Post on 21-Apr-2023
Fall 2012
Signals and Systems
Chapter SS-4The Continuous-Time Fourier Transform
Shou shui Wei
SDU-BME
Sep08 – Dec08
Figures and images used in these lecture notes are adopted from“Signals & Systems” by Alan V. Oppenheim and Alan S. Willsky, 1997
SDUBME-SS4-CTFT-2Shou shui Wei©2012Outline
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
NTUEE-SS4-CTFT-3Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
CT Fourier Transform of an Aperiodic Signal:Page 193, Ex 3.5
SDUBMESS4-CTFT-4Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
SDUBME-SS4-CTFT-5Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
SDUBME-SS4-CTFT-6Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Shou shui Wei[-281.30]<00A9>[-280.60]2008Shou shui Wei[-281.30]<00A9>[-280.60]2008
SDUBME-SS4-CTFT-7Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
SDUBME-SS4-CTFT-8Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Shou shui Wei[-281.30]<00A9>[-280.60]2008Shou shui Wei[-281.30]<00A9>[-280.60]2008
SDUBME-SS4-CTFT-9Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Sufficient conditions for the convergence of FT
SDUBME-SS4-CTFT-10Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
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 minimawithin any finite interval
3.x(t) have a finite number of discontinuities within any finite intervalFurthermore, each of these discontinuities must be finite
Shou shui Wei[-281.30]<00A9>[-280.60]2008Shou shui Wei[-281.30]<00A9>[-280.60]2008Shou shui Wei[-281.30]<00A9>[-280.60]2008Shou shui Wei[-281.30]<00A9>[-280.60]2008
SDUBME-SS4-CTFT-11Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Example 4.1:
SDUBME-SS4-CTFT-12Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Example 4.1:
SDUBME-SS4-CTFT-13Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Example 4.2:
SDUBME-SS4-CTFT-14Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Example 4.3:
SDUBME-SS4-CTFT-15Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Example 4.4:
SDUBME-SS4-CTFT-16Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
Example 4.5:
SDUBME-SS4-CTFT-17Shou shui Wei©2012Representation of Aperiodic Signals: CT Fourier Transform
sinc functions:
SDUBME-SS4-CTFT-18Shou shui Wei©2012Outline
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
SDUBME-SS4-CTFT-19Shou shui Wei©2012Fourier Transform for Periodic Signals
Fourier Transform from Fourier Series:
SDUBME-SS4-CTFT-20Shou shui Wei©2012Fourier Transform for Periodic Signals
Example 4.6:
SDUBME-SS4-CTFT-21Shou shui Wei©2012Fourier Transform for Periodic Signals
Example 4.7:
SDUBME-SS4-CTFT-22Shou shui Wei©2012Fourier Transform for Periodic Signals
Example 4.8:
SDUBME-SS4-CTFT-23Shou shui Wei©2012Outline
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
SDUBME-SS4-CTFT-24Shou shui Wei©2012Outline
Differentiation in Time4.3.4
Multiplication4.5
Convolution4.4
Time and Frequency Scaling4.3.5
Time Reversal4.3.5
Conjugation4.3.3
Frequency Shifting4.3.6
Time Shifting4.3.2
Symmetry for Real and Even Signals4.3.3
Conjugate Symmetry for Real Signals4.3.3
Differentiation in Frequency4.3.6
Integration4.3.4
Even-Odd Decomposition for Real Signals4.3.3
Symmetry for Real and Odd Signals4.3.3
Parseval’s Relation for Aperiodic Signals4.3.7
Linearity4.3.1
PropertySection
SDUBME-SS4-CTFT-25Shou shui Wei©2012Properties of CT Fourier Transform
Fourier Transform Pair:
• Synthesis equation:
• Analysis equation:
• Notations:
SDUBME-SS4-CTFT-26Shou shui Wei©2012Properties of CT Fourier Transform
Linearity:
SDUBME-SS4-CTFT-27Shou shui Wei©2012Properties of CT Fourier Transform
Time Shifting:
SDUBME-SS4-CTFT-28Shou shui Wei©2012Properties of CT Fourier Transform
Time Shift → Phase Shift:
SDUBME-SS4-CTFT-29Shou shui Wei©2012Properties of CT Fourier Transform
Example 4.9:
SDUBME-SS4-CTFT-30Shou shui Wei©2012Properties of CT Fourier Transform
Conjugation & Conjugate Symmetry:
SDUBME-SS4-CTFT-31Shou shui Wei©2012Properties of CT Fourier Transform
Conjugation & Conjugate Symmetry:
SDUBME-SS4-CTFT-32Shou shui Wei©2012Properties of CT Fourier Transform
Conjugation & Conjugate Symmetry:
SDUBME-SS4-CTFT-33Shou shui Wei©2012Properties of CT Fourier Transform
Example 4.10:
SDUBME-SS4-CTFT-34Shou shui Wei©2012Properties of CT Fourier Transform
Differentiation & Integration:
SDUBME-SS4-CTFT-35Shou shui Wei©2012Properties of CT Fourier Transform
= +
SDUBME-SS4-CTFT-36Shou shui Wei©2012Properties of CT Fourier Transform
Example 4.11:
SDUBME-SS4-CTFT-37Shou shui Wei©2012Properties of CT Fourier Transform
Example 4.12:
SDUBME-SS4-CTFT-38Shou shui Wei©2012Properties of CT Fourier Transform
Time & Frequency Scaling:
SDUBME-SS4-CTFT-39Shou shui Wei©2012Properties of CT Fourier Transform
Duality:
Example 4.4
Example 4.5
SDUBME-SS4-CTFT-40Shou shui Wei©2012Properties of CT Fourier Transform
Duality:
SDUBME-SS4-CTFT-41Shou shui Wei©2012Properties of CT Fourier Transform
Duality:
SDUBME-SS4-CTFT-42Shou shui Wei©2012Properties of CT Fourier Transform
Parseval’s relation:
SDUBME-SS4-CTFT-43Shou shui Wei©2012Outline
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
SDUBME-SS4-CTFT-44Shou shui Wei©2012Convolution Property & Multiplication Property
Convolution Property:
Multiplication Property:X
SDUBME-SS4-CTFT-45Shou shui Wei©2012Convolution Property
From Superposition (or Linearity):
Linear System
SDUBME-SS4-CTFT-46Shou shui Wei©2012Convolution Property
From Superposition (or Linearity):
SDUBME-SS4-CTFT-47Shou shui Wei©2012Convolution Property
From Convolution Integral:
SDUBME-SS4-CTFT-48Shou shui Wei©2012Convolution Property
Equivalent LTI Systems:
SDUBME-SS4-CTFT-49Shou shui Wei©2012Convolution Property
Example 4.15: Time Shift
SDUBME-SS4-CTFT-50Shou shui Wei©2012Convolution Property
Examples 4.16 & 17: Differentiator & Integrator
SDUBME-SS4-CTFT-51Shou shui Wei©2012Convolution Property
Example 4.18: Ideal Lowpass Filter
SDUBME-SS4-CTFT-52Shou shui Wei©2012Convolution Property
Filter Design:
FilterLTI System
SDUBME-SS4-CTFT-53Shou shui Wei©2012Convolution Property
Filter Design:
FilterLTI System
SDUBME-SS4-CTFT-54Shou shui Wei©2012Convolution Property
Example 4.19:
Filter
LTI System
SDUBME-SS4-CTFT-55Shou shui Wei©2012Convolution Property
Example 4.19:
SDUBME-SS4-CTFT-56Shou shui Wei©2012Convolution Property
Example 4.20:
Filter
LTI System
SDUBME-SS4-CTFT-57Shou shui Wei©2012Outline
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
SDUBME-SS4-CTFT-58Shou shui Wei©2012Multiplication Property
Convolution & Multiplication:
Multiplication of One Signal by Another:• Scale or modulate the amplitude of the other signal
• Modulation
X
SDUBME-SS4-CTFT-59Shou shui Wei©2012Multiplication Property
Example 4.21:
SDUBME-SS4-CTFT-60Shou shui Wei©2012Multiplication Property
Example 4.22:
SDUBME-SS4-CTFT-61Shou shui Wei©2012Multiplication Property
Example 4.23:
SDUBME-SS4-CTFT-62Shou shui Wei©2012Multiplication Property
Bandpass Filter Using Amplitude Modulation:
SDUBME-SS4-CTFT-65Shou shui Wei©2012
NTUEE-SS4-CTFT-66Shou shui Wei©2012Outline
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
SDUBME-SS4-CTFT-67Shou shui Wei©2012Systems Characterized by Linear Constant-Coefficient Differential Equations
A useful class of CT LTI systems:
LTI System
NTUEE-SS4-CTFT-68Shou shui Wei©2012Systems Characterized by Linear Constant-Coefficient Differential Equations
SDUBME-SS4-CTFT-69Shou shui Wei©2012Systems Characterized by Linear Constant-Coefficient Differential Equations
Examples 4.24 & 4.25:
SDUBME-SS4-CTFT-70Shou shui Wei©2012Systems Characterized by Linear Constant-Coefficient Differential Equations
Example 4.26:
LTI System
SDUBME-SS4-CTFT-71Shou shui Wei©2012
Periodic Aperiodic
FS– CT
– DT(Chap 3)
FT– DT
– CT (Chap 4)
(Chap 5)
Bounded/Convergent
LT
zT – DT
Time-Frequency
(Chap 9)
(Chap 10)
Unbounded/Non-convergent
– CT
CT-DT
Communication
Control
(Chap 6)
(Chap 7)
(Chap 8)
(Chap 11)
Signals & Systems LTI & Convolution(Chap 1) (Chap 2)
Flowchart