积分变换总结

正弦变换式
Fs ( w ) = ∫
+∞
0
f ( t )sin wtdt = FS [ f ( t )]
1
2 +∞ 正弦逆变换式 f (t ) = ∫ Fs ( w)sin wtdw = Fs−1[Fs ( w)]s 0 π +∞ 余弦变换式 Fc ( w ) = ∫ f ( t )cos wtdt = Fc [ f ( t )]
1. F[ f ( t ± t 0 )] = e
± jwt 0
F[ f ( t )] = e
± jwt 0
ຫໍສະໝຸດ Baidu
F (w)
2. F[e ± jw0 t f ( t )] = F ( w ∓ w 0 )
F[ f (t )] = F(w)
3. F[ f ( n ) ( t )] = ( jw ) n F[ f ( t )] = ( jw ) n F ( w ) 1 d nF (w) 4. F[t n f ( t )] = ( − j ) n dw n 1 w 5. F[ f (at )] = F ( ), ( a ≠ 0) |a| a 1 6. F[ f ( t ) sin w 0 t ] = [ F ( w − w 0 ) − F ( w + w 0 )] 2j 1 7. F[ f ( t ) cos w0 t ] = [ F ( w − w0 ) + F ( w + w0 )] 2
+∞
1 1) L[ f ′( t )] = sF ( s ) − f (0) L[u( t )] = ,Re( s ) > 0 s 2) L[t n f ( t )] = ( −1)n F ( n ) ( s ) 1 kt L[e ] = ,Re( s ) > k 3) L[e at f ( t )] = F ( s − a ) s−k t 1 k L[sin kt ] = 2 ,Re( s ) > 0 4) L[ ∫0 f ( t )dt ] = F ( s ) s s + k2 5) L[ f ( t − τ )] = e − sτ F ( s ) t < 0, f ( ) = s L[cos kt ] = 2 ,Re( s ) > 0 2 ∞ f (t ) s +k 6) L[ ] = ∫ F ( s )ds s m! t m L[t ] = m +1 ,(Re( s ) > 0) 1 s s 7) L[ f (at )] = F ( ) a a t f1 ( t ) * f 2 ( t ) = ∫ f1(τ) f2(t −τ)dτ
2
0
+∞
2
+∞
f (τ )sinw dτ ]sinwtdw τ
f (τ )cos w dτ ]cos wtdw τ
∫ [∫ π
−∞
+∞
0
F (w) = ∫
f ( t )e − jwt dt = F[ f ( t )]
+∞
1 f (t ) = 2π
∫
−∞
F ( w )e jwt dw = F −1 [ F ( w )]
1 积分表达式： 积分表达式： f ( t ) = 2π
∫
0
+∞ −∞
[∫
+∞ −∞
f (τ )e − jwτ dτ ]e jwt dw
f (t) =
∫ [∫ π
+∞ 0 0
+∞
1
+∞
+∞
−∞
f (τ )cos w(t −τ )dτ ]dw
f (t) =
f (t) =
傅氏变换式 傅氏逆变换式
∫ [∫ π
0
2 +∞ −1 余弦逆变换式 f (t ) = ∫ Fc ( w)cos wtdw = Fc [Fc ( w)]s π 0
1
∫
δ ( t )dt = 1 −∞
−∞
+∞
∫
δ ( t ) f ( t )dt = f (0) −∞
−∞
+∞
∫
+∞
−∞
δ ( t − t0 ) f ( t )dt = f ( t 0 )
f 1 (t) * f 2 (t) = ∫
+∞ −∞
f 1 (τ ) ⋅ f 2 ( t − τ )dτ
F[ f 1 (t) * f 2 (t)] = F1 (w) ⋅ F2 (w)
1 F[ f1 ( t ) ⋅ f 2 ( t )] = F1 ( w ) * F2 ( w ) 2π
F ( s) = ∫
0
0
f ( t )e − st dt
L[ f1 ( t ) * f 2 ( t )] = F1 ( s ) ⋅ F2 ( s )
F[δ ( t )] = 1
F[δ ( t − t0 )] = e − jwt0
1 F[ u( t )] = + πδ ( w ). jw
F[sinω 0 t] = jπ [δ ( w + w0 ) − δ ( w − w0 )]
F[cosω 0 t] = π [δ ( w + w 0 ) + δ ( w − w 0 )]