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Engineering Mathematics 2

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(1)

2008_Fourier Transform(2)

Naval Architecture & Ocean Enginee ring

Engineering Mathematics 2

Prof. Kyu-Yeul Lee

Department of Naval Architecture and Ocean Engineering, Seoul National University of College of Engineering

[2008][13-1]

December, 2008

(2)

2008_Fourier Transform(2)

Naval Architecture & Ocean Enginee ring

Fourier Transform(2)

: Fourier Transform Analysis

Basic Fourier Transform Analysis

Fourier Transform

(3)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

(4)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

(5)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis

(6)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis

Problem statement

z

y  x

(7)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis

Problem statement

z

y  x

(8)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis

Complex analysis

Long-hand division Problem statement

z

y  x

(9)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis

Complex analysis

Long-hand division Problem statement

z

y  x

(10)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis

Problem solution Complex analysis

Long-hand division Problem statement

z

y  x

(11)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution Complex analysis

Long-hand division Problem statement

z

y  x

(12)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution Complex analysis

Long-hand division Problem statement

z

y  x

(13)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution Complex analysis

Long-hand division Problem statement

z y  x

Transform

z x

y log log

log  

(14)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution Complex analysis

Long-hand division Problem statement

z y  x

Transform

z x

y log log

log  

(15)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution

Simplified analysis Table look-up subtraction Complex analysis

Long-hand division Problem statement

z y  x

Transform

z x

y log log

log  

(16)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution

Simplified analysis Table look-up subtraction Complex analysis

Long-hand division Problem statement

z y  x

Transform

z x

y log log

log  

(17)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution

Simplified analysis Table look-up subtraction Complex analysis

Long-hand division

Inverse transform Anti-logarithm Table look-up Problem statement

z y  x

Transform

z x

y log log

log  

(18)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution

Simplified analysis Table look-up subtraction Complex analysis

Long-hand division

Inverse transform Anti-logarithm Table look-up Problem statement

z y  x

Transform

z x

y log log

log  

(19)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Relationship of Conventional and Transform analysis

conventional analysis transform analysis

Problem solution

Simplified analysis Table look-up subtraction Complex analysis

Long-hand division

Inverse transform Anti-logarithm Table look-up Problem statement

z y  x

Transform

z x

y log log

log  

complexity

reduced

(20)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

What kind of Transformation ?

(21)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis What kind of Transformation ?

z

y  x log log y  log x  log z

x 2

y  y x

dx

d  2

dx d

x 2

y   dx  ydx  3 1 x 3  c

(22)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis What kind of Transformation ?

z

y  x log log y  log x  log z

x 2

y  y x

dx

d  2

dx d

x 2

y   dx  ydx  3 1 x 3  c

) (t

f  0   e st dt ( ) { ( )}

0 e st f t dt  L f t

   Laplace Transform

(23)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis What kind of Transformation ?

z

y  x log log y  log x  log z

x 2

y  y x

dx

d  2

dx d

x 2

y   dx  ydx  3 1 x 3  c

) (t

f  0   e st dt ( ) { ( )}

0 e st f t dt  L f t

  

) (t

s     e i t dt

 

0

( ) ˆ ( ) e i t  f t dt f 

  

Laplace Transform

Fourier Transform

(24)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series* ( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(25)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined From -∞ to+ ∞

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(26)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Series

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(27)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined

From -∞ to+ ∞ Fourier Transform

frequency 3

 T 3

T 2 T 2

 T

1/ 2  1/ 4 

1/ 2  1/ 4 

1 T 1

 T

Fourier Series

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(28)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined From -∞ to+ ∞

6

 T

6

T t

1 2 

2

 T

2 T t

 1

Fourier Transform

frequency 3

 T 3

T 2 T 2

 T

1/ 2  1/ 4 

1/ 2  1/ 4 

1 T 1

 T

Fourier Series

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(29)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Transform

Synthesize a summation of sinusoids which add to give the waveform

6

 T

6

T t

1 2 

2

 T

2 T t

 1

Fourier Transform

frequency 3

 T 3

T 2 T 2

 T

1/ 2  1/ 4 

1/ 2  1/ 4 

1 T 1

 T

Fourier Series

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(30)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Transform

Synthesize a summation of sinusoids which add to give the waveform

+

6

 T

6

T t

1 2 

2

 T

2 T t

 1

Fourier Transform

frequency 3

 T 3

T 2 T 2

 T

1/ 2  1/ 4 

1/ 2  1/ 4 

1 T 1

 T

Fourier Series

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(31)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Transform

Synthesize a summation of sinusoids which add to give the waveform

Construct a diagram which displays amplitude and frequency of each sinusoid

+

6

 T

6

T t

1 2 

2

 T

2 T t

 1

Fourier Transform

frequency 3

 T 3

T 2 T 2

 T

1/ 2  1/ 4 

1/ 2  1/ 4 

1 T 1

 T

Fourier Series

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(32)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Transform

Synthesize a summation of sinusoids which add to give the waveform

Construct a diagram which displays amplitude and frequency of each sinusoid

+

6

 T

6

T t

1 2 

2

 T

2 T t

 1

Fourier Transform

frequency 3

 T 3

T 2 T 2

 T

1/ 2  1/ 4 

1/ 2  1/ 4 

1 T 1

 T

Fourier Series

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(33)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Series*

2

 T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Transform

Synthesize a summation of sinusoids which add to give the waveform

Construct a diagram which displays amplitude and frequency of each sinusoid

+

6

 T

6

T t

1 2 

2

 T

2 T t

 1

Fourier Transform

frequency 3

 T 3

T 2 T 2

 T

1/ 2  1/ 4 

1/ 2  1/ 4 

1 T 1

 T

Fourier Series

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

(34)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Transform* ( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

2

T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Transform

Synthesize a summation of sinusoids which add to give the waveform

+ 6

T

6

T

t

1 2

2

T

2 T t

1

Fourier Series

(35)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Transform*

The essence of the Fourier transform of a waveform is

to decompose or separate the waveform into a sum of sinusoids of different frequencies

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

2

T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Transform

Synthesize a summation of sinusoids which add to give the waveform

+ 6

T

6

T

t

1 2

2

T

2 T t

1

Fourier Series

(36)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Interpretation of the Fourier Transform*

The essence of the Fourier transform of a waveform is

to decompose or separate the waveform into a sum of sinusoids of different frequencies The pictorial representation of the Fourier transform is a diagram which displays

the amplitude and frequencies of each of the determined sinusoids

Construct a diagram which displays amplitude and frequency of each sinusoid

Fourier Transform

frequency 3

 T 3

T 2 T 2

 T

1/ 2  1/ 4 

1/ 2  1/ 4 

1 T 1

 T

Usual convention : displaying both positive and negative frequency sinusoids for each frequencies; the amplitude has been halved accordingly

( ) n in t p /

n

f t c e 



  c n  2 1 p   p p f t e ( )  in t p  / dt

2 2

2

T p p

  

   

2

T

2 T t

Waveform defined From -∞ to+ ∞

Fourier Transform

Synthesize a summation of sinusoids which add to give the waveform

+ 6

T

6

T

t

1 2

2

T

2 T t

1

Fourier Series

(37)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

t

( ) f t

1

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T

1/ 2 1/ 4

1/ 2 1/ 4

1 T 1

T

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Series

(38)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

t

( ) f t

1

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T

1/ 2 1/ 4

1/ 2 1/ 4

1 T 1

T

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Series

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(39)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T

1/ 2 1/ 4

1/ 2 1/ 4

1 T 1

T

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Series

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(40)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

t 

ˆ( ) f 

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(41)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

t 

ˆ( ) f 

cos(2  f t

0

)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(42)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

t

0

1 cos(6 ) 3  f t

ˆ( ) f 

cos(2  f t

0

)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(43)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

t

0

1 cos(6 ) 3  f t

ˆ( ) f 

cos(2  f t

0

)

1 ( ) s t 

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(44)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

t

0

1 cos(6 ) 3  f t

ˆ( ) f 

1/ 2 1/ 2

f0

f0

cos(2  f t

0

)

1 ( ) s t 

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(45)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

t

0

1 cos(6 ) 3  f t

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

cos(2  f t

0

)

1 ( ) s t 

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(46)

2008_Fourier Transform(2)

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

t

0

1 cos(6 ) 3  f t

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

cos(2  f t

0

)

1 ( ) s t 

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(47)

2008_Fourier Transform(2)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

t

0 0

cos(2 ) 1 cos(6 )

f t 3 f t

  

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(48)

2008_Fourier Transform(2)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

t

0

1 cos(10 )

5  f t

0 0

cos(2 ) 1 cos(6 )

f t 3 f t

  

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(49)

2008_Fourier Transform(2)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

t

0

1 cos(10 )

5  f t

2 ( ) 

s t 

0

1

0

cos(2 ) cos(6 )

f t 3 f t

  

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(50)

2008_Fourier Transform(2)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

t

0

1 cos(10 )

5  f t

1/10 1/10

2 ( )

s t 

0

1

0

cos(2 ) cos(6 )

f t 3 f t

  

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(51)

2008_Fourier Transform(2)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

t

0

1 cos(10 )

5  f t

1/10 1/10 1/10 1/10

2 ( )

s t 

0

1

0

cos(2 ) cos(6 )

f t 3 f t

  

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(52)

2008_Fourier Transform(2)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

t

1/10 1/10

0 0 0

1 1

cos(2 ) cos(6 ) cos(10 )

3 5

f t f t f t

    

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(53)

2008_Fourier Transform(2)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

t

0

1 cos(14 )

7  f t

1/10 1/10

0 0 0

1 1

cos(2 ) cos(6 ) cos(10 )

3 5

f t f t f t

    

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(54)

2008_Fourier Transform(2)

2

T

2 T t Waveform defined From -∞ to+ ∞

Fourier Transform Synthesize a summation of sinusoids

which add to give the waveform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

+ 6

T 6

T t

1 2

2

T

2 T t

1

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T Fourier Transform

Construct a diagram which displays amplitudeand frequencyof each sinusoid

Fourier Transform

frequency 3

T 3

T 2 T 2

T 1/ 2

1/ 4

1/ 2

1/ 4

1 T 1

T

Basic Fourier Transform Analysis

Ex.) Fourier Transform of square wave function

ˆ( ) f  t

( ) f t

1

ˆ( ) f 

1/ 2 1/ 2

1/ 6

1/ 6

f0

f0

t

0

1 cos(14 )

7  f t

1/10 1/10

0 0 0

1 1

cos(2 ) cos(6 ) cos(10 )

3 5

f t f t f t

    

3 ( ) s t 

Fourier Series

( ) n in t p /

n

f t c e 



 

1 /

2 ( )

p in t p

n p

c f t e dt

p

  

(5

참조

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