VALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF INFORMATION TECHNOLOGY. Academic Year

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1 VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur DEPARTMENT OF INFORMATION TECHNOLOGY Academic Year QUESTION BANK-ODD SEMESTER NAME OF THE SUBJECT SUBJECT CODE SEMESTER YEAR DEPARTMENT HANDLED & PREPARED BY IT6502 / DIGITAL SIGNAL PROCESSING V III INFORMATION TECHNOLOGY A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE UNIT I PART A 1 What is meant by aliasing? How can it be avoided? 2 Define Nyquist rate. Compute Nyquist rate of the signal x(t) = A sin(250πt)+b sin(500πt)? 3 Find the energy and power of x(n) = Ae jωn u(n)? 4 Find the signal energy of (1/2) n u(n)? 5 What is the necessary and sufficient condition on the impulse response for stability? 6 What is correlation? What are it types? 7 Classify the discrete time systems 8 Show the graphical and mathematical representation of CT and DT unit impulse, unit step and unit ramp function. 9 Compare energy signal and power signals 10 Give the properties of convolution 11 Experiment whether the following sinusoid is periodic; if periodic then compute their fundamental period. (a) cos 0.01πn (b) sin(π62n/10) 12 Experiment whether the system y (n) = e x (n) is linear A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 1 of 16

2 13 Identify the autocorrelation of the sequence x(n) = {1, 1, -2, -2} 14 Examine which of the following sequences is periodic, and compute their fundamental period. (a) Ae j7πn (b) sin(3n) 15 Inspect the system y (n) = ln [x (n)] is linear and time invariant? 16 List the properties of Z-Transform 17 Explain low pass sampling theorem 18 A discrete time signal x (n) = {0, 0, 1, 1, 2, 0, 0,,,, }. Evaluate the x (n) and x (- n + 2) signals. 19 Estimate the Z transform of x(n) = 5 n u(n) 20 Estimate the Z transform of x(n) = { 1,0,2,0,3} PART-B 1 (i) Find the convolution of the signals x(n) = and h(n) = u(n).(8) (ii) Find the transfer function and impulse response of the system y(n) + y(n 1) = x(n) + x(n 1). (8) 2 (i) Find the inverse Z transform of X(Z) = if (1)ROC: Z > 1, (2) ROC: Z < 0.5, (3) ROC: 0.5 < Z <1 (10) (ii) Show the expressions to relate Z transform and DFT.(6) 3 (i)determine the transfer function, and impulse response of the system A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 2 of 16

3 y(n) y(n 1) + y(n 2) = x(n) + x(n 1).(8) (ii) Find the convolution sum of and h(n) = δ(n) δ(n 1) + δ(n 2) δ(n 3).(8) 4 (i) Find the Z transform of 1. x(n) = 2 n u(n 2) 2. x(n) = n 2 u(n) (8) (ii) Explain the scaling and time delay properties of Z transform.(8) 5 (i) Explain the different types of digital signal representation. (8) (ii) A causal system is represented by the following difference Equation Demonstrate the system function H (z) and find the unit sample response of the system in analytical form. (8) 6 (i) Compute the normalized autocorrelation of the signal x(n ) =(0.5) n u(n ).(6) (ii) Determine the impulse response for the cascade of two LTI system having impulse responses, (10) 7 (i) Find the inverse Z-Transform of using A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 3 of 16

4 a. Residue method and b. Convolution method. (10) (ii) State and prove circular convolution. (6) 8 (i) Prove the sampling theorem with suitable expressions. (8) (ii) (ii) Find the Z-transform of the following sequences: x(n)= cos(wn) u(n).(8) 9 Determine the causual signal x(n)for the following Z- transform (i) X(z) = (z 2 +z) / ((z-0.5) 3 (z-0.25)) (8) (ii) X(z) = (1+z -1 ) / (1-z z -2 ) (8) 10 Suppose LTI system with input x (n) and output y (n) is characterize by unit sample response. Find the response y (n) of such a system to the input signal. (16) 11 concentric circle method, compute circular convolution of the sequences h(n) = {1, 2, 3, 4 } & x(n) = {1, 2, 3}. (16) 12 Find the convolution of given signals x(n)= 3 n u(- n) and h(n)= (1/3) n u(n - 2).(16) 13 Compute the convolution of the signal x(n)={1,2,3,4,5,3-1,-2} and h(n)={3,2,1,4} using tabulation method. (16) 14 Check whether the following systems are static or dynamic, linear or nonlinear, time invariant,causal or non-causal,stable or unstable i)y(n)=cos [(x(n)] ii) y(n)=x(-n+2) iii) y(n)= x(2n) iv) y(n)=x(n)cosw o (n).(4*4=16) UNIT II PART A 1 Find DFT of the sequence x(n) = {1, 1, -2, -2} 2 Define phase factor or twiddle factor? A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 4 of 16

5 3 List any four properties of DFT 4 Find the DTFT of x(n)= -b n.u(-n-1). 5 Find DFT of sequence x(n) = {1, 1, -2, -2} 6 What is meant by radix 4 FFT? 7 Contrast DFT from DTFT 8 llustrate the basic butterfly structure of DIT and DIF FFT Explain the relation between DFT and Z-Transform? 9 10 Using the definition W = and the euler identity, the value of? 11 Calculate the computational saving (both complex multiplication and complex addition) in using N point FFT algorithm? 12 Calculate the value of W K N when N = 8 and K = 2 and also k = 3? 13 Calculate the DFT of the four point sequence x(n ) = {0,1,2,3}. 14 List the uses of FFT in linear filtering? 15 Explain transform pair equation of DFT? 16 Determine the IDFT of Y(k)={1,0,1,0}. 17 Explain and prove Parseval's theorem 18 Evaluate DFT of x(n) = {1, -1, 1, -1} 19 Estimate % saving in computing through radix 2, DFT algorithm of DFT coefficients. Assume N = Design DFT of the signal x(n) = δ(n) PART-B Q.N o. 1 (i)discuss the properties of DFT. (8) Question Competence Level (ii) State and prove the circular convolution property of DFT. (8) 2 (i) Solve DFT of following sequence (1) x(n) = {1, 0, -1, 0} (2) x(n) = {j, 0, j, 1}.(8) (ii) Make use of DFT and IDFT method, perform circular convolution of the sequence x(n) = {1, 2, 2, 1} and h(n) = A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 5 of 16

6 {1, 2, 3}.(8) 3 Find DFT of the sequence x(n) = { 1, 1, 1, 1, 1, 1, 0, 0} using radix -2 DIF FFT algorithm.(16) 4 Solve the eight point DFT of the given sequence x(n) = { ½, ½, ½, ½, 0, 0, 0, 0} using radix 2 DIT - DFT algorithm.(16) 5 (i) Explain, how linear convolution of two finite sequences are obtained via DFT. (8) (ii) Compute 8 point DFT of the following sequenceusing radix 2 DIT FFT algorithm x(n) = {1,-1,-1,-1,1,1,1,- 1}.(8) 6 Illustrate the flow chart for N = 8 using radix-2, DIF algorithm for finding DFT coefficients.(16) 7 By means of the DFT and IDFT, determine the response at the FIR filter with the impulse response h (n ) ={1,2,3}and the input sequence x(n )={1,2,2,1}. (16) 8 (i) Evaluate the 8-point for the following sequences using DIT-FFT algorithm Understandin g (ii) Calculate the percentage of saving in calculations in a 1024-point radix -2 FFT, when compared to direct DFT. (8) 9 Determine the response of LTI system when the input sequence x (n)={1, 1, 2, 1}by radix 2 DIT FFT. The impulse response of the system is h(n)={1, 1, -1, 1}.(16) 10 (i)find 8-point DFT for the following sequence using direct method {1, 1, 1, 1, 1, 1, 0, 0} (ii)list out the properties of DFT.(16) 11 Compute the eight point DFT of the following sequence using radix 2 DIT FFT algorithm. x(n)={1,-1,-1,-1,1,1,1,-1}.(16) 12 State and prove convolution and correlation property of DFT.(16) 13 Discuss the uses of FFT algorthim in linear filtering and correlation.(16) 14 Compute the 8-point DFT of the equation x(n) = n+1 using radix-2 DIF- FFT algorithm.(16) (8) A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 6 of 16

7 UNIT III PART A 1 What is meant by warping? 2 What is aliasing? 3 What are the characteristics of Chebyshev filter? 4 Define Phase Delay and Group Delay 5 Why IIR filters do not have linear phase? 6 What are the limitations of invariance method? 7 Outline the limitations of Impulse invariant method of designing digital filters? 8 Illustrate the ideal gain Vs frequency characteristics of: HPF and BPF 9 Compare bilinear and impulse invariant transformation? 10 Show the structure of cascade form. 11 Show the various tolerance limits to approximate an ideal low pass and high pass filter 12 Explain the importance of poles in filter design? 13 List the properties of Butterworth filter 14 Compare digital filter vs analog filter? 15 Use the backward difference for the derivative and convert the analog filter to digital filter given H(s)=1/(s Compare the Butterworth and Chebyshev Type-1 filters 17 Explain the drawbacks of impulse invariant mapping? 18 What is bilinear transformation with expressions? 19 Draw the direct form structure of IIR filter 20 Use the backward difference for the derivative and convert the analog filter to digital filter given H(s)=1/(s 2 +16) A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 7 of 16

8 PART-B 1 (i) Find the H (z ) corresponding to the impulse invariance design using a sample rate of 1/T samples/sec for an analog filter H (s) specified as follows. (8) (ii) Design a digital low pass filter using the bilinear transform to satisfy the following characteristics (1) Monotonic stop band and pass band (2) -3 dbcutoff frequency of 0.5 πrad (3) Magnitude down at least -15 db at 0.75πrad. (8) 2 Design digital low pass filter using Bilinear transformation, Given that Assume sampling frequency of 100 rad/sec.(16) 3 Design FIR filter using impulse invariance technique. Given that and implement the resulting digital filter by adder, multipliers and delays Assume sampling period T = 1 sec.(16) 4 Build an IIR filter using impulse invariance technique for the given Assume T = 1 sec. Realize this filter using direct A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 8 of 16

9 form I and direct form II.(16) 5 The specification of the desired lowpass filter is Construct a Butterworth digital filter using bilinear transformation. (16) 6 The specification of the desired low pass filter is Construct a Chebyshev digital filter using impulse invariant transformation.(16) 7 Design an IIR digital low pass butterworth filter to meet the following requirements: Pass band ripple (peak to peak): 0.5dB, Pass band edge: 1.2kHz, Stop band attenuation: 40dB, Stop band edge: 2.0 khz, Sampling rate: 8.0 khz. Use bilinear transformation technique.(16) 8 (i) Discuss the limitation of designing an IIR filter using impulse invariant method.(8) (ii) Convert the analog filter with system transfer function using bilinear transformation H a (S) =(S+0.3) / ((S+0.3) 2 +16).(8) 9 The specification of the desired low pass filter is 0.8 H(ω) 1.0; 0 ω 0.2ᴨ H(ω) 0.2; 0.32ᴨ ᴨ Design butterworth digital filter using impulse invariant transformation.(16) 10 Analyze briefly the different structures of IIR filter.(16) 11 Realize the system given by difference equation in direct form-i (16) A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 9 of 16

10 12 Design a Butterworth digital filter using bilinear transformations that satisfy the following specifications.(16) 13 Draw the ideal gain vs. frequency characteristics of HPF & BPF and also how the above filter specified.(16) 14 Write a short note on frequency translation in both analog and digital domain.(16) UNIT IV PART A 1 What is Gibb s phenomenon or Gibb s oscillation? 2 Write the equation of Bartlett window 3 What are the advantages and disadvantages of FIR filter? 4 What is the reason that FIR filter is always stable? 5 What are called symmetric and anti symmetric FIR filters? 6 List out the steps involved in designing FIR filter using windows 7 Give the equations for rectangular window and hamming window 8 Give the equations for blackman window and hanning window 9 Distinguish between FIR and IIR filters 10 Give the desirable properties of windowing technique? 11 Illustrate the Direct form I structure of the FIR filter 12 Solve the direct form implementation of the FIR system having difference equation y(n) = x(n) 2x(n- 1) + 3x(n-2) 10x(n-6) 13 Solve direct cascade realization of the system H(Z) = (1+5Z -1 +6Z -2 )(1+Z -1 ) 14 Compare FIR filters and FIR filters with regard to A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 10 of 16

stability and complexity 15 List out the conditions for the FIR filter to be linear phase 16 Compare the digital and

18 Explain the condition for a digital filter to be causal and stable?

PART-B 1 Design a high pass filter with a frequency response Find the values of h(n) for N = 11 using hamming window.

(16) 2 (i) Realize the following FIR system using minimum number of multipliers 1. Η(Ζ) = 1 + 2Ζ 1 + 0.5Ζ 2 0.5Ζ 3 0.

Η(Ζ) = 1 + 2Ζ 1 + 3Ζ 2 + 4Ζ 3 + 3Ζ 4 + 2Ζ 5 (8) (ii) Using a rectangular window technique, design a low pass filter with

11 stability and complexity 15 List out the conditions for the FIR filter to be linear phase 16 Compare the digital and analog filter 17 Explain the necessary and sufficient condition for linear phase characteristic in FIR filter? 18 Explain the condition for a digital filter to be causal and stable? 19 What do you understand by linear phase response in filters? 20 What is the reason that FIR filter is always stable? PART-B 1 Design a high pass filter with a frequency response Find the values of h(n) for N = 11 using hamming window. Find H(z) and determine the magnitude response. (16) 2 (i) Realize the following FIR system using minimum number of multipliers 1. Η(Ζ) = 1 + 2Ζ Ζ 2 0.5Ζ 3 0.5Ζ 4 2. Η(Ζ) = 1 + 2Ζ 1 + 3Ζ 2 + 4Ζ 3 + 3Ζ 4 + 2Ζ 5 (8) (ii) Using a rectangular window technique, design a low pass filter with pass band gain of unity cut off frequency of 1000Hz and working at a sampling frequency of 5 khz. The length of the impulse response should be 5.(8) 3 Determine the coefficients of a linear phase FIR filter of length M = 15 which has a symmetric unit sample response and a frequency response that satisfies the conditions (16) A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 11 of 16

4 Develop an ideal high pass filter using hanning window with a frequency response Assume N = 11.

(16) 6 Design an FIR low pass digital filter using the frequency sampling method for the following

(8) (ii) Construct a digital FIR band pass filter with lower cut off frequency 2000Hz and upper cut off

(8) 8 (i)determine the frequency response of FIR filter defined by y(n) = 0.25x(n) + x(n 1) + 0.25x(n 2).

(8) 9 Design an FIR filter using hanning window with the following specification Assume N = 5.

12 4 Develop an ideal high pass filter using hanning window with a frequency response Assume N = 11. (16) 5 Construct a FIR low pass filter having the following specifications using Blackman window Assume N = 7.(16) 6 Design an FIR low pass digital filter using the frequency sampling method for the following specifications Cut off frequency = 1500Hz Sampling frequency = 15000Hz Order of the filter N = 10 Filter Length required L = N+1 = 11.(16) 7 (i) Explain with neat sketches the implementation of FIR filters in direct form and Lattice form.(8) (ii) Construct a digital FIR band pass filter with lower cut off frequency 2000Hz and upper cut off frequency 3200 Hz using Hamming window of length N = 7. Sampling rate is 10000Hz.(8) 8 (i)determine the frequency response of FIR filter defined by y(n) = 0.25x(n) + x(n 1) x(n 2).(8) (ii) Discuss the design procedure of FIR filter using frequency sampling method.(8) 9 Design an FIR filter using hanning window with the following specification Assume N = 5. (16) 10 (i) Explain briefly how the zeros in FIR filter is located. (8) (ii) Using a rectangular window technique, design a low pass filter with pass band gain of unity cut off frequency of 1000Hz and working at a sampling frequency of 5 khz. The length of A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 12 of 16

13 the impulse response should be 7.(8) 11 Design a FIR band stop filter to reject frequencies in the range 1.2 to 1.8 rad/sec using hamming window, with length N=6. Also realize the linear phase structure of the band stop FIR filter.(16) 12 Design the first 15 coefficients of FIR filters of magnitude specification is given below 13 = 0, otherwise.(16) Realize a direct form and linear phase FIR filter structures with the following impulse response..(16) 14 Discuss the deign procedure of FIR filter using frequency sampling method.(16) UNIT V PART A 1 What is truncation? 2 Define product quantization error? 3 What is meant by fixed point arithmetic? Give example 4 What is overflow oscillations? 5 What is dead band of a filter? 6 What is rounding and what is the range of rounding? 7 Give the advantages of floating point arithmetic? 8 What do you understand by input quantization error? 9 Discuss the two types of quantization employed in a digital system? 10 Comparison of fixed point and floating point arithmetic? 11 Explain the meaning of limit cycle oscillator? 12 Compare truncation with rounding errors? A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 13 of 16

14 13 Will you recommend quantization step size to be small or large? 14 How are limit cycles are created in a digital system? Analysing 15 What are zero input limit cycle oscillations? Analysing 16 Explain Noise transfer function? Analysing 17 What is meant by block floating point representation? What are its advantages? 18 Explain coefficient quantization error? What is its effect? 19 Why rounding is preferred to truncation in realizing digital filter? 20 Explain the three-quantization errors to finite word length registers in digital filters? PART-B 1 (i) Discuss in detail the errors resulting from rounding and truncation.(8) (ii) Explain the limit cycle oscillations due to product round off and overflow errors.(8) 2 (i)explain the characteristics of a limit cycle oscillation with respect to the system described by the equation 3 y(n)=0.85y(n-2)+0.72y(n-1)+x(n).(8) (ii)determine the dead band of the filter x(n) = (3/4)δ(n), b=4.(8) (i) Explain the characteristics of limit cycle oscillations with respect to the system described by the difference equation y(n)=0.95y(n-1)+x(n). x(n)=0; y(n-1) =13. Determine the dead band of the system (ii) Define zero input limit cycle oscillation and Explain 4 Compare fixed point and floating point representations. What is overflow? Why do they occur? (16) A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 14 of 16

15 5 (i ) Explain the effects of co-efficient quantization in FIR filters.(8) (ii) Distinguish between fixed point and floating point arithmetic.(8) 6 Discuss the following with respect to finite word length effects in digital filters. a. Over flow limit cycle oscillation (8) b. Signal scaling.(8) 7 (i) Consider a second order IIR filter with Analysing Find the effect of quantization on pole locations of the given system function in direct form and in cascade form. Assume b = 5 bits. (8) (ii) The output of A/D converter is applied to digital filter with the system function H(Z) = 0.5Z / (Z- 0.5) Estimate the output noise power from the digital filter when the input signal is quantized to have 8 bits. (8) 8 (i) What is called quantization noise? Derive the expression for quantization noise power.(8) (ii) How to prevent limit cycle oscillations? Explain.(8) 9 (i) Compare the truncation and rounding errors using fixed point and floating point representation. (8) (ii) Show the following numbers in floating point format with five bits for mantissa and three bits for exponent. (a) 7 10 (b) (8) 10 (i) The output of a 12 bit A/D converter is passed thought a digital filter which is described by the difference equation y(n)= x(n)+ 0.2y(n-1). Calculate the steady state output noise power due to A/D converter quantization.(8) (ii) For the system H(Z) = (1+0.75Z -1 ) / (1-0.4Z -1 ) find the scale factor to prevent overflow.(8) 11 Consider the transfer function where H(z) = H 1 (z)h 2 (z) Let H(z) =H 1 (Z)H 2 (Z), i.e., Analysing A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 15 of 16

16 Find the output roundoff noise power. (16) 12 Briefly explain the product quantization error.(16) Analysing 13 Explain in detail about finite word length effects in digital filters.(16) 14 In the IIR system given below the products are rounded 4 bits (including sign bits). The system function is Find the output roundoff noise power in (a) direct form realization and (b) cascade form realization.(16) Verified by : [ ] [ ] [ ] [ ] Forwarded by Year Coordinator : Approved by HOD : *************** A.Anbarasan A.P (O.G)/ECE, R.Dhananjeyan A.P (O.G)/ECE Page 16 of 16

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A Classification of systems : Continuous and Discrete