Tabla de Propiedades y algunas Transformadas de Fourier +∞
1
f(t) =
∫
F (ω)·e
d ω
2 π −∞
+∞
F( ω ω ) =
jωt
∫ f (t )·e
− jωt
dt
−∞
1
a1 f 1 ( t ) + a2 f 2 ( t )
2
f ( at ) , a ≠ 0
3
f ( t m t 0 )
4
e
a1 F 1 ( ω ) + a2 F 2 ( ω )
ω a
1
F
a e
± jω0 t
m j ωt 0
F ( ω )
F ( ω m ω0 )
f ( t )
5
f ( t ) ⋅ cos( ω0 t )
1 2
6
f ( t ) ⋅ sen( ω0 t )
1 2 j
F ( ω − ω0 ) + 12 F ( ω + ω0 ) F ( ω − ω0 ) −
F ( ω + ω0 )
1 2 j
F ( t )
2 π f ( −ω )
d f ( t ) , n ∈ ℵ n dt
( jω ) F ( ω )
7 n
8
n
t
1
−∞
jω
∫ f ( t ′ )d t ′
9
F ( ω ) + πF ( 0 )δ( ω ) n
d F ( ω )
n
( − jt ) f ( t ) , n ∈ ℵ
10 11
d ω
f ( t )
ω
− jt
−∞
∫ F ( ω′ )d ω′
12
f 1 ( t )* f 2 ( t )
13
f 1 ( t ) ⋅ f 2 ( t )
F 1 ( ω ) ⋅ F 2 ( ω ) 1 2π
∞
F 1 ( ω )* F 2 ( ω )
∞
∫ f 1 ( t ) f 2 ( t )dt
14
1 2π
−∞
∫ F 1 ( ω ) ⋅ F 2 ( ω )d ω
−∞
1
− at
e u( t )
15 16
e − at 2
a + jω 2a
− a t 2
a +ω
17
e
18
A ⋅ p2T ( t )
19
A(1 − T t ) , t < T ∆( t ) = t > T 0 ,
, a ≠ 0
π a
2
sen ω0 t ⋅ u( t )
− at
cos ω0 t ⋅ u( t )
e
22
e
4a
ωT 2
1
e u( t )
− at
21
ω2
ATsinc
− at
(n − 1) !
e
−
2
2 ATsinc( ωT )
n −1
t
20
n
( jω + a )
n
ω0 (a + jω)2 + ω02 a + jω
(a + jω) + ω 2
2
0
23
k δ( t )
k
24
k
2πk δ( ω )
25
sgn( t )
26
u( t )
27
cos ω0t
π[δ(ω − ω0 ) + δ(ω + ω0 )]
28
sen ω0 t
jπ[δ(ω + ω0 ) − δ(ω − ω0 )]
29
e ∞
30
∑ C e n
n= −∞
1
jnω0t
, C n =
2 jω 1 jω
2πδ( ω m ω0 )
± j ω0t
T 2
∫
f (t )·e
− jnω0 t
T −T 2
dt , ω0 =
2π
∞
2 π ∑ C n δ(ω − nω0 )
T
n= −∞
∞
31
+ πδ( ω )
∞
∑ δ (t − nT )
ω S
S
n = −∞
∑ δ (ω − nω ), ω S
n = −∞
d δ(ω) n
32
n
t , n ∈ ℵ
2 π j MSc. Ing. Franco Martin Pessana E-mail:
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n
d ωn
S
=
2π T S
Tabla de Propiedades y algunas Transformadas de Laplace f(t) =
σ + j∞
1
∫
2π j
+∞
F (s )·e ds , s = σ + jω st
∫ f (t )·e
F(s) =
σ − j∞
− st
dt
0
1
a1 f 1 ( t ) + a2 f 2 ( t )
a1 F 1 ( s ) + a2 F 2 ( s )
2
f ( at ) , a ≠ 0
s a a
3
f ( t − τ )u( t − τ )
4
e ± at f ( t )
F ( s m a )
5
f 1 ( t )* f 2 ( t )
F 1 ( s ) ⋅ F 2 ( s )
6
f 1 ( t ) ⋅ f 2 ( t )
1
e
F
− τs
F ( s )
c + j∞
∫ F 1 ( τ )F 2 ( s − τ )d τ c − j∞
n
d f ( t ) , n ∈ ℵ , t ≥ 0 n dt
7
−
−
−
s n F ( s ) − s n 1 f ( 0 ) − s n 2 f ′( 0 ) − L − f ( n 1 ) ( 0 ) 0
∫ f ( t )dt
t
∫ f ( t ′ )d t ′
8
F ( s )
−∞
s
−∞
+
s
n
d F ( s )
n
( −t ) f ( t ) , n ∈ ℵ
9 10
ds
f ( t )
∞
t
s
∫ F ( u )du
1
f ( t ) = f ( t + T )
11
n
T
1− e
− st ∫ f ( t )e dt
− sT
0
12
f ( 0 )
lím sF ( s )
13
lím f ( t )
lím sF ( s )
t → ∞
( n −1 )
lím f
14
P(α k )
∑ Q′(α k =1
n
( t )
lím s F ( s )
k
s →0
P( s ) , gr (P ) < gr (Q) = n Q( s )
e , α k / Q(α k ) = 0 αk t
)
1
±
16
e at u( t )
17
t e u( t ) , n ∈ ℵ
18
u( t )
n
sma n!
± at
19
e
(s m a ) +
n 1
1 s +∞
2a
− a t 2
a −s
2
−∞
− at
e
− at
∫ f (t )·e
, con F b (s) = a
(1 − e ) ⋅ u( t )
20 21
s →0
t →∞
n
15
s →∞
s (s + a )
ω0
sen ω0 t ⋅ u( t )
(s + a ) + ω 2
2
0
22
e
− at
s+a
cos ω0 t ⋅ u( t )
(s + a ) + ω 2
2
0
s
23
cos ω0 t ⋅ u( t )
24
sen ω0 t ⋅ u( t )
ω0 2 s + ω0
25
k δ( t )
k
2
s + ω02 2
n
d δ( t )
26
s
n
dt
1
∞
27
∑ δ(t − nT )
1 − e − sT s
n = −∞
28
cosh ω0 t ⋅ u( t )
29
senh ω0 t ⋅ u( t )
n
s 2 − ω02
ω0 2 s − ω0 2
MSc. Ing. Franco Martin Pessana E-mail:
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− st
dt
Tabla de Propiedades de la Transformada Z Propiedades Comunes para TZB y TZU #
1
f [ n ] =
∫
2π j
F ( z )· z
n −1
∞
dz , n ∈ Ζ
∑
F b (z) =
f [n z ]
−n
∞
−n ; F(z) = ∑ f [n z ]
n=−∞
C
n=0
1
a1 f 1[n ] + a2 f 2 [n]
a1 F 1 ( z ) + a2 F 2 ( z )
2
f [n ]
F ( z )
[F ( z ) + F ( z )] [F ( z ) − F ( z )]
3
Re f [n ]
1 2
4
Im f [n ]
1 2 j
5
k
n f [n ]
d ( −1 ) z F ( z ) , k ∈ ℵ ∪ {0} dz
6
±n a f [n ] , a ≠ 0
F (a 1 z )
7
{ f [ ]}* {g[ ]}
F ( z ) ⋅ G( z )
k
8
9
k
n
1
f [n ] =
f [nT S ] =
2π j
n
n −1 ∫ F ( z ) z dz , n ∈ Ζ
F ( z )
γ
π T S
T S
2π
m
∫ F (e
jω T S
)⋅e
jnω T S
d ω
F ( z )
− π T S
1
f [n ] ⋅ g [n]
10
z d ω ω
1
∫
F (ω) ⋅ G
2π j γ ω
Propiedades Exclusivas de TZB #
f [ n ] =
1 2π j
∫
F ( z )· z
n −1
∞
dz , n ∈ Ζ
∑ f [ z]
F b (z) =
−n
n
n= −∞
C
1
f [n+a ]
z F ( z )
2
f [n−a ]
z F ( z )
3
f [−n ]
1 F z
a
−a
f [n ]
4
− z a ∫ z −1−a F ( z )dz
n+a
1 F z
f [−n ]
5
Propiedades Exclusivas de TZU #
f [ n ] =
1
∫ F ( z )· z 2 j π
n −1
∞
F(z) = ∑ f [n z ]
dz , n ∈ ℵ ∪ {0}
−n
n=0
C
1
f [n+a ]
z [F ( z ) − f [0 ] − z f [1] − L − z
2
f [n−a ]
z F ( z ) , a ∈ ℵ
3
f [n−a ]
4
f [0 ]
5
lim f [n ]
6
∇ f [n ]
−1
a
f [a −1] ] , a ∈ ℵ
− a +1
−a
z F ( z ) + f [−a ] + z f [−a +1] + z f [−a+ 2 ] + L + z −a
−1
−2
lim F ( z )
z →∞
(
)
lim 1 − z − ·F ( z )
n→∞
z →1
1
z − 1 z
F ( z ) m
7
∇ f [n ]
z − 1 F ( z ) z
8
∆ f [n ]
( z − 1)F ( z ) − zf [ ]
m
9
∆m f [n ]
10
∑ f [k ]
11
( z − 1)
m
F ( z ) − z ∑ ( z − 1)
m − k −1
k =0
n
z
k =0
z − 1
f [n ] n
0
m −1
∞
∫
, n ∈ ℵ
z
MSc. Ing. Franco Martin Pessana E-mail:
[email protected]
F ( z ′) z ′
F ( z )
d z ′ + lím n→∞
f [n ] n
∆k f [0 ]
− a +1
f [−1]
Identidad de Parseval ∞ 1 1 −1 F (ω) ⋅ G ω d ω ∑ f [n ] g [n ] = ∫ n = −∞ 2π j γ ω
∇ f [n ] = f [n ] − f [n−1] ∆ f [n ] = f [n+1 ] − f [n]
Siendo:
∞
1 −1 ∫ F (ω) ⋅ F ω d ω 2π j γ ω 1
2
∑ f [n ] =
n = −∞
Pares de Transformadas Z Unilateral #
f [ n] =
1
2 j ∫
F ( z )· z
n −1
∞
dz , n ∈ ℵ ∪ {0}
π
C
1
δ[n]
2
δ[n − m]
3
u[n ]
4
n·u[n ]
5
n ·u[n ]
6
a ·u[n ] n −1
(n + 1)·a
8 9
m!
z
a
12
a ·cos(Ω 0 n )·u[n]
13
a ·sen(Ω 0 n )·u[n ]
n
z
·u[n ]
16 17
senh(Ω0 n )·u[n ]
18
cosh(Ω 0 n )·u[n ]
z > a z > 1
z sen Ω0
z > 1
2
z − 2 z cos Ω0 + 1 z ( z − a cos Ω0 ) 2
z − 2az cos Ω 0 + a
2
z > a
2
z > a
az sen Ω0 2
z − 2az cos Ω 0 + a z
u[n ]
·sen(nω0T )·u [n ]
m+1
2
− anT
·cos(nω0T )·u [n ]
e
z > a
z − 2 z cos Ω0 + 1
n
− anT
2
( z − a )m+1 z ( z − cos Ω0 )
n
e
z > a
( z − a )2
sen(Ω 0 n )·u[n ]
15
z > a
( z − a )2
11
− anT
z > 1
z − a z
cos(Ω0 n )·u[n]
e
z > 1
z
10
14
z > 1
( z − 1)2 z ( z + 1) ( z − 1)3
·u[n ]
(n + 1)(n + 2)L(n + m)
∀ z ∈ Ζ − {0}
z − 1 z
·u [n ] n
∀ z ∈ Ζ
m
n
n·a
1
n
z − z
2
7
n=0
Región de Convergencia
− F(z) = ∑ f [n z ]
z − e
(
z z − e 2
z − 2 ze
−aT
ze 2
z − 2 ze
− aT
−aT
aT
z > e −
aT
− 2 aT
− 2 aT
− aT z > e
)
cos ω0T
cos ω0T + e
− aT
z > e −
−aT
sen ω0T
cos ω0T + e
z senh Ω 0 2
z − 2 z cosh Ω 0 + 1 z ( z − cosh Ω0 ) 2
z − 2 z cosh Ω 0 + 1
z > e
− Ω0
z > e
− Ω0
Expresiones útiles 2 2 sen θ + cos θ = 1
cos(α ± β ) = cos α ·cos β m sen α ·sen β cos(α − β ) − 12 cos(α + β )
cos θ − sen θ = cos 2θ
sen α ·sen β =
(1 + cos 2θ ) sen 2 θ = 12 (1 − cos 2θ ) sen(α ± β ) = sen α ·cos β ± cos α ·sen β
cos α ·cos β = 12 cos(α − β ) + 12 cos(α + β )
2
cos 2 θ =
2
1 2
sen α ·cos β =
MSc. Ing. Franco Martin Pessana E-mail:
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1 2
1 2
sen(α − β ) + 12 sen(α + β )
Tabla de Propiedades Transformada de Fourier de una Secuencia (TFS) f [ n ] =
#
π
1
∫ F (e ) e jω
2π
∞
jnω
F(e
∑ f [ ] ⋅ e
) =
a1 f 1[n ] + a 2 f 2 [n]
2
f [n m n0 ] f [n] ⋅ e
− jnω
n
n = −∞
−π
1
3
d ω
jω
a1F 1 (e jω ) + a2 F 2 (e jω )
( ) jn ω j (ω ω ) ) F (e
F e jω ⋅ e
± jnω 0
m
m
0
4
f [n]
F e − jω
5
f [− n]
F e − jω
6
f [− n]
F e
7
f [n L ] si n = kL , k ∈ Ζ Sobremuestrestreador: f ( L ) [n ] = , 0 si n ≠ kL , k ∈ Ζ
8
Submuestreador: g [n ] = f [nM ]
jω
(
F e jω L
( jω ) =
Ge
0
M −1
)
(( F (e ∑ M
1
j ω −2π l ) M )
)
l =0
x[n ]* h[n]
9
X (e jω ) ⋅ H (e jω ) ( ) )d θ X (e )⋅ H (e ∫ 2π
x[n ]⋅ h[n ]
10
π
1
jθ
j ω −θ
−π
f [n] − f [n − 1]
11
1 − e − jω F e jω
n
∑ f [k ]
12
(1 − e− jω )−1 F (e jω )
k = −∞
13
nf [n ]
14
f [n] = δ [n ]
15
f [n ] = δ [n − n0 ]
j
f [n ] =
d ω
F e
jω
=1
( )
F e jω = e
∞
16
( )
dF e jω
∑δ [n − kN ]
( )=
F e
jω
k = −∞
− jn0ω
∞
δ ω − ∑ N N 2π
2π k
k =−∞
∞
f [n ] = e
17
( ) = 2π ∑δ (ω − ω 0 − 2π k )
jω 0n
F e
jω
k = −∞ N −1
18
Serie Discreta de Fourier: f [n ] =
∑a e k
j
2π nk
∞
( ) = 2π ∑ ak δ ω − 2π k
N
F e
jω
∞
19
k = −∞
k =0
∑
E =
Relación de Parseval para señales aperiódicas:
2
f [n ] =
n =−∞
π
1 2π
∫
( jω )
F e
N
2
d ω
−π
Tabla de Propiedades de Transformada Discreta de Fourier (TDF) #
f [ n ] =
N −1
∑ F [ ] ⋅ e N 1
k
j
2π ⋅ n ⋅ k
N −1
; n = 0,1,2,3, L , N − 1
N
F [ k ] =
k = o
∑ f [ ] ⋅ e
− j
n
2π ⋅ n ⋅ k N
; k = 0,1,2,3, L , N − 1
n =o
1
a1 f 1[n ] + a 2 f 2 [n ]
2
f ( [n m n0 ]) N
3
f [n ] ⋅ W N
a1 F 1 [k ] + a2 F 2 [k ] ± n0 k
F [k ] ⋅ W N
± k 0n
con W N = e
F ( [k ± k 0 ]) N con W N = e
− j
− j
2π N 2π N
N −1
4
x[n ] ⊗ h[n ] =
∑ x[l ]⋅ h([n − l ])
N
X [k ]⋅ H [k ]
, con n = 0,1,2,3, L , N − 1
l =0
1
N −1
∑ X [l ]⋅ H ([k − l ])
5
x[n ]⋅ h[n ]
6
f [n]
F ( [− k ]) N
7
f ( [− n ]) N
F [k ]
N l =0
MSc. Ing. Franco Martin Pessana E-mail:
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N
, con k = 0,1,2,3, L , N − 1
