Download - Fourier Transforms of Special Functions
![Page 1: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/1.jpg)
Fourier Transforms of Special Functions
主講者:虞台文http://www.google.com/search?hl=en&sa=X&oi=spell&resnum=0&ct=result&cd=1&q=unit+step+fourier+transform&spell=1
![Page 2: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/2.jpg)
Content Introduction More on Impulse Function Fourier Transform Related to Impulse Function Fourier Transform of Some Special Functions Fourier Transform vs. Fourier Series
![Page 3: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/3.jpg)
Introduction Sufficient condition for the existence of a
Fourier transform
dttf |)(|
That is, f(t) is absolutely integrable. However, the above condition is not the
necessary one.
![Page 4: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/4.jpg)
Some Unabsolutely Integrable Functions
Sinusoidal Functions: cos t, sin t,…Unit Step Function: u(t).
Generalized Functions:– Impulse Function (t); and– Impulse Train.
![Page 5: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/5.jpg)
Fourier Transforms of Special Functions
More onImpulse Function
![Page 6: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/6.jpg)
Dirac Delta Function
000
)(tt
t and 1)(
dtt
0 t
Also called unit impulse function.
![Page 7: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/7.jpg)
Generalized Function The value of delta function can also be defined
in the sense of generalized function:
)0()()(
dttt (t): Test Function
We shall never talk about the value of (t). Instead, we talk about the values of integrals
involving (t).
![Page 8: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/8.jpg)
Properties of Unit Impulse Function
)()()( 00 tdtttt
Pf)
dtttt )()( 0
Write t as t + t0
dtttt )()( 0
)( 0t
![Page 9: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/9.jpg)
Properties of Unit Impulse Function
)0(||
1)()(
adttat
Pf)
dttat )()(
Write t as t/aConsider a>0
dt
att
a)(1
)0(||
1
a
dttat )()(
Consider a<0
dt
att
a)(1
)0(||
1
a
![Page 10: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/10.jpg)
Properties of Unit Impulse Function
)()0()()( tfttf
Pf)
dttttf )()]()([
dtttft )]()()[(
)0()0( f
dtttf )()()0(
dtttf )()]()0([
![Page 11: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/11.jpg)
Properties of Unit Impulse Function
)()0()()( tfttf
Pf)
dttat )()(
)(||
1)( ta
at
)0(||
1
a
dttt
a)()(
||1
dttt
a)()(
||1
![Page 12: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/12.jpg)
Properties of Unit Impulse Function
)()0()()( tfttf )(
||1)( ta
at
0)( tt )()( tt
![Page 13: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/13.jpg)
Generalized Derivatives
The derivative f’(t) of an arbitrary generalized function f(t) is defined by:
dtttfdtttf )(')()()('
Show that this definition is consistent to the ordinary definition for the first derivative of a continuous function.
dtttf )()(' dtttfttf
)(')()()(
=0
![Page 14: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/14.jpg)
Derivatives of the -Function
)0(')(')()()('
dtttdttt
0
)()0(' ,)()('
tdttd
dttdt
)0()1()()( )()( nnn dttt
0
)()( )()0( ,)()(
tn
nn
n
nn
dttd
dttdt
![Page 15: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/15.jpg)
Product Rule)(')()()(')]'()([ ttfttfttf
dttttf )(')]()([
Pf)dttttf )(')]()([
dtttft )](')()[(
dtttfttft )}()(')]'()(){[(
dtttftdtttft )]()'()[()]'()()[(
dtttftdtttft )]()'()[()]()()[('
dtttfttft )()](')()()('[
![Page 16: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/16.jpg)
Product Rule)()0(')(')0()(')( tftfttf
)()'()]'()([)(')( ttfttfttf
Pf)
)]'()0([ tf )(')0( tf
)()0(' tf
![Page 17: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/17.jpg)
Unit Step Function u(t)
Define
0)()()( dttdtttu
0 t
u(t)
0001
)(tt
tu
![Page 18: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/18.jpg)
Derivative of the Unit Step Function
Show that )()(' ttu
dtttu )()('
0)(' dtt
)]0()([ )0(
dtttu )(')(
dttt )()(
![Page 19: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/19.jpg)
Derivative of the Unit Step Function
0 t
u(t)Derivative
0 t
(t)
![Page 20: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/20.jpg)
Fourier Transforms of Special Functions
Fourier Transform Related toImpulse Function
![Page 21: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/21.jpg)
Fourier Transform for (t)
1)( Ft
dtett tj)()]([F 10
t
tje
0 t
(t)
0
1
F(j)
F
![Page 22: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/22.jpg)
Fourier Transform for (t)
Show that
det tj
21)(
]1[)( 1 Ft
de tj121
de tj
21
de tj
21
The integration converges to
in the sense of generalized function.
)(t
![Page 23: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/23.jpg)
Fourier Transform for (t)
Show that
0
cos1)( tdt
det tj
21)(
dtjt )sin(cos
21
tdjtd sin
2cos
21
0
cos1 td Converges to (t) in the sense of generalized function.
![Page 24: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/24.jpg)
Two Identities for (t)
dxey jxy
21)(
0cos1)( xydxy
These two ordinary integrations themselves are meaningless.
They converge to (t) in the sense of generalized function.
![Page 25: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/25.jpg)
Shifted Impulse Function
0)( 0tjett F
0)()]([ 0tjejFttf F
0
1
|F(j)|
F
Use the fact
0 t
(t t0)
t0
![Page 26: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/26.jpg)
Fourier Transforms of Special Functions
Fourier Transform of a Some Special Functions
![Page 27: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/27.jpg)
Fourier Transform of a Constant
)(2)()( AjFAtf F
dAeAjF tj][)( F
dteA tj )(
212
)(2 A
![Page 28: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/28.jpg)
Fourier Transform of a Constant
)(2)()( AjFAtf F
F
0 t
A A2()
0
F(j)
![Page 29: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/29.jpg)
Fourier Transform of Exponential Wave
)(2)()( 00 jFetf tj F
)(2]1[ F
)]([])([ 00 jFetf tjF
)(2][ 00 tjeF
![Page 30: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/30.jpg)
Fourier Transforms of Sinusoidal Functions
)()(cos 000 Ft
)()(sin 000 jjt F
F
(+0)
0
F(j)(0)
0 0
t
f(t)=cos0t
![Page 31: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/31.jpg)
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?
![Page 32: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/32.jpg)
Fourier Transform of Unit Step Function
)(2)()( jFjF
Guess )()()( BkjF
)()()()()()( BBkkjFjF
)()()(2 BBk
k
0B() must be odd
![Page 33: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/33.jpg)
Fourier Transform of Unit Step Function
Guess )()()( BkjF k
)()(' ttu
)()]([ jFtuF1)]([)]('[ ttu FF
)()]('[ jFjtuF)]()([ Bj
)()( Bjj
0
jB 1)(
![Page 34: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/34.jpg)
Fourier Transform of Unit Step Function
Guess )()()( BkjF k
jB 1)(
jtu 1)()( F
![Page 35: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/35.jpg)
Fourier Transform of Unit Step Function
jtu 1)()( F
F()
0
|F(j)|
0 t
1
f(t)
![Page 36: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/36.jpg)
Fourier Transforms of Special Functions
Fourier Transform vs. Fourier Series
![Page 37: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/37.jpg)
Find the FT of a Periodic Function
Sufficient condition --- existence of FT
dttf |)(|
Any periodic function does not satisfy this condition.
How to find its FT (in the sense of general function)?
![Page 38: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/38.jpg)
Find the FT of a Periodic Function
We can express a periodic function f(t) as:
Tectf
n
tjnn
2 ,)( 00
n
tjnnectfjF 0)]([)( FF
n
tjnn ec ][ 0F
n
n nc )(2 0
n
n nc )(2 0
![Page 39: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/39.jpg)
Find the FT of a Periodic Function
We can express a periodic function f(t) as:
Tectf
n
tjnn
2 ,)( 00
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.
![Page 40: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/40.jpg)
Example:Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()( Find the FT of the impulse train.
![Page 41: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/41.jpg)
Example:Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()( Find the FT of the impulse train.
n
tjnT e
Tt 0
1)(
c n
![Page 42: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/42.jpg)
Example:Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()( Find the FT of the impulse train.
n
tjnT e
Tt 0
1)(
c n
n
T nT
t )(2)]([ 0F 0
![Page 43: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/43.jpg)
Example:Impulse Train
0 tT 2T 3TT2T3T
n
T nT
t )(2)]([ 0F 0
0 0 20 3002030
2/T
F
![Page 44: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/44.jpg)
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
)(10 jnF
Tc on
![Page 45: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/45.jpg)
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
)(10 jnF
Tc on
Sampling the Fourier Transform of fo(t) with period 2/T, we can find the Fourier Series of f (t).
![Page 46: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/46.jpg)
Example:The Fourier Series of a Rectangular Wave
0
f(t)
d1
t0
t
fo(t)1
dtejFd
d
tjo
2/
2/)(
2sin2 d
n
tjnnectf 0)(
)(10 jnF
Tc on
2sin2 0
0
dnTn
2sin1 0dn
n
![Page 47: Fourier Transforms of Special Functions](https://reader034.vdocuments.site/reader034/viewer/2022042707/586cb6ef1a28ab08198bc69a/html5/thumbnails/47.jpg)
Example:The Fourier Transform of a Rectangular Wave
0
f(t)
d1
t
n
tjnnectf 0)(
)(10 jnF
Tc on
2sin2 0
0
dnTn
2sin1 0dn
n
F [f(t)]=?
n
n ncjF )(2)( 0
)(2
sin2)( 00
ndnn
jFn