Tabela - Transformada de Fourier

Prévia do material em texto

Transformadas de Fourier
f(t) F (w) = F [f(t)]
a1f1(t) + a2f2(t) a1F1(w) + a2F2(w)
f(at) 1|a|F
(
w
a
)
f(−t) F (−w)
f(t− t0) F (w)e−wt0i
f(t)ew0ti F (w − w0)
f(t)cos(w0t)
1
2F (w − w0) + 12F (w + w0)
f(t)sen(w0t)
1
2iF (w − w0)− 12iF (w + w0)
F (t) 2pif(−w)
f ′(t) iwF (w)
f (n)(t) (iw)nF (w)∫ t
−∞ f(x)dx
1
iwF (w) + piF (0)δ(w)
−itf(t) F ′(w)
(−it)nf(t) F (n)(w)
f1(t) ∗ f2(t) F1(w)F2(w)
f1(t)f2(t)
1
2piF1(w) ∗ F2(w)
e−atu(t) 1a+wi
e−a|t| 2a
a2+w2
pa(t) 2a
sen(wa2 )
wa
sen(at)
pit p2a(w)
f(t) F (w) = F [f(t)]
te−atu(t) 1
(a+wi)2
δ(t) 1
δ(t− t0) e−wt0i
δ′(t) wi
δ(n)(t) (wi)n
u(t) piδ(w) + 1wi
u(t− t0) piδ(w) + 1wie−wt0i
1 2piδ(w)
t 2piiδ′(w)
tn 2piinδ(n)(w)
ew0ti 2piδ(w − w0)
cos(w0t) pi[δ(w − w0) + δ(w + w0)]
sen(w0t) −ipi[δ(w − w0)− δ(w + w0)]
sen(w0t)u(t)
w0
w20−w2
+ pi2i [δ(w − w0)− δ(w + w0)]
cos(w0t)u(t)
wi
w20−w2
+ pi2 [δ(w − w0) + δ(w + w0)]
tu(t) ipiδ′(w)− 1
w2
1
t pii− 2piiu(w)
1
tn
(−wi)n−1
(n−1)! [pii− 2piiu(w)]
sgn(t) 2wi
Lembre-se que:
1. F (w) = F [f(t)] =
∫ ∞
−∞
f(t)e−wtidt
2. f(t) = F−1[F (w)] = 1
2pi
∫ ∞
−∞
F (w)ewtidw
3. f1(t) ∗ f2(t) =
∫ ∞
−∞
f1(x)f2(t− x)dx
4. pa(t) =
{
1, |t| < a2
0, |t| > a2
5. u(t) =
{
1, t > 0
0, t < 0
6. sgn(t) =
{
1, t > 0
−1, t < 0