Tabla de relaciones entre los dominios temporal y frecuencial

7.19. Tablas útiles
7.19.
157
Tablas útiles
Muestreo
dominio frecuencial ( f )
dominio temporal
dominio frecuencial (Ω)
Xc ( f )
xc (t)
Xc ( Ω )
PT ( f ) =
1
T
∑ δ( f
1
T
PT ( f )
∑ k Xc ( f
=
∑
k
=
1
T
2π
F
(ω
∑Xc 2π
k
Xs ( Ω ) =
p T (t)
= ∑ xc (nT )δ(t nT )
=
n
Fs
Xc (ω 2π
h
xs (t) = xc (t)
kFs )
X (e jω ) = Xs ( f )j f =ω Fs =
1
T
PT (Ω) =
nT )
n
k
Xs ( f ) = Xc ( f )
=
p T (t) = ∑ δ(t
kFs )
s
2π
T
∑ δ(Ω
1
2π Xc ( Ω )
1
T
kΩs )
n
∑ k Xc ( Ω
PT (Ω)
kΩs )
ω
2πT
Xs (Ω)jΩ=ω Ωs = ω
kFs )
1
T
x [n] = xc (t)jt=nT
=
∑
Ωs
Xc (ω 2π
T
2π
kΩs )
k
i
2πk )
=
1
T
∑Xc T1 (ω
2πk )
k
Reconstrucción
dominio frecuencial ( f )
dominio temporal
dominio frecuencial (Ω)
Y (e jω )
y[n]
Y (e jω )
ys (t) = ∑ y[n]δ(t nT )
Ys ( f ) = Y (e jω )jω = f 2π = f 2πT
Fs
Hr ( f ) =
(
T, j f j < F2s
hr ( t ) =
0, j f j > F2s
ω = f 2π
Fs = f 2πT
si j f j <
Ωs
sen
πt
T
πt
T
= sinc
t
T
yr (t) = ∑n y[n]hr (t nT )
Yr ( f ) = Hr ( f )Ys ( f )
= TY (e jω )
Ys (Ω) = Y (e jω )jω =Ω 2π =ΩT
n
Fs
2
Procesamiento Digital de Señales
= ∑n y[n] sinc
t nT
T
Hr (Ω) =
(
T, jΩj < Ω2s = πT
0, jΩj > Ω2s = πT
Yr (Ω) = Hr (Ω)Ys (Ω)
= TY (e jω )jω =Ω 2π =ΩT
Ωs
si Ωj <
Ωs
2
= πT
U.N.S. 2011
7. Muestreo de señales
158
Procesamiento discreto de señales continuas
dominio frecuencial ( f )
dominio temporal
dominio frecuencial (Ω)
Xc ( f )
xc (t)
Xc ( Ω )
X (e jω ) =
1
T
h
i
Fs
X
(
ω
2πk
)
c
∑ 2π
X (e jω ) =
x [n] = xc (t)jt=nT
k
1
T
∑Xc T1 (ω
2πk )
k
Y (e jω ) = H (e jω ) X (e jω )
Y (e jω ) = H (e jω ) X (e jω )
y[n] = h[n] x [n]
Ys ( f ) = Y (e jω )jω = f 2π = f 2πT
ys (t) = ∑y[n]δ(t nT )
Ys (Ω) = Y (e jω )jω =Ω 2π =ΩT
Yr ( f ) = Hr ( f )Ys ( f )
yr (t) = ∑n y[n]hr (t nT )
Yr (Ω) = Hr (Ω)Ys (Ω)
Fs
Ωs
n
= ∑n y[n] sinc
= TH (e jω) X (e jω)jω = f 2π
Fs
t nT
T
= TH (e jω) X (e jω)jω =Ω 2π
Ωs
=ΩT
= f 2πT
8 jω
H (e )jω = f 2π
>
>
Fs
>
>
= f 2πT
>
<
si j f j < F2s
Hc ( f ) =
>
>
>
0,
>
>
:
si j f j > F2s
8 jω
H (e )jω =Ω 2π
>
>
Ωs
>
>
=ΩT
>
<
si jΩj < Ω2s
Hc (Ω) =
>
>0,
>
>
>
:
si jΩj > Ω2s
Procesamiento continuo de señales discretas
dominio frecuencial ( f )
Xc ( f ) = TX (e jω)jω = f 2π
Fs
dominio temporal
xc (t) = ∑n x [n]hr (t nT ) Xc (Ω) = TX (e jω)jω =ΩT
Yc ( f ) = Hc ( f ) Xc ( f )
y c ( t ) = hr ( t ) x c ( t )
Y (e jω ) = T1 Yc ( f )j f =ω Fs
y[n] = yc (t)jt=nT
2π
H (e jω ) = Hc ( f )j f =ω Fs , jω j < π
2π
Procesamiento Digital de Señales
dominio frecuencial (Ω)
Yc (Ω) = Hc (Ω) Xc (Ω)
Y (e jω ) = T1 Yc (Ω)jΩ=ω Ωs = ω
2π
T
H (e jω ) = Hc (Ω)jΩ= ωT , jω j < π
U.N.S. 2011
7.19. Tablas útiles
159
Decimación
x f [n] = x [n] h PB [n]
= ∑ x [`]h PB [n
, X f e jω = HPB e jω X e jω
`]
`
, Xd e jω =
xd [n] = x f [nM ]
1
M
M 1
j
∑ Xf e (
ω 2πr
M
)
r =0
Interpolación
xe [n] =
(
x [n/L],
si n = 0,
0,
en caso contrario.
xi [ n ] = x e [ n ] h I [ n ] =
= ∑ x [`]h[n
L,
∑ xi [`]h[n
2L, . . .
,
`]
`
` L]
,
Xe e jω = X e jωL
Xi e jω = H I e jω Xe e jω
`
= L X e jωL , jω j < π/L
Cambio de la frecuencia de muestreo por un factor no entero
Intercambio de filtrado y sub/sobre muestreo
Procesamiento Digital de Señales
U.N.S. 2011
160
Procesamiento Digital de Señales
7. Muestreo de señales
U.N.S. 2011