1. Chapter 6 Discrete-Time System

2. Discrete time system Discrete time system where and are multiplier
(6-1)
where and are multiplier
D is delay element
Fig. 6-1.

3. Difference equation Difference equation for discrete-time system If
(6-2)
where and are constants or functions of n
(6-3)

4. Example 6-1
Moving average If
(6-4)

5. Example 6-2
Integration If
(6-5)
(6-6)

6. Linear time-invariant system
Functional relationship of discrete-time system
(6-7)
where is impulse response of system
Fig. 6-2.

7. Time-invariant system
Linear system
Principle of superposition
Time-invariant system
(6-8)
(6-9)

8. Linear system
Non-linear system
(6-10)
(6-11)

9. Transfer function for discrete-time system
Difference equation of discrete-time system
Applying z-transform gives
(6-12)
(6-13)

10. Transfer function If where is impulse response of system (6-14) (6-15)
(6-16)
where is impulse response of system

11. Impulse response of discrete-time system
Inverse z-transform
If is expressed in power series expansion, i.e.,
then coefficients of z-transform become impulse response.
(6-17)

12. For
(6-19)

13. Example 6-3
Using power series expansion

14. Difference equation from transfer function
Impulse response with initial condition y(-1) = 0

15. Example 6-4
z-transform
Inverse z-transform
(6-20)
(6-21)

16. If initial value Transfer function Inverse z-transform (6-22) (6-23)
(6-24)

17. Stability BIBO(bounded input, bounded output) Proof is bounded
Necessary and sufficient condition for bounded
(6-25)
(6-26)
(6-27)

18. Raible’s tabulation to
(6-28)
Table 6-1. Raible’s tabulation

19. Singular case and for If part or all factors in the first row is 0,
replace z by
and for
(6-29)
n-th order case, use approximation
(6-30)

20. Example 6-5
and
Two poles exists inside the unit circle

21. Example 6-6
Poles are located at :
(6-31)

22. Singular case
and
 One pole is outside unit circle

23. Example 6-7
 Three poles exist on the unit circle
(6-32)

24. Description of Pole-Zero
Transfer function for discrete-time system
(6-33)
where K is gain
Poles are located at:
Zeros are located at:

25. Description of poles and zeros in z-plane
Fig. 6-3.

26. Example 6-8
Fig. 6-4.

27. Example 6-9
Fig. 6-5.
K=0.2236

28. Frequency response
Frequency response of discrete-time system
(6-34)

29. Example 6-10
Using Euler’s formula
Substituting gives

30. For

31. Fig. 6-6.

32. Using pole-zero plot Pole is located at : Zero is located at :
Magnitude response
Phase response
Fig. 6-7.

33. Realization of system Unit delay (6-35) Adder/subtractor (6-36)
Constant multiplier
(6-37)
Branching
(6-38)
Signal multiplier
(6-39)
Fig. 6-8.

34. Direct form
Direct form 1
Difference equation
(6-40)
(6-41)

35. (a)
(b)
Fig. 6-9.

36. Direct form 2 Inverse transform for Eq (6-43) and (6-44) (6-42) (6-43)
(6-45)
(6-46)

37. (a)
(b)
Fig

38. Example 6-11
Direct form 1
Direct form 2

39. (a)
(b)
Fig

40. Finite wordlength effect
Quantization errors from parameters
Input/output signal quantization
Arithmetic roundoff errors
Coefficient quantization

41. Effect of coefficient quantization
Transfer function of IIR filter
Converting into quantized coefficients and
(6-47)
(6-48)
where and
additive noise

42. Coefficient quantization error
Transfer function of digital filter
After coefficient quantization
Quantization step :
 It’s unstable system

43. Cascade and parallel forms
Cascade form or series form
First-order
Second-order
(6-49)
(6-50)
(6-51)

44. Fig
(a)
(b)
Fig

45. Parallel form
First-order
Second-order
(6-52)
(6-53)
(6-54)

46. Fig

47. (a)
(b)
Fig

48. Example 6-12
Cascade form
Quantization error of parameters is decreased

49. Parallel form
Calculating coefficients gives
Multiplying each term by

50. (a)
(b)
Fig

51. Example 6-13
Fig

52. Example 6-14
Calculating coefficients gives
(6-55)
Substituting

53. Multiplying each term by
Fig

54. Realization of FIR system
Direct form
FIR difference equation
Tapped delay line or transversal filter
(6-56)
Fig

55. Cascade form
(6-57)
where
Fig

56. Linear phase FIR system
N is even : type 1 and type 3
(6-58) or
(6-59)

57. N is odd : type 2 and type 4 Type 1 Type 3 Type 2 Type 4 (6-60) (6-61)
(6-62)
(6-63)

58. (a)
(b)
Fig

59. Lattice structure Lattice structure for FIR filter Difference equation
Linear prediction
(6-64)
(6-65)
(6-66)
Error between and
FIR filter uses linear predictor
(6-67)
where is prediction coefficient

60. For N=1 Single-stage FIR lattice structure
(6-68)
(6-69)
where is reflection coefficient
Fig

61. For N=2
(6-70)
Fig

62. For N=3
Substituting
(6-71)
Fig

63. For N=M If
(6-72)
where
(6-73)
(6-74)

64. Fig

65. Example 6-15
(1)

66. (2) Calculation of coefficients
From Fig and Eq. (6-73)
(6-75)
Fig

67. Calculation of filter coefficient From Eq. (6-75)
m-stage
(6-76)
(6-77)
Substituting
(6-78)

68. For
(6-79)
Substituting in Eq. (6-74)
(6-80)

69. Example 6-16
from example 6-15

70. General form of calculating filter coefficient
From Eq. (6-73)
Substituting M=m
Substituting z=1/z
(6-81)
(6-82)
(6-83)
(6-84)

71. For

72. In Eq. (6-80)
Substituting and
(6-85)

73. Dividing by
Coefficient
(6-86)

74. Example 6-17

75. Calculating coefficient of
For

76. For

77. from Eq. (6-86)

78. Fig

79. Lattice structure of IIR filter
All-pole system
Difference equation
(6-87)
(6-88)
(6-89)
(6-90)

80. For N=1
(6-91)
Fig

81. For N=2
(6-92)
Fig

82. For N th order
(6-93)
Fig

83. General form of IIR system
Lattice-ladder structure
(6-94)
All-pole lattice structure
Ladder structure

84. Lattice-ladder structure System output
Transfer function
(6-95)
(6-96)

85. From Eq. (6-94) and (6-96)
(6-97)
where
(6-98)

86. Fig

87. Example 6-18
All-pole system
Fig

88. Example 6-19
Equation for each node

89. Comparison of coefficients

90. Fig

91. Introduction of digital filter
Real-time digital filter with analog input and output signal
Sampling the bandlimited analog signal periodically
Converting into a series of digital samples
Implementation of the filtering operation in digital processor
Mapping the input sequence into the output sequence
Converting the digitally filtered output into analog values by using DAC
Smoothing and removing unwanted high frequency components
Fig

92. Advantage of digital filter compared with analog filters
A truly linear phase response
The performance of digital filters does not vary with environmental change
The frequency response of a digital filter can be automatically adjusted if it is implemented using a programmable processor
Several input signals or channels can be filtered by one digital filter without the need to replicate the hardware
Both filtered and unfiltered data can be saved for further use

93. Miniaturization and low power consumption by VLSI
The precision achievable with analog filters is restricted
The performance of digital filters is repeatable from unit to unit
Digital filters can be used at very low frequencies

94. Disadvantage of digital filters compared with analog filters
Speed limitation
Finite wordlength effect
Long design and development times

95. Types of digital filters : FIR and IIR filters
FIR filter
(6-99)
(6-100)

96. IIR filtering equation of recursive form
Alternative representations for FIR and IIR filters
Transfer function
where and are coefficients of filter
IIR is feedback system of some sort
(6-102)
(6-103)

97. Choosing between FIR and IIR filters
Relative advantages of the two filter types
FIR filters can have an exactly linear phase response
FIR filters are always stable. The stability of IIR filters cannot always be guaranteed
The effect of using a limited number of bits to implement filters
Roundoff noise and coefficient quantization errors are much less severe in FIR than in IIR
FIR requires more coefficients for sharp cutoff filters than IIR
Analog filters can be readily transformed into equivalent IIR digital filters meeting similar specifications
In general, FIR is algebraically more difficult to synthesize

98. Guideline on when to use FIR or IIR
Use IIR when the only important requirements are sharp cutoff filters and high throughput
Use FIR if the number of filter coefficients is not too large and, in particular, if little or no phase distortion is desired