IT6502 DIGITAL SIGNAL PROCESSING Unit I - SIGNALS AND SYSTEMS Basic elements of DSP concepts of frequency in Analo and Diital Sinals samplin theorem Discrete time sinals, systems Analysis of discrete time LTI systems Z transform Convolution Correlation. Part A Q.No Questions BT Level Competence 1. What is meant by aliasin? How can it be avoided? Rememberin 2. Give the raphical and mathematical representation of CT and DT unit impulse, unit step and unit ramp function. 3. Explain low pass samplin theorem Evaluatin 4. Classify discrete time sinals 5. Define Nyquist rate. Compute Nyquist rate of the sinal x(t) = A sin(250πt)+b sin(500πt) Rememberin 6. Find the enery and power of x(n) = Ae jωn u(n). Rememberin 7. Compare enery sinal and power sinals 8. Examine which of the followin sequences is periodic, and compute their fundamental period. (a) Ae j7πn (b) sin(3n) Analyzin 9. Inspect the system y (n) = ln [x (n)] is linear and time invariant? Analyzin 10. Estimate the Z transform of x(n) = 5 n u(n) Creatin 11. Find the sinal enery of (1/2) n u(n) Rememberin 12. Experiment whether the followin sinusoid is periodic, if periodic then compute their fundamental period. (a) cos 0.01πn (b) sin(π62n/10) 13. Experiment whether the system y (n) = e x (n) is linear 14. Classify the discrete time systems 15. Estimate the Z transform of x(n) = a n u(n) Creatin 16. Give the properties of convolution 17. List the properties of Z-Transform Analyzin 18. Write down the Z transform pairs Rememberin 19. Find the inverse Z transform of the sequence What is Rememberin

correlation? What are it types? Z -5 X(Z) 20. Identify the autocorrelation of the sequence x(n) = {1, 1, -2, -2} 1. (i) Find the convolution of the sinals x(n) = and h(n) = u(n) (ii) Find the transfer function and impulse response of the system y(n) + y(n 1) = x(n) + x(n 1). Rememberin Rememberin 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 (ii) Outline the expressions to relate Z transfer and DFT 3. (i)determine the transfer function, and impulse response of the system y(n) y(n 1) + y(n 2) = x(n) + x(n 1) (ii) Solve the convolution sum of Rememberin Evaluatin And h(n) = δ(n) δ(n 1) + δ(n 2) δ(n 3) 4. (i) Find the Z transform of (8) 1. x(n) = 2 n u(n 2) 2. x(n) = n 2 u(n) (ii) Explain the scalin and time delay properties of Z transform.(8) Analyzin Analyzin 5. (i) Suppose a LTI system with input x(n ) and output y(n ) is characterized by its unit sample response h(n ) = (0.8) n u(n ). Find the response y(n ) of such a system to the input sina lx(n ) = u(n ). (ii) A causal system is represented by the followin difference Equation Evaluatin Creatin Compute the system function H (z )and find the unit sample

response of the system in analytical form. 6. (i) Compute the normalized autocorrelation of the sinal x(n ) = a n u(n ),0 < a <1 (ii) Determine the impulse response for the cascade of two LTI system havin impulse responses, Evaluatin 7. 1. (i) Find the inverse Z-Transform of usin a. Residue method and b. Convolution method. (ii) State and prove circular convolution. 8. (i) Prove samplin theorem with suitable expressins. (ii) Find the Z-transform of the followin sequences: x(n) = (0.5) n u(n) + u(n 1) x(n) = δ(n 5). 9. LTI system is described by the difference equation y n a y n 1 b x n. Find the impulse response, manitude function and phase function. Solve b, if Sketch the manitude and phase response for α=0.6 10. 1. Determine the causual sinal x(n)for the followin Z- transform Analyzin Evaluatin Evaluatin (i) X(z) = (z 2 +z) / ((z-0.5) 3 (z-0.25)) (ii) X(z) = (1+z -1 ) / (1-z -1 +0.5z -2 ) UNIT II FREQUENCY TRANSFORMATIONS Introduction to DFT Properties of DFT Circular Convolution - Filterin methods based on DFT FFT Alorithms - Decimation in time Alorithms, Decimation in frequency Alorithms Use of FFT in Linear Filterin DCT Use and Application of DCT PART A Q.No Questions BT Level Competence 1. Contrast DFT from DTFT

2. Find DFT of the sequence x(n) = {1, 1, -2, -2} Rememberin 3. Calculate the computational savin (both complex multiplication and complex addition) in usin N point FFT alorithm 4. Define phase factor or twiddle factor? Rememberin 5. Compare DIT FFT alorithm with DIF FFT alorithm 6. Give DFT pair of equation 7. List any four properties of DFT Rememberin 8. llustrate the basic butterfly structure of DIT and DIF FFT 9. Evaluate DFT of x(n) = {1, -1, 1, -1} BTL Evaluatin 10. Estimate % savin in computin throuh radix 2, DFT Creatin alorithm of DFT coefficients. Assume N = 512. 11. Calculate the value of W K N when N = 8 and K = 2 and also k = 3 12. Explain and prove Parseval's theorem Evaluatin 13. Calculate the DFT of the four point sequence x(n ) = {0,1,2,3}. 14. Explain the relation between DFT and Z-Transform? 15. List the uses of FFT in linear filterin? Analyzin 16. Find the DTFT of x(n)= -b n.u(-n-1). Rememberin 17. Determine the IDFT of Y(k)={1,0,1,0}. Evaluatin 18. Find DFT of sequence x(n) = {1, 1, -2, -2} Rememberin 19. What is meant by radix 4 FFT? Rememberin 20. Discuss transform pair equation of DCT? Rememberin PART B 1. (i) Discuss the properties of DFT. (ii) State and prove the circular convolution property of DFT. 2. (i) Solve DFT of followin sequence (1) x(n) = {1, 0, -1, 0} (2) x(n) = {j, 0, j, 1} (ii) Make use of DFT and IDFT method, perform circular convolution of the sequence x(n) = {1, 2, 2, 1} and h(n) = {1, 2, 3} Evaluatin

3. Find DFT of the sequence x(n) = { 1, 1, 1, 1, 1, 1, 0, 0} usin radix -2 DIF FFT alorithm 4. Solve the eiht point DFT of the iven sequence x(n) = { ½, ½, ½, ½, 0, 0, 0, 0} usin radix 2 DIT - DFT alorithm 5. (i) Explain, how linear convolution of two finite sequences are obtained via DFT. (ii) Compute 8 point DFT of the followin sequenceusin radix 2 DIT FFT alorithm x(n) = {1,-1,-1,-1,1,1,1,-1} 6. Illustrate the flow chart for N = 8 usin radix-2, DIF alorithm for findin DFT coefficients 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 sequencex(n ) = [1,2,2,1]. 8. (i) Evaluate the 8-point for the followin sequences usin DIT-FFT alorithm BTL Rememberin Evaluatin Evaluatin Evaluatin Evaluatin (ii) Calculate the percentae of savin in calculations in a 1024- point radix -2 FFT, when compared to direct DFT 9. Determine the response of LTI system when the input sequence x (n ) = { 1, 1, 2, 1, 1 } by radix 2 DIT FFT. The impulse response of the system is h(n) = { 1, 1, 1, 1}. 10. (i)find 8-point DFT for the followin sequence usin direct method {1, 1, 1, 1, 1, 1, 0, 0} (ii)list out the properties of DFT. BTL Evaluatin Rememberin Analyzin UNIT III IIR FILTER DESIGN Structures of IIR Analo filter desin Discrete time IIR filter from analo filter IIR filter desin by Impulse Invariance, Bilinear transformation, Approximation of derivatives (LPF, HPF, BPF, BRF) filter desin usin frequency translation PART A Q.No Questions BT Level Competence 1. Outline the limitations of Impulse invariant method of desinin diitalfilters? 2. Illustrate the ideal ain Vs frequency characteristics of: HPF and

BPF 3. What is meant by warpin? Rememberin 4. Give the limitations of impulse invariance method? 5. Show the various tolerance limits to approximate an ideal low pass and hih pass filter 6. Explain the importance of poles in filter desin? Analyzin 7. Compare bilinear and impulse invariant transformation 8. What is aliasin? Rememberin 9. Define Bilinear transformation with expressions. Rememberin 10. List the properties of Butterworth filter Analyzin 11. What are the characteristics of Chebyshev filter? Rememberin 12. Formulate the transformation equation to convert low pass filter into band stop filter Creatin 13. Define Phase Delay and Group Delay Rememberin 14. Why IIR filters do not have linear phase? Rememberin 15. Use the backward difference for the derivative and convert the analo filter to diital filter iven H(s)=1/(s 2 +16) 16. Formulate the relationship between the analo and diital frequencies when convertin an analo filter usin bilinear transformation Creatin 17. Explain the advantae and drawback of bilinear transformation Evaluatin 18. Compare the Butterworth and Chebyshev Type-1 filters 19. Explain the drawbacks of impulse invariant mappin? Evaluatin 20. Compare diital filter vs analo filter Analyzin PART B 1. Desin diital low pass filter usin Bilinear transformation, Given that Creatin

Assume samplin frequency of 100 rad/sec 2. 3. Desin FIR filter usin impulse invariance technique. Given that Creatin and implement the resultin diital filter by adder, multipliers and delays Assume samplin period T = 1 sec. 4. (i) Find the H (z ) correspondin to the impulse invariance desin usin a sample rate of 1/T samples/sec for an analo filter H (s) specified as follows : Rememberin (ii) Desin a diital low pass filter usin the bilinear transform to satisfy the followin characteristics (1) Monotonic stop band and pass band (2) -3 dbcutoff frequency of 0.5 πrad (3) Manitude down at least -15 db at 0.75πrad. 5. Build an IIR filter usin impulse invariance technique for the iven Creatin Assume T = 1 sec. Realize this filter usin direct form I and direct form II. 6. 7. The specification of the desired lowpass filter is Construct a Butterworth diital filter usin bilinear transformation. 8. 9. The specification of the desired low pass filter is Construct a Chebyshev diital filter usin impulse invariant transformation 10. Desin an IIR diital low pass butterworth filter to meet the followin requirements: Pass band ripple (peak to peak): 0.5dB, Pass band ede: 1.2kHz, Stop band attenuation: 40dB, Stop band ede: 2.0 khz, Samplin rate: 8.0 khz. Use bilinear transformation technique 11. (i) Discuss the limitation of desinin an IIR filter usin impulse invariant method (ii) Convert the analo filter with system transfer function usin bilinear transformation Creatin

H a (S) =(S+0.3) / ((S+0.3) 2 +16) 12. The specification of the desired low pass filter is 0.8 H(ω) 1.0; 0 ω 0.2 H(ω) 0.2; 0.32 Desin butterworth diital filter usin impulse invariant transformation uu UNIT IV FIR FILTER DESIGN Structures of FIR Linear phase FIR filter Fourier Series - Filter desin usin windowin techniques (Rectanular Window, Hammin Window, Hannin Window), Frequency samplin techniques. PART A BLT 4 Creatin Q.No Questions BT Level Competence 1. Compare FIR filters and FIR filters with reard to stability and complexity Analyzin 2. List out the conditions for the FIR filter to be linear phase Analyzin 3. What is Gibb s phenomenon or Gibb s oscillation? Rememberin 4. Give the equations for rectanular window and hammin window 5. Give the equations for blackman window and hannin window 6. Distinuish between FIR and IIR filters 7. Compare the diital and analo filter Analyzin 8. Give the desirable properties of windowin technique? 9. Write the equation of Bartlett window Rememberin 10. Illustrate the Direct form I structure of the FIR filter 11. Discuss the steps involved in FIR filter desin 12. Solve the direct form implementation of the FIR system havin 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. What are the advantaes and disadvantaes of FIR filter? Rememberin 15. What is the reason that FIR filter is always stable? Rememberin

16. Explain the necessary and sufficient condition for linear phase characteristic in FIR filter? Evaluatin 17. State the properties of FIR filter? 18. What are called symmetric and anti symmetric FIR filters? Rememberin 19. Explain the condition for a diital filter to be causal and stable? Evaluatin 20. List out the steps involved in desinin FIR filter usin windows Rememberin PART B 1. Desin a hih pass filter with a frequency response Creatin Find the values of h(n) for N = 11 usin hammin window. Find H(z) and determine the manitude response. 2. (i) Realize the followin FIR system usin minimum number of multipliers 1. Η(Ζ) = 1 + 2Ζ 1 + 0.5Ζ 2 0.5Ζ 3 0.5Ζ 4 2. Η(Ζ) = 1 + 2Ζ 1 + 3Ζ 2 + 4Ζ 3 + 3Ζ 4 + 2Ζ 5 (ii) Usin a rectanular window technique, desin a low pass filter with pass band ain of unity cut off frequency of 1000Hz and workin at a samplin frequency of 5 khz. The lenth of the impulse response should be 5. 3. Determine the coefficients of a linear phase FIR filter of lenth M = 15 which has a symmetric unit sample response and a frequency response that satisfies the conditions Analyzin 4. Develop an ideal hih pass filter usin hannin window with a frequency response Creatin

Assume N = 11. 5. Construct a FIR low pass filter havin the followin specifications usin Blackman window Assume N = 7 6. Desin an FIR low pass diital filter usin the frequency samplin method for the followin specifications Cut off frequency = 1500Hz Samplin frequency = 15000Hz Order of the filter N = 10 Filter Lenth required L = N+1 = 11 7. (i) Explain with neat sketches the implementation of FIR filters in direct form and Lattice form (ii) Construct a diital FIR band pass filter with lower cut off frequency 2000Hz and upper cut off frequency 3200 Hz usin Hammin window of lenth N = 7. Samplin rate is 10000Hz 8. (i)determine the frequency response of FIR filter defined by y(n) = 0.25x(n) + x(n 1) + 0.25x(n 2) Rememberin Analyzin Evaluatin (ii) Discuss the desin procedure of FIR filter usin frequency samplin method 9. Desin an FIR filter usin hannin window with the followin specification Creatin Assume N = 5. 10. (i) Explain briefly how the zeros in FIR filter is located. (ii) Usin a rectanular window technique, desin a low pass filter with pass band ain of unity cut off frequency of 1000Hz and workin at a samplin frequency of 5 khz. The lenth of the impulse response should be 7. Analyzin Creatin

UNIT V FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS Binary fixed point and floatin point number representations Comparison Quantization noise truncation and roundin quantization noise power- input quantization error- coefficient quantization error limit cycle oscillations-dead bandoverflow error-sinal scalin PART A Q.No Questions BT Level Competence 1. What is truncation? Rememberin 2. Define product quantization error? Rememberin 3. What is meant by fixed point arithmetic? Give example Rememberin 4. Explain the meanin of limit cycle oscillator Evaluatin 5. What is overflow oscillations? Rememberin 6. Give the advantaes of floatin point arithmetic? 7. Compare truncation with roundin errors. Analyzin 8. What is dead band of a filter? Rememberin 9. What do you understand by input quantization error? 10. Describe the methods used to prevent overflow? 11. Compare fixed point and floatin point arithmetic? Analyzin 12. Discuss the two types of quantization employed in a diital system? 13. What is roundin and what is the rane of roundin? Rememberin 14. Will you recommend quantization step size to be small or lare Evaluatin 15. Define Noise transfer function? Rememberin 16. How are limit cycles are created in a diital system Evaluatin

17. What is meant by block floatin point representation? What are its advantaes? 18. Explain the three-quantization errors to finite word lenth reisters in diital filters? Analysin Analyzin 19. Explain coefficient quantization error? What is its effect? Evaluatin 20. Why roundin is preferred to truncation in realizin diital filter? Rememberin PART B 1. (i) Discuss in detail the errors resultin from roundin and truncation (ii) Explain the limit cycle oscillations due to product round off and overflow errors Creatin 2. Explain the characteristics of a limit cycle oscillation with respect to the system described by the equation y(n)=0.85y(n-2)+0.72y(n-1)+x(n) Determine the dead band of the filter x(n) = (3/4)δ(n), b=4 Analyzin 3. (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 Analyzin (ii) Define zero input limit cycle oscillation and Explain Evaluatin 4. Compare fixed point and floatin point representations. What is overflow? Why do they occur? 5. (i)explain the effects of co-efficient quantization in FIR filters? (ii)distinuish between fixed point and floatin point arithmetic 6. Discuss the followin with respect to finite word lenth effects in diital filters, a. Over flow limit cycle oscillation b. Sinal scalin Rememberin Evaluatin Creatin 7. (i) Consider a second order IIR filter with Analysin

Find the effect of quantization on pole locations of the iven system function in direct form and in cascade form. Assume b = 5 bits. (ii) The output of A/D converter is applied to diital filter with the system function H(Z) = 0.5Z / (Z- 0.5) Estimate the output noise power from the diital filter when the input sinal is quantized to have 8 bits. 8. (i) What is called quantization noise? Derive the expression for quantization noise power (ii) How to prevent limit cycle oscillations? Explain 9. (i) Compare the truncation and roundin errors usin fixed point and floatin point representation. (ii) Show the followin numbers in floatin point format with five bits for mantissa and three bits for exponent. (a) 7 10 (b) 0.25 10 (c) -7 10 (d) -0.25 10 BTL3 Rememberin Rememberin 10. (i) The output of a 12 bit A/D converter is passed thouht a diital 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 (ii) For the system H(Z) = (1+0.75Z -1 ) / (1-0.4Z -1 ) find the scale factor to prevent overflow Analyzin Evaluatin