1. Discrete-Time Signals and Systems
主講人：虞台文

2. Content Introduction Discrete-Time Signals---Sequences
Linear Shift-Invariant Systems
Stability and Causality
Linear Constant-Coefficient Difference Equations
Frequency-Domain Representation of Discrete-Time Signals and Systems
Representation of Sequences by Fourier Transform
Symmetry Properties of Fourier Transform
Fourier Transform Theorems
The Existence of Fourier Transform
Important Transform Pairs

3. Discrete-Time Signals and Systems
Introduction

4. The Taxonomy of Signals
Signal: A function that conveys information
Time
Amplitude
analog signals
continuous-time
signals
discrete-time
digital signals
Continuous
Discrete

5. Signal Process Systems
Facilitate the extraction of desired information e.g.,
Filters
Parameter estimation
Signal
Processing
System
signal
output

6. Signal Process Systems
analog
system
signal
output
continuous-time signal
discrete-
time
system
signal
output
discrete-time signal
digital
system
signal
output
digital signal

7. Signal Process Systems
A important class of systems
Linear Shift-Invariant Systems.
In particular, we’ll discuss
Linear Shift-Invariant Discrete-Time Systems.

8. Discrete-Time Signals and Systems
Discrete-Time Signals---Sequences

9. Representation by a Sequence
Discrete-time system theory
Concerned with processing signals that are represented by sequences. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n
x(n)

10. Important Sequences Unit-sample sequence (n) Sometime call (n)
a discrete-time impulse; or
an impulse 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n
(n)

11. Important Sequences Unit-step sequence u(n) Fact: u(n) n 1 2 3 4 5 6 78 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n
u(n)

12. Important Sequences Real exponential sequence . . . x(n) n 1 2 3 4 5 67 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n
x(n)
. . .

13. Important Sequences
Sinusoidal sequence n
x(n)

14. Important Sequences
Complex exponential sequence

15. Important Sequences
A sequence x(n) is defined to be periodic with period N if
Example: consider
must be a rational number

16. Energy of a Sequence
Energy of a sequence is defined by

17. Operations on Sequences
Sum
Product
Multiplication
Shift

18. Sequence Representation Using delay unit1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 n
x(n) a1 a2 a7
a-3

19. Discrete-Time Signals and Systems
Linear Shift-Invariant Systems

20. Mathematically modeled as a unique transformation or operator.
Systems
T [ ]
y(n)=T[x(n)]
x(n)
Mathematically modeled as a unique transformation or operator.

21. Linear Systems
T [ ]
x(n)
y(n)=T[x(n)]

22. Examples: y(n)=T[x(n)] x(n) T [ ] Ideal Delay System Moving Average
Accumulator

23. Examples: Are these system linear? y(n)=T[x(n)] x(n) T [ ]
Ideal Delay System
Accumulator
Moving Average
T [ ]
x(n)
y(n)=T[x(n)]
Are these system linear?

24. Examples: y(n)=T[x(n)] x(n) Is this system linear? T [ ]
A Memoryless System
Is this system linear?

25. Linear Systems
T [ ]
x(n)
y(n)=T[x(n)]
時間k之impulse
於時間n時之輸出值

26. Shift-Invariant Systems
x(n)
y(n)=T[x(n)]
T [ ]
x(nk)
y(nk)
y(n)
x(n)
y(n-1)
x(n-1)
x(n-2)
y(n-2)

27. Shift-Invariant Systems
x(n)
y(n)=T[x(n)]
T [ ]
x(n-k)
y(n-k)
y(n)
x(n-1)
y(n-1)
x(n-2)
y(n-2)
輸入/輸出關係僅與時間差有關

28. Linear Shift-Invariant Systems
x(n)
y(n)=T[x(n)]
時間k之impulse
於時間n時之輸出值
僅與時間差有關

29. Impulse Response
h(n)=T[(n)]
x(n)=(n)
T [ ]

30. Convolution Sum h(n) (n) x(n) y(n) T [ ] convolution
A linear shift-invariant system is completely characterized by its impulse response.

31. Characterize a System
h(n)
x(n)
x(n)*h(n)

32. Properties of Convolution Math

33. Properties of Convolution Math
h1(n)
x(n)
h2(n)
y(n)
h2(n)
x(n)
h1(n)
y(n)
h1(n)*h2(n)
x(n)
y(n)
These systems are identical.

34. Properties of Convolution Math
h1(n)
x(n)
h2(n)
y(n) +
h1(n)+h2(n)
x(n)
y(n)
These two systems are identical.

35. Example
y(n)=? 1 2 3 4 5 6 1 2 3 4 5 6

36. Example 1 2 3 4 5 6 k
x(k) 1 2 3 4 5 6 k
h(k) 1 2 3 4 5 6 k
h(0k)

37. Example compute y(0) compute y(1) How to computer y(n)? x(k) k h(0k)1 2 3 4 5 6 k
x(k)
compute y(0) 1 2 3 4 5 6 k
h(0k)
compute y(1) 1 2 3 4 5 6 k
h(1k)
How to computer y(n)?

38. Example Two conditions have to be considered. n<N and nN.1 2 3 4 5 6 k
x(k)
h(0k)
h(1k)
compute y(0)
compute y(1)
How to computer y(n)?
n<N and nN.

39. Example
n < N
n  N

40. Example
n < N
n  N

41. Impulse Response of the Ideal Delay System
By letting x(n)=(n) and y(n)=h(n),
(n nd) 1 2 3 4 5 6 nd

42. Impulse Response of the Ideal Delay System
你必須知道
(n nd)扮演如下功能：
Shift; or
Copy
(n nd) 1 2 3 4 5 6 nd

43. Impulse Response of the Moving Average
M1  0  M2
. . .
你能以(n k)解釋嗎?

44. Impulse Response of the Accumulator
. . .
你能解釋嗎?

45. Discrete-Time Signals and Systems
Stability and Causality

46. Stability
Stable systems --- every bounded input produce a bounded output (BIBO)
Necessary and sufficient condition for a BIBO

47. Prove Necessary Condition for Stability
Show that if x is bounded and S < , then y is bounded.
where M = max x(n)

48. Prove Sufficient Condition for Stablility
Show that if S = , then one can find a bounded sequence x such that y is unbounded.
Define

49. Example:
Show that the linear shift-invariant system with impulse response h(n)=anu(n) where |a|<1 is stable.

50. Causality
Causal systems --- output for y(n0) depends only on x(n) with n n0.
A causal system whose impulse response h(n) satisfies

51. Discrete-Time Signals and Systems
Linear Constant-Coefficient Difference Equations

52. N-th Order Difference Equations
Examples:
Ideal Delay System
Moving Average
Accumulator

53. Compute y(n)

54. The Ideal Delay System x(n) y(n) y(n) x(n) . . . Delay
nd sample delays
x(n)
y(n)

55. The Moving Average

56. The Moving Average
Attenuator +
M+1 sample
delay
Accumulator
system _

57. Discrete-Time Signals and Systems
Frequency-Domain Representation of
Discrete-Time Signals and Systems

58. Sinusoidal and Complex Exponential Sequences
Play an important role in DSP
LTI
h(n)

59. Frequency Response
eigenvalue
eigenfunction

60. Frequency Response
phase
magnitude

61. Example: The Ideal Delay System
magnitude
phase

62. Example: The Ideal Delay System

63. Periodic Nature of Frequency Response

64. Periodic Nature of Frequency Response 
2
3
4 2 3 4

65. Periodic Nature of Frequency Response
Generally, we choose

To represent one period in frequency domain.  
2
3
4 2 3 4

66. Periodic Nature of Frequency Response 
High
Frequency
Low
Frequency

67. Ideal Frequency-Selective Filters  c
c 1 a
a b
b
Lowpass Filter
Bandstop Filter
Highpass Filter

68. Moving Average
h(n) M

69. Moving Average

70. M=4
Lowpass
Try larger M
Moving Average

71. Discrete-Time Signals and Systems
Representation of Sequences by
Fourier Transform

72. Fourier Transform Pair
Synthesis
Inverse Fourier Transform
(IFT)
Analysis
Fourier Transform
(FT)

73. Prove
n = m

74. Prove
n  m

75. Prove
= x(n)

76. Inverse Fourier Transform
Notations
Synthesis
Inverse Fourier Transform
(IFT)
Analysis
Fourier Transform
(FT)

77. Real and Imaginary Parts
Fourier Transform (FT)
is a complex-valued function

78. Magnitude and Phase
magnitude
phase

79. Discrete-Time Signals and Systems
Symmetry Properties of Fourier Transform

80. Conjugate-Symmetric and Conjugate-Antisymmetric Sequences
Conjugate-Symmetric Sequence
Conjugate-Antisymmetric Sequence
an even sequence if it is real.
an odd sequence if it is real.

81. Sequence Decomposition
Any sequence can be expressed as the sum of a conjugate-symmetric one and a conjugate-antisymmetric one, i.e.,
Conjugate
Symmetric
Conjugate
Antisymmetric

82. Function Decomposition
Any function can be expressed as the sum of a conjugate-symmetric one and a conjugate-antisymmetric one, i.e.,
Conjugate
Symmetric
Conjugate
Antiymmetric

83. Conjugate-Symmetric and Conjugate-Antiymmetric Functions
Conjugate-Symmetric Function
Conjugate-Antisymmetric Function
an even function if it is real.
an odd function if it is real.

84. Symmetric Properties  
magnitude
phase  
magnitude
phase

85. Symmetric Properties  
magnitude
phase  
magnitude
phase

86. Symmetric Properties  
magnitude
phase  
magnitude
phase

87. Symmetric Properties

88. Symmetric Properties

89. Symmetric Properties for Real Sequence x(n)
Facts:
1. real part is even
2. Img. part is odd
3. Magnitude is even
4. Phase is odd  
magnitude
phase

90. Discrete-Time Signals and Systems
Fourier Transform Theorems

91. Linearity

92. Time Shifting  Phase Change

93. Frequency Shifting Signal Modulation

94. Time Reversal

95. Differentiation in Frequency

96. The Convolution Theorem

97. The Modulation or Window Theorem

98. Parseval’s Theorem
Facts:
Letting =0, then proven.

99. Parseval’s Theorem Energy Preserving

100. Example: Ideal Lowpass Filter

101. Example: Ideal Lowpass Filter
The ideal lowpass fileter
Is noncausal.

102. Example: Ideal Lowpass Filter
The ideal lowpass fileter
Is noncausal.
To approximate the ideal lowpass filter using a window.

103. Example: Ideal Lowpass Filter-4 -3 -2 -1 1 2 3 4 M =3 =5
=19

104. Discrete-Time Signals and Systems
The Existence of Fourier Transform

105. Key Issue Synthesis Analysis Does X(ej) exist for all ?
We need that |X(ej)| <  for all 
Analysis

106. Sufficient Condition for Convergence

107. More On Convergence
Define
Uniform Convergence
Mean-Square Convergence

108. Discrete-Time Signals and Systems
Important Transform Pairs

109. Fourier Transform Pairs
Sequence
Fourier Transform

110. Fourier Transform Pairs
Sequence
Fourier Transform

111. Fourier Transform Pairs
Sequence
Fourier Transform