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

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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





  

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

1/14

1/14

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





  

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

1/14

1/14

1/14

1/14

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





  

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

^{1/ 2 1/ 2}

1/ 6

1/ 6

f0

f0



0 0 0 0

1 1 1

( ) ( ) cos(2 ) cos(6 ) cos(10 ) cos(14 )

3 5 7

s t  s t

_

  f t   f t   f t   f t 

1/10 1/10

1/14

1/14 1/18 1/18

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





  

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

^{1/ 2 1/ 2}

1/ 6

1/ 6

f0

f0



0 0 0 0

1 1 1

( ) ( ) cos(2 ) cos(6 ) cos(10 ) cos(14 )

3 5 7

s t  s t

_

  f t   f t   f t   f t 

1/10 1/10

1/14

1/14 1/18 1/18

n

 

 

  

문서에서 Engineering Mathematics 2 (페이지 37-58)