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DiscreteTime Signals and Systems

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Title: DiscreteTime Signals and Systems


1
Discrete-Time Signals and Systems
  • ??????

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

5
Signal Process Systems
Facilitate the extraction of desired information
e.g.,
  • Filters
  • Parameter estimation

Signal Processing System
signal
output
6
Signal Process Systems
7
Signal Process Systems
  • A important class of systems

Linear Shift-Invariant Systems.
  • In particular, well 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.

10
Important Sequences
  • Unit-sample sequence ?(n)
  • Sometime call ?(n)
  • a discrete-time impulse or
  • an impulse

11
Important Sequences
  • Unit-step sequence u(n)
  • Fact

12
Important Sequences
  • Real exponential sequence

13
Important Sequences
  • Sinusoidal sequence

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 RepresentationUsing delay unit
19
Discrete-Time Signals and Systems
  • Linear Shift-Invariant Systems

20
Systems
y(n)Tx(n)
x(n)
Mathematically modeled as a unique transformation
or operator.
21
Linear Systems
22
Examples
Ideal Delay System
Moving Average
Accumulator
23
Examples
Are these system linear?
24
Examples
A Memoryless System
Is this system linear?
25
Linear Systems
??k?impulse ???n?????
26
Shift-Invariant Systems
27
Shift-Invariant Systems
??/???????????
28
Linear Shift-Invariant Systems
??k?impulse ???n?????
???????
29
Impulse Response
h(n)T?(n)
x(n)?(n)
T
30
Convolution Sum
y(n)
x(n)
convolution
A linear shift-invariant system is completely
characterized by its impulse response.
31
Characterize a System
x(n)
x(n)h(n)
32
Properties of Convolution Math
33
Properties of Convolution Math
These systems are identical.
34
Properties of Convolution Math
These two systems are identical.
35
Example
y(n)?
36
Example
37
Example
compute y(0)
compute y(1)
How to computer y(n)?
38
Example
Two conditions have to be considered.
nltN and n?N.
39
Example
40
Example
41
Impulse Response ofthe Ideal Delay System
Ideal Delay System
By letting x(n)?(n) and y(n)h(n),
42
Impulse Response ofthe Ideal Delay System
?????
  • ?(n? nd)??????
  • Shift or
  • Copy

43
Impulse Response ofthe Moving Average
Moving Average
????(n? k)????
44
Impulse Response ofthe Accumulator
Accumulator
??????
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
ProveNecessary Condition for Stability
  • Show that if x is bounded and S lt ?, then y is
    bounded.

where M max x(n)
48
ProveSufficient Condition for Stablility
  • Show that if S ?, then one can find a bounded
    sequence x such that y is unbounded.

49
Example
  • Show that the linear shift-invariant system with
    impulse response h(n)anu(n) where alt1 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
55
The Moving Average
56
The Moving Average
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

59
Frequency Response
eigenvalue
eigenfunction
60
Frequency Response
phase
magnitude
61
ExampleThe Ideal Delay System
magnitude
phase
62
ExampleThe Ideal Delay System
63
Periodic Nature ofFrequency Response
64
Periodic Nature ofFrequency Response
65
Periodic Nature ofFrequency Response
Generally, we choose ?????? To represent one
period in frequency domain.
66
Periodic Nature ofFrequency Response
Low Frequency
67
Ideal Frequency-Selective Filters
Lowpass Filter
Bandstop Filter
Highpass Filter
68
Moving Average
69
Moving Average
70
Moving Average
  • M4
  • Lowpass
  • Try larger M

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
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 andConjugate-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-antisymmet
    ric 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-antisymmet
    ric one, i.e.,

Conjugate Antiymmetric
Conjugate Symmetric
83
Conjugate-Symmetric andConjugate-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
85
Symmetric Properties
86
Symmetric Properties
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
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
Parsevals Theorem
Facts
Letting ?0, then proven.
99
Parsevals TheoremEnergy Preserving
100
Example Ideal Lowpass Filter
101
Example Ideal Lowpass Filter
The ideal lowpass fileter Is noncausal.
102
Example Ideal Lowpass Filter
To approximate the ideal lowpass filter using a
window.
103
Example Ideal Lowpass Filter
104
Discrete-Time Signals and Systems
  • The Existence of Fourier Transform

105
Key Issue
Does X(ej?) exist for all ??
Synthesis
We need that X(ej?) lt ? 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
Fourier Transform
Sequence
110
Fourier Transform Pairs
Fourier Transform
Sequence
111
Fourier Transform Pairs
Fourier Transform
Sequence
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