Fourier变换的性质
线性性质
设
F
1
(
ω
)
=
F
[
f
1
(
t
)
]
F_1(\omega)=\mathscr{F}[f_1(t)]
F1(ω)=F[f1(t)]，
F
2
(
ω
)
=
F
[
f
2
(
t
)
]
F_2(\omega)=\mathscr{F}[f_2(t)]
F2(ω)=F[f2(t)]，
α
\alpha
α，
β
\beta
β是常数，则
F
[
α
f
1
(
t
)
+
β
f
2
(
t
)
]
=
α
F
[
f
1
(
t
)
]
+
β
F
[
f
2
(
t
)
]
=
α
F
1
(
ω
)
+
β
F
2
(
ω
)
\mathscr{F}[\alpha f_1(t)+\beta f_2(t)]=\alpha\mathscr{F}[f_1(t)]+\beta\mathscr{F}[f_2(t)]=\alpha F_1(\omega)+\beta F_2(\omega)
F[αf1(t)+βf2(t)]=αF[f1(t)]+βF[f2(t)]=αF1(ω)+βF2(ω)
F − 1 [ α F 1 ( ω ) + β F 2 ( ω ) ] = α F − 1 [ F 1 ( ω ) ] + β F − 1 [ F 2 ( ω ) ] = α F 1 ( ω ) + β F 2 ( ω ) \mathscr{F}^{-1}[\alpha F_1(\omega)+\beta F_2(\omega)]=\alpha\mathscr{F}^{-1}[F_1(\omega)]+\beta\mathscr{F}^{-1}[F_2(\omega)]=\alpha F_1(\omega)+\beta F_2(\omega) F−1[αF1(ω)+βF2(ω)]=αF−1[F1(ω)]+βF−1[F2(ω)]=αF1(ω)+βF2(ω)
位移性质
F
[
f
(
t
±
t
0
)
]
=
e
±
j
ω
t
0
F
[
f
(
t
)
]
\mathscr{F}[f(t\pm t_0)]=e^{\pm j\omega t_0}\mathscr{F}[f(t)]
F[f(t±t0)]=e±jωt0F[f(t)]
F
−
1
[
F
(
ω
∓
ω
0
)
]
=
f
(
t
)
e
±
j
ω
0
t
\mathscr{F}^{-1}[F(\omega \mp \omega_0)]=f(t)e^{\pm j\omega_0t}
F−1[F(ω∓ω0)]=f(t)e±jω0t
微分性质
如果
f
(
t
)
f(t)
f(t)在
(
−
∞
,
+
∞
)
(-\infty,+\infty)
(−∞,+∞)上连续或只有有限个可去间断点，且当
∣
t
∣
→
0
|t|\to 0
∣t∣→0时，
f
(
t
)
→
0
f(t)\to 0
f(t)→0，则
F
[
f
′
(
t
)
]
=
j
ω
F
[
f
(
t
)
]
\mathscr{F}[f'(t)]=j\omega\mathscr{F}[f(t)]
F[f′(t)]=jωF[f(t)]
推论
若
f
(
k
)
(
t
)
f^{(k)}(t)
f(k)(t)在
(
−
∞
,
+
∞
)
(-\infty,+\infty)
(−∞,+∞)上连续或只有有限个可去间断点，且
lim
∣
t
∣
→
+
∞
f
(
k
)
(
t
)
=
0
,
k
=
0
,
1
,
2
,
⋅
⋅
⋅
，
n
−
1
\lim\limits_{|t|\to +\infty}f^{(k)}(t)=0,k=0,1,2,···，n-1
∣t∣→+∞limf(k)(t)=0,k=0,1,2,⋅⋅⋅，n−1，则有
F
[
f
(
n
)
(
t
)
]
=
(
j
ω
)
n
F
[
f
(
t
)
]
\mathscr{F}[f^{(n)}(t)]=(j\omega)^n\mathscr{F}[f(t)]
F[f(n)(t)]=(jω)nF[f(t)]
同样
d
n
d
ω
n
F
(
ω
)
=
(
−
j
)
n
F
[
t
n
f
(
t
)
]
\frac{d^n}{d\omega^n}F(\omega)=(-j)^n\mathscr{F}[t^nf(t)]
dωndnF(ω)=(−j)nF[tnf(t)]
积分性质
如果当 t → + ∞ t\to +\infty t→+∞时， g ( t ) = ∫ − ∞ t f ( t ) d t → 0 g(t)=\displaystyle\int_{-\infty}^{t}f(t)\,dt\to0 g(t)=∫−∞tf(t)dt→0，那么 F [ ∫ − ∞ t f ( t ) ] = 1 j ω F [ f ( t ) ] \mathscr{F}\left[\int_{-\infty}^{t}f(t)\right]=\frac{1}{j\omega}\mathscr{F}[f(t)] F[∫−∞tf(t)]=jω1F[f(t)]
乘积定理
若
F
1
(
ω
)
=
F
[
f
1
(
t
)
]
F_1(\omega)=\mathscr{F}[f_1(t)]
F1(ω)=F[f1(t)]，
F
2
(
ω
)
=
F
[
f
2
(
t
)
]
F_2(\omega)=\mathscr{F}[f_2(t)]
F2(ω)=F[f2(t)]，则
∫
−
∞
+
∞
f
1
(
t
)
‾
f
2
(
t
)
d
t
=
1
2
π
∫
−
∞
+
∞
F
1
(
ω
)
‾
F
2
(
ω
)
d
ω
\int_{-\infty}^{+\infty}\overline{f_1(t)}f_2(t)\,dt=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\overline{F_1(\omega)}F_2(\omega)d\omega
∫−∞+∞f1(t)f2(t)dt=2π1∫−∞+∞F1(ω)F2(ω)dω
∫
−
∞
+
∞
f
2
(
t
)
‾
f
1
(
t
)
d
t
=
1
2
π
∫
−
∞
+
∞
F
2
(
ω
)
‾
F
1
(
ω
)
d
ω
\int_{-\infty}^{+\infty}\overline{f_2(t)}f_1(t)\,dt=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\overline{F_2(\omega)}F_1(\omega)d\omega
∫−∞+∞f2(t)f1(t)dt=2π1∫−∞+∞F2(ω)F1(ω)dω
其中
f
1
(
t
)
‾
\overline{f_1(t)}
f1(t)，
f
2
(
t
)
‾
\overline{f_2(t)}
f2(t)，
F
1
(
ω
)
‾
\overline{F_1(\omega)}
F1(ω)及
F
2
(
ω
)
‾
\overline{F_2(\omega)}
F2(ω)分别为
f
1
(
t
)
f_1(t)
f1(t),
f
2
(
t
)
f_2(t)
f2(t),
F
1
(
ω
)
F_1(\omega)
F1(ω)及
F
2
(
ω
)
F_2(\omega)
F2(ω)的共轭函数
能量积分
若
F
(
ω
)
=
F
[
f
(
t
)
]
F(\omega)=\mathscr{F}[f(t)]
F(ω)=F[f(t)]，则有
∫
−
∞
+
∞
[
f
(
t
)
]
2
d
t
=
1
2
π
∫
−
∞
+
∞
∣
F
(
ω
)
∣
2
d
ω
\int_{-\infty}^{+\infty}[f(t)]^2dt=\frac{1}{2\pi}\int_{-\infty}^{+\infty}|F(\omega)|^2d\omega
∫−∞+∞[f(t)]2dt=2π1∫−∞+∞∣F(ω)∣2dω
称为Parseval等式
能量密度函数（或称能量谱密度）
S
(
ω
)
=
∣
F
(
ω
)
∣
2
S(\omega)=|F(\omega)|^2
S(ω)=∣F(ω)∣2
显然
S
(
ω
)
=
S
(
−
ω
)
S(\omega)=S(-\omega)
S(ω)=S(−ω)
—————————————————————————————————
若
F
(
ω
)
=
F
[
f
(
t
)
]
F(\omega)=\mathscr{F}[f(t)]
F(ω)=F[f(t)]
F
[
f
(
t
)
cos
ω
0
t
]
=
1
2
[
F
(
ω
)
+
F
(
ω
0
)
]
\mathscr{F}[f(t)\cos\omega_0t]=\frac{1}{2}[F(\omega)+F(\omega_0)]
F[f(t)cosω0t]=21[F(ω)+F(ω0)]
F
[
f
(
t
)
sin
ω
0
t
]
=
1
2
j
[
F
(
ω
−
ω
0
)
−
F
(
ω
+
ω
0
)
]
\mathscr{F}[f(t)\sin\omega_0t]=\frac{1}{2j}[F(\omega-\omega_0)-F(\omega+\omega_0)]
F[f(t)sinω0t]=2j1[F(ω−ω0)−F(ω+ω0)]