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Ex.) Fourier Transform of square wave function

문서에서 Engineering Mathematics 2 (페이지 37-58)

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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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( ) ft

( ) 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 ts 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( ) ft

( ) 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 ts 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)

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