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Comparison of Laplace Transform and Fourier

t (real word) s (Laplace domain) ω(Fourier domain) )

(t

f 定義 F s f t e

−st

dt

∞

∫

=

0

) ( )

( F ( ω ) =

∞

∫ f ( t ) e

−iωt

dt

∞

−

積分： ∫

t

f d

0

)

( τ τ ( )

1 F s

s ( ω )

ω F

− i

微分： f ' t ( ) sF ( s ) − f ( 0 ) i ω F ( ω ) 二次微分： f " t ( ) s

2

F ( s ) − sf ( 0 ) − f ' ( 0 ) − ω

2

F ( ω ) 向右平移 a： f ( t − a ) e

−sa

F (s ) e

−iωa

F ( ω )

乘以e

at

e

at

f ( t ) → F ( s − a ) e

iat

f ( t ) → F ( ω − a )

褶積： h ( t ) = f ( t ) * g ( t ) ∫

t

f u g t − u du

0

) ( )

( ∫

∞

f ( u ) g ( t − du u )

∞

−

) (t

f F(s) F(ω)

g(t) G(s) G(ω)

) (

* ) ( )

( t f t g t

h = H ( s ) = F ( s ) G ( s ) H ( ω ) = F ( ω ) G ( ω )

海大河工系 陳正宗特聘教授 廖倩綾製表(Tansform.doc)Jan,10,2005

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