1. Structures for Discrete-Time Systems
主講人：虞台文

2. Content Introduction Block Diagram Representation Signal Flow Graph
Basic Structure for IIR Systems
Transposed Forms
Basic Structure for FIR Systems
Lattice Structures

3. Structures for Discrete-Time Systems
Introduction

4. Characterize an LTI System
Impulse Response
z-Transform
Difference Equation

5. Example
Noncomputable
Computable

6. Basic Operations Addition Multiplication Delay
In fact, there are unlimited variety of computational structures.
Computable

7. Why Implement Using Different Structures?
Finite-precision number representation of a digital computer.
Truncation or rounding error.
Modeling methods:
Block Diagram
Signal Flow Graph

8. Block Diagram Representation+
x1(n)
x2(n)
x1(n) + x2(n)
Adder
x(n) a
ax(n)
Multiplier
x(n)
x(n1)
z1
Unit Delay

9. Example + b a1
z1 a2
x(n)
y(n)
y(n1)
y(n2)

10. Higher-Order Difference Equations

11. Block Diagram Representation (Direct Form I)+
z1 b0 b1
bM1 bM
x(n)
x(n1)
x(n2)
x(nM) a1
aN1 aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)

12. Block Diagram Representation (Direct Form I)+
z1 b0 b1
bM1 bM
x(n)
x(n1)
x(n2)
x(nM) a1
aN1 aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)

13. Block Diagram Representation (Direct Form I)+
z1 b0 b1
bM1 bM
x(n)
x(n1)
x(n2)
x(nM) a1
aN1 aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)

14. Block Diagram Representation (Direct Form I)
Implementing zeros
Implementing poles
Block Diagram Representation (Direct Form I) +
z1 b0 b1
bM1 bM
x(n)
x(n1)
x(n2)
x(nM) a1
aN1 aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)

15. Block Diagram Representation (Direct Form I)
How many Adders?
How many multipliers?
How many delays?
Block Diagram Representation (Direct Form I) +
z1 b0 b1
bM1 bM
x(n)
x(n1)
x(n2)
x(nM) a1
aN1 aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)

16. Block Diagram Representation (Direct Form II)+
z1 b0 b1
bN1 bN
x(n) a1
aN1 aN
y(n)
w(n1)
w(n2)
w(nN)
w(n)
Assume
M = N

17. Block Diagram Representation (Direct Form II)+
z1 b0 b1
bN1 bN
x(n) a1
aN1 aN
y(n)
w(n1)
w(n2)
w(nN)
w(n)
Assume
M = N

18. Block Diagram Representation (Direct Form II)
Implementing poles
Implementing zeros
Block Diagram Representation (Direct Form II) +
z1 b0 b1
bN1 bN
x(n) a1
aN1 aN
y(n)
w(n1)
w(n2)
w(nN)
w(n)
Assume
M = N

19. Block Diagram Representation (Direct Form II)
How many Adders?
How many multipliers?
How many delays?
Block Diagram Representation (Direct Form II) +
z1 b0 b1
bN1 bN
x(n) a1
aN1 aN
y(n)
w(n1)
w(n2)
w(nN)
w(n)
Assume
M = N

20. Block Diagram Representation (Canonic Direct Form)+ b0 b1
bN1 bN
x(n)
z1 a1
aN1 aN
y(n)
Assume
M = N

21. Block Diagram Representation (Canonic Direct Form)
How many Adders?
How many multipliers?
How many delays? max(M, N)
Block Diagram Representation (Canonic Direct Form) + b0 b1
bN1 bN
x(n)
z1 a1
aN1 aN
y(n)
Assume
M = N

22. Structures for Discrete-Time Systems
Signal Flow Graph

23. Nodes And Branches wj(n) wk(n)
Associated with each node is a variable or node value.
wj(n)
wk(n)

24. Nodes And Branches Input wj(n) wj(n) wk(n) Brach (j, k)
Output: A linear transformation of input, such as constant gain and unit delay.
wj(n)
wk(n)
Brach (j, k)
Each branch has an input signal and an output signal.

25. More on Nodes wj(n) wk(n)
An internal node serves as a summer, i.e., its value is the sum of outputs of all branches entering the node.
wj(n)
wk(n)

26. Source Nodes
Nodes without entering branches
xj(n)
wk(n)
Source node j

27. Sink Nodes yk(n) wj(n) Nodes that have only entering branches
Sink node k

28. Example
x(n)
y(n)
w1(n)
w2(n) a b c d e
Source
Node
Sink
Node

29. Block Diagram vs. Signal Flow Graph
x(n)
w(n)
y(n) + a
z1 b1 b0 a b1 b0
z1 1 2 3 4
w1(n)
x(n)
y(n)
w2(n)
w3(n)
w4(n)

30. Block Diagram vs. Signal Flow Graph
x(n) + a
z1 b1 b0
w(n)
y(n)
w1(n)
w2(n)
w3(n) 1 2 3 4
w4(n)

31. Block Diagram vs. Signal Flow Graph

32. Structures for Discrete-Time Systems
Basic Structure for IIR Systems

33. Criteria Reduce the number of constant multipliers
Increase speed
Reduce the number of delays
Reduce the memory requirement
Modularity: VLSI design
The effects of finite register length and finite-precision arithmetic.

34. Basic Structures
Direct Forms
Cascade Form
Parallel Form

35. Direct Forms

36. Direct Form I x(n) v(n) y(n) b0 b1 x(n1) x(n2) x(nN) b2 bN-1 bNa1 a2
aN-1 aN
y(n1)
y(n2)
y(nN)
y(nN+1)
z1
v(n)

37. Direct Form I x(n) v(n) y(n) b0 b1 x(n1) x(n2) x(nN) b2 bN-1 bNa1 a2
aN-1 aN
y(n1)
y(n2)
y(nN)
y(nN+1)
z1
v(n)

38. Direct Form II
x(n)
y(n)
w(n) b0 b1 b2
bN-1 bN a1 a2
aN-1 aN
z1

39. Direct Form II
x(n)
y(n)
w(n) b0 b1 b2
bN-1 bN a1 a2
aN-1 aN
z1

40. Example x(n) y(n) x(n) y(n) Direct Form I Direct Form II z1 2 0.75
0.125
x(n)
y(n)
z1 2
Direct Form II
0.75
0.125

41. Cascade Form

42. Cascade Form

43. Cascade Form
2nd Order
System

44. Cascade Form x(n) y(n) z1 a11 a21 b11 b21 b01 z1 a12 a22 b12 b22 b01

45. Another Cascade Form

46. Parallel Form

47. Parallel Form
Group
Real Poles
Complex Poles
Poles at zero
Real Poles

48. Parallel Form
z1
a1k
a2k
e0k
e1k

49. Parallel Form
x(n)
y(n)

50. Example 8
x(n)
y(n)
z1
0.75
0.125 7

51. Example
z1
0.5 18 8
x(n)
y(n)
0.25
25

52. Structures for Discrete-Time Systems
Transposed Forms

53. Signal Flow Graph Transformation
To transform signal graphs into different forms while leaving the overall system function between input and output unchanged.

54. Transposition of Signal Flow Graph
Reverse the directions of all arrows.
Changes the roles of input and output.
z1 a
z1 a
x(n)
y(n)
y(n)
x(n)

55. Transposition of Signal Flow Graph
Are there any relations between the two systems?
x(n)
y(n)
z1 a

56. Example:
x(n)
y(n)
z1 a
y(n)
x(n)
x(n)
y(n)
z1 a
z1 a

57. Transposition of Signal Flow Graph
Reverse the directions of all arrows.
Changes the roles of input and output.
x(n)
y(n)
z1 a
Detail proof see reference

58. Structures for Discrete-Time Systems
Basic Structure for FIR Systems

59. FIR
For causal FIR systems, the system function has only zeros.

60. Direct Form
x(n)
y(n)
z1
h(0)
h(1)
h(2)
h(M1)
h(M)

61. Direct Form x(n) y(n) y(n) x(n) z1 h(0) h(1) h(2) h(M1) h(M) z1

62. Direct Form x(n) y(n) y(n) x(n) z1 h(0) h(1) h(2) h(M1) h(M) z1

63. Cascade Form

64. Cascade Form
x(n)
y(n)
z1
b01
b11
b21
b02
b12
b22
b1Ms
b2Ms
b0Ms

65. Structures for Linear Phase Systems
A generalized linear phase system satisfies:
h(Mn) = h(n) for n = 0,1,…,M or
h(Mn) = h(n) for n = 0,1,…,M
M is even
M is odd
h(Mn) = h(n)
h(Mn) = h(n)
Type I
Type II
Type III
Type VI

66. Type I

67. Type I
x(n)
y(n)
z1
h(M/2)
h(M/21)
h(0)
h(1)
h(2)

68. Type II, III and VI
Construct them in a similar manner by yourselves.

69. Structures for Discrete-Time Systems
Lattice Structures

70. Consider x(n)=(n), one will see
FIR Lattice

71. Consider x(n)=(n), one will see
FIR Lattice

72. Consider x(n)=(n), one will see
FIR Lattice

73. Define
Consider x(n)=(n), one will see
FIR Lattice

74. Define
FIR Lattice
Show that

75. FIR Lattice
FIR Lattice
i=1:
Show that

76. FIR Lattice FIR Lattice i = n: Assumed true i = n+1 also true.
Prove
i = n+1 also true.
Show that

77. FIR Lattice
FIR Lattice =

78. FIR Lattice
FIR Lattice

79. FIR Lattice FIR Lattice Given the lattice, to find A(z). m=0 k1 k2 k3

80. FIR Lattice FIR Lattice Given A(z), to find the lattice. m=0 m=1 m=2

81. FIR Lattice FIR Lattice Given A(z), to find the lattice. m=0 m=1 m=2

82. Example 1
m=0
m=1
m=2
m=3
0.6728
0.7952
0.9
0.1820
0.64
0.576

83. Example
0.576
0.1820  1
m=0
m=1
m=2
m=3
0.6728
0.7952
0.1820
0.9
0.64
0.576

84. Inverse Filter

85. All-Pole Filter

86. All-Pole Filter

87. All-Pole Filter

88. All-Pole Filter

89. Example
0.576
0.1820 
0.6728
0.6728
0.1820
0.1820
0.576
 0.576

90. Example
0.9
 0.64
0.576
0.6728
0.6728
0.1820
0.1820
0.576
 0.576

91. Stability of All-Pole Filter
All zeros of A(z) have to lie within the unit circle.
Necessary and sufficient conditions:
All of k-parameters ki’s satisfy |ki| < 1.

92. Normalized Lattice

93. Normalized Lattice

94. Normalized Lattice
Section i

95. Normalized Lattice
Section i
Section N
N1 1

96. Normalized Lattice
Section i
Three-Multiplier Form

97. Normalized Lattice Four-Multiplier, Kelly-Lochbaum Form
Three-Multiplier Form
Four-Multiplier, Normalized Form

98. Normalized Lattice
Three-Multiplier Form
Section N
N1 1

99. Normalized Lattice
Four-Multiplier, Normalized Form
Section N
N1 1

100. Normalized Lattice Four-Multiplier, Kelly-Lochbaum Form Section N N1

101. Lattice Systems with Poles and Zeros
Section
N1 1 N c0 c1
cN2
cN1 cN

102. Lattice Systems with Poles and Zeros
Section
N1 1 N c0 c1
cN2
cN1 cN

103. Lattice Systems with Poles and Zeros

104. Example
0.6728
0.6728
0.1820
0.1820
0.576
 0.576 c3 c2 c1 c0

105. Example 1 c3 c2 c1 c0 m=0 m=1 m=2 m=3 0.6728 0.7952 0.1820 0.9 0.64
0.6728
0.1820
0.1820
0.576
 0.576 c3 c2 c1 c0
Example 1
m=0
m=1
m=2
m=3
0.6728
0.7952
0.1820
0.9
0.64
0.576

106. Example
0.6728
0.6728
0.1820
0.1820
0.576
 0.576 1
3.9
5.4612
4.5404