信號與系統 Signals and Systems
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1 Spring 2011 信號與系統 Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform Feng-Li Lian NTU-EE Feb11 Jun11 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan V. Oppenheim and Alan S. Willsky, 1997
2 Outline NTUEE-SS4-CTFT-2 Representation of Aperiodic Signals: the Continuous-Time Fourier Transform The Fourier Transform for Periodic Signals Properties of the Continuous-Time Fourier Transform The Convolution Property The Multiplication Property Systems Characterized by Linear Constant-Coefficient Differential Equations
3 Fourier Series Representation of CT Periodic Signals NTUEE-SS4-CTFT-3 Example 3.5:
4 Fourier Series Representation of CT Periodic Signals NTUEE-SS4-CTFT-4 Example 3.5:
5 Fourier Series Representation of CT Periodic Signals Example 3.5: NTUEE-SS4-CTFT-5
6 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-6 Page 193, Ex 3.5 CT Fourier Transform of an Aperiodic Signal:
7 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-7
8 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-8
9 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-9
10 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-10
11 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-11
12 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-12 Sufficient conditions for the convergence of FT
13 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-13 Sufficient conditions for the convergence of FT Dirichlet conditions: 1.x(t) be absolutely integrable; that is, 2.x(t) have a finite number of maxima and minima within any finite interval 3.x(t) have a finite number of discontinuities within any finite interval Furthermore, each of these discontinuities must be finite
14 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-14 Example 4.1:
15 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-15 Example 4.1:
16 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-16 Example 4.2:
17 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-17 Example 4.3:
18 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-18 Example 4.4:
19 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-19 Example 4.5:
20 Representation of Aperiodic Signals: CT Fourier Transform NTUEE-SS4-CTFT-20 sinc functions:
21 Outline NTUEE-SS4-CTFT-21 Representation of Aperiodic Signals: the Continuous-Time Fourier Transform The Fourier Transform for Periodic Signals Properties of the Continuous-Time Fourier Transform The Convolution Property The Multiplication Property Systems Characterized by Linear Constant-Coefficient Differential Equations
22 Fourier Transform for Periodic Signals NTUEE-SS4-CTFT-22 Fourier Transform from Fourier Series:
23 Fourier Transform for Periodic Signals NTUEE-SS4-CTFT-23 Example 4.6:
24 Fourier Transform for Periodic Signals NTUEE-SS4-CTFT-24 Example 4.7:
25 Fourier Transform for Periodic Signals NTUEE-SS4-CTFT-25 Example 4.8:
26 Outline NTUEE-SS4-CTFT-26 Representation of Aperiodic Signals: the Continuous-Time Fourier Transform The Fourier Transform for Periodic Signals Properties of the Continuous-Time Fourier Transform The Convolution Property The Multiplication Property Systems Characterized by Linear Constant- Coefficient Differential Equations
27 Outline NTUEE-SS4-CTFT-27 Section Property Linearity Time Shifting Frequency Shifting Conjugation Time Reversal Time and Frequency Scaling Convolution Multiplication Differentiation in Time Integration Differentiation in Frequency Conjugate Symmetry for Real Signals Symmetry for Real and Even Signals Symmetry for Real and Odd Signals Even-Odd Decomposition for Real Signals Parseval s Relation for Aperiodic Signals
28 Outline NTUEE-SS4-CTFT-28 Property CTFS DTFS CTFT DTFT LT zt Linearity Time Shifting Frequency Shifting (in s, z) Conjugation Time Reversal Time & Frequency Scaling (Periodic) Convolution Multiplication Differentiation/First Difference , , , , Integration/Running Sum (Accumulation) Conjugate Symmetry for Real Signals Symmetry for Real and Even Signals Symmetry for Real and Odd Signals Even-Odd Decomposition for Real Signals Parseval s Relation for (A)Periodic Signals Initial- and Final-Value Theorems
29 Properties of CT Fourier Transform NTUEE-SS4-CTFT-29 Fourier Transform Pair: Synthesis equation: Analysis equation: Notations:
30 Properties of CT Fourier Transform NTUEE-SS4-CTFT-30 Linearity:
31 Properties of CT Fourier Transform NTUEE-SS4-CTFT-31 Time Shifting:
32 Properties of CT Fourier Transform Time Shift Phase Shift: NTUEE-SS4-CTFT-32
33 Properties of CT Fourier Transform NTUEE-SS4-CTFT-33 Example 4.9:
34 Properties of CT Fourier Transform NTUEE-SS4-CTFT-34 Conjugation & Conjugate Symmetry:
35 Properties of CT Fourier Transform NTUEE-SS4-CTFT-35 Conjugation & Conjugate Symmetry:
36 Properties of CT Fourier Transform NTUEE-SS4-CTFT-36 Conjugation & Conjugate Symmetry:
37 Properties of CT Fourier Transform NTUEE-SS4-CTFT-37 Example 4.10: Ex 4.1 Ex 4.2
38 Properties of CT Fourier Transform NTUEE-SS4-CTFT-38 Differentiation & Integration:
39 Properties of CT Fourier Transform NTUEE-SS4-CTFT-39 = +
40 Properties of CT Fourier Transform NTUEE-SS4-CTFT-40 Example 4.11:
41 Properties of CT Fourier Transform NTUEE-SS4-CTFT-41 Example 4.12:
42 Properties of CT Fourier Transform NTUEE-SS4-CTFT-42 Time & Frequency Scaling:
43 Properties of CT Fourier Transform NTUEE-SS4-CTFT-43 Duality: Example 4.4 Example 4.5
44 Properties of CT Fourier Transform NTUEE-SS4-CTFT-44 Duality:
45 Properties of CT Fourier Transform NTUEE-SS4-CTFT-45 Duality:
46 Properties of CT Fourier Transform NTUEE-SS4-CTFT-46 Parseval s relation:
47 Outline NTUEE-SS4-CTFT-47 Representation of Aperiodic Signals: the Continuous-Time Fourier Transform The Fourier Transform for Periodic Signals Properties of the Continuous-Time Fourier Transform The Convolution Property The Multiplication Property Systems Characterized by Linear Constant-Coefficient Differential Equations
48 Convolution Property & Multiplication Property NTUEE-SS4-CTFT-48 Convolution Property: Multiplication Property: X
49 Convolution Property NTUEE-SS4-CTFT-49 From Superposition (or Linearity): Linear System
50 Convolution Property NTUEE-SS4-CTFT-50 From Superposition (or Linearity):
51 Convolution Property NTUEE-SS4-CTFT-51 From Convolution Integral:
52 Convolution Property NTUEE-SS4-CTFT-52 Equivalent LTI Systems:
53 Convolution Property NTUEE-SS4-CTFT-53 Example 4.15: Time Shift
54 Convolution Property NTUEE-SS4-CTFT-54 Examples 4.16 & 17: Differentiator & Integrator
55 Convolution Property NTUEE-SS4-CTFT-55 Example 4.18: Ideal Lowpass Filter
56 Convolution Property NTUEE-SS4-CTFT-56 Filter Design: Filter LTI System
57 Convolution Property NTUEE-SS4-CTFT-57 Filter Design: Filter LTI System RC circuit
58 Convolution Property NTUEE-SS4-CTFT-58 Example 4.19: Filter LTI System
59 Convolution Property NTUEE-SS4-CTFT-59 Example 4.19:
60 Convolution Property NTUEE-SS4-CTFT-60 Example 4.20: Filter LTI System
61 Outline NTUEE-SS4-CTFT-61 Representation of Aperiodic Signals: the Continuous-Time Fourier Transform The Fourier Transform for Periodic Signals Properties of the Continuous-Time Fourier Transform The Convolution Property The Multiplication Property Systems Characterized by Linear Constant-Coefficient Differential Equations
62 Multiplication Property NTUEE-SS4-CTFT-62 Convolution & Multiplication: Multiplication of One Signal by Another: Scale or modulate the amplitude of the other signal Modulation X
63 Multiplication Property NTUEE-SS4-CTFT-63
64 Multiplication Property NTUEE-SS4-CTFT-64 Example 4.21:
65 Multiplication Property NTUEE-SS4-CTFT-65 Example 4.22:
66 Multiplication Property NTUEE-SS4-CTFT-66 Example 4.23:
67 Multiplication Property Bandpass Filter Using Amplitude Modulation: NTUEE-SS4-CTFT-67
68 Multiplication Property Bandpass Filter Using Amplitude Modulation: NTUEE-SS4-CTFT-68
69 Multiplication Property Bandpass Filter Using Amplitude Modulation: NTUEE-SS4-CTFT-69 On Page , Problem 4.46 X X X X +
70 NTUEE-SS4-CTFT-70
71 NTUEE-SS4-CTFT-71
72 NTUEE-SS4-CTFT-72
73 Outline NTUEE-SS4-CTFT-73 Representation of Aperiodic Signals: the Continuous-Time Fourier Transform The Fourier Transform for Periodic Signals Properties of the Continuous-Time Fourier Transform The Convolution Property The Multiplication Property Systems Characterized by Linear Constant-Coefficient Differential Equations
74 Systems Characterized by Linear Constant-Coefficient Differential Equations NTUEE-SS4-CTFT-74 A useful class of CT LTI systems: LTI System
75 Systems Characterized by Linear Constant-Coefficient Differential Equations NTUEE-SS4-CTFT-75
76 Systems Characterized by Linear Constant-Coefficient Differential Equations NTUEE-SS4-CTFT-76 Examples 4.24 & 4.25:
77 Systems Characterized by Linear Constant-Coefficient Differential Equations NTUEE-SS4-CTFT-77 Example 4.26: LTI System
78 Chapter 4: The Continuous-Time Fourier Transform NTUEE-SS4-CTFT-78 Representation of Aperiodic Signals: the CT FT The FT for Periodic Signals Properties of the CT FT Linearity Time Shifting Frequency Shifting Conjugation Time Reversal Time and Frequency Scaling Convolution Multiplication Differentiation in Time Integration Differentiation in Frequency Conjugate Symmetry for Real Signals Symmetry for Real and Even Signals & for Real and Odd Signals Even-Odd Decomposition for Real Signals Parseval s Relation for Aperiodic Signals The Convolution Property The Multiplication Property Systems Characterized by Linear Constant- Coefficient Differential Equations
79 Questions for Chapter 4 NTUEE-SS4-CTFT-79 Why to study FT In order to analyze or represent aperiodic signals How to develop FT From FS and let T -> infinity Do periodic signals have FT Yes, their FT is function of isolated impulses Why to know the properties of FT Avoid using the fundamental formulas of FT to compute the FT What the duality of FT and why FT and IFT have almost identical integration formulas Why to know the convolution property To analyze system response and/or design proper circuits To simplify computation Why to know the multiplication property For signal modulation with different-frequency carriers To simplify computation
80 X(jw) of Aperiodic Signals and a_k of Periodic Signals NTUEE-SS4-CTFT-80
81 X(jw) of Aperiodic Signals and a_k of Periodic Signals NTUEE-SS4-CTFT-81
82 X(jw) of Aperiodic Signals and a_k of Periodic Signals NTUEE-SS4-CTFT-82
83 X(jw) of Aperiodic Signals and a_k of Periodic Signals NTUEE-SS4-CTFT-83
84 Flowchart Signals & Systems (Chap 1) LTI & Convolution (Chap 2) NTUEE-SS4-CTFT-84 Bounded/Convergent Periodic Aperiodic FS (Chap 3) CT DT FT CT (Chap 4) DT (Chap 5) Unbounded/Non-convergent LT zt CT DT (Chap 9) (Chap 10) Time-Frequency (Chap 6) Communication (Chap 8) CT-DT (Chap 7) Control (Chap 11)
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