Tabela Transformada de Fourier

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Análise de Sinais e Sistemas - PSMP
TRANSFORMADAS DE FOURIER
Considerar:
x(t)↔ X(ω)
onde X(ω) é a transformada de Fourier de x(t), dada por:
X(ω) = F [x(t)] =
∫ ∞
−∞
x(t) exp[−jωt]dt
e x(t) é a tranformada inversa de Fourier de X(ω), dada por:
x(t) = F−1[X(ω)] = 1
2π
∫ ∞
−∞
X(ω) exp[jωt]dω
Transformadas de Fourier
x(t) X(ω) = F [x(t)] x(t) X(ω) = F [x(t)]
N∑
n=1
αnxn(t)
N∑
n=1
αnXn(ω) 1 2πδ(ω)
x∗(t) X∗(−ω) u(t) πδ(ω) + 1
jω
x(t− t0) X(ω) exp(−jωt0) δ(t) 1
exp[jω0t]x(t) X(ω − ω0) δ(t− t0) exp[−jωt0]
x(αt)
1
|α|
X(
ω
α
) sgn(t)
2
jω
dnx(t)
dtn
(jω)nX(ω) exp[jω0t] 2πδ(ω − ω0)∫ t
−∞
x(τ)dτ
X(ω)
jω
+ πX(0)δ(ω) cosω0t π[δ(ω − ω0) + δ(ω + ω0)]∫ ∞
−∞
|x(t)|2dt − 1
2π
∫ ∞
−∞
|X(ω)|2dω sinω0t
π
j
[δ(ω − ω0)− δ(ω + ω0)]
x(t) ∗ h(t) X(ω)H(ω) (cosω0t)u(t)
π
2
[δ(ω − ω0) + δ(ω + ω0)] +
jω
ω20 − ω2
X(t) 2πx(−ω) (sinω0t)u(t)
π
2j
[δ(ω − ω0)− δ(ω + ω0)] +
ω0
ω20 − ω2
(−jt)nx(t) d
nX(ω)
dωn
exp[−at], a > 0 1
a+ jω
x(t)m(t)
1
2π
X(ω) ∗M(ω) |t| exp[−a|t|], Re[a] > 0 4ajω
(a2 + ω2)
Série exponencial de Fourier:
x(t) =
∞∑
n=−∞
cn exp[jnω0t] =
∞∑
n=−∞
cn exp[jωnt]
onde
cn =
1
T
∫ t0+T
t0
x(t) exp[−jnω0t]dt =
1
T
∫ t0+T
t0
x(t) exp[−jωnt]dt
1