formulario capitulo 4

TABLA DE TRANSFORMADAS DE FOURIER (TIEMPO
SERIES DE FOURIER (SEÑALES CONTINUAS )
F{1} = δ(f)
x(t ) =
∞
∑ Cne jnω t
F{ δ(t)} = 1
0
−∞
Cn =
F{ δ(t-t0)} = e-jωt0
1
− jnω 0t
x(t )e
dt
T∫
⎧∞
⎫ ∞
F⎨∑ C n e jnω0t ⎬ =∑ C n δ(f − nf 0 )
⎩ −∞
⎭ −∞
T
TRANSFORMADA DE FOURIER (SEÑALES CONTINUAS )
F{ A∏ (t/τ) } = Aτ Sinc fτ
∞
V(f) =
∫ v(t)e-jωt dt
-∞
F{ u(t)} = 1/jω + δ( f ) /2
∞
v(t) =
∫ V(f)ejωt df
F{ sgn(t)} = 2/ jω
-∞
F { Cos ( ω0t + θ) } = 0.5 δ(f-f0) ejθ+0.5 δ(f+f0) e-jθ
PROPIEDADES:
F{ v(t)} = V(f)
F{ Sinc2Wt} = (1/2W) ∏ (f/2W)
F{ w(t} =W(f)
F {e−at u(t )} =
F{ av(t) + bw(t)} = aV(f) + bW(f)
F{ v(t-t0)} = V(f)e-jωt0
F {te
⎛
⎞⎟
1
u( t)} = ⎜
⎝ (a + j2 πf ) ⎠
− at
F{ v(t)ejω0t} = V(f-f0)
⎧ dnv(t ) ⎫
F⎨
= ( jω)n V(f )
n ⎬
⎩ dt ⎭
⎫
⎧
1
F ⎨ ∫∫∫ v( τ)dτ ⎬ =
n V(f )
(
jω)
⎭
⎩ ...n
F {e
si V(0 ) = 0
⎧ dnV(f ) ⎫
F t nv(t ) = ( − j 2 π )−n ⎨
n ⎬
⎩ df ⎭
{
}
1
(a + j2 πf)
−a t
}=
2
2a
2
2
(a + (2 πf ) )
F {e−at Senω 0tu(t )} =
F {e−at Cosω 0tu(t )} =
ω0
((a + j2 πf)2 + ω 0 2 )
a + jω
((a + j2 πf)2 + ω 0 2 )
F{ v(at)} =(|a|)-1V(f/a)
Si F{ v(t)} =V(f)
entonces F{ V(t)} =v(-f)
F {e
−at 2
ω2
}=
π − 4a
e
a
CONVOLUCION
∞
y(t ) = ∫ x( τ) h( t − τ)dτ = x(t)*h(t)
−∞
v(t) * w(t) = w(t)*v(t)
F{
F{
F{ v(t) *w(t)} =V(f). W(f)
F{
F-1{ V(f) *W(f)} =v(t) . w(t)
dv (t)
dw( t)
d
* w (t ) = v(t ) *
[v( t) * w( t) ] =
dt
dt
dt
F{
tn −1
1
− at
e u(t )} =
(n − 1)!
( jω + a )n
1
2
2
a +t
}=
π −a ω
e
a
Cosbt
π
−a ω−b
−a ω+b
}=
(e
+e
)
2
2
a +t
2a
Senbt
2
2
a +t
}=
π
− a ω−b
−a ω+b
(e
−e
)
2 ja