Signals and Systems

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


Shou shui Wei[-281.30]<00A9>[-280.60]2008Shou shui Wei[-281.30]<00A9>[-280.60]2008



Shou shui Wei[-281.30]<00A9>[-280.60]2008Shou shui Wei[-281.30]<00A9>[-280.60]2008


Sufficient conditions for the convergence of FT


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


Example 4.1:


Example 4.1:


Example 4.2:


Example 4.3:


Example 4.4:


Example 4.5:


sinc functions:








SDUBME-SS4-CTFT-19Shou shui Wei©2012Fourier Transform for Periodic Signals

Fourier Transform from Fourier Series:


Example 4.6:


Example 4.7:


Example 4.8:









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:


Linearity:


Time Shifting:


Time Shift → Phase Shift:


Example 4.9:


Conjugation & Conjugate Symmetry:


Example 4.10:


Differentiation & Integration:


= +


Example 4.11:


Example 4.12:


Time & Frequency Scaling:


Duality:

Example 4.4

Example 4.5


Duality:


Duality:


Parseval’s relation:








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


From Superposition (or Linearity):


From Convolution Integral:


Equivalent LTI Systems:


Example 4.15: Time Shift


Examples 4.16 & 17: Differentiator & Integrator


Example 4.18: Ideal Lowpass Filter


Filter Design:

FilterLTI System


Filter Design:

FilterLTI System


Example 4.19:

Filter

LTI System


Example 4.19:


Example 4.20:

Filter

LTI System








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


Example 4.21:


Example 4.22:


Example 4.23:


Bandpass Filter Using Amplitude Modulation:

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


Examples 4.24 & 4.25:


Example 4.26:

LTI System


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

Signals and Systems

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