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
2008_Fourier Transform(2)
Naval Architecture & Ocean Enginee ring
Fourier Transform(2)
: Fourier Transform Analysis
Basic Fourier Transform Analysis
Fourier Transform
2008_Fourier Transform(2)
Basic Fourier Transform Analysis
2008_Fourier Transform(2)
Basic Fourier Transform Analysis
Relationship of Conventional and Transform analysis
2008_Fourier Transform(2)
Basic Fourier Transform Analysis
Relationship of Conventional and Transform analysis
conventional analysis
2008_Fourier Transform(2)
Basic Fourier Transform Analysis
Relationship of Conventional and Transform analysis
conventional analysis
Problem statement
z
y x
2008_Fourier Transform(2)
Basic Fourier Transform Analysis
Relationship of Conventional and Transform analysis
conventional analysis
Problem statement
z
y x
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
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
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
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
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
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
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
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
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
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
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
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
2008_Fourier Transform(2)
Basic Fourier Transform Analysis
What kind of Transformation ?
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
01
0cos(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
01
0cos(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
01
0cos(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
