HOLYCROSS ENGINEERING COLEEGEELECTRONICS AND COMMUNICATION ENGINEERING

IT6502-DIGITAL SIGNAL PROCESSING QUESTION BANK

UNIT-I SIGNALS AND SYSTEMS

PART-A Discrete – time signals

1. What do you understand by the term signal processing? [May 14]2. Let x(n) = [1, 2,-3, 4, 5,-6] .sketch x (n/2) and x (3n). [May 11]3. Let x(n) = [6, 1, 5, 7, 2, 1] .sketch x (n/2) and x (2n). [Nov 11]4. A discrete time signal x (n) = {0, 0, 1, 1, 2, 0, 0…}. Sketch the x(n) and x(-n+2) signals. [Nov 14]5. Define energy and power signals. [May 13][Nov 12]6. Distinguish between power and energy signal with an example. [May 11]7. Define impulse signal. [Nov 2009]8. Determine whether the following sinusoids are periodic; if periodic then compute their fundamental period

(a) cos (0.01πn); (b) . [Nov 14]

Systems – Analysis of discrete time LTI systems9. What is time invariant system? [May 14]10. Determine whether the system described by the input output relation y (n) = Ax (n) + B is linear or non-

linear. [May 2010]11. Define an LTI system.12. Define and express the transfer function of Nth order LTI system. [Nov 11]

Z transform13. What is Z- transform?14. State the convolution property of Z transforms. [Nov 13]

Sampling theorem15. What are the effects aliasing?16. What is meant by aliasing?17. State Sampling theorem. [Nov 2012][May 11][May 15][Nov 13]18. Define Nyquist rate19. Calculate the minimum sampling frequency required for x (t) = 0.5 sin 50π t +0.25 sin 25π t, so as to avoid

aliasing. [Nov 10]Convolution & Correlation

20. What is correlation? What are its types? [May 13]21. Compute the autocorrelation of the signal x(n) =(0.5)nu (n). [May 12]22. State any two properties of auto correlation function. [Nov 10]

UNIT-II FREQUENCY TRANSFORMATIONS PART-A

Introduction to DFT – Properties of DFT23. Write down DFT Pair of equations. [May 2013][Nov 10]24. Compute the DFT of the sequence x(n) ={1,1, 1, 1} [N/D15-R13]25. Calculate the DFT of x(n) = {1,2,3,4}26. Compute DFT of the signal x (n) = ∂ (n). [May 15]27. Compute the DFT of the four point sequence x(n) ={0,1,2,3} [May 11]28. What is the relation between DFT and Z-Transform? [Nov 11]29. Compute the IDFT of Y(k) ={1,0,1,0}. [Nov 13]

30. State and prove parseval’s Theorem. [May 11]31. List any two properties of DFT. [May 14]

Decimation in time Algorithms (FFT)32. Draw the basic butterfly diagram of DIT- FFT.33. What is meant by radix-2 FFT algorithm? [May 2013][May 14][May15]34. What is phase factor or twiddle factor? [Nov 2011] [Nov 2012]35. List the uses of FFT in linear filtering. [Nov 2012]

36. Using the definition , and the Euler identity = the value of

is — . [Nov 2014]37. Calculate % saving in computing through radix- 2 DFT algorithm of DFT co-efficients. Assume

N=512. [Nov 10]38. In the direct computation of N-point DFT of a sequence, how many multiplications and

additions are required? [ Nov 14] 39. What is FFT? [May 12]40. Draw the basic butterfly diagram of DIF- FFT.

Circular Convolution, DCT 41. Find circular convolution of the sequences x(n) = {1,2,3} and h(n) = {4,5,6} [ N/D 15-R13]42. Find the circular convolution of the sequences x(n) = {1,2,2,1} and h(n) = {1,1,1,1}43. Find the DTFT of x (n) = -bn u(-n-1). [Nov 13]44. Compare linear and circular convolution. [Nov 11]45. Write the formula of Discrete Time Cosine Transform (DCT) Pair [May 12]

UNIT-III IIR FILTER DESIGN PART-A

IIR filter design by Bilinear transformation46. What is meant by warping? [May 11]47. Distinguish between FIR and IIR filters.48. Give any two properties of Butterworth filter and chebyshev filter.49. Mention the properties of Butterworth filter. [Nov 13]50. Compare Butterworth and Chebyshev filters.51. What are the characteristics of chebyshev filter? [May 13]52. Write the transformation equation to convert low pass filter into band stop filter. [May 13]53. Draw the ideal gain Vs frequency characteristics of HPF and BPF. [Nov 10]54. Compare analog filters with digital filters. [Nov 14]55. Write frequency translation for BPF from LPF56. Distinguish between the frequency response of chebyshev type I and Type II filter57. Define bilinear transformation with expressions. [Nov 13]

58. Given the low pass transfer function Ha(s) = .find the High pass transfer function having a cut off

frequency 10 rad/sec. [May 12]59. What is meant by bilinear transformation method of designing IIR filter? [May 15]

IIR filter design by impulse invariance method60. What are the limitations of impulse invariance method of designing digital filters?

[May11][May12][Nov 10]61. What are the properties of impulse invariant transformation? [May 14]

62. Compare bilinear and impulse invariant transformation. [Nov 12]63. What is the importance of poles in filter design? [Nov 11]64. Sketch the mapping of s-plane and z-plane in approximation of derivatives. [Nov 14]65. What are the requirements for converting a stable analog filter into a stable digital filter?

Structures of IIR66. Draw the direct form structure of IIR filter. [May 14][May 15]

UNIT-IV FIR FILTER DESIGN PART-A

Filter design using windowing techniques (Rectangular Window, Hamming Window, and Hanning Window)67. Write the equation for Blackman window and Hanning window. [May 13]68. What is Gibb’s Phenomenon? [Nov 12][Nov 13]69. What is a window and why it is necessary?70. Write the equation for Rectangular window and Hamming window.71. What are the desirable characteristics of the window? [May 15]

Linear phase FIR filter72. What is the reason that FIR filter is always stable? [May 14]73. List out conditions for the FIR filter to be linear phase. [May 11]74. What are the properties of FIR filters? [May 15]75. List few applications, where in linear phase is preferred. [May 12]76. What do you understand by linear phase response in filters? [May 14]

What is linear phase response of filter? [May 15]77. What are the characteristic features of FIR filters? [Nov 14]78. Distinguish between FIR and IIR filters. [Nov 13]

What are the advantages of FIR filters?Structures of FIR

79. Determine the transversal structure of the system function

80. Realize the system function by linear phase FIR structure

81. What are the techniques of designing FIR filters?UNIT-V FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS

PART-AFixed point and floating point number representations

82. Compare the digital signal processing systems with fixed point and floating representation. [May 12]83. Represent decimal number 0.69 in fixed point representation of length N =6. [Nov 10]84. What do you understand by a fixed point number? Give example85. What are the advantages of floating point arithmetic?86. What are the three quantization errors due to finite word length registers in digital filter? (OR)

Define finite word length effects. [Nov 14]Quantization- Truncation and Rounding errors - Quantization noise

87. What is truncation?88. What are the different quantization methods? OR What are two quantization method employed in digital

system?89. Explain briefly Quantization noise. (OR)

What do you understand by input quantization error?Coefficient quantization error – Product quantization error

90. What is input quantization error?91. What is product quantization error? (OR) What is product round off noise?

Limit cycle oscillations due to product round off and overflow errors – Principle of scaling

91. What is zero input limit cycle oscillation? [May 13]92. What are limit cycles? [Nov 12]

93. Define the dead band of the filter94. List the type of limit cycle oscillation.95. What is overflow oscillations?96. State the methods to prevent overflow.97. State the need for scaling in filter implementation. 98. What is scaling?

UNIT-I SIGNALS AND SYSTEMS

Basic elements of DSP – concepts of frequency in Analog and Digital Signals – sampling theorem – Discrete – time signals, systems – Analysis of discrete time LTI systems – Z transform – Convolution– Correlation.

PART-B Discrete – time signals

1. Determine whether the following signals are power ,energy or neither energy nor power signals (i) x(n) =

;(ii) x(n) = (1/3)n u(n) ; (iii) x(n) = ;(iv) x(n) = [16 Mark]

2. Determine the fundamental period of the following signals ,if they are periodic (i) x(n) =

(ii) x(n) = ; (iii) x(n) = [12 Mark]

3. Determine the Nyquist rate for (a) cosπt + 3sin2πt + sin4πt ,(b) 1 + cos2000πt + sin4000πt

4. Consider the analog signal x (t) = 3 cos 2000πt + 5 sin 6000πt + 10 cos 12000πt.[8 Mark] [May 2013]

i) What is the Nyquist rate for this signal?

ii) Assume that we sample the signal at a rate Fs = 5000 samples/s. What is the discrete time signal

obtained after sampling?

Systems – Analysis of discrete time LTI systems5. Determine whether the system is time invariant, linear, causal, dynamic, stable.[16 Mark] [May 2013]

(i) y(n) = x(4n + 1) ; (ii) y(n) = ex(n); (iii) y(n) = 2x(n) + ; (iv) y(n) =cos[x(n)]

6. Find whether the following systems are linear , Time invariant

(i) y (n) = e-x (n). [8 Mark][May 14](ii) y (n) =x (n) cosωn. [8 Mark] [May 14]

7. Check whether the following system are static or dynamic, time variant or invariant, linear or nonlinear,

causal or non causal , stable or unstable. (i) y (n) = x (n) cosωo (n); (ii) y(n)=sgn[x(n)]; (iii) y (n) = |x(n)|

[16 Mark][Nov 2014]

8. Check whether the following system are static or dynamic, time variant or invariant, linear or nonlinear,

causal or non causal , stable or unstable.[10 Mark] [Nov 2013]

(i) y(n) = x(- n + 2) ; (ii) y(n) = x(2n); (iii) y(n) = x(n) cosωo(n); (iv) y(n) =cos[x(n)]

Sampling theorem9. Explain the process of analog to digital conversion of signal in terms of sampling, quantization and coding.

[16 Mark] [May 15]10. State sampling theorem and explain aliasing graphically. [8 Mark][May 12][8 Mark][Nov 10]11. What is Nyquist rate? Explain its significance while sampling the analog signals.[8 Mark] [Nov 13]

Z transform12. Determine the response y(n), n ≥ 0 of a system described by the second order difference equation y(n) – 4

y(n – 1) + 4 y(n – 2) = x(n) – x( n – 1) when the input is (-1)n u(n) and the initial conditions are y(-2) = y(-1)

= 0.[16 Mark]

13. Determine the impulse response for the cascade of two LTI system having impulse responses h1(n) = (1/2)n

u(n) and h2(n) =(1/4)n u(n).[8 Mark] [May 11]

14. LTI system is described by the difference equation y (n) =a y (n-1) + b x (n). Find the impulse response,

magnitude function and phase function. Solve b, if | H (ω)| =1. Sketch the magnitude and phase response for

a = 0.6. [16 Mark][Nov 11]

15. A causal system is represented by the difference equation y (n) + ¼ y (n-1) = x (n) + ½ x (n – 1).Find the

system transfer function H(z), unit sample response, magnitude and phase function of the system.

[16 Mark] [May 11][Nov 2012]

16. Suppose a LTI system with input x (n) and output y (n) is characterized by its unit sample response h (n)

= (0.8) nu (n). Find the response y (n) of such a system to the input signal x (n) =u (n).[8 Mark][May2011]

17. A LTI system is characterized by the system function H(z) = Specify the ROC of H (z)

and determine h (n) for the following conditions: i) System is stable ii) System is causal iii) System is anti-causal [16 Mark] [May 10]

18. Determine the causal signal x(n) for the z-transform X(z) = [8 Mark] [Nov 12]

19. Determine the causal signal x(n) for the z-transform X(z) = [8 Mark] [Nov 12]

20. Find the z-transform and ROC of x (n) = -an u (-n-1).[6 Mark] [May 2013]

21. Find the Z-Transform of the given sequence x (n) =δ(n-5) +enu(n-2) +u (n).[4 Mark] [May 12]

22. Determine the z- transform and ROC of the signal x(n) = [3(2n) – 4(3n)] u(n).[8 Mark ][May 2010]

23. Find the Z-Transform of the following sequence: (a)x (n) =(0.5)nu (n) +u (n-1) ; (b)x (n)=δ(n-5)

[8 Mark] [Nov 10]

24. Find the Z-transform of the following discrete time signals and find ROC.

(i) x (n) = [-1/5]n u (n) + 5[1/2]-n u (-n-1). [8 Mark] [May

14]

(ii) x (n) = u (n-2). [8 Mark] [May

14]

25. Find the inverse Z-Transform of X(z) = , ROC : |z|>3, using (1) Residue Method;

(2) Convolution Method. [8 Mark] [Nov 11]

26. Find the inverse z Transform of the function X (z) = Using partial fraction method for

ROC (a) |Z| > 3,(b) 3 > |Z| > 2, (c) |Z| < 1.

Convolution & Correlation

27. Find the convolution of the signals x (n) = (3) nu (-n) and h (n) = [1/3]nu (n-2).[8 Mark] [May 15]

28. Find the convolution x(n)*h(n), where x(n) = αn u (n), h(n) = βnu (n), [8 Mark] [Nov 10]

29. Find the linear convolution of x (n)*h (n) through circular convolution .Assume the suitable length, M x (n)

= ; h (n) = .[8 Mark] [May 12]

30. Compute the linear convolution of the following sequence using mathematical equation, Multiplication and

Tabulation Methods. x (n) = {0, 2, 2, 3} and h (n) = , 0≤ n≤ 4.[16 Mark] [Nov 2014]

31. Compute the convolution of the signals x(n) = {1,2,3,4,5,3,-1,-2} and h(n) ={3,2,1,4}using tabulation

method .[6 Mark] [Nov 13]

32. Find the circular convolution of x(n)*h(n) given that x(n) = 1, 0≤ n≤ 99, h(n) =

[4 Mark][May 12]

33. Applying concentric circle method ,compute circular convolution of the sequences h(n) ={1,2,3,4} & x(n) =

{1,2,3}.[8 Mark][May 15]

34. Compute and plot the convolution x(n) * h(n) for x(n) = {1, 1, 1, 1} &

[8 Mark][May 2010]

35. Find the Z- Transform of Auto correlation of the signal. [4 Mark][May 12][8 Mark][Nov 10]

36. Compute the normalized autocorrelation of the signal x (n) = an u (n), 0<a<1.[8 Mark][May 11]

UNIT-II FREQUENCY TRANSFORMATIONS

Introduction to DFT – Properties of DFT – Circular Convolution - Filtering methods based on DFT – FFT Algorithms - Decimation – in – time Algorithms, Decimation – in – frequency Algorithms – Use of FFT in Linear Filtering – DCT – Use and Application of DCT.

PART-B

Introduction to DFT – Properties of DFT

37. Compute the DFT of the following sequence:[8 Mark][Nov 10]

(1) x= [1, 0,-1, 0] ; (2) x =[ j,0,j,1] when j =

38. Compute the 4-point DFT of the following sequence:[8 Mark][ Nov 14] (1) x (n) = [0, 1,0, -1] ; (2) x (n) = 2n

39. Compute the DFT of the following sequence:[8 Mark][May 12] (1) x (n) = [1, 0,-1, 0] ; (2) x (n) = ,n = 0, 1, 2….7

40. Find 8-point DFT of x(n)={1,1,1,1,1,1,0,0}using direct method.[10 Mark][May 2013]41. State and prove periodicity and time reversal properties of DFT.[8 Mark][ Nov 14]42. State any six properties of DFT.[6 Mark][May 2013][8 Mark][Nov 13]43. State and prove convolution property of DFT.[6 Mark][Nov 12]44. State and prove any two properties of DFT.[7 Mark][May 12]

Decimation in time Algorithms45. Find the FFT of x(n) = n2 + 1for 0 ≤ n ≤ N-1, where N=8using DIT.[16Mark][Nov 2012]46. Find 8-point DFT of the sequence using radix -2 DIT Algorithm x(n) ={1,-1,1,-1,0,0,0,0}

[16 Mark][May 14]47. Compute the eight point DFT of the sequence x(n)={1, -1, -1, -1, 1, 1, 1, -1} using radix2

decimation in time [16 Mark][May 13][OR] Compute the DFT of the following sequence x(n)={1, -1, -1, -1, 1, 1, 1, -1} using decimation in time FFT algorithm.[16][May11]

48. Find DFT for {1, 1, 2, 0, 1, 2, 0, 1} using FFT DIT butterfly algorithm and plot the spectrum. [16 Mark][Nov 13]

49. Using radix 2 DIT –FFT algorithm, determine DFT of the given sequence for N=8

x (n) = [16Mark][May 14]

50. Find 8-point DFT of the sequence using radix -2 DIT Algorithm x (n)= [16

Mark][May15]51. Compute the IDFT of X(k)={8,1+j2,1-j,0,1,0,1+j,1-j2} using DIT algorithm 52. Evaluate the 8-point for the following sequences using DIT-FFT algorithm [8Mark][Nov 11]

x (n) =

Decimation in frequency Algorithms53. Find 8-point DFT of the sequence using radix -2 DIF Algorithm x(n) ={1,-1,1,-1,0,0,0,0} [16

Mark][May 15]54. Compute the FFT of the sequence x (n) =n+1 where N=8 using the in place radix 2 decimation

in frequency algorithm.[16 Mark][Nov 14]55. Find IDFT for {1, 4, 3, 1} using FFT-DIF method56. Calculate % of saving in calculations in a 1024-point radix- 2 FFT when compared to direct

DFT [8Mark][Nov 11]57. Derive the key equation of radix 2 DIF FFT algorithm and draw the relevant flow graph taking

the computation of an 8 point DFT for your illustration.[16 Mark] [OR]Draw the flow chart for N = 8 using radix -2, DIF algorithm for finding DFT coefficients[16 Mark][Nov 10]

Circular Convolution - Filtering methods based on DFT

58. 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}. [16 Mark][Nov 11]

59. Find the circular convolution of the following two sequences using concentricCircle method x1(n) = {1, 2, 3, 4} and x2(n) = {1, 1, 1, 1}

60. By means of DFT and IDFT, determine the sequence x3(n) corresponding to the circular convolution of the sequence x1(n)={2,1,2,1} and x2(n)={1,2,3,4}.

61. Find the output y (n) of a filter whose impulse response is h (n) = {1, 1, 1} and input signal x (n) = {3, -1, 0, 1, 3, 2, 0, 1, 2, 1} using i) Overlap-add ii) Overlap-save method.

62. By means of DFT and IDFT, determine the response at the FIR filter with the impulse responseh (n)={1,2,3} and the input sequence x (n)={1,2,2,1}.[16 Mark][May11]

63. Write shot note on filtering methods using DFT.[8 Mark][May 12]64. The input x (n) and impulse response h(n) of a system are given by

Determine the response of the system using DFT. [10 Mark][Nov 12]

UNIT-III IIR FILTER DESIGN

Structures of IIR – Analog filter design – Discrete time IIR filter from analog filter – IIR filter design by Impulse Invariance, Bilinear transformation, Approximation of derivatives – (LPF, HPF, BPF, BRF) filter design using frequency translation.

PART- B

IIR filter design by Bilinear transformation

65. Compare analog filters with digital filters.[4 Mark][May 12]

66. Differentiate between bilinear transformationwith frequency translation of filter transfer function.[ 4 Mark]

[May 12]

67. Write short notes on frequency translation in both analog and digital domain.[8 Mark][May 12]

68. Convert the analog filter with system transfer function Ha(s) = using bilinear transformation.[10

Mark][Nov 13]

69. A digital filter with a 3dB bandwidth of 0.25π is to be designed from the analog filter whose system

response is Ha(s) = using bilinear transformation and obtain H (Z). [16 Mark][Nov 14]

70. Use bilinear transformation method to obtain H(Z) if T= 1 sec and H(s) is 1/(s2+2s +1).[8 Mark]

71. Design a digital low pass filter using bilinear transformation, given that Ha(s) = .

Assume sampling frequency of 100 rad/sec. [16 Mark][Nov 10]

72. Design a digital low pass filter using the bilinear transform to satisfy the following characteristics (1)

Monotonic stop band and pass band; (2) -3dB cutoff frequency of 0.5π rad ; (3)magnitude down at least -

15dB at 0.75π rad. [10 Mark][May 11]

73. Explain the method of design of IIR filters using bilinear transform method.[10 Mark]

74. A low pass filter meeting the following specifications is required : pass band ― 0-500Hz ; stop band ― 2-

4kHz ;pass band ripple ― 3dB ;stop band attenuation ― 20dB; sampling frequency ― 8kHz ; Determine

the following :(i) Pass and stop band edge frequencies for a suitable analog prototype low pass filter;

(ii)Order N of the prototype low pass filter ;(iii) Coefficients and hence the transfer function of the discrete

time filter using the bilinear z-transform .Assume Butterworth characteristics of the filter.[16 Mark]

[May13]

75. The specification of the desired low pass filter is

0.8≤ | H (w)| ≤ 1; 0 ≤ ω ≤ 0.2π; |H (w)| ≤ 0.2; 0.32 π ≤ ω ≤ π . Design a Butterworth filter using bilinear

transformation. [16Mark][Nov13]

76. Design a digital Butterworth filter using bilinear transformation satisfying the following constraints.

Assume T = 1 Sec.0.75≤ | H (ejω)| ≤ 1; 0 ≤ ω ≤ π/2 ;|H (ejω)| ≤ 0.2; 3 π /4 ≤ ω ≤ π.[16 Mark][May 15]

77. Design a digital chebyshev filter using bilinear transformation satisfying the following constraints. Assume

T = 1 Sec.0.75≤ | H (ejω)| ≤ 1; 0 ≤ ω ≤ π/2 ;|H (ejω)| ≤ 0.2; 3 π /4 ≤ ω ≤ π.[16 Mark][May 14]

78. Design a Butterworth digital filter using bilinear transformation that satisfy the following specifications

0.89≤ | H (w)| ≤ 1; 0 ≤ ω ≤ 0.2π; |H (w)| ≤ 0.18; 0.3π ≤ ω ≤ π. [16Mark][Nov12]

79. The specification of the desired low pass filter is

1/ ≤ | H (w)| ≤ 1; 0 ≤ ω ≤ 0.2 π; |H (w)| ≤ 0.08; 0.4 π ≤ ω ≤ π .Design a Butterworth digital filter using

bilinear transformation [16Mark][Nov11]

80. Determine the order of the analog Butterworth filter that has a -2 dB pass band attenuation at a frequency of

20 rad/sec and atleast -10 dB stop band attenuation at 30 rad/sec.[6 Mark]

81. Design a Chebyshev filter with a maximum pass band attenuation of 2.5dB;at Ωp = 20 rad/sec and the stop

band attenuation of 30dB at Ωs = 50 rad/sec.[8 Mark]

82. Design a low pass Butterworth filter that has a 3 dB cut off frequency of 1.5 KHz and an attenuation of 40

dB at 3.0 kHz. [6 Mark]

IIR filter design by impulse invariance method

83. Discuss the limitation of designing an IIR filter using impulse invariant method.[6Mark][Nov13]

84. 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 :H(s) = .[6 Mark][May 11]

85. By Impulse Invariant method, obtain the digital filter transfer function and differential equation of the

analog filter H(s) = 1 / (s+1).[10 Mark]

86. Design an IIR filter using impulse invariance technique for the given Ha(s) = . Assume

T=1sec .Realize this filter using direct form I and direct form II. [16 Mark][May 11]

87. For the analog transfer function H(s) = .Determine H (z) using impulse invariance method.

Assume T=1 sec. [8 Mark][May 13]

88. Determine the system function of the IIR filter for the analog transfer function Ha(s) = with T

=0.2sec using impulse invariance method.[16 Mark][Nov 14]

89. Design FIR filter using impulse invariance technique .Given that Ha(s) = and implement the

resulting digital filter by adder, multipliers and delays assume sampling period T= 1sec.[16Mark][Nov10]

90. Design a Chebyshev low pass filter with the specifications αp=1 dB ripple in the pass band 0 ≤ ω ≤ 0.2π,

αs=15 dB ripple in the stop band 0.3π ≤ ω ≤ π using impulse invariance method[8 Mark]

91. The specification of the desired low pass filter is

0.9≤ | H (w)| ≤ 1; 0 ≤ ω ≤ 0.25π; |H (w)| ≤ 0.24; 0.5 π ≤ ω ≤ π .Design a chebyshevfilter using impulse

invariant transformation [16Mark][Nov11]Nov12]

Structures of IIR

92. Realize the following FIR system with difference equation y(n) = 3/4 y(n-1) – 1/8 y(n-2) + x(n) + 1/3 x(n-1)

in direct form I. [6 Mark][May 14]

93. Determine the direct form realization of the following system y (n)=-0.1y (n-1) +0.72y (n-2) +0.7x (n)-

0.252x(n-2).[8 Mark]

94. Obtain the direct form I, direct form II, cascade and parallel form realization for the system y (n)

= -0.1y (n-1) + 0.2y (n-2) + 3x (n) + 3.6 x (n-1) + 0.6 x (n-2).[16 Mark][May15][8][May 13]

95. Analyze briefly the different structures of IIR filter.[10 Mark][May 14]

UNIT-IV FIR FILTER DESIGN

Structures of FIR – Linear phase FIR filter – Fourier Series - Filter design using windowing techniques (Rectangular Window, Hamming Window, Hanning Window), Frequency sampling techniques.

PART-B

Fourier series

96. Design the first 15 coefficients of FIR filters of magnitude specification is given below.

H(ejω) = [16 Mark][Nov 10]

97. Derive the frequency response of a linear phase FIR filter when impulse responses symmetric& anti

symmetric when N is EVEN.[16 Mark][Nov 13]

98. Derive the frequency response of a linear phase FIR filter when impulse responses symmetric& anti

symmetric when N is odd.[16 Mark]

Filter design using windowing techniques (Rectangular Window, Hamming Window, Hanning Window)

99. Design an ideal high pass filter with a frequency response Hd(ejω) = find the

value of h (n) for N =11 using hamming window .Find H (z) and compute magnitude response. [16

Mark][May13]

100. Design an ideal band reject filter using hamming window for the given frequency response. Assume N=11.

Hd (ejω) = .[16 Mark][May14]

101. Design an FIR filter for the ideal frequency response using hamming window with N=7Hd(ejω) =

.[16 Mark][May14]

102. Design an ideal differentiator with frequency response H (ejω) = j ; - using Hamming window

with N =7.[16 Mark][May15]

103. Design an FIR filter for the ideal frequency response using hamming window with N=7

Hd(ejω) = .[16 Mark][May14]

104. 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 Mark]

[Nov 12]

105. By using rectangular window function of length N=11,design an ideal band pass digital FIR filter with

desired frequency response H (ejω) = .[16 Mark]

106. Using a rectangular window technique design a low pass filter with pass band gain of unity, cut off

frequency of 1000 Hz and working at a sampling frequency of 5 KHz. The length of the impulse response

should be 7.[10 Mark]

107. Design a single tier notch filter to reject frequencies in the range 1 to 2 rad/sec using rectangular window

with N =7.[8 Mark][Nov 11]

108. Design the symmetric FIR low pass filter whose desired frequency response is given as Ha (ω)

= the length of the filter should be 5 and ωc = 1 radians/sampleusingrectangular window.

[16 Mark][Nov 14]

109. The hamming window is given by ω (n) = 0.54 – 0.46 , 0 ≤ n≤ m-1. Compute the first 10

coefficients using the above window functions having the magnitude response. [16 Mark][May 12]

Frequency sampling techniques.

110. Design and obtain the coefficients of a 15 tap linear phase FIR low pass filter using Hamming window to

meet the given frequency response . Hd (w) = [16 Mark][May 11]

111. 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 Hr ( ) =

[8 Mark] [May 11]

112. 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 Hr ( ) =

[8 Mark] [May 13]

113. Design a 15-tap linear phase filter to the following discrete frequency response (N=15) using frequency

sampling method H(k) = 1 for 0≤ k≤ 4 ; H(k) = 0.5 for k=5; H(k) = 0.25 for k=6 ;

H(k) = 0.1 for k=7; H(k) =0elsewhere.[16 Mark]

114. Discuss the design procedures of FIR filter using frequency sampling method.[16 Mark][May15]

115. A low pass filter has the desired response as given below Hd(ejω) = Determine

the filter coefficients h (n) for N=7 using frequency sampling technique. [16 Mark]

Structures of FIR

116. Draw the three different FIR structure for H (z). H(z) = [ 16] [Nov 10]

117. Draw the three different structure of H (z). H(z) = [8 Mark] [May 12]

118. Realize a direct form and linear phase FIR filter structures with the following impulse response. Which is the

best realization? Why? h(n) = δ (n) + δ (n-1) - δ (n-2) + δ (n-3) + δ (n-4). [16][Nov 14]

UNIT-V FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS

Binary fixed point and floating point number representations – Comparison - Quantization noise – truncation and rounding – quantization noise power- input quantization error- coefficient quantization error – limit cycle oscillations-dead band- Overflow error-signal scaling.

PART-B

Fixed point and floating point number representations

119. Distinguish between fixed point and floating point arithmetic.(6)120. Explain the various formats of the fixed point representation of binary numbers (8)121. What is meant by finite word length effects on digital filters? List them. (8)

Quantization- Truncation and Rounding errors - Quantization noise

122. Explain quantization noise Error.123. Discuss the various methods of quantization (8)124. Explain finite word length effects in FIR digital filters.(8)125. Draw the product quantization noise model of second order IIR system.126. Discuss in detail the errors resulting from rounding and truncation.(16M) (or)

What is meant by truncation? Explain the error that arises due to truncation in floating point number.Explain the quantization error in digital filter (OR) Explain the quantization process and errors introduced due to quantization. (16M)

127. Compute the steady state noise power in the output due to input quantization for a first order discrete time system having difference equation. y(n)=a y(n-1) + x(n). (OR)Find the steady state variance of the noise in the output due to quantization of input for the first order filter. y(n) = a y(n-1) + x(n).

128. Find the output noise power for second order system.(16M).

129. Find the output round-off noise power for the system having transfer function which is

realized in cascade form. Assume word length is 4bits.130. Derive the equation for quantization noise power (8)

Coefficient quantization error – Product quantization error

131. Explain the effects of coefficient Quantization in FIR filters.132. Consider a second order IIR filter with find the effect on quantization on pole locations of the given

function in direct form and cascade form. Assume b=3 bits.133. Given H (z) compute the truncated H’(Z) with coefficients represented by (1) 4-bit word length ;

(2) 6-bit word length at frequency ω = /3. H (z) = [8 Mark] [May 12]

134. Explain in detail about finite word length effects in digital filters. [16 Mark][Nov 13]

Limit cycle oscillations due to product round off and overflow errors – Principle of scaling

135. Write the behavior of zero- input limit cycle oscillation for the system having difference equation y(n)=0.95 y(n-1) + x(n) for the word length of 4 bits. (16M)

136. Determine the dead band of the system assume 4 bits are used

for a signal representation. (16M)137. Explain the characteristics of a limit cycle oscillation with respect to the system described by the equation

y (n) = 0.85 y (n-2) +0.72 y (n-1) + x (n). Determine the dead band of the filter x (n) = 3/4 δ (n).[16 Mark][Nov 12]

138. Explain the limit cycle oscillations due to product round off and overflow errors.(16M)139. With respect to finite word length effects in digital filters, with examples discuss about (i) overflow limit

cycle oscillation (ii) signal scaling (16M)