表 6.3 f (t )= 1 2π +∞ ∞

常用的连续傅里叶变换对及其对偶关系 dω F (ω )= +∞ ∞

∫ F (ω )e

jωt

∫ f (t )e

jωt

dt

重要

连续傅里叶变换对连续时间函数 f (t )傅里叶变换 F (ω )

相对偶的连续傅里叶变换对连续时间函数 f (t )傅里叶变换 F (ω ) 1 t tk

重要

√√

δ (t ) dδ (t ) dt

1 jω

2πδ (ω ) j 2π dδ (ω ) dω

√

dkδ (t ) dt k

( jω ) k 1+πδ (ω ) jω jπ

2πj k

dkδ (ω ) dω k u (ω )

√

u (t ) tu (t )

1 1δ (t ) 2 j 2πt

1 dδ (ω ) 2 dωω 2 jω e jωt 0

√

1, t> 0 sgn(t )= 1, t< 0δ (t t 0 ) cosω 0t sinω 0t

1

π

,t≠ 0 0

π[δ (ω+ω 0 )+δ (ω ω 0 )] jπ[δ (ω+ω 0 ) δ (ω ω 0 )]

e jω tδ (t+ t0 )+δ (t t0 )

j,ω> 0 F (ω )= j,ω< 0 2πδ (ω ω 0 ) 2 cosωt 0 j 2 sinωt0

√

δ (t+ t 0 ) δ (t t0 ) W

√

1, t<τ f (t )= 0, t>τ 1 tτ, t<τ f (t )= 0, t>τ

τSa (

ωτ 2

)

π

Sa (Wt )

1,ω< W F (ω )= 0,ω> W 1 ω W,ω< W F (ω )= 0,ω> W

√

√

τSa 2 (

ωτ 2

)

W Wt Sa 2 ( ) 2π 2

√

e at u (t ), Re{a}> 0 e a t

1 a+ jω 2aω+ a2 2

1

τ jt τ t+τ 2 2

2πe τω u (ω ),τ> 0

, Re{a}> 0

πe τω,τ> 0

√√

e at cosω 0tu (t ), Re{a}> 0

a+ jω (a+ jω ) 2+ω 02

e at sinω 0tu (t ), Re{a}> 0 te at u (t ), Re{a}> 0 t k 1e at u (t ), Re{a}> 0 (k 1)!

ω0 (a+ jω ) 2+ω 02 1 ( a+ jω ) 2 1 ( a+ jω ) k 1,τ> 0 (τ jt ) 2 2πωe τω u (ω )

√√

δ T (t )=

l= ∞

∑δ (t lT ) t ( )2

+∞

2π T

k= ∞

∑δ (ω k πτe (

+∞

2π ) T

ωτ 2

e

τ

)2

√

[u (t+

τ 2

) u (t

τ

)] cosω t 0 2

τ 2

[ Sa

(ω+ω )τ 0 2 +∞

+ Sa

(ω ω )τ 0 2

]

k= ∞

∑F e k

+∞

jkω 0t

2π

k= ∞

∑ Fδ (ω kω k

0

)