Sources of error in IIR digital filters. b) Notch and Comb filters. c) Architecture of the TMS series (any one). d) Pipelining.

--END--

THE MAHARAJA SAYAJIRAO UNIVERSITY OF BARODA First Semester of BE IV (Electronics) examination 2005 Tuesday, April 12 3 PM to 6 PM

Digital Signal Processing

N.B.: i) Answer the two sections in separate answer-books . ii) Numerals to the right of each question indicate maximum marks. iii) All symbols carry usual significance. Make suitable assumptions wherever required. Maximum marks: 100

SECTION I

Q1 a) What are discrete time systems? Is the system given by y(n) = x( −n+2) i) static or dynamic ii) linear or nonlinear iii)Time variant or time invariant iv)causal or noncausal and v)stable or unstable. Discuss with proof. (8) b) Determine the causal signal x(n) whose z-transform X(z) is given by 1− 1 z −1 X z)( = 2 1− z −1 + z −2 (8)

Q1 a) Find the Z-transform of the following and give their ROC i) x(n) = -αn U(-n-1) ii) x(n) = cos (n π/2)U(n) (8) b) (i) Explain the term “LTI System” and “Causal System”. (ii) Find the impulse response of the system described by x[n] = y[n]-(1/4)y[n-2]. Assume the system is initially at rest (8)

Q2 a) Using the first order sections obtain the cascade realization for the following system (9) 1 − 1 − 1( + z 1 1)( + z 1 ) H z)( = 2 4 1 − 1 − 1 − 1( − z 1 1)( − z 1 1)( − z 1 ) 2 4 8 b) Obtain the parallel form realization for the system (9) − 1 − 2 + z 1 + z 2 H z)( = 4  1 −  − 1 −  1+ z 1 1+ z 1 + z 2   2  2 

Q3 a) Discuss the following (i) DFT symmetry (ii) DFT Linearity (iii) DFT Shifting and (iv) DFT leakage (8) b) Find the linear convolution of sequences x 1[n] and x 2[n] given below using circular convolution (8) 2 1 0 ≤ n ≤ 2  −n ≤ ≤ = = 2 0 n 3 x1 n][  and x2 n][  0 otherwise  0 otherwise

Q3 a) Describe fully Radix-2 DIF FFT algorithm. Write necessary equations and draw the flow diagram for N = 8 (8) b) Using DFT and IDFT find the convolution of the following sequences x1(n) = {1, 2, 3, 4} and x 2(n) = { 2, 1, 2, 1 } (8)

SECTION II

Q4 a) Explain the relationship between single and double pole locations in the z-plane and the impulse response (8) b) If g[n] is an even sequence and h[n] is an odd sequence state with proof which of the following sequences v[n] and x[n] is odd and which is even i) v[n] = h[n]h[n] ii) x[n] = g[n]h[n] (8)

Q4 a) Discuss the digital resonator with zeros at the origin. (8) b) Describe a test for stability of a given digital filter transfer function. (8) . Q5 a) Write a note on the geometric evaluation of the frequency response from the z-plane pole zero diagram illustrate your answer with a suitable example. (8) b) Discuss properties of the rectangular window and Hanning window. (8)

OR Q5 a) Write a note on the Comb filter. (8) b) Obtain the transfer function of a filter which rejects a signal in a narrow-band centred at 500 Hz. and passes the signal without attenuation at 0 and 1000 Hz. The bandwidth of the filter is to be 40 Hz. Use a sampling frequency is 2000 Hz. (8)

Q6 Write short notes on any two (18)

a) Linear phase filter. b) Computer aided design of FIR filter. c) Differences between conventional microprocessor architecture and DSP processor architecture .

--END--