傅里叶变换性质及常用变换

jπ f
e −α t u (t )(α > 0)
δ ( n ) (t )
( jπ f )
n
n = −∞
∑ δ (t − nT )
+∞
+∞
e
−π t 2
e
+∞
−t 2
πe
+∞
−
ω2
4
∫
−∞
x(t ) y *(t )dt = ∫
−∞
X ( f )Y *( f )df =
1 2π
∫
+∞
−∞
X (ω )Y *(ω )d ω （帕斯瓦尔关系） ∫ | x(t ) |2 dt = ∫ | X ( f ) |2 df
f T 1 f − j 2π a X ( )e |a| a
α X 1 (ω ) + β X 2 (ω )
2π X (−ω )
X (ω )e − jωT
x(t )e j 2π f0t x(t ) sin 2π f 0t x(t ) cos 2π f 0t
j [ X ( f + f0 ) − X ( f − f0 )] 2
2
e − j 2π f 0 t sinc 2 (t ) 1 t
δ ( f − f0 )
Λ( f )
sinc( f ) 1 [δ ( f + f0 ) + δ ( f − f0 )] 2 1 1 δ( f )+ 2 j 2π f 1 α + j 2π f 1 +∞ n δ( f − ) ∑ T n = −∞ T
ω 1 X( ) a |a|
1 ω − jωT X ( )e a |a| a 1 [ X (ω ) ∗ Y (ω )] 2π jn dn X (ω ) dω n
1 [ X ( f − f0 ) + X ( f + f0 )] 2
π [ X (ω − ω0 ) + X (ω + ω0 )]
X (ω )Y (ω )
( jω ) n X (ω ) X (ω ) + π X (0)δ (ω ) jω
t x(t )
+∞
n
∫
T
t
−∞
x(τ )dτ
n = −∞
∑ δ (t − nT ) = ∑e
1 +∞ T n = −∞
n j 2π t T
n 1 +∞ δ( f − ) ∑ T n = −∞ T
+∞ 2π ω0 ∑ δ (ω − nω0 ) (ω0 = ) T n = −∞
X *( f )Y ( f ) = Exy ( f )
X *(ω )Y (ω )
X(f )
2
= Ex ( f
)
X (ω )
2
∫
+∞
−∞
x(τ ) y (t − τ= )dτ
∫
+∞ −∞
+∞
−∞
y (τ ) x(t − τ )dτ
∫
+∞
−∞
x(t + τ ) y (τ )= dτ
∫
+∞
−∞
x(τ ) y (τ − t )dτ
常见傅立叶变换
δ (t )
rect (t ) cos(2π f 0t )
1
1
δ( f )
rect ( f ) j [δ ( f + f0 ) − δ ( f − f0 )] 2
δ (t − t0 )
Λ (t ) sgn(t )
e − j 2π ft0 sinc 2 ( f ) 1 jπ f ( jπ f ) n 2α α + (2π f ) 2
−∞ −∞
=
1 2π
∫
+∞
−∞
X (ω ) d ω
2
（帕斯瓦尔定理）
1
rect ( t )
1
1
Λ (t )
sinc ( t )
1 − 2
0
1 2
t
−1
0
1 t
−2
−1
0
1
2 t
n = −∞
∑ δ (t − nT )
Rx (τ ) x(t ) * y (= −t ) Rxy (τ ) =
+∞
n = −∞
T ∑ e − jn 2π Tf ∑ δ( f − ) =
n = −∞
+∞
n T
+∞
2π
n = −∞
∑ δ (ω − nω )
0
+∞
Rxy (τ )
卷积 x(t ) *= y (t ) 相关 Rxy (τ ) =
傅立叶变换的性质 正反变换
X(f ) = ∫
+∞
−∞
x(t )e − j 2π ft dt
X (ω ) = ∫
+∞
−∞
x(t )e − jωt dt
x(t ) = ∫ x *(t )
+∞
−∞
X ( f )e j 2π ft df X *(− f ) X ( f − f0 )
x(t ) =
1 2π
∫
+∞
+∞
sinc(t ) sin(2π f 0t ) j 1 δ (t ) + 2 2π t te −α t u (t )(α > 0)
− jπ sgn( f )
u (t )
sgn( f )
1 (α + j 2π f ) 2 e
−π f 2
δ ( n ) (t )
e −α |t| (α > 0)
δ '(t )
−∞
X (ω )e jωt d ω X *(−ω ) X (ω − 2π f 0 )
jπ [ X (ω + ω0 ) − X (ω − ω0 ) ]
α x1 (t ) + β x2 (t )
X (t ) x(t − T )
x(at )
α X1 ( f ) + β X 2 ( f )
x(− f )
X ( f )e − j 2π fT f 1 X( ) a |a|
∫
+∞
−∞
x *(t ) y (t + τ )dt=
∫
y (t + τ ) x *(t )dt ＝ y (τ ) ∗ x∗ (−τ )
(∫
+∞
−∞
= R* y *(t + τ ) x(t )dt yxBaidu Nhomakorabea( −τ )
)
∗
Rx (τ )=
∫
+∞
−∞
x *(t ) x(t + τ )dt=
∫
+∞
−∞
x(t + τ ) x *(t )dt ＝ x(τ ) ∗ x∗ (−τ )
x(at − T )
x(t ) ∗ y (t )
dn x(t ) dt n
X ( f )Y ( f )
x(t ) y (t )
X ( f ) ∗Y ( f )
n j d X(f ) n 2π df n
( j 2π f ) n X ( f ) X(f ) 1 + X (0)δ ( f ) j 2π f 2