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Fourier Transforms of Special Functions
主講者:虞台文
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Content Introduction
More on Impulse Function Fourier Transform Related to Impulse Function
Fourier Transform of Some Special Functions
Fourier Transform vs. Fourier Series
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Introduction Sufficient condition for the existence of a
Fourier transform
dt t f |)(|
That is, f (t ) is absolutely integrable. However, the above condition is not the
necessary one.
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Some Unabsolutely Integrable Functions
Sinusoidal Functions: cos t , sin t ,…
Unit Step Function: u(t ).
Generalized Functions:
– Impulse Function (t ); and
– Impulse Train.
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Fourier Transforms of Special Functions
More on
Impulse Function
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Dirac Delta Function
0
00
)( t
t
t and 1)(
dt t
0t
Also called unit impulse function.
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Generalized Function The value of delta function can also be defined
in the sense of generalized function:
)0()()(
dt t t (t): Test Function
We shall never talk about the value of (t ).
Instead, we talk about the values of integrals
involving (t ).
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Properties of Unit Impulse Function
)()()( 00 t dt t t t
Pf)
dt t t t )()( 0
Write t as t + t 0
dt t t t )()( 0
)( 0t
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Properties of Unit Impulse Function
)0(
||
1)()(
a
dt t at
Pf)
dt t at )()(
Write t as t /a
Consider a>0
dt a
t t
a)(
1
)0(||
1
a
dt t at )()(
Consider a<0
dt a
t t
a)(
1
)0(||
1
a
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Properties of Unit Impulse Function
)()0()()( t f t t f
Pf)
dt t t t f )()]()([
dt t t f t )]()()[(
)0()0( f
dt t t f )()()0(
dt t t f )()]()0([
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Properties of Unit Impulse Function
)()0()()( t f t t f
Pf)
dt t at )()(
)(||
1
)( t aat
)0(||
1
a
dt t t
a)()(
||
1
dt t t
a)()(
||
1
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Properties of Unit Impulse Function
)()0()()( t f t t f
)(||
1
)( t aat
0)( t t )()( t t
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Generalized Derivatives
The derivative f ’ (t ) of an arbitrary
generalized function f (t ) is defined by:
dt t t f dt t t f )(')()()('
Show that this definition is consistent to the ordinarydefinition for the first derivative of a continuous function.
dt t t f )()(' dt t t f t t f
)(')()()(
=0
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Derivatives of the -Function
)0(')(')()()('
dt t t dt t t
0
)()0(' ,
)()('
t dt
t d
dt
t d t
)0()1()()( )()( nnn dt t t
0
)()( )()0( ,
)()(
t n
nn
n
nn
dt
t d
dt
t d t
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Product Rule)(')()()(')]'()([ t t f t t f t t f
dt t t t f )(')]()([
Pf) dt t t t f )(')]()([
dt t t f t )](')()[(
dt t t f t t f t )}()(')]'()(){[(
dt t t f t dt t t f t )]()'()[()]'()()[(
dt t t f t dt t t f t )]()'()[()]()()[('
dt t t f t t f t )()](')()()('[
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Product Rule)()0(')(')0()(')( t f t f t t f
)()'()]'()([)(')( t t f t t f t t f
Pf)
)]'()0([ t f )(')0( t f
)()0(' t f
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Unit Step Function u(t)
Define
0
)()()( dt t dt t t u
0t
u(t )
0001)(
t t t u
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Derivative of the Unit Step Function
Show that )()(' t t u
dt t t u )()('
0)(' dt t
)]0()([ )0(
dt t t u )(')(
dt t t )()(
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Derivative of the Unit Step Function
0t
u(t )
Derivative
0t
(t )
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Fourier Transforms of Special Functions
Fourier Transform
Related to
Impulse Function
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Fourier Transform for (t)
1)(
F t
dt et t t j)()]([F 10
t
t je
0 t
(t )
0
1
F ( j)
F
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Fourier Transform for (t)
Show that
d et t j
2
1)(
]1[)( 1 F t
d e t j1
2
1
d e t j
2
1
d e t j
2
1The integration converges to
in the sense of generalized function.
)(t
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Fourier Transform for (t)
Show that
0
cos1
)( td t
d et t j
2
1)(
d t jt )sin(cos
2
1
td
jtd sin
2cos
2
1
0
cos1
td Converges to (t ) in the sense of
generalized function.
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Two Identities for (t)
dxe y
jxy
2
1
)(
0
cos1
)( xydx y
These two ordinary integrations themselves are meaningless.
They converge to (t ) in the sense of generalized function.
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Shifted Impulse Function
0)(0
t jet t
F
0)()]([ 0
t je j F t t f
F
0
1
|F ( j)|
F
Use the fact
0 t
(t t 0)
t 0
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Fourier Transforms of Special Functions
Fourier Transform of a
Some Special Functions
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Fourier Transform of a Constant
)(2)()( A j F At f F
d Ae A j F t j][)( F
dt e At j )(
2
1
2
)(2 A
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Fourier Transform of a Constant
)(2)()( A j F At f F
F
0t
A A2()
0
F ( j)
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Fourier Transform of Exponential Wave
)(2)()( 00
j F et f t j F
)(2]1[ F
)]([])([ 00
j F et f t j
F
)(2][ 00 t j
eF
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Fourier Transforms of Sinusoidal Functions
)()(cos 000 F t
)()(sin 000 j jt F
F
(+0)
0
F ( j)
(0)
0
0
t
f (t )=cos0t
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Fourier Transform of Unit Step Function
)()]([ j F t uF Let )()]([ j F t uF
)0for (except1)()( t t ut u
]1[)]()([ F F t ut u
)(2)]([)]([ t ut u F F
)(2)()( j F j F
F ( j)=?
Can you guess it?
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Fourier Transform of Unit Step Function
)(2)()( j F j F
Guess )()()( Bk j F
)()()()()()( B Bk k j F j F
)()()(2 B Bk
k
0 B() must be odd
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Fourier Transform of Unit Step Function
Guess )()()( Bk j F k
)()(' t t u
)()]([ j F t uF
1)]([)]('[ t t u F F
)()]('[ j F jt uF
)]()([ B j
)()( B j j
0
j B
1
)(
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Fourier Transform of Unit Step Function
Guess )()()( Bk j F k
j B
1
)(
j
t u1
)()( F
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Fourier Transform of Unit Step Function
j
t u1
)()( F
F ()
0
| F ( j)|
0t
1
f (t )
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Fourier Transforms of Special Functions
Fourier Transform vs.
Fourier Series
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Find the FT of a Periodic Function
Sufficient condition --- existence of FT
dt t f |)(|
Any periodic function does not satisfy this
condition.
How to find its FT (in the sense of general
function)?
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Find the FT of a Periodic Function
We can express a periodic function f (t ) as:
T ect f
n
t jnn
2 ,)( 00
n
t jn
nect f j F 0
)]([)(F F
n
t jn
n ec ][0
F
n
n nc )(2 0
n
n nc )(2 0
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Find the FT of a Periodic Function
We can express a periodic function f (t ) as:
T ect f
n
t jnn
2 ,)( 00
nn nc j F )(2)( 0
The FT of a periodic function consists of a sequence of
equidistant impulses located at the harmonic frequencies
of the function.
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Example:
Impulse Train
0t
T 2T 3T T 2T 3T
n
T nT t t )()( Find the FT of the
impulse train.
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Example:
Impulse Train
0t
T 2T 3T T 2T 3T
n
T nT t t )()( Find the FT of the
impulse train.
n
t jn
T eT
t 01
)(
cn
2
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Example:
Impulse Train
0t
T 2T 3T T 2T 3T
n
T nT t t )()( Find the FT of the
impulse train.
n
t jn
T eT
t 01
)(
cn
n
T nT
t )(2
)]([ 0F
0
2
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Example:
Impulse Train
0t
T 2T 3T T 2T 3T
n
T nT
t )(2
)]([ 0F
0
0 0 20 30020 30
2/T
F
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Find Fourier Series Using
Fourier Transform
n
t jn
nect f 0)(
2/
2/
0)(1 T
T
t jn
n et f T
c
T /2 T /2
f (t )
t
T /2 T /2
f o(t )
t
t j
oo et f j F )()(
2/
2/)(
T
T
t jet f
)(
10 jn F T c on
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Find Fourier Series Using
Fourier Transform
n
t jn
nect f 0)(
2/
2/
0)(1 T
T
t jn
n et f T
c
T /2 T /2
f (t )
t
T /2 T /2
f o(t )
t
t j
oo et f j F )()(
2/
2/)(
T
T
t jet f
)(
10 jn F T c on
Sampling the Fourier Transform of f o(t ) with period
2/T , we can find the Fourier Series of f (t ).
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Example:
The Fourier Series of a Rectangular Wave
0
f (t )
d
1
t 0
t
f o(t )1
dt e j F d
d
t j
o
2/
2/)(
2sin
2 d
n
t jn
nect f 0)(
)(1
0 jn F T
c on
2
sin2 0
0
d n
Tn
2
sin1 0d n
n