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Fourier Transforms of Special Functions
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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
Introduction
Sufficient condition for the existence of a Fourier transform
dttf |)(|
dttf |)(|
That is, f(t) is absolutely integrable. However, the above condition is not the
necessary one.
Some Unabsolutely Integrable Functions
Sinusoidal Functions: cos t, sin t,…Unit Step Function: u(t).
Generalized Functions:– Impulse Function (t); and– Impulse Train.
Fourier Transforms of Special Functions
More on
Impulse Function
Dirac Delta Function
0
00)(
t
tt and 1)(
dtt
0 t
Also called unit impulse function.
Generalized Function
The value of delta function can also be defined in the sense of generalized function:
)0()()(
dttt )0()()(
dttt (t): Test Function
We shall never talk about the value of (t). Instead, we talk about the values of integrals
involving (t).
Properties of Unit Impulse Function
)()()( 00 tdtttt
)()()( 00 tdtttt
Pf)
dtttt )()( 0
Write t as t + t0
dtttt )()( 0
)( 0t
Properties of Unit Impulse Function
)0(||
1)()(
adttat )0(
||
1)()(
adttat
Pf)
dttat )()(
Write t as t/a
Consider a>0
dt
a
tt
a)(
1
)0(||
1
a
dttat )()(
Consider a<0
dt
a
tt
a)(
1
)0(||
1
a
Properties of Unit Impulse Function
)()0()()( tfttf )()0()()( tfttf
Pf)
dttttf )()]()([
dtttft )]()()[(
)0()0( f
dtttf )()()0(
dtttf )()]()0([
Properties of Unit Impulse Function
)()0()()( tfttf )()0()()( tfttf
Pf)
dttat )()(
)(||
1)( t
aat )(
||
1)( t
aat
)0(||
1
a
dttt
a)()(
||
1
dttt
a)()(
||
1
Properties of Unit Impulse Function
)()0()()( tfttf )()0()()( tfttf )(
||
1)( t
aat )(
||
1)( t
aat
0)( tt 0)( tt )()( tt )()( tt
Generalized Derivatives
The derivative f’(t) of an arbitrary generalized function f(t) is defined by:
dtttfdtttf )(')()()('
dtttfdtttf )(')()()('
Show that this definition is consistent to the ordinary definition for the first derivative of a continuous function.
dtttf )()(' dtttfttf
)(')()()(
=0
Derivatives of the -Function
)0(')(')()()('
dtttdttt )0(')(')()()('
dtttdttt
0
)()0(' ,
)()('
tdt
td
dt
tdt
)0()1()()( )()( nnn dttt
)0()1()()( )()( nnn dttt
0
)()( )()0( ,
)()(
t
n
nn
n
nn
dt
td
dt
tdt
Product Rule
)(')()()(')]'()([ ttfttfttf )(')()()(')]'()([ ttfttfttf
dttttf )(')]()([
Pf)dttttf )(')]()([
dtttft )](')()[(
dtttfttft )}()(')]'()(){[(
dtttftdtttft )]()'()[()]'()()[(
dtttftdtttft )]()'()[()]()()[('
dtttfttft )()](')()()('[
Product Rule
)()0(')(')0()(')( tftfttf )()0(')(')0()(')( tftfttf
)()'()]'()([)(')( ttfttfttf
Pf)
)]'()0([ tf )(')0( tf
)()0(' tf
Unit Step Function u(t)
Define
0)()()( dttdtttu
0)()()( dttdtttu
0 t
u(t)
00
01)(
t
ttu
Derivative of the Unit Step Function
Show that )()(' ttu
dtttu )()('
0)(' dtt
)]0()([ )0(
dtttu )(')(
dttt )()(
Derivative of the Unit Step Function
0 t
u(t)
DerivativeDerivative
0 t
(t)
Fourier Transforms of Special Functions
Fourier Transform Related to
Impulse Function
Fourier Transform for (t)
1)( Ft 1)( Ft
dtett tj)()]([F 10
t
tje
0 t
(t)
0
1
F(j)
F
Fourier Transform for (t)
Show that
det tj
2
1)(
det tj
2
1)(
]1[)( 1 Ft
de tj12
1
de tj
2
1
de tj
2
1The integration converges to
in the sense of generalized function.
)(t
Fourier Transform for (t)
Show that
0
cos1
)( tdt
0
cos1
)( tdt
det tj
2
1)(
dtjt )sin(cos
2
1
td
jtd sin
2cos
2
1
0
cos1
td Converges to (t) in the sense of generalized function.
Two Identities for (t)
dxey jxy
2
1)(
dxey jxy
2
1)(
0cos
1)( xydxy
0cos
1)( xydxy
These two ordinary integrations themselves are meaningless.
They converge to (t) in the sense of generalized function.
Shifted Impulse Function
0)( 0tjett F 0)( 0tjett F
0)()]([ 0tjejFttf F
0
1
|F(j)|
F
Use the fact
0 t
(t t0)
t0
Fourier Transforms of Special Functions
Fourier Transform of a Some Special Functions
Fourier Transform of a Constant
)(2)()( AjFAtf F )(2)()( AjFAtf F
dAeAjF tj][)( F
dteA tj )(
2
12
)(2 A
Fourier Transform of a Constant
)(2)()( AjFAtf F )(2)()( AjFAtf F
F
0 t
A A2()
0
F(j)
Fourier Transform of Exponential Wave
)(2)()( 00 jFetf tj F )(2)()( 0
0 jFetf tj F
)(2]1[ F
)]([])([ 00 jFetf tjF )]([])([ 0
0 jFetf tjF
)(2][ 00 tjeF
Fourier Transforms of Sinusoidal Functions
)()(cos 000 Ft )()(cos 000 Ft
)()(sin 000 jjt F )()(sin 000 jjt F
F
(+0)
0
F(j)
(0)
0 0
t
f(t)=cos0t
Fourier Transform of Unit Step Function
)()]([ jFtuFLet )()]([ jFtuF
)0for (except 1)()( ttutu
]1[)]()([ FF tutu
)(2)]([)]([ tutu FF
)(2)()( jFjF
F(j)=?
Can you guess it?
Fourier Transform of Unit Step Function
)(2)()( jFjF
Guess )()()( BkjF
)()()()()()( BBkkjFjF
)()()(2 BBk
k
0B() must be odd
Fourier Transform of Unit Step Function
Guess )()()( BkjF k
)()(' ttu
)()]([ jFtuF
1)]([)]('[ ttu FF
)()]('[ jFjtuF
)]()([ Bj
)()( Bjj
0
jB
1)(
Fourier Transform of Unit Step Function
Guess )()()( BkjF k
jB
1)(
jtu
1)()( F
jtu
1)()( F
Fourier Transform of Unit Step Function
jtu
1)()( F
jtu
1)()( F
F()
0
|F(j)|
0 t
1
f(t)
Fourier Transforms of Special Functions
Fourier Transform vs. Fourier Series
Find the FT of a Periodic Function
Sufficient condition --- existence of FT
dttf |)(|
dttf |)(|
Any periodic function does not satisfy this condition.
How to find its FT (in the sense of general function)?
Find the FT of a Periodic Function
We can express a periodic function f(t) as:
Tectf
n
tjnn
2 ,)( 0
0
Tectf
n
tjnn
2 ,)( 0
0
n
tjnnectfjF 0)]([)( FF
n
tjnn ec ][ 0F
n
n nc )(2 0
n
n nc )(2 0
Find the FT of a Periodic Function
We can express a periodic function f(t) as:
Tectf
n
tjnn
2 ,)( 0
0
Tectf
n
tjnn
2 ,)( 0
0
n
n ncjF )(2)( 0
n
n ncjF )(2)( 0
The FT of a periodic function consists of a sequence of equidistant impulses located at the harmonic frequencies of the function.
Example:Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()(
n
T nTtt )()( Find the FT of the impulse train.
Example:Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()(
n
T nTtt )()( Find the FT of the impulse train.
n
tjnT e
Tt 0
1)(
n
tjnT e
Tt 0
1)(
c n
Example:Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()(
n
T nTtt )()( Find the FT of the impulse train.
n
tjnT e
Tt 0
1)(
n
tjnT e
Tt 0
1)(
c n
n
T nT
t )(2
)]([ 0F
n
T nT
t )(2
)]([ 0F
0
Example:Impulse Train
0 tT 2T 3TT2T3T
n
T nT
t )(2
)]([ 0F
n
T nT
t )(2
)]([ 0F
0
0 0 20 3002030
2/T
F
Find Fourier Series Using Fourier Transform
n
tjnnectf 0)(
2/
2/
0)(1 T
T
tjnn etf
Tc
T/2 T/2
f(t)t
T/2 T/2
fo(t)
t
tjoo etfjF )()(
2/
2/)(
T
T
tjetf
)(1
0 jnFT
c on
Find Fourier Series Using Fourier Transform
n
tjnnectf 0)(
2/
2/
0)(1 T
T
tjnn etf
Tc
T/2 T/2
f(t)t
T/2 T/2
fo(t)
t
tjoo etfjF )()(
2/
2/)(
T
T
tjetf
)(1
0 jnFT
c on
Sampling the Fourier Transform of fo(t) with period 2/T, we can find the Fourier Series of f (t).
Example:The Fourier Series of a Rectangular Wave
0
f(t)
d
1
t0
t
fo(t)1
dtejFd
d
tjo
2/
2/)(
2sin
2 d
n
tjnnectf 0)(
)(1
0 jnFT
c on
2sin
2 0
0
dn
Tn
2sin
1 0dn
n
Example:The Fourier Transform of a Rectangular Wave
0
f(t)
d
1
t
n
tjnnectf 0)(
)(1
0 jnFT
c on
2sin
2 0
0
dn
Tn
2sin
1 0dn
n
F [f(t)]=?
n
n ncjF )(2)( 0
n
n ncjF )(2)( 0
)(2
sin2
)( 00
ndn
njF
n
)(2
sin2
)( 00
ndn
njF
n
