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2008_Fourier Transform(1)
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
Engineering Mathematics 2
Prof. Kyu-Yeul Lee
Department of Naval Architecture and Ocean Engineering,
Seoul National University of College of Engineering
[2008][12-2]
November, 2008
2008_Fourier Transform(1)
Naval A
rch
itectu
re &
Ocean
En
gin
eerin
g
Fourier Transform(1) : Fourier Series and Transform
Fourier Series & Fourier Transform
Fourier Series- Review
2008_Fourier Transform(1)
Fourier Series and Fourier Transform
3/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
4/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
x
( )f x
5/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
x
( )f x
2 22
2f
T p p
6/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
x
( )f x
2 22
2f
T p p
7/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
-in series of sine, cosine, and complex**
Fourier Series
x
( )f x
2 22
2f
T p p
8/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
Fourier Series
x
( )f x
2 22
2f
T p p
9/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
Fourier Series
x
( )f x
2 22
2f
T p p
10/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Fourier Series
x
( )f x
2 22
2f
T p p
11/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Fourier Series
x
( )f x
2 22
2f
T p p
12/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
0, 1, 2,...n
......
Fourier Series
x
( )f x
2 22
2f
T p p
13/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Q1. is it possible to represent a function which is not periodic by something analogues to a Fourier series?
nc
2 3023
0, 1, 2,...n
......
Fourier Series
x
( )f x
2 22
2f
T p p
14/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Q1. is it possible to represent a function which is not periodic by something analogues to a Fourier series?
nc
2 3023
0, 1, 2,...n
......
Q2. can we somehow extend or modify Fourier series to cover the case of a continuous set of frequencies?
Fourier Series
x
( )f x
2 22
2f
T p p
15/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Q1. is it possible to represent a function which is not periodic by something analogues to a Fourier series?
nc
2 3023
0, 1, 2,...n
......
Q2. can we somehow extend or modify Fourier series to cover the case of a continuous set of frequencies?
Fourier Series
x
( )f x
2 22
2f
T p p
16/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Q1. is it possible to represent a function which is not periodic by something analogues to a Fourier series?
nc
2 3023
0, 1, 2,...n
......
Q2. can we somehow extend or modify Fourier series to cover the case of a continuous set of frequencies?
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series
x
( )f x
2 22
2f
T p p
17/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Q1. is it possible to represent a function which is not periodic by something analogues to a Fourier series?
nc
2 3023
0, 1, 2,...n
......
Q2. can we somehow extend or modify Fourier series to cover the case of a continuous set of frequencies?
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series
Fourier Series
x
( )f x
2 22
2f
T p p
18/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Q1. is it possible to represent a function which is not periodic by something analogues to a Fourier series?
nc
2 3023
0, 1, 2,...n
......
Q2. can we somehow extend or modify Fourier series to cover the case of a continuous set of frequencies?
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral
Fourier Series
x
( )f x
2 22
2f
T p p
19/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
-expand a periodic function
Function expansion
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
Q1. is it possible to represent a function which is not periodic by something analogues to a Fourier series?
nc
2 3023
0, 1, 2,...n
......
Q2. can we somehow extend or modify Fourier series to cover the case of a continuous set of frequencies?
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral
Fourier Series
x
( )f x
2 22
2f
T p p
20/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series Fourier Integral
-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
21/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series Fourier Integralextend
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
22/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
23/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
24/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-integral of sine, cosine, and complex**
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
25/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
26/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
27/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
28/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
29/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
30/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
interval-valued variable n
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
31/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
interval-valued variable n
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
32/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
interval-valued variable n
continuous variable
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
33/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
interval-valued variable n
ncthe set of coefficients
continuous variable
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
34/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
interval-valued variable n
ncthe set of coefficients
continuous variable
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
35/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
interval-valued variable n
ncthe set of coefficients
continuous variable
become a function ˆ( )f
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
36/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
interval-valued variable n
ncthe set of coefficients
continuous variable
become a function ˆ( )f
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
37/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Integralextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
ˆ( )f
interval-valued variable n
ncthe set of coefficients
continuous variable
become a function ˆ( )f
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
38/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Series
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
extend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
ˆ( )f
interval-valued variable n
ncthe set of coefficients
continuous variable
become a function ˆ( )f
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
39/152
2008_Fourier Transform(1)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Seriesextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
Fourier Transform
-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
ˆ( )f
interval-valued variable n
ncthe set of coefficients
continuous variable
become a function ˆ( )f
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral
2 22
2f
T p p
40/152
2008_Fourier Transform(1)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Seriesextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
Fourier Transform
-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
ˆ( )f
interval-valued variable n
ncthe set of coefficients
continuous variable
become a function ˆ( )f
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral
Fourier Transform of ( )f x
2 22
2f
T p p
41/152
2008_Fourier Transform(1)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*
A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
Function expansion
Fourier Seriesextend
-represent nonperiodic functions
-integral of sine, cosine, and complex**
-continuous set of frequencies
Fourier Transform
-expand a periodic function
-in series of sine, cosine, and complex**
-infinite but discrete set of frequencies
corresponding
ˆ( )f
interval-valued variable n
ncthe set of coefficients
continuous variable
become a function ˆ( )f
nc
2 3023
2 2
2T p p
0, 1, 2,...n
......
t
( )f t
n
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integraln
recall, an ‘integral’ is a ‘limit of a sum’
Fourier Series Fourier Integral
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f 2 2
22
fT p p
42/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64843/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64844/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
n
n
p
If we call and 1n n
p
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64845/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
n
n
p
If we call and 1n n
p
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
then, 1
2 2p
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64846/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
n
n
p
If we call and 1n n
p
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
then, 1
2 2p
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( ) ni x
nn
f x c e
( )2
np
i u
np
c f u e du
Integration value changed xu to avoid confusion
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64847/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( ) ni x
nn
f x c e
( )2
np
i u
np
c f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64848/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( ) ni x
nn
f x c e
( )2
np
i u
np
c f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64849/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( ) ni x
nn
f x c e
( )2
np
i u
np
c f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64850/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( ) ni x
nn
f x c e
( )2
np
i u
np
c f u e du
( ) ( )2
n np
i u i x
pn
f x f u e du e
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64851/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( ) ni x
nn
f x c e
( )2
np
i u
np
c f u e du
( ) ( )2
n np
i u i x
pn
f x f u e du e
( )( )2
np
i x u
pn
f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64852/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( ) ni x
nn
f x c e
( )2
np
i u
np
c f u e du
( ) ( )2
n np
i u i x
pn
f x f u e du e
( )( )2
np
i x u
pn
f u e du
( )1( )
2n
pi x u
pn
f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64853/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( )1( ) ( )
2n
pi x u
pn
f x f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64854/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( )1( ) ( )
2n
pi x u
pn
f x f u e du
1( ) ( )
2n
n
f x F
( ), ( ) ( ) np
i x u
np
where F f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64855/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( )1( ) ( )
2n
pi x u
pn
f x f u e du
1( ) ( )
2n
n
f x F
( ), ( ) ( ) np
i x u
np
where F f u e du
0 ,as pp
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64856/152
2008_Fourier Transform(1)
( ) ( )nn
F F d
n continuous variable
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( )1( ) ( )
2n
pi x u
pn
f x f u e du
1( ) ( )
2n
n
f x F
( ), ( ) ( ) np
i x u
np
where F f u e du
0 ,as pp
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64857/152
2008_Fourier Transform(1)
( ) ( )nn
F F d
n continuous variable
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
( )1( ) ( )
2n
pi x u
pn
f x f u e du
1( ) ( )
2n
n
f x F
( ), ( ) ( ) np
i x u
np
where F f u e du
0 ,as pp
also,
( )( ) ( ) i x uF f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64858/152
2008_Fourier Transform(1)
( ) ( )nn
F F d
n continuous variable
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1( ) ( )
2n
n
f x F
( ), ( ) ( ) n
pi x u
np
where F f u e du
0 ,as pp
also,
( )( ) ( ) i x uF f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64859/152
2008_Fourier Transform(1)
( ) ( )nn
F F d
n continuous variable
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1( ) ( )
2n
n
f x F
( ), ( ) ( ) n
pi x u
np
where F f u e du
0 ,as pp
also,
( )( ) ( ) i x uF f u e du
1( ) ( )
2f x F d
( ), ( ) ( ) i x uwhere F f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64860/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1( ) ( )
2f x F d
( ), ( ) ( ) i x uwhere F f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64861/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1( ) ( )
2f x F d
( ), ( ) ( ) i x uwhere F f u e du
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64862/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1( ) ( )
2f x F d
( ), ( ) ( ) i x uwhere F f u e du
( )1( ) ( )
2
i x uf x f u e dud
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64863/152
2008_Fourier Transform(1)
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
Fourier Transform of ( )f x
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1( ) ( )
2f x F d
( ), ( ) ( ) i x uwhere F f u e du
( )1( ) ( )
2
i x uf x f u e dud
1 1( )
2 2
i u i xf u e du e d
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64864/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
1ˆ( ) ( )2
i xf f x e dx
Fourier Transform of ( )f x
1 ˆ( ) ( )2
i xf x f e d
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1 1( ) ( )
2 2
i u i xf x f u e du e d
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64865/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
1ˆ( ) ( )2
i xf f x e dx
Fourier Transform of ( )f x
1 ˆ( ) ( )2
i xf x f e d
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1 1( ) ( )
2 2
i u i xf x f u e du e d
If we define
1ˆ( ) ( ) )2
1(
2
i x i uf u e duf f x e dx
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64866/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
1ˆ( ) ( )2
i xf f x e dx
Fourier Transform of ( )f x
1 ˆ( ) ( )2
i xf x f e d
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1 1( ) ( )
2 2
i u i xf x f u e du e d
If we define
1ˆ( ) ( ) )2
1(
2
i x i uf u e duf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
then,
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64867/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
1ˆ( ) ( )2
i xf f x e dx
Fourier Transform of ( )f x
1 ˆ( ) ( )2
i xf x f e d
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1 1( ) ( )
2 2
i u i xf x f u e du e d
If we define
1ˆ( ) ( ) )2
1(
2
i x i uf u e duf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
then,
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64868/152
2008_Fourier Transform(1)
Fourier Series and Fourier Transform*Function expansion
Fourier Series Fourier Transform
1ˆ( ) ( )2
i xf f x e dx
Fourier Transform of ( )f x
1 ˆ( ) ( )2
i xf x f e d
Inverse Fourier Transform of ˆ( )f
Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)
/( ) in x pn
n
f x c e
/1( )
2
pin x p
np
c f x e dxp
1 1( ) ( )
2 2
i u i xf x f u e du e d
If we define
1ˆ( ) ( ) )2
1(
2
i x i uf u e duf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
then,
2 22
2f
T p p
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64869/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
70/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
71/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
1ˆ( ) ( )2
i xf f x e dx
1( ) cos sin
2f x x i x dx
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
72/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
1ˆ( ) ( )2
i xf f x e dx
1( ) cos sin
2f x x i x dx
1 1( )cos ( )sin
2 2f x xdx i f x xdx
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
73/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
1ˆ( ) ( )2
i xf f x e dx
1( ) cos sin
2f x x i x dx
1 1( )cos ( )sin
2 2f x xdx i f x xdx
even odd
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
74/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
1ˆ( ) ( )2
i xf f x e dx
1( ) cos sin
2f x x i x dx
1 1( )cos ( )sin
2 2f x xdx i f x xdx
even odd
if, ( ) :oddf x
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
75/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
1ˆ( ) ( )2
i xf f x e dx
1( ) cos sin
2f x x i x dx
1 1( )cos ( )sin
2 2f x xdx i f x xdx
even odd
if, ( ) :oddf x
1 1( )cos ( )sin
2 2f x xdx i f x xdx
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
76/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
1ˆ( ) ( )2
i xf f x e dx
1( ) cos sin
2f x x i x dx
1 1( )cos ( )sin
2 2f x xdx i f x xdx
even odd
if, ( ) :oddf x
1 1( )cos ( )sin
2 2f x xdx i f x xdx
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
evenodd odd odd
77/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
1ˆ( ) ( )2
i xf f x e dx
1( ) cos sin
2f x x i x dx
1 1( )cos ( )sin
2 2f x xdx i f x xdx
even odd
if, ( ) :oddf x
1 1( )cos ( )sin
2 2f x xdx i f x xdx
0
20 ( )sin
2i f x xdx
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
evenodd odd odd
78/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
cos sini xe x i x By the Euler formula
1ˆ( ) ( )2
i xf f x e dx
1( ) cos sin
2f x x i x dx
1 1( )cos ( )sin
2 2f x xdx i f x xdx
even odd
if, ( ) :oddf x
1 1( )cos ( )sin
2 2f x xdx i f x xdx
0
20 ( )sin
2i f x xdx
0
2ˆ( ) ( )sinf i f x xdx
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
:
:
:
odd even
odd odd
even even
f f
f f
f f
odd
even
even
evenodd odd odd
79/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64880/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
81/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
0
0
2ˆ( ) ( )sin( )
2 ˆ( )sin( ) ( )
f i f x x dx
i f x x dx f
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
82/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
0
0
2ˆ( ) ( )sin( )
2 ˆ( )sin( ) ( )
f i f x x dx
i f x x dx f
odd function!
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
83/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
0
0
2ˆ( ) ( )sin( )
2 ˆ( )sin( ) ( )
f i f x x dx
i f x x dx f
odd function!
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
84/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
0
0
2ˆ( ) ( )sin( )
2 ˆ( )sin( ) ( )
f i f x x dx
i f x x dx f
odd function!
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
1 1ˆ ˆ( ) ( )cos ( )sin2 2
f x f x xd i f x xd
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
85/152
2008_Fourier Transform(1)
Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
0
0
2ˆ( ) ( )sin( )
2 ˆ( )sin( ) ( )
f i f x x dx
i f x x dx f
odd function!
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
1 1ˆ ˆ( ) ( )cos ( )sin2 2
f x f x xd i f x xd
oddodd
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
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Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
0
0
2ˆ( ) ( )sin( )
2 ˆ( )sin( ) ( )
f i f x x dx
i f x x dx f
odd function!
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
1 1ˆ ˆ( ) ( )cos ( )sin2 2
f x f x xd i f x xd
evenodd oddodd
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
87/152
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Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2 ˆ0 ( )sin2
i f x xd
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
0
0
2ˆ( ) ( )sin( )
2 ˆ( )sin( ) ( )
f i f x x dx
i f x x dx f
odd function!
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
1 1ˆ ˆ( ) ( )cos ( )sin2 2
f x f x xd i f x xd
evenodd oddodd
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
88/152
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Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2 ˆ0 ( )sin2
i f x xd
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
0
0
2ˆ( ) ( )sin( )
2 ˆ( )sin( ) ( )
f i f x x dx
i f x x dx f
odd function!
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
1 1ˆ ˆ( ) ( )cos ( )sin2 2
f x f x xd i f x xd
evenodd oddodd
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
ˆ( ) :f cf, odd function?
0
2 ˆ( )sini f x xd
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Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2 ˆ( ) ( )sinf x i f x xd
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
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Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2 ˆ( ) ( )sinf x i f x xd
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
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Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2 ˆ( ) ( )sinf x i f x xd
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
2 2 2( )f x i i
The numerical factor
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
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Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2 ˆ( ) ( )sinf x i f x xd
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
2 2 2( )f x i i
The numerical factor
Thus the imaginary factors are not needed, the factor may multiply
either of the two integrals, or each integral may be multiplied by .
Let us make the latter choice in giving the following definition
2
2
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
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Fourier Cosine and Sine Transform*
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
0
2 ˆ( ) ( )sinf x i f x xd
0
2ˆ( ) ( )sinf i f x xdx
if, ( ) :oddf x
2 2 2( )f x i i
The numerical factor
Thus the imaginary factors are not needed, the factor may multiply
either of the two integrals, or each integral may be multiplied by .
Let us make the latter choice in giving the following definition
2
2
*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
0
2 ˆ( ) ( ) (sin )f x f x d
0
2ˆ( ) ( )sinf f x xdx
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Fourier Cosine and Sine Transform
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
Fourier Sine Transform
The Fourier transform of an odd function on the interval is the sine
transform
Where,
),(
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
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Fourier Cosine and Sine Transform
Fourier Transform
Fourier Transform of : ( )f x
Inverse Fourier Transform of : ˆ( )f
1ˆ( ) ( )2
i xf f x e dx
1 ˆ( ) ( )2
i xf x f x e d
is even, if f )()( xfxf
dxxfdxxfaa
a 0
)(2)(
is odd, if f )()( xfxf
0)( dxxfa
a
In a similar way
Fourier Cosine Transform
The Fourier transform of an even function on the interval is the cosine
transform
Where,
),(
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
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Fourier Transform
Convergence of a Fourier Transform
Conditions for Convergence
Let and be piecewise continuous on every finite interval, and let be
absolutely integrable on . Then the Fourier transform of on the
interval converges to at a point of continuity. At a point of discontinuity,
the Fourier integral will converge to the average
Where and denote the limit of at from the right and from
the left, respectively.
Theorem 15.1
2
)()( xfxf
f f f*),( f
)(xf
)( xf )( xf xf
* This means that the integral converges.
dxxf |)(|
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1 if 0,
1 if ,1)(
x
xxf
Find the Fourier integral representation of the function
1
x
)(xf
1 0 1
Example 1 an impulse
Fourier Cosine and Sine Transform
1) Fourier Transform
1
1
1 1ˆ( ) ( ) 12 2
i x i xf f x e dx e dx
1
1
1 1 1 1( )
2 2
1 1( 2 sin )
2
i x i ie e ei i
ii
2 sin
cos sin
cos sin
i
i
e i
e i
By the Euler formula
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
When f : odd
When f : even
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1 if 0,
1 if ,1)(
x
xxf
Find the Fourier integral representation of the function
1
x
)(xf
1 0 1
Example 1 an impulse
Fourier Cosine and Sine Transform
1) Fourier Transform
2 sinˆ( )f
1 2 sin 1 sin
( ) cos sin2
i xf x e d x i x d
even
odd
sin
oddeven
odd
0
1 sin 2 sin cos( ) cos
xf x xd d
even
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
When f : odd
When f : even
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1 if 0,
1 if ,1)(
x
xxf
Find the Fourier integral representation of the function
1
x
)(xf
1 0 1
Example 1 an impulse
Fourier Cosine and Sine Transform
2) Fourier Sine Transform
1
1
1
1
ˆ( ) ( )sin 1 sin
cos cos cos( )0
f f x xdx xdx
x
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
When f : odd
When f : even
f : even can’t apply Fourier Sine Transform formula0
2ˆ( ) ( )sinf f x xdx
Bur if try to integrate f with sine over x
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1 if 0,
1 if ,1)(
x
xxf
0 0
2 2 sin 2 sin cos( ) cos
xf x xd d
Find the Fourier integral representation of the function
1
x
)(xf
1 0 1
Example 1 an impulse
Fourier Cosine and Sine Transform
3) Fourier Cosine Transform
11
0 00
2 2 2 sin 2 sinˆ( ) ( )cos 1 cosx
f f x xdx xdx
Because f(x) is even function, the result of Fourier transform and Fourier Cosine transform are identical
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
When f : odd
When f : even
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0
2 cos sin( )
xf x d
1 if 0,
1 if ,1)(
x
xxf
1
x
)(xf
1 0 1
Fourier Cosine and Sine Transform
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The average of the left- and right-hand limits of f (x) at x = 1 is equal to (1+0)/2, that is, 1/2. (Theorem 15.1)
Furthermore, multipling by we obtain by2/
0
2 cos sin( )
xf x d
1 if 0,
1 if ,1)(
x
xxf
1
x
)(xf
1 0 1
Fourier Cosine and Sine Transform
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The average of the left- and right-hand limits of f (x) at x = 1 is equal to (1+0)/2, that is, 1/2. (Theorem 15.1)
Furthermore, multipling by we obtain by
0
1 , 12 2
cos sin 1( ) , 1
2 2 2 4
0 0, 12
x
xf x d x
x
2/
0
2 cos sin( )
xf x d
1 if 0,
1 if ,1)(
x
xxf
1
x
)(xf
1 0 1
Fourier Cosine and Sine Transform
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The average of the left- and right-hand limits of f (x) at x = 1 is equal to (1+0)/2, that is, 1/2. (Theorem 15.1)
Furthermore, multipling by we obtain by
0
1 , 12 2
cos sin 1( ) , 1
2 2 2 4
0 0, 12
x
xf x d x
x
2/
We mention that this integral is called Dirichlet’s
discontinuous factor.
0
2 cos sin( )
xf x d
1 if 0,
1 if ,1)(
x
xxf
1
x
)(xf
1 0 1
Fourier Cosine and Sine Transform
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Example 2 Cosine and Sine integral
Represent
Fourier Cosine and Sine Transform
( ) , 0xf x e x
(a) By a cosine integral
(b) By a sine integralx
y
1
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
When f : odd
When f : even
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Example 2 Cosine and Sine integral
Represent
Fourier Cosine and Sine Transform
( ) , 0xf x e x
(a) By a cosine integral
(b) By a sine integral
(a) By a cosine integral
20
2 2 1ˆ( ) cos1
xf e x dx
x
y
1
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
When f : odd
When f : even
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Example 2 Cosine and Sine integral
Represent
Fourier Cosine and Sine Transform
( ) , 0xf x e x
(a) By a cosine integral
(b) By a sine integral
(a) By a cosine integral
20
2 2 1ˆ( ) cos1
xf e x dx
x
y
1
0 00
0
2 200
2 2 0
20
sin sincos ( )
sin0
( cos ) ( cos )( )
1 1cos
1cos
1
x x x
x
x x
x
x
x xe x dx e e dx
xe dx
x xe e dx
e xdx
e xdx
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
When f : odd
When f : even
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Example 2 Cosine and Sine integral
Represent
Fourier Cosine and Sine Transform
( ) , 0xf x e x
(a) By a cosine integral
(b) By a sine integral
(a) By a cosine integral
20
2 2 1ˆ( ) cos1
xf e x dx
20
2 cos( )
1
xf x d
x
y
1
x
y
1
Cosine integral
0 00
0
2 200
2 2 0
20
sin sincos ( )
sin0
( cos ) ( cos )( )
1 1cos
1cos
1
x x x
x
x x
x
x
x xe x dx e e dx
xe dx
x xe e dx
e xdx
e xdx
1 ˆ( ) ( )2
i xf x f e d
1ˆ( ) ( )2
i xf f x e dx
0
2 ˆ( ) ( )sinf x f xd
0
2ˆ( ) ( )sinf f x xdx
0
2 ˆ( ) ( )cosf x f xd
0
2ˆ( ) ( )cosf f x xdx
When f : odd
When f : even
109/152
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Example 4 Cosine and Sine integral Representation
Represent
Fourier Cosine and Sine Transform
( ) , 0xf x e x
(a) By a cosine integral
(b) By a sine integralx
y
1
Fourier Cosine and Sine Integrals
(i) The Fourier Integral of an even function on the interval is the cosine integral
Where,
(ii) The Fourier Integral of an odd function on the interval is the sine integral
Where,
Definition 15.2
),(
),(
0( ) ( ) (cos )f x A x d
0
2( ) ( )cos ,A f x xdx
0( ) ( ) (sin )f x B x d
0
2( ) ( )sinB f x xdx
110/152
2008_Fourier Transform(1)
Example 4 Cosine and Sine integral Representation
Represent
Fourier Cosine and Sine Transform
( ) , 0xf x e x
(a) By a cosine integral
(b) By a sine integral
(b) By a sine integral
20
2 2ˆ( ) sin1
xf e xdx
x
y
1
0 00
2 200
2 0
20
( cos ) ( cos )sin ( )
1 sin (sin )( )
1 10 sin
sin1
x x x
x x
x
x
x xe x dx e e dx
x xe e dx
e xdx
e xdx
Fourier Cosine and Sine Integrals
(i) The Fourier Integral of an even function on the interval is the cosine integral
Where,
(ii) The Fourier Integral of an odd function on the interval is the sine integral
Where,
Definition 15.2
),(
),(
0( ) ( ) (cos )f x A x d
0
2( ) ( )cos ,A f x xdx
0( ) ( ) (sin )f x B x d
0
2( ) ( )sinB f x xdx
111/152
2008_Fourier Transform(1)
Example 4 Cosine and Sine integral Representation
Represent
Fourier Cosine and Sine Transform
( ) , 0xf x e x
(a) By a cosine integral
(b) By a sine integral
(b) By a sine integral
20
2 2ˆ( ) sin1
xf e xdx
20
2 sin( )
1
xf x d
x
y
1
x
y
1
sine integral
0 00
2 200
2 0
20
( cos ) ( cos )sin ( )
1 sin (sin )( )
1 10 sin
sin1
x x x
x x
x
x
x xe x dx e e dx
x xe e dx
e xdx
e xdx
Fourier Cosine and Sine Integrals
(i) The Fourier Integral of an even function on the interval is the cosine integral
Where,
(ii) The Fourier Integral of an odd function on the interval is the sine integral
Where,
Definition 15.2
),(
),(
0( ) ( ) (cos )f x A x d
0
2( ) ( )cos ,A f x xdx
0( ) ( ) (sin )f x B x d
0
2( ) ( )sinB f x xdx
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Background of Fourier Series
113/152
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Orthogonal Set/ Weight Function
A set of real-valued functions is said to be
orthogonal with respect to a weight function on an interval if
Definition 12.4
nmdxxxxwb
anm ,0)()()(
),(),(),( 210 xxx
],[ ba)(xw
Sturm-Liouville Problem
Solve(Equation):
0)]()([])([ yxpxqyxrdx
d
Subject to (Boundary Condition): 0)()(
0)()(
22
11
byBbyA
ayBayA
Boundary Value Problem
Background of Fourier Series①
② Solutions of Sturm-Liouville equation are
Eigenfunctions (All of the solutions are
linearly independent and Orthogonal)
③ Solutions are Basis functions.
ex)
0 yy
kxBkxAxy sincos)(
)()(
)()(
yy
yyboundary
condition
,2,1,0k
solution
‘Basis functions’ and ‘Orthogonal Set’
1
0 )sincos()(n
nn nxbnxaaxf
④ Series solution using Orthogonal Set
Fourier Sine/Cosine Series
By utilizing the inner product
Properties of the Regular Sturm-Liouville Problem
(a) There exist an infinite number of real eigenvalues that can be arranged in increasing
order such that as
(b) For each eigenvalues there is only one eigenfunction (except for nonzero constant multiples)
(c) Eigenfunctions corresponding to different eigenvalues are linearly independent
(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with
respect to the weight function on interval
Theorem 12.3
)(xp
n 321 n n
],[ ba
( )f x :a given function
Fourier-Bessel Series
Fourier-Legendre Series
1
)()(i
ini xJcxf
0
)()(n
nn xPcxf
Sturm-Liouville Problem Boundary Value Problem
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Fourier Series
Fourier Series
The Fourier series of a function defined on the interval
Is given by
Definition 12.5
p
pn
p
pn
p
p
nnn
xdxp
nxf
pb
xdxp
nxf
pa
dxxfp
a
where
xp
nbx
p
na
axf
sin)(1
cos)(1
)(1
,
sincos2
)(
0
1
0
f ),( pp
115/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
Fourier Series
For some engineering problem, it is actually more convenient to representa real function in an infinite series of complex-valued function of a real variables such as x
xixe
xixeix
ix
sincos
sincos
Recall,
i
eex
eex
ixixixix
2sin,
2cos
Then,
n
pxinnecxf /)(
A real function is represented by a complex series ; a series in which the coefficients are complex numbers*
( )f x
nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
,( 1,2,3..)inxe n
2 22
2f
T p p
116/152
2008_Fourier Transform(1)
Complex Fourier Series
Complex Fourier Series
The complex Fourier series of functions defined on an interval
is given by
Definition 12.7
),( pp
,...2,1,0,)(2
1
,
/ ndxexf
pc
where
p
p
pxinn
f
n
pxinnecxf /)(
2 22
2f
T p p
117/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
i
eex
p
n
eex
p
n
pxinpxin
pxinpxin
2sin
2cos
//
//
1
/
1
/0
1
//0
1
////0
][2
1][
2
1
2
222
n
pxinn
n
pxinn
n
pxinnn
pxinnn
n
pxinpxin
n
pxinpxin
n
ececc
eibaeibaa
i
eeb
eea
a
0 0
,
1
2
1[ ]
2
1[ ]
2
n n n
n n n
where
c a
c a ib
c a ib
Derivation of complex Fourier series
A real function is represented by a complex series ; a series in which the coefficients are complex numbers*
( )f x
nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670118/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
p
pdxxf
pac )(
1
2
1
2
100
p
p
pxin
p
p
p
p
p
pn
dxexfp
dxxp
nix
p
nxf
p
xdxp
nxf
pixdx
p
nxf
pc
/)(2
1
sincos)(1
2
1
sin)(1
cos)(1
2
1
p
pn xdx
p
nxf
pb
sin)(
1
p
pdxxf
pa )(
10
p
pn xdx
p
nxf
pa
cos)(
1
xixe
xixeix
ix
sincos
sincos
Recall,
,
1[ ]
2n n n
where
c a ib
A real function is represented by a complex series ; a series in which the coefficients are complex numbers*
( )f x
nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670119/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
p
pdxxf
pac )(
1
2
1
2
100
p
p
pxin
p
p
p
p
p
pn
dxexfp
dxxp
nix
p
nxf
p
xdxp
nxf
pixdx
p
nxf
pc
/)(2
1
sincos)(1
2
1
sin)(1
cos)(1
2
1
xixe
xixeix
ix
sincos
sincos
Recall,
p
pn xdx
p
nxf
pb
sin)(
1
p
pdxxf
pa )(
10
p
pn xdx
p
nxf
pa
cos)(
1
,
1[ ]
2n n n
where
c a ib
A real function is represented by a complex series ; a series in which the coefficients are complex numbers*
( )f x
nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670120/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
0 0
,
1
2
1 1[ ], [ ]
2 2n n n n n n
where
c a
c a ib c a ib
p
p
pxinn dxexf
pc /)(
2
1
p
p
pxinn dxexf
pc /)(
2
1
p
pdxxf
pc )(
2
10
Complex Fourier Series
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
121/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
0 0
,
1
2
1 1[ ], [ ]
2 2n n n n n n
where
c a
c a ib c a ib
p
p
pxinn dxexf
pc /)(
2
1
p
p
pxinn dxexf
pc /)(
2
1
p
pdxxf
pc )(
2
10
/ / / /00
1 1 1 1
0
1
1 1( ) [ ] [ ]
2 2 2
1[ ](cos sin ) [ ](cos sin )
2 2
in x p in x p in x p in x pn n n n n n
n n n n
n n n nn
af x c c e c e a ib e a ib e
a n n n na ib x i x a ib x i x
p p p p
Complex Fourier Series
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
122/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
0 0
,
1
2
1 1[ ], [ ]
2 2n n n n n n
where
c a
c a ib c a ib
p
p
pxinn dxexf
pc /)(
2
1
p
p
pxinn dxexf
pc /)(
2
1
p
pdxxf
pc )(
2
10
/ / / /00
1 1 1 1
0
1
1 1( ) [ ] [ ]
2 2 2
1[ ](cos sin ) [ ](cos sin )
2 2
in x p in x p in x p in x pn n n n n n
n n n n
n n n nn
af x c c e c e a ib e a ib e
a n n n na ib x i x a ib x i x
p p p p
Complex Fourier Series
0
1
1cos sin sin cos cos sin sin cos
2 2n n n n n n n n
n
a n n n n n n n na x b x i a x b x a x b x i a x b x
p p p p p p p p
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
123/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
0 0
,
1
2
1 1[ ], [ ]
2 2n n n n n n
where
c a
c a ib c a ib
p
p
pxinn dxexf
pc /)(
2
1
p
p
pxinn dxexf
pc /)(
2
1
p
pdxxf
pc )(
2
10
/ / / /00
1 1 1 1
0
1
1 1( ) [ ] [ ]
2 2 2
1[ ](cos sin ) [ ](cos sin )
2 2
in x p in x p in x p in x pn n n n n n
n n n n
n n n nn
af x c c e c e a ib e a ib e
a n n n na ib x i x a ib x i x
p p p p
Complex Fourier Series
0
1
1cos sin sin cos cos sin sin cos
2 2n n n n n n n n
n
a n n n n n n n na x b x i a x b x a x b x i a x b x
p p p p p p p p
imaginary partimaginary part
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
124/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
0 0
,
1
2
1 1[ ], [ ]
2 2n n n n n n
where
c a
c a ib c a ib
p
p
pxinn dxexf
pc /)(
2
1
p
p
pxinn dxexf
pc /)(
2
1
p
pdxxf
pc )(
2
10
/ / / /00
1 1 1 1
0
1
1 1( ) [ ] [ ]
2 2 2
1[ ](cos sin ) [ ](cos sin )
2 2
in x p in x p in x p in x pn n n n n n
n n n n
n n n nn
af x c c e c e a ib e a ib e
a n n n na ib x i x a ib x i x
p p p p
Complex Fourier Series
0
1
1cos sin sin cos cos sin sin cos
2 2n n n n n n n n
n
a n n n n n n n na x b x i a x b x a x b x i a x b x
p p p p p p p p
0
1
( ) cos sin2
n nn
a n nf x a x b x
p p
imaginary partimaginary part
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
125/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
0 0
,
1
2
1 1[ ], [ ]
2 2n n n n n n
where
c a
c a ib c a ib
p
p
pxinn dxexf
pc /)(
2
1
p
p
pxinn dxexf
pc /)(
2
1
p
pdxxf
pc )(
2
10
A real function is represented by a complex series ; a series in which the coefficients are complex numbers*
( )f x
nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ / / /00
1 1 1 1
0
1
1 1( ) [ ] [ ]
2 2 2
1[ ](cos sin ) [ ](cos sin )
2 2
in x p in x p in x p in x pn n n n n n
n n n n
n n n nn
af x c c e c e a ib e a ib e
a n n n na ib x i x a ib x i x
p p p p
Complex Fourier Series
0
1
1cos sin sin cos cos sin sin cos
2 2n n n n n n n n
n
a n n n n n n n na x b x i a x b x a x b x i a x b x
p p p p p p p p
0
1
( ) cos sin2
n nn
a n nf x a x b x
p p
real part
imaginary partimaginary part
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
126/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
//
1
// )(2
1)(
2
1)(
2
1
n
pxinp
p
pxin
n
pxinp
p
pxinp
pedxexf
pedxexf
pdxxf
p
1
/
1
/0
n
pxinn
n
pxinn ececc
Complex Fourier Series
127/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
//
1
// )(2
1)(
2
1)(
2
1
n
pxinp
p
pxin
n
pxinp
p
pxinp
pedxexf
pedxexf
pdxxf
p
1
/
1
/0
n
pxinn
n
pxinn ececc
0
0
//)(2
1
n
pxinp
p
pxin edxexfp
1
//)(2
1
n
pxinp
p
pxin edxexfp
nn , 0when n
Complex Fourier Series
128/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
//
1
// )(2
1)(
2
1)(
2
1
n
pxinp
p
pxin
n
pxinp
p
pxinp
pedxexf
pedxexf
pdxxf
p
1
/
1
/0
n
pxinn
n
pxinn ececc
0
0
//)(2
1
n
pxinp
p
pxin edxexfp
1
//)(2
1
n
pxinp
p
pxin edxexfp
nn , 0when n
n
pxinp
p
pxin edxexfp
//)(2
1
Complex Fourier Series
129/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
//
1
// )(2
1)(
2
1)(
2
1
n
pxinp
p
pxin
n
pxinp
p
pxinp
pedxexf
pedxexf
pdxxf
p
1
/
1
/0
n
pxinn
n
pxinn ececc
0
0
//)(2
1
n
pxinp
p
pxin edxexfp
1
//)(2
1
n
pxinp
p
pxin edxexfp
nn , 0when n
n
pxinp
p
pxin edxexfp
//)(2
1
,...2,1,0,)(2
1, //
ndxexfp
cecp
p
pxinn
n
pxinn
Complex Fourier Series
in a more compact manner
130/152
2008_Fourier Transform(1)
Complex Fourier Series
Example 1 Complex Fourier SeriesExpandin a complex Fourier series
xexf x ,)(
pWith
][)1(2
1
2
1
2
1
)1()1(
)1(
inin
xininxxn
eein
dxedxeec
eninee
enineenin
nin
)1()sin(cos
)1()sin(cos)1(
)1(
0sin)1(cos nandn nsince
2
( ) ( ) 1 1( 1) ( 1)
2( 1) 2 ( 1)
sinh 1( 1)
1
n nn
n
e e e ec
in in
in
n
then
inx
n
n en
inxf
1
1)1(
sinh)(
2
/ /1( ) , ( ) , 0, 1, 2,...
2
pin x p in x p
n np
n
f x c e c f x e dx np
xixe
xixeix
ix
sincos
sincos
131/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
Expand in a complex Fourier series
132/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
2
1( 1)
1
nn
inc
n
133/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
134/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex
135/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
136/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
from
137/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
(1 )(cos sin ) cos sin (sin cos )in nx i nx nx n nx i nx n nx
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
from
138/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
(1 )(cos sin ) cos sin (sin cos )in nx i nx nx n nx i nx n nx
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
from
For positive integer : n=m (m>0): cos sin (sin cos )mx m mx i mx m mx
139/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
(1 )(cos sin ) cos sin (sin cos )in nx i nx nx n nx i nx n nx
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
from
For positive integer : n=m (m>0): cos sin (sin cos )mx m mx i mx m mx
For negative integer : n=-m (m>0): cos sin ( sin cos )mx m mx i mx m mx
140/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
(1 )(cos sin ) cos sin (sin cos )in nx i nx nx n nx i nx n nx
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
from
For positive integer : n=m (m>0): cos sin (sin cos )mx m mx i mx m mx
For negative integer : n=-m (m>0): cos sin ( sin cos )mx m mx i mx m mx :opposite in sign
141/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
(1 )(cos sin ) cos sin (sin cos )in nx i nx nx n nx i nx n nx
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
from
2
1( 1)
1
n inxine
n
real part of will be remained in the summation
For positive integer : n=m (m>0): cos sin (sin cos )mx m mx i mx m mx
For negative integer : n=-m (m>0): cos sin ( sin cos )mx m mx i mx m mx :opposite in sign
142/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
(1 )(cos sin ) cos sin (sin cos )in nx i nx nx n nx i nx n nx
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
from
2
1( 1)
1
n inxine
n
real part of will be remained in the summation
For positive integer : n=m (m>0): cos sin (sin cos )mx m mx i mx m mx
For negative integer : n=-m (m>0): cos sin ( sin cos )mx m mx i mx m mx
( ) :f x real function
:opposite in sign
143/152
2008_Fourier Transform(1)
Complex Fourier Series
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
1
/
1
/0
n
pxinn
n
pxinn ececc
A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc
*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670
/ /1, ( )
2
pin x p in x p
n np
n
c e c f x e dxp
in a more compact manner
Complex Fourier Series
Example)
xexf x ,)(
sinh( ) inx
nn
f x c e
Expand in a complex Fourier series
:complex
2
1( 1)
1
nn
inc
n
:complex( )f x : complex too?
(1 )(cos sin ) cos sin (sin cos )in nx i nx nx n nx i nx n nx
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
from
2
1( 1)
1
n inxine
n
real part of will be remained in the summation
For positive integer : n=m (m>0): cos sin (sin cos )mx m mx i mx m mx
For negative integer : n=-m (m>0): cos sin ( sin cos )mx m mx i mx m mx
( ) :f x real function
:opposite in sign
144/152
2008_Fourier Transform(1)
Complex Fourier Series
Fundamental Frequency
n
pxinn ecxf /)(
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
145/152
2008_Fourier Transform(1)
Complex Fourier Series
Fundamental Frequency
2T pFundamental period of the function
n
pxinn ecxf /)(
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
146/152
2008_Fourier Transform(1)
Complex Fourier Series
Fundamental Frequency
2T pFundamental period of the function
Fundamental angular frequency
n
pxinn ecxf /)(
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
,2
whereT
147/152
2008_Fourier Transform(1)
Complex Fourier Series
Fundamental Frequency
2T pFundamental period of the function
Fundamental angular frequency
n
pxinn ecxf /)(
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
,2
whereT
2 2
2T p p
148/152
2008_Fourier Transform(1)
Complex Fourier Series
Fundamental Frequency
n
xinn ecxf )(
2T pFundamental period of the function
Fundamental angular frequency
n
pxinn ecxf /)(
1
0 sincos2
)(n
nn xnbxnaa
xf
1
0 sincos2
)(n
nn xp
nbx
p
na
axf
,2
whereT
2 2
2T p p
149/152
2008_Fourier Transform(1)
Complex Fourier Series
Frequency Spectrum
nc
2 3023
If is periodic and fundamental period
the plot of the points is called
frequency spectrum
where is the fundamental angular frequency,
are the coefficient.
f T
( , )nn c
( ) in x
nnx cf e
nC
150/152
2008_Fourier Transform(1)
Complex Fourier Series
Example 2 Frequency SpectrumIn Example 1, so that take on the values Using
22 2
2 2 2
sinh 1 sinh 1
1 1 1n
nc
n n n
1 n,2,1,0
22 i
162.1644.1599.2676.3599.2644.1162.1nc
n 3 2 1 0 1 2 3
xexf x ,)(
2
sinh 1( ) ( 1)
1
n inx
n
inf x e
n
Example 1
2 21
2T p p
nc
1 2 30123
0.5
1.0
1.5
2.0
2.5
3.0
3.5
sinh3.676
xexf x ,)(
/ /1( ) , ( ) , 0, 1, 2,...
2
pin x p in x p
n np
n
f x c e c f x e dx np
p
151/152
2008_Fourier Transform(1)
Complex Fourier Series
Example 3 Frequency Spectrum
Find the frequency spectrum of the periodic square wave
2
1
4
1,0
4
1
4
1,1
4
1
2
1,0
)(
x
x
x
xf
2
121 psopT
1/2 1/42 2
1/2 1/4
1/42 /2 /2 /2 /2
1/4
( ) 1
1 1
2 2 2
in x in xn
in x in in in in
c f x e dx e dx
e e e e e
in n i n i
2sin
1
n
ncn
1/4
01/4
11
2c dx
5
10
3
10
1
2
110
3
10
5
1nc
n 3 2 1 0 1 2 345 4 5x
y
11
/ /1( ) , ( ) , 0, 1, 2,...
2
pin x p in x p
n np
n
f x c e c f x e dx np
2 2
2T p p
nc
2 40 3 5
2435
0.1
0.2
0.3
0.4
0.5
/2
/2
cos / 2 sin / 2
cos / 2 sin / 2
in
in
e n i n
e n i n
2 22
2T p p
152/152