This document provides an outline for a course on digital signal processing. It discusses several topics including: - Realization of digital filters, which focuses on representing linear constant-coefficient difference equations using block diagrams. This includes basic structures for IIR and FIR systems. - Digital filters, including analog filter approximations, design of IIR and FIR digital filters, and comparisons between IIR and FIR filters. - Multirate digital signal processing, including topics like decimation, interpolation, and applications like filter banks. The unit on realization of digital filters covers representing systems using Z-transforms, system functions, basic IIR structures like direct forms I and II, cascade, and parallel forms. It includes

1. DIGITAL SIGNAL PROCESSING
(DSP-7CC10)
Unit-IV REALIZATION OF DIGITAL
FILTERS
Faculty: V.RAJENDRA CHARY(VRC),
Assistant Professor,ECE
Class: III B.Tech,II Semester, ECE

2. RAJENDRA CHARY 2
SYLLABUS
UNIT-
1. INTRODUCTION
2. DISCRETE FOURIER TRANSFORM
3. FAST FOURIER TRANSFORMS
…..mid 1
4. REALIZATION OF DIGITAL FILTERS
5. DIGITAL FILTERS
6. MULTIRATE DIGITAL SIGNAL PROCESSING
…..mid 2

3. RAJENDRA CHARY 3
UNIT-IV REALIZATION OF DIGITAL
FILTERS
• Review of Z-transforms, Applications of Z–transforms, Block
diagram representation of linear constant-coefficient difference
equations, Basic structures of IIR systems, Transposed forms,
Basic structures of FIR systems, System function.
• Applications: Design of digital system function to meet the given
specifications.

4. RAJENDRA CHARY 4
UNIT-V DIGITAL FILTERS
• Analog Filter Approximations – Butterworth and Chebyshev
Approximations.
• IIR digital filters: Design of IIR Digital filters from analog filters-
Impulse Invariance, Step invariance and Bilinear Transformation
methods, Design Examples, Analog-Digital transformations.

5. RAJENDRA CHARY 5
UNIT-V DIGITAL FILTERS(contd.)
• FIR digital filters: Characteristics of FIR Digital Filters, frequency
Windowing Techniques-Rectangular,
Design of FIR Digital Filters using Fourier series
Triangular,
response,
method,
Hamming, Hanning and Bartlett’s Windows, Steps in Kaiser
windowing method, Frequency Sampling technique, Comparison
of IIR and FIR filters.
• Applications: Design of IIR/FIR digital filter conforming to
given specifications.

6. RAJENDRA CHARY 6
UNIT-VI MULTIRATE DIGITAL SIGNAL
PROCESSING
• Decimation, interpolation, sampling rate conversion. Introduction
to DSP Processors.
• Applications of Multirate Digital Signal processing: Design of
digital filter banks and quadrature mirror filters etc.

7. REFERENCE TEXTBOOKS(R1)
RAJENDRA CHARY 7

8. REFERENCE TEXTBOOKS(R2)
RAJENDRA CHARY 8

9. REFERENCE TEXTBOOKS(R3)
RAJENDRA CHARY 9

10. RAJENDRA CHARY 10
Outline Of The Unit
• Review of Z-transform, Applications of Z–transforms(3)
• System function(1)
• Block diagram representation of linear constant-coefficient
difference equations(1)
• Basic structures of IIR systems, Transposed forms
• Basic structures of FIR systems
• Applications

11. RAJENDRA CHARY 11
Lecture-1

12. RAJENDRA CHARY 12
Review of Z-transforms
• Z-transform
• Region of Convergence
• Properties of Z-transform
• Inverse Z-transform

13. RAJENDRA CHARY 13
Z-transform
• The Z-transform is the discrete-time counterpart of the Laplace
transform.
• The Z-transform plays an important role in the analysis and
representation of discrete-time LTI systems.
• The Z-transform may be one-sided(unilateral) or two-
sided(bilateral).

14. Z-transform(contd.)
• The bilateral or two-sided Z-transform of a discrete-time signal or
sequence x(n) is given by:
where z is a complex variable
RAJENDRA CHARY 14

15. Z-transform(contd.)
• The unilateral or one-sided Z-transform of a discrete-time signal or
sequence x(n) is given by:
RAJENDRA CHARY 15

16. RAJENDRA CHARY 16
Region of Convergence(ROC)
• The set of values of z or equivalently the set of points in z-plane,
for which X(z) converges is called the region of
convergence(ROC) of X(z).
• If there is no value of z( i.e no point in the z-plane) for which X(z)
converges, then the sequence x(n) is said to be having no Z-
transform.

17. Properties of ROC
RAJENDRA CHARY 17

18. Properties of ROC
RAJENDRA CHARY 18

19. Summary of ROC of discrete-time signals
RAJENDRA CHARY 19

20. Summary of ROC of discrete-time
signals(contd.)
RAJENDRA CHARY 20

21. Summary of ROC of discrete-time
signals(contd.)
RAJENDRA CHARY 21

22. RAJENDRA CHARY 22
Properties of Z-transform
• Linearity
• Time shifting
• Multiplication by an exponential sequence
• Time reversal
• Time expansion
• Multiplication by n or differentiation in z-domain
• Convolution

23. RAJENDRA CHARY 23
Properties of Z-transform(contd.)
• Multiplication or complex convolution
• Correlation
• Parseval’s theorem
• Initial value theorem
• Final value theorem

24. Linearity Property
RAJENDRA CHARY 24

25. Time Shifting Property
RAJENDRA CHARY 25

26. Time Shifting Property(contd.)
RAJENDRA CHARY 26

27. Multiplication by an exponential sequence
Property
RAJENDRA CHARY 27

28. Time Reversal Property
RAJENDRA CHARY 28

29. Time expansion Property
RAJENDRA CHARY 29

30. Multiplication by n or differentiation in z-
domain Property
RAJENDRA CHARY 30

31. Convolution Property
RAJENDRA CHARY 31

32. Multiplication or complex convolution
Property
RAJENDRA CHARY 32

33. Correlation Property
RAJENDRA CHARY 33

34. Parseval’s theorem
RAJENDRA CHARY 34

35. Initial value theorem
RAJENDRA CHARY 35

36. Final value theorem
RAJENDRA CHARY 36

37. RAJENDRA CHARY 37
Applications of Z-transform
• The Z-transform converts the difference equations of a discrete
time system into linear algebraic equations so that the analysis
becomes easy and simple.
• Frequency response can be determined.
• Pole-zero description of the discrete-time system.
• It also helps in system design, analysis and also checks the system
stability.
• Used to obtain impulse response.

38. RAJENDRA CHARY 38
Advantage of Z-transform
• Z-transform exists for most of the signals for which DTFT does not
exist.

39. RAJENDRA CHARY 39
Lecture-2

40. Problem
• (R2 eg 3.6)
RAJENDRA CHARY 40

41. Problem
•
(R2 eg 3.8)
RAJENDRA CHARY 41

42. Problem
• Find the Z-transform and ROC of the sequence (R2 eg3.9)
RAJENDRA CHARY 42

43. Problem
• Find the Z-transform of the following sequences (R2 eg3.10)
RAJENDRA CHARY 43

44. RAJENDRA CHARY 44
Problem
• Find the Z-transform of
a)unit impulse sequence δ(n), b)unit step sequence u(n), c) anu(n)
d)a-nu(n) e) a-nu(-n-1) f) -anu(-n-1) g) ∑ δ(n-k) (summation of k
limits-o to ∞)

45. RAJENDRA CHARY 45
Lecture-3

46. Inverse Z-transform
RAJENDRA CHARY 46

47. RAJENDRA CHARY 47
Inverse Z-transform(contd.)
• This direct method of finding inverse Z-transform is quite
complex. So inverse Z-transform is normally found using indirect
methods. Some of those methods are:
Power series/long division method
Partial fraction expansion method
Complex inversion integral/residue/contour integration method
Convolution method

48. System function
• Consider a discrete-time LTI system having an impulse response
h(n) as shown in figure
x(n)
RAJENDRA CHARY 48
y(n)

49. System function(contd.)
RAJENDRA CHARY 49

50. RAJENDRA CHARY 50
Block diagram representation of Linear Constant-
coefficient Difference Equations(LCCDE)
• To realize a discrete-time system, the given difference equation in
time domain is to be converted into an algebraic equation in z-
domain and each term of that equation is to be represented by a
suitable element.
• The symbols of the basic elements used for constructing the block
diagram of a discrete time system are tabulated below
(contd..in next slide)

51. Block diagram representation of LCCDE
(contd.)
RAJENDRA CHARY 51

52. RAJENDRA CHARY 52
Lecture-4

53. Problems
• Find the inverse Z-transform of
a) X(z)=z3+2z2+z+1-2z-1-3z-2+4z-3( R2 eg 3.22)
b) ( R2 eg 3.23,b)
RAJENDRA CHARY 53

54. RAJENDRA CHARY 54
Problems
• Find the system function and impulse response of the system
described by the difference equation
y(n)=1/5*y(n-1)+x(n) (R1 eg 2.15)
• Find the system function and impulse response of the system
described by the difference equation
y(n)=x(n)+2x(n-1)-4x(n-2)+x(n-3) (R1 eg 2.16)

55. Problem
• Using long division, find the inverse Z-transform of (R2 eg 3.29)
RAJENDRA CHARY 55

56. Problem
• Using long division, find the inverse Z-transform of (R2 eg 3.30)
RAJENDRA CHARY 56

57. RAJENDRA CHARY 57
Lecture-5

58. Problem
• Using Partial fraction expansion, find the inverse Z-transform of
(R2 eg 3.31)
RAJENDRA CHARY 58

59. Problems
• Using residue method, find the inverse Z-transform of (R2 eg 3.36)
RAJENDRA CHARY 59

60. Problem
• Determine the inverse Z-transform of the following equation using
Long division, Partial fraction method and Residue method .
( R3 eg 3.7)
RAJENDRA CHARY 60

61. Problems
• Find the inverse Z-transform of the following equation using
Convolution method. (R2 eg 3.40)
RAJENDRA CHARY 61

62. RAJENDRA CHARY 62
Problem
• Construct the block diagram for the discrete-time systems whose
input-output relations are described by the following difference
equations: (R2 eg4.1)
a)y(n)=0.7x(n)+0.3x(n-1)
b)y(n)=0.5y(n-1)+0.8x(n)+0.4x(n-1)

63. RAJENDRA CHARY 63
Canonical and Non-Canonical Structure
• When the number of delays in a structure is equal to the order of
the system, the structure is called canonical structure.
• When the number of delays in a structure is not equal to the order
of the system, the structure is called non-canonical structure.

64. Basic Structures of IIR Systems
• If the impulse response of the system is of infinite duration, then
the system is called IIR system.
• The convolution formula for IIR systems is given by:
RAJENDRA CHARY 64

65. Basic Structures of IIR systems(contd.)
• In general, an IIR system is described by the difference equation:
RAJENDRA CHARY 65

66. Basic Structures of IIR systems(contd.)
RAJENDRA CHARY 66

67. RAJENDRA CHARY 67
Basic Structures of IIR systems(contd.)
• For each of these equations, we can construct a block diagram
consisting of delays, adders and multipliers. Such block diagrams
are referred to as structure for realization of system.
• The main advantage of re-arranging the sets of difference equation
is to reduce the computational complexity, memory requirements
and finite length effects in computations.

68. RAJENDRA CHARY 68
Basic Structures of IIR systems(contd.)
• The basic structures for realizing IIR systems are:
Direct form-I
Direct form-II
Cascade form
Parallel form

69. RAJENDRA CHARY 69
Lecture-6&7

70. Direct form-I
RAJENDRA CHARY 70

71. RAJENDRA CHARY 71
Direct form-I(contd.)
• The equation of Y(z) can be directly represented by a block
diagram as shown in figure(next slide).This structure is called
direct form-I structure.
• The direct form-I requires separate delays for input and output.
Hence for realizing the structure, more memory is required.

72. Direct form-I(contd.) (R3 fig3.15,pg3.75)
RAJENDRA CHARY 72

73. RAJENDRA CHARY 73
Direct form-I(contd.)
• From the direct form-I structure, it is observed that the realization
of Nth order discrete-time system with M number of zeros and N
number of poles involves M+N+1 multiplications ,M+N additions
and M+N+1 memory locations .
• The direct form-I is a non-canonical structure.

74. RAJENDRA CHARY 74
Limitations of Direct form-I
• Since the number of delay elements used in direct form-I is more
than the order of difference equation, it is not effective.
• It lacks hardware flexibility.
• There are chances of instability due to the quantization noise.

75. Direct form-II
RAJENDRA CHARY 75

76. Direct form-II(contd.)
RAJENDRA CHARY 76

77. RAJENDRA CHARY 77
Direct form-II(contd.)
• The equations of W(z) and Y(z) represent the IIR system and the
structure realized is called direct form-II structure.

78. Direct form-II(contd.) (R3 pg 3.77)
RAJENDRA CHARY 78

79. RAJENDRA CHARY 79
Direct form-II(contd.)
• From the direct form-II structure, it is observed that the realization
of Nth order discrete-time system with M number of zeros and N
number of poles involves M+N+1 multiplications ,M+N additions
and max(M,N) memory locations .
• The direct form-II is a canonical structure.

80. RAJENDRA CHARY 80
Limitations of Direct form-II
• It also lacks hardware flexibility.
• There are chances of instability due to the quantization noise.

81. Cascade form
• Consider an IIR system with system function
H(z)=H1(z)H2(z)……Hk(z)
This can be represented using block diagram as:
RAJENDRA CHARY 81

82. Cascade form(contd.)
• Now realize each Hk(z) in direct form-II(or direct form-I) and
cascade all structures.
• For example let us consider a system whose transfer function is
RAJENDRA CHARY 82

83. Cascade form(contd.)
• Realizing H1(z) and H2(z) in direct form-II and cascading we
obtain a structure of cascade form.
RAJENDRA CHARY 83

84. RAJENDRA CHARY 84
Cascade form(contd.)
The difficulties in cascade structure are:
• Decision of pairing poles and zeros
• Deciding the order of cascading the first and second order sections.
• Scaling multipliers should be provided between individual sections
to prevent the filter variables from becoming too large or too small.

85. Parallel form
• The transfer function H(z) of a discrete time system can be
expressed as a sum of first and second-order sections using partial
fraction expansion as:
RAJENDRA CHARY 85

86. RAJENDRA CHARY 86
Parallel form(contd.)
• Each first and second order section is realized either in direct form-
I or direct form-II structure and the individual sections are
connected in parallel to obtain the overall system as shown in
figure(next slide).
• This structure is used for high speed filtering applications.
• The difficulty with this structure is expressing the transfer function
in partial fraction form is not easy for higher order systems.

87. Parallel form(contd.) (R2 pg4.13)
RAJENDRA CHARY 87

88. RAJENDRA CHARY 88
Problem
• Obtain the direct form-I realization for the system described by
difference equation
y(n)=0.5y(n-1)-0.25y(n-2)+x(n)+0.4x(n-1) (R1 eg5.21)
• Obtain the direct form-II realization for the following system
y(n)=-0.1y(n-1)+0.72y(n-2)+0.7x(n)-0.252x(n-2) (R1 eg5.23)

89. RAJENDRA CHARY 89
Problem
• An LTI system is described by the equation
y(n)+2y(n-1)-y(n-2)=x(n).
Determine the cascade and parallel realization structures of the
system. (R2 eg4.10)

90. RAJENDRA CHARY 90
Lecture-8

91. Problem
• Obtain the direct form-I, direct form-II, cascade and parallel form
realizations of the LTI system governed by the equation (R2 eg4.6)
RAJENDRA CHARY 91

92. Problem
• Determine
structures of:
the direct form-I,direct form-II,cascade,parallel
(DTSP Oppenheim)
RAJENDRA CHARY 92

93. Problem
• Obtain the direct form-I, direct form-II, cascade and parallel form
realizations of the LTI system governed by the equation.(R3
eg3.26)
RAJENDRA CHARY 93

94. Problem
• Obtain the direct form-I, direct form-II, cascade and parallel form
realizations of the LTI system governed by the equation.(R3
E3.15)
RAJENDRA CHARY 94

95. RAJENDRA CHARY 95
Lecture-9

96. Transposed form realization of IIR System
RAJENDRA CHARY 96

97. Transposed form realization of IIR with
direct form-I system(R2 fig 4.10a pg.288)
RAJENDRA CHARY 97

98. Transposed form realization of IIR with
direct form-II system(R2 fig 4.10b pg.288)
RAJENDRA CHARY 98

99. RAJENDRA CHARY 99
Problem
• Determine the direct form-II and transposed direct form-II for the
given system y(n)=1/2*y(n-1)-1/4*y(n-2)+x(n)+x(n-1) (R1 eg5.24)
(A12 supply)

100. Direct form-II
RAJENDRA CHARY 100

101. Transpose form
RAJENDRA CHARY 101

102. Basic Structures of FIR systems
• If the impulse response of the system is of finite duration, then
the system is called FIR system.
• The convolution formula for FIR systems is given by:
RAJENDRA CHARY 102

103. Basic Structures of FIR systems(contd.)
RAJENDRA CHARY 103

104. Basic Structures of FIR systems(contd.)
RAJENDRA CHARY 104

105. RAJENDRA CHARY 105
Basic Structures of FIR systems(contd.)
The basic structures of FIR system are:
• Direct form/Transversal form/tapped delay line filter form
• Cascade form

106. Direct form realization
• Since there are no poles in an FIR system, the direct form has only
one structure. The direct form structure can be obtained from the
general equation for Y(z) or y(n) governing the FIR system.
RAJENDRA CHARY 106

107. Direct form realization (contd.)
RAJENDRA CHARY 107

108. RAJENDRA CHARY 108
Direct form realization (contd.)
• From the direct form structure, it is observed that the realization of
an Nth order FIR discrete time system involves N number of
multiplications ,N-1 number of additions and N-1 delays.
• The direct form structure is a canonical structure.

109. Cascade form realization
• The block diagram representation of cascade form is
RAJENDRA CHARY 109

110. Cascade form realization(contd.)
• The transfer function of FIR system, H(z) is an (N-1)th
polynomial in z.
order
• When N is odd,(N-1) will be even and so H(z) will have (N-1)/2
second order factors.
RAJENDRA CHARY 110

111. RAJENDRA CHARY 111
Cascade form realization(contd.)
• When N is odd, each second order factor of H(z) can be realized in
direct form and all the second order systems are connected in
cascade to realize H(z) as shown in figure(next slide).

112. Cascade structure of FIR System when
N=odd
RAJENDRA CHARY 112

113. Cascade form realization(contd.)
• When N is even,(N-1) will be odd and so H(z) will have one first
order factor and (N-2)/2 second order factors.
• When N is even,
• Each one of them can be realized in direct form and all of them are
connected in cascade as shown in figure(next slide)
RAJENDRA CHARY 113

114. Cascade structure of FIR System when
N=even
RAJENDRA CHARY 114

115. Problem
• Determine the direct form realization of system function
H(z)=1+2z-1-3z-2-4z-3+5z-4 (R1 eg6.22)
RAJENDRA CHARY 115

116. Problem
• Draw the direct form structure of the FIR system described by the
transfer function (R3 eg3.34)
RAJENDRA CHARY 116

117. Direct form structure of
problem
previous slide
RAJENDRA CHARY 117

118. Problem
• Obtain the cascade realization of system function (R1 eg6.23)
H(z)=(1+2z-1-z-2)(1+z-1-z-2)
RAJENDRA CHARY 118

119. Problem
• Obtain the cascade realization of system function (R1 eg6.24)
H(z)= 1+5/2*z-1+2z-2+2z-3
RAJENDRA CHARY 119

120. Transposed form structure realization of FIR
System
RAJENDRA CHARY 120

121. Transposed form structure realization of FIR
System(contd.)
RAJENDRA CHARY 121

122. RAJENDRA CHARY 122
Problem
• Realize the second order FIR system y(n)=2x(n)+4x(n-1)-3x(n-2)
by using the transposed form structure. (R2 eg4.13)

123. Direct form-I v/s Direct form-II
RAJENDRA CHARY 123

124. IIR System v/s FIR System
RAJENDRA CHARY 124

125. RAJENDRA CHARY 125
Application
• Design of digital system function to meet the given specifications.

126. Appendix(Z-transforms)
RAJENDRA CHARY 126

127. Appendix(Z-transforms)
RAJENDRA CHARY 127

128. RAJENDRA CHARY 128
Conclusion of the Unit
In this unit we studied about:
• Review of Z-transform, Applications of Z–transforms
• System function
• Block diagram representation of linear constant-coefficient difference
equations
• Basic structures of IIR systems, Transposed forms
• Basic structures of FIR systems
• Applications
• Appendix

129. RAJENDRA CHARY 129
Thank you
For Your Attention !
Any Questions?