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Spring 2011
信號與系統Signals and Systems
Chapter SS-3Fourier Series Representation
of Periodic Signals
Feng-Li LianNTU-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
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Feng-Li Lian © 2011NTUEE-SS3-FS-2Outline
A Historical Perspective The Response of LTI Systems
to Complex Exponentials Fourier Series Representation
of Continuous-Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation
of Discrete-Time Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
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Feng-Li Lian © 2011NTUEE-SS3-FS-3A Historical Perspective
• L. Euler’s study on the motion of a vibrating string in 1748
Leonhard Euler1707-1783
Born in SwitzerlandPhoto from wikipedia
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Feng-Li Lian © 2011NTUEE-SS3-FS-4A Historical Perspective
• L. Euler showed (in 1748)–The configuration of a vibrating string
at some point in time is a linear combinationof these normal modes
• D. Bernoulli argued (in 1753)–All physical motions of a string could be represented
by linear combinations of normal modes–But, he did not pursue this mathematically
• J.L. Lagrange strongly criticized (in 1759)–The use of trigonometric series
in examination of vibrating strings–Impossible to represent signals with corners
using trigonometric series
Daniel Bernoulli1700-1782
Born in DutchPhoto from wikipedia
Joseph-Louis Lagrange1736-1813Born in Italy
Photo from wikipedia
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Feng-Li Lian © 2011NTUEE-SS3-FS-5A Historical Perspective
• In 1807, Jean Baptiste Joseph Fourier– Submitted a paper of using trigonometric series to represent
“any” periodic signal– It is examined by
S.F. Lacroix, G. Monge, P.S. de Laplace, and J.L. Lagrange, – But Lagrange rejected it!
• In 1822, Fourier published a book “Theorie analytique de la chaleur”– “The Analytical Theory of Heat”
Jean Baptiste Joseph Fourier1768-1830
Born in FrancePhoto from wikipedia
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Feng-Li Lian © 2011NTUEE-SS3-FS-6A Historical Perspective
Pierre-Simon, Marquis de Laplace1749-1827
Born in FrancePhoto from wikipedia
Gaspard Monge, Comte de Péluse1746-1818
Born in FrancePhoto from wikipedia
Sylvestre François de Lacroix1765-1843
Born in FrancePhoto from
A short biography of Silvestre-François Lacroix In Science Networks. Historical Studies, V35, Lacroix and the Calculus, Birkhäuser Basel
2008, ISBN 978-3-7643-8638-2
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Feng-Li Lian © 2011NTUEE-SS3-FS-7A Historical Perspective
• Fourier’s main contributions:– Studied vibration, heat diffusion, etc.
– Found series of harmonically related sinusoidsto be useful in representing the temperature distribution through a body
– Claimed that “any” periodic signal could be represented by such a series (i.e., Fourier series discussed in Chap 3)
– Obtained a representation for aperiodic signals(i.e., Fourier integral or transform discussed in Chap 4 & 5)
– (Fourier did not actually contribute to the mathematical theory of Fourier series)
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Feng-Li Lian © 2011NTUEE-SS3-FS-8A Historical Perspective
• Impact from Fourier’s work:– Theory of integration, point-set topology,
eigenfunction expansions, etc.
– Motion of planets, periodic behavior of the earth’s climate, wave in the ocean, radio & television stations
– Harmonic time series in the 18th & 19th centuries> Gauss etc. on discrete-time signals and systems
– Faster Fourier transform (FFT) in the mid-1960s> Cooley (IBM) & Tukey (Princeton) reinvented in 1965> Can be found in Gauss’s notebooks (in 1805)
Carl Friedrich Gauss (Gauß)1777-1855
Born in GermanyPhoto from wikipedia
James W. Cooley & John W. Tukey (1965): "An algorithm for the machine calculation of complex Fourier series", Math. Comput. 19, 297–301.
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Feng-Li Lian © 2011NTUEE-SS3-FS-9
Periodic Aperiodic
FS – CT– DT
(Chap 3)FT
– DT
– CT (Chap 4)
(Chap 5)
Bounded/Convergent
LTzT – 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
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Feng-Li Lian © 2011NTUEE-SS3-FS-10Outline
A Historical Perspective The Response of LTI Systems
to Complex Exponentials Fourier Series Representation
of Continuous-Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation
of Discrete-Time Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
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Feng-Li Lian © 2011NTUEE-SS3-FS-11Response of LTI Systems to Complex Exponentials
Basic Idea:• To represent signals
as linear combinations of basic signals
Key Properties:
1. The set of basic signals can be used to construct a broad and useful class of signals
2. The response of an LTI system to each signalshould be simple enoughin structure to provide us with a convenient representationfor the response of the system to any signals constructed as linear combination of basic signals
LTI
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Feng-Li Lian © 2011NTUEE-SS3-FS-12Response of LTI Systems to Complex Exponentials
One of Choices:
• The set of complex exponential signals
The Response of an LTI System:
LTI
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Feng-Li Lian © 2011NTUEE-SS3-FS-13Response of LTI Systems to Complex Exponentials
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Feng-Li Lian © 2011NTUEE-SS3-FS-14Response of LTI Systems to Complex Exponentials
Eigenfunctions and Superposition Properties:
LTI
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Feng-Li Lian © 2011NTUEE-SS3-FS-15Outline
A Historical Perspective The Response of LTI Systems
to Complex Exponentials Fourier Series Representation
of Continuous-Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation
of Discrete-Time Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
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Feng-Li Lian © 2011NTUEE-SS3-FS-16Fourier Series Representation of CT Periodic Signals
Harmonically related complex exponentials
The Fourier Series Representation:
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Feng-Li Lian © 2011NTUEE-SS3-FS-17Fourier Series Representation of CT Periodic Signals
Example 3.2:
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Feng-Li Lian © 2011NTUEE-SS3-FS-18Fourier Series Representation of CT Periodic Signals
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Feng-Li Lian © 2011NTUEE-SS3-FS-19Fourier Series Representation of CT Periodic Signals
Procedure of Determining the Coefficients:
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Feng-Li Lian © 2011NTUEE-SS3-FS-20Fourier Series Representation of CT Periodic Signals
Procedure of Determining the Coefficients:
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Feng-Li Lian © 2011NTUEE-SS3-FS-21Fourier Series Representation of CT Periodic Signals
In Summary:
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Feng-Li Lian © 2011NTUEE-SS3-FS-22Fourier Series Representation of CT Periodic Signals
Fourier Series of Real Periodic Signals:
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Feng-Li Lian © 2011NTUEE-SS3-FS-23Fourier Series Representation of CT Periodic Signals
Alternative Forms of the Fourier Series:
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Feng-Li Lian © 2011NTUEE-SS3-FS-24Fourier Series Representation of CT Periodic Signals
Alternative Forms of the Fourier Series:
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Feng-Li Lian © 2011NTUEE-SS3-FS-25Fourier Series Representation of CT Periodic Signals
Example 3.4:
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Feng-Li Lian © 2011NTUEE-SS3-FS-26Fourier Series Representation of CT Periodic Signals
Example 3.4:
-0.4636
>> a1 = 1-0.5j>> abs(a1)>> angle(a1)
0.5000
0.7854
1.1180
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Feng-Li Lian © 2011NTUEE-SS3-FS-27Fourier Series Representation of CT Periodic Signals
Example 3.5:
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Feng-Li Lian © 2011NTUEE-SS3-FS-28Fourier Series Representation of CT Periodic Signals
Example 3.5:
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Feng-Li Lian © 2011NTUEE-SS3-FS-29Fourier Series Representation of CT Periodic Signals
Example 3.5:
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Feng-Li Lian © 2011NTUEE-SS3-FS-30Fourier Series Representation of CT Periodic Signals
Example 3.5:
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Feng-Li Lian © 2011NTUEE-SS3-FS-31Fourier Series Representation of CT Periodic Signals
Example 3.5:
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Feng-Li Lian © 2011NTUEE-SS3-FS-32Fourier Series Representation of CT Periodic Signals
Example 3.5:
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Feng-Li Lian © 2011NTUEE-SS3-FS-33Fourier Series Representation of CT Periodic Signals
Example 3.5:
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Feng-Li Lian © 2011NTUEE-SS3-FS-34Outline
A Historical Perspective The Response of LTI Systems
to Complex Exponentials Fourier Series Representation
of Continuous-Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation
of Discrete-Time Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
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Feng-Li Lian © 2011NTUEE-SS3-FS-35Convergence of the Fourier Series
• Fourier maintained that“any” periodic signal could be represented by a Fourier series
• The truth is thatFourier series can be used to represent an extremely large class of periodic signals
• The question is thatwhen a periodic signal x(t) does in fact have a Fourier series representation?
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Feng-Li Lian © 2011NTUEE-SS3-FS-36Convergence of the Fourier Series
One class of periodic signals:• Which have finite energy over a single period:
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Feng-Li Lian © 2011NTUEE-SS3-FS-37Convergence of the Fourier Series
The other class of periodic signals:• Which satisfy Dirichlet conditions: • Condition 1:
– Over any period, x(t) must be absolutely integrable,i.e.,
Johann Peter Gustav Lejeune Dirichlet1805-1859
Born in GermanyPhoto from wikipedia
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Feng-Li Lian © 2011NTUEE-SS3-FS-38Convergence of the Fourier Series
The other class of periodic signals:• Which satisfy Dirichlet conditions: • Condition 2:
– In any finite interval, x(t) is of bounded variation; i.e.,– There are no more than
a finite number of maxima and minima during any single period of the signal
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Feng-Li Lian © 2011NTUEE-SS3-FS-39Convergence of the Fourier Series
The other class of periodic signals:• Which satisfy Dirichlet conditions: • Condition 3:
– In any finite interval, x(t) has only finite number of discontinuities.
– Furthermore, each of these discontinuities is finite
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Feng-Li Lian © 2011NTUEE-SS3-FS-40Convergence of the Fourier Series
How the Fourier series converges for a periodic signal with discontinuities
• In 1898,Albert Michelson (an American physicist) used his harmonic analyzerto compute the truncated Fourier series approximation for the square wave
Albert Abraham Michelson1852-1931
Polish-born German-AmericanPhoto from wikipedia
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Feng-Li Lian © 2011NTUEE-SS3-FS-41Convergence of the Fourier Series
• Michelson wrote to Josiah Gibbs
• In 1899, Gibbs showed that
– the partial sum near discontinuity exhibits ripples &– the peak amplitude remains constant
with increasing N
• The Gibbs phenomenonJosiah Willard Gibbs1839-1903Born in USAPhoto from wikipedia
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Feng-Li Lian © 2011NTUEE-SS3-FS-42Outline
A Historical Perspective The Response of LTI Systems
to Complex Exponentials Fourier Series Representation
of Continuous-Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation
of Discrete-Time Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
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Feng-Li Lian © 2011NTUEE-SS3-FS-43Properties of CT Fourier Series
CT Fourier Series Representation:
![Page 44: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/44.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-44Outline
Section Property3.5.1 Linearity3.5.2 Time Shifting
Frequency Shifting3.5.6 Conjugation3.5.3 Time Reversal3.5.4 Time Scaling
Periodic Convolution3.5.5 Multiplication
DifferentiationIntegration
3.5.6 Conjugate Symmetry for Real Signals3.5.6 Symmetry for Real and Even Signals3.5.6 Symmetry for Real and Odd Signals
Even-Odd Decomposition for Real Signals3.5.7 Parseval’s Relation for Periodic Signals
![Page 45: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/45.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-45Properties of CT Fourier Series
Linearity:
Add
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Feng-Li Lian © 2011NTUEE-SS3-FS-46Properties of CT Fourier Series
Time Shifting:
![Page 47: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/47.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-47Properties of CT Fourier Series
Time Reversal:
![Page 48: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/48.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-48Properties of CT Fourier Series
Time Scaling:
![Page 49: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/49.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-49Properties of CT Fourier Series
Multiplication:
Add
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Feng-Li Lian © 2011NTUEE-SS3-FS-50Properties of CT Fourier Series
Differentiation:
![Page 51: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/51.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-51Properties of CT Fourier Series
Integration:
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Feng-Li Lian © 2011NTUEE-SS3-FS-52Properties of CT Fourier Series
Conjugation & Conjugate Symmetry:
![Page 53: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/53.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-53Properties of CT Fourier Series
Conjugation & Conjugate Symmetry:
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Feng-Li Lian © 2011NTUEE-SS3-FS-54Properties of CT Fourier Series
Parseval’s relation for CT periodic signals:• As shown in Problem 3.46:
• Parseval’s relation states thatthe total average power in a periodic signalequalsthe sum of the average powers in all of its harmonic components
![Page 55: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/55.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-55
![Page 56: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/56.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-56Properties of CT Fourier Series
Example 3.6:
![Page 57: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/57.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-57Properties of CT Fourier Series
Example 3.7:
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Feng-Li Lian © 2011NTUEE-SS3-FS-58Properties of CT Fourier Series
Example 3.8:
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Feng-Li Lian © 2011NTUEE-SS3-FS-59Properties of CT Fourier Series
Example 3.8:
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Feng-Li Lian © 2011NTUEE-SS3-FS-60Outline
A Historical Perspective The Response of LTI Systems
to Complex Exponentials Fourier Series Representation
of Continuous-Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation
of Discrete-Time Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
![Page 61: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/61.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-61Fourier Series Representation of DT Periodic Signals
Harmonically related complex exponentials
The Fourier Series Representation:
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Feng-Li Lian © 2011NTUEE-SS3-FS-62Fourier Series Representation of DT Periodic Signals
Procedure of Determining the Coefficients:
![Page 63: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/63.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-63Fourier Series Representation of DT Periodic Signals
Procedure of Determining the Coefficients:
![Page 64: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/64.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-64Properties of DT Fourier Series
In Summary:
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Feng-Li Lian © 2011NTUEE-SS3-FS-65Fourier Series Representation of DT Periodic Signals
Example 3.11:
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Feng-Li Lian © 2011NTUEE-SS3-FS-66Fourier Series Representation of DT Periodic Signals
Example 3.11:
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Feng-Li Lian © 2011NTUEE-SS3-FS-67Fourier Series Representation of DT Periodic Signals
Example 3.12:
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Feng-Li Lian © 2011NTUEE-SS3-FS-68Fourier Series Representation of DT Periodic Signals
Example 3.12:
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Feng-Li Lian © 2011NTUEE-SS3-FS-69
-40 -30 -20 -10 0 10 20 30 40-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-40 -30 -20 -10 0 10 20 30 40-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Fourier Series Representation of DT Periodic Signals
Example 3.12:
-15 -10 -5 0 5 10 15-0.5
0
0.5
1
1.5
2
0 - 2 0 0 2 0 4 0 6 0
-30 -20 -10 0 10 20 30-0.5
0
0.5
1
1.5
2
-15 -10 -5 0 5 10 15-0.5
0
0.5
1
1.5
2
-10 -5 0 5 10 15
-30 -20 -10 0 10 20 30-0.5
0
0.5
1
1.5
2
20 30-40 -30 -20 -10 0 10 20 30 40
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
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Feng-Li Lian © 2011NTUEE-SS3-FS-70Fourier Series Representation of DT Periodic Signals
Example 3.12:
-40 -30 -20 -10 0 10 20 30 40-2
-1
0
1
2
3
4
5
-40 -30 -20 -10 0 10 20 30 40-2
-1
0
1
2
3
4
5
-15 -10 -5 0 5 10 15-0.5
0
0.5
1
1.5
2
0 - 2 0 0 2 0 4 0 6 0
-30 -20 -10 0 10 20 30-0.5
0
0.5
1
1.5
2
-15 -10 -5 0 5 10 15-0.5
0
0.5
1
1.5
2
-10 -5 0 5 10 15
-30 -20 -10 0 10 20 30-0.5
0
0.5
1
1.5
2
20 30 -40 -30 -20 -10 0 10 20 30 40-2
-1
0
1
2
3
4
5
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Feng-Li Lian © 2011NTUEE-SS3-FS-71Fourier Series Representation of DT Periodic Signals
Example 3.12:
-80 -60 -40 -20 0 20 40 60 80-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
-80 -60 -40 -20 0 20 40 60 80-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-80 -60 -40 -20 0 20 40 60 80-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-80 -60 -40 -20 0 20 40 60 80-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-60 -40 -20 0 20 40 60-0.5
0
0.5
1
1.5
2
-60 -40 -20 0 20 40 60-0.5
0
0.5
1
1.5
2
-60 -40 -20 0 20 40 60-0.5
0
0.5
1
1.5
2
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Feng-Li Lian © 2011NTUEE-SS3-FS-72Fourier Series Representation of CT Periodic Signals
Examples 3.5 (CT) & 3.12 (DT):
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Feng-Li Lian © 2011NTUEE-SS3-FS-73Fourier Series Representation of DT Periodic Signals
Partial Sum:
![Page 74: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/74.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-74Outline
A Historical Perspective The Response of LTI Systems to Complex
Exponentials Fourier Series Representation of Continuous-
Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation of Discrete-Time
Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
![Page 75: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/75.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-75Outline
Section PropertyLinearity
Time ShiftingFrequency Shifting
ConjugationTime ReversalTime Scaling
Periodic Convolution3.7.1 Multiplication3.7.2 First Difference
Running SumConjugate Symmetry for Real SignalsSymmetry for Real and Even SignalsSymmetry for Real and Odd Signals
Even-Odd Decomposition for Real Signals3.7.3 Parseval’s Relation for Periodic Signals
![Page 76: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/76.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-76Properties of DT Fourier Series
![Page 77: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/77.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-77Fourier Series Representation of DT Periodic Signals
In Summary:
![Page 78: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/78.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-78Properties of DT Fourier Series
Linearity:
Time Shifting:
![Page 79: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/79.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-79Properties of DT Fourier Series
Multiplication:
Add
![Page 80: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/80.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-80Properties of DT Fourier Series
First Difference:
![Page 81: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/81.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-81Properties of DT Fourier Series
Parseval’s relation for DT periodic signals:• As shown in Problem 3.57:
• Parseval’s relation states thatthe total average power in a periodic signalequalsthe sum of the average powers in all of its harmonic components (only N distinct harmonic components in DT)
![Page 82: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/82.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-82Properties of DT Fourier Series
Example 3.13:
![Page 83: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/83.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-83CT & DT Fourier Series
CT & DT Fourier Series Representation:
![Page 84: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/84.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-84Outline
A Historical Perspective The Response of LTI Systems to Complex
Exponentials Fourier Series Representation of Continuous-
Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation of Discrete-Time
Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
![Page 85: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/85.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-85Fourier Series & LTI Systems
The Response of an LTI System:
LTI
– On pages 12-14
![Page 86: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/86.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-86Fourier Series & LTI Systems
In Summary:
LTIH(s/z/w)
Examples 3.16 & 3.17
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Feng-Li Lian © 2011NTUEE-SS3-FS-87Outline
A Historical Perspective The Response of LTI Systems to Complex
Exponentials Fourier Series Representation of Continuous-
Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation of Discrete-Time
Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
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Feng-Li Lian © 2011NTUEE-SS3-FS-88Filtering
Filtering:
• Change the relative amplitudes of the frequency components in a signal, – Frequency-shaping filters
• OR, significantly attenuate or eliminatesome frequency components entirely– Frequency-selective filters
filter
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Feng-Li Lian © 2011NTUEE-SS3-FS-89Filtering: Frequency-Shaping Filters
Frequency-Shaping Filters:• Audio System:
Bass Treble EqualizerMicrophone
or Tape Speaker
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Feng-Li Lian © 2011NTUEE-SS3-FS-90Filtering
Frequency-Shaping Filters:• Differentiating filter on enhancing edges:
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Feng-Li Lian © 2011NTUEE-SS3-FS-91Filtering
Frequency-Shaping Filters:• Differentiating filter:
Differentiating Filter
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Feng-Li Lian © 2011NTUEE-SS3-FS-92Filtering
Frequency-Shaping Filters:• A simple DT filter: Two-point average
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Feng-Li Lian © 2011NTUEE-SS3-FS-93Filtering: Frequency-Selective Filters
Frequency-Selective Filters:• Select some bands of frequencies and reject others
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Feng-Li Lian © 2011NTUEE-SS3-FS-94Filtering
Frequency-Selective Filters:• Select some bands of frequencies and reject others
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Feng-Li Lian © 2011NTUEE-SS3-FS-95Outline
A Historical Perspective The Response of LTI Systems to Complex
Exponentials Fourier Series Representation of Continuous-
Time Periodic SignalsConvergence of the Fourier SeriesProperties of Continuous-Time Fourier Series Fourier Series Representation of Discrete-Time
Periodic SignalsProperties of Discrete-Time Fourier Series Fourier Series & LTI Systems Filtering & Examples of CT & DT Filters
![Page 96: Chapter SS-3 Fourier Series Representation of Periodic Signals …cc.ee.ntu.edu.tw/~fengli/Teaching/SignalsSystems/992_ss... · 2012-03-27 · Spring 2011 信號與系統 Signals](https://reader033.vdocuments.site/reader033/viewer/2022042709/5f429d9ca65b6f46a670e388/html5/thumbnails/96.jpg)
Feng-Li Lian © 2011NTUEE-SS3-FS-96CT Filters by Differential Equations
A Simple RC Lowpass Filter:
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Feng-Li Lian © 2011NTUEE-SS3-FS-97CT Filters by Differential Equations
A Simple RC Lowpass Filter:
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Feng-Li Lian © 2011NTUEE-SS3-FS-98CT Filters by Differential Equations
A Simple RC Highpass Filter:
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Feng-Li Lian © 2011NTUEE-SS3-FS-99CT Filters by Differential Equations
A Simple RC Highpass Filter:
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Feng-Li Lian © 2011NTUEE-SS3-FS-100DT Filters by Difference Equations
First-Order Recursive DT Filters:
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Feng-Li Lian © 2011NTUEE-SS3-FS-101DT Filters by Difference Equations
First-Order Recursive DT Filters:
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Feng-Li Lian © 2011NTUEE-SS3-FS-102DT Filters by Difference Equations
First-Order Recursive DT Filters:
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Feng-Li Lian © 2011NTUEE-SS3-FS-103DT Filters by Difference Equations
Nonrecursive DT Filters:• An FIR nonrecursive difference equation:
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Feng-Li Lian © 2011NTUEE-SS3-FS-104DT Filters by Difference Equations
Nonrecursive DT Filters:• Three-point moving average (lowpass) filter:
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Feng-Li Lian © 2011NTUEE-SS3-FS-105DT Filters by Difference Equations
Nonrecursive DT Filters:• N+M+1 moving average (lowpass) filter:
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Feng-Li Lian © 2011NTUEE-SS3-FS-106DT Filters by Difference Equations
Nonrecursive DT Filters:• N+M+1 moving average
(lowpass) filter:
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Feng-Li Lian © 2011NTUEE-SS3-FS-107DT Filters by Difference Equations
Lowpass Filtering on Dow Jones Weekly Stock Market Index:
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Feng-Li Lian © 2011NTUEE-SS3-FS-108DT Filters by Difference Equations
Nonrecursive DT Filters:• Highpass filters:
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Feng-Li Lian © 2011NTUEE-SS3-FS-109Correction
On page 235, Eq. 3.139
On page 249, Eq. 3.164
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Feng-Li Lian © 2011NTUEE-SS3-FS-110Chapter 3: Fourier Series Representation of Periodic Signals
A Historical Perspective The Response of LTI Systems to Complex Exponentials FS Representation of CT Periodic Signals Convergence of the FS Properties of CT FS
• Linearity Time Shifting Frequency Shifting Conjugation• Time Reversal Time Scaling Periodic Convolution Multiplication• Differentiation Integration 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 Periodic Signals
FS Representation of DT Periodic Signals Properties of DT FS
• Multiplication First Difference Running Sum
FS & LTI Systems Filtering
• Frequency-shaping filters & Frequency-selective filters
Examples of CT & DT Filters
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Feng-Li Lian © 2011NTUEE-SS3-FS-111
Periodic Aperiodic
FS – CT– DT
(Chap 3)FT
– DT
– CT (Chap 4)
(Chap 5)
Bounded/Convergent
LTzT – 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