Tabla De Transformadas 4n2i5e

Tabla de Propiedades y algunas Transformadas de Fourier +∞

1 +∞ F (ω)·e jωt dω 2 π −∫∞

f(t) =

F(ω)= ∫ f (t )·e − jωt dt −∞

1

a1 f 1 ( t ) + a 2 f 2 ( t )

a1 F1 ( ω ) + a2 F2 ( ω )

2

f ( at ), a ≠ 0

1  ω F  a a

3

f ( t m t0 )

e m jωt0 F ( ω )

4

e ± jω0t f ( t )

5

f ( t ) ⋅ cos( ω0 t )

1 2

6

f ( t ) ⋅ sen( ω0 t )

1 2j

F ( ω m ω0 ) F ( ω − ω0 ) + 12 F ( ω + ω0 ) F ( ω − ω0 ) −

1 2j

F ( ω + ω0 )

F( t )

2 πf ( −ω )

8

d f (t ) , n ∈ℵ dt n

( jω ) n F ( ω )

9

∫ f ( t ′ )dt ′

1 F ( ω ) + πF ( 0 )δ( ω ) jω

( − jt )n f ( t ), n ∈ ℵ

d n F( ω ) dωn

7 n

t

−∞

10

12

f (t ) − jt f 1 ( t )* f 2 ( t )

13

f1( t ) ⋅ f 2 ( t )

11

ω

∫ F ( ω′ )dω′

−∞

F1 ( ω ) ⋅ F2 ( ω ) 1 2π

∞

∫ f 1 ( t ) f 2 ( t )dt

14

F1 ( ω )* F2 ( ω )

∞ 1 2π

−∞

∫ F1 ( ω ) ⋅ F 2 ( ω )dω

−∞

1 a + jω 2a 2 a + ω2

e − at u( t )

15 16

e

−a t

2

−

ω2 4a

17

e − at , a ≠ 0

18

A ⋅ p2T ( t )

2 ATsinc( ωT )

 A 1 − , t < T ∆( t ) =  t >T 0 , t n −1 e −at u( t ) (n − 1) !

 ωT  ATsinc 2    2 

(

19 20

t T

π a

)

e

1

( jω + a )

n

21

e − at sen ω0 t ⋅ u( t )

ω0 (a + jω)2 + ω02

22

e − at cos ω0 t ⋅ u( t )

(a +

23

kδ( t )

24

k

25

sgn( t )

2 jω

26

u( t )

1 + πδ( ω ) jω

27

cos ω0 t

28

sen ω0 t

jπ[δ(ω + ω0 ) − δ(ω − ω0 )]

29

e ± jω0t

2πδ( ω m ω0 )

∞

30

k 2 πkδ( ω )

π[δ(ω − ω0 ) + δ(ω + ω0 )]

T 2

∑ Cn e jnω t , Cn = T ∫ f (t )·e − jnω t dt , ω0 = n = −∞ 1

0

0

−T 2

∞

31

a + jω 2 jω) + ω02

∑

2π T

δ (t − nTS )

∞

n = −∞

ωS

n = −∞

32

2 π ∑ Cn δ(ω − nω0 ) ∞

∑ δ (ω − nω

n = −∞

t n , n ∈ℵ

2 πj n MSc. Ing. Franco Martin Pessana E-mail: [email protected]

S

), ω S

d n δ(ω) dω n

=

2π TS

Tabla de Propiedades y algunas Transformadas de Laplace f(t) =

σ + j∞

1 F (s )·e st ds , s = σ + jω 2πj σ−∫j∞

F(s) =

+∞

∫ f (t )·e

− st

dt

0

1

a1 f 1 ( t ) + a 2 f 2 ( t )

a1 F1 ( s ) + a2 F2 ( s )

2

f ( at ), a ≠ 0

1 s F  a a

3

f ( t − τ )u( t − τ )

e − τs F ( s )

4

e ± at f ( t )

F( s m a )

5

f 1 ( t )* f 2 ( t )

6

f1( t ) ⋅ f2 ( t )

7

dn f (t ) , n ∈ ℵ, t ≥ 0 dt n

F1 ( s ) ⋅ F2 ( s ) c + j∞

∫ F1 ( τ )F2 ( s − τ )dτ

c − j∞

s n F ( s ) − s n−1 f ( 0 ) − s n−2 f ′( 0 ) − L − f ( n−1 ) ( 0 ) 0

f ( t )dt F ( s ) −∫∞ + s s n d F( s ) ds n

t

∫ f ( t ′ )dt ′

8

−∞

9

( −t )n f ( t ), n ∈ ℵ

10

f (t ) t

∞

∫ F ( u )du s

11

f (t ) = f (t +T )

12

f (0 )

1 T − st ∫ f ( t )e dt 1 − e −sT 0 lím sF ( s )

13

lím f ( t )

lím sF ( s )

14

( n −1 ) (t ) lím f

n lím s F ( s )

P(α ) ∑ Q′(αk ) e α t , α k / Q(α k ) = 0

P( s ) , gr (P ) < gr (Q ) = n Q( s )

s →∞

t →∞

s→0

t →∞

s→0

n

15

k

k =1

k

16

e ± at u( t )

17

t n e ± at u( t ), n ∈ ℵ

18

u( t )

19

e

1 sma n! (s m a )n+1 1 s +∞ 2a , con Fb (s) = ∫ f (t )·e −st dt 2 a −s −∞

−a t

2

20

(1 − e ) ⋅ u( t )

21

e − at sen ω0 t ⋅ u( t )

22

e − at cos ω0 t ⋅ u( t )

23

cos ω0 t ⋅ u( t )

s s + ω02

24

sen ω0 t ⋅ u( t )

ω0 s + ω02

25

kδ( t )

k

26

d δ( t ) dt n

sn

27

∑ δ(t − nT )

a s (s + a ) ω0 (s + a )2 + ω02

− at

s+a 2 + ω02

(s + a ) 2

2

n

∞

1 1 − e − sT s 2 s − ω02

n = −∞

28

cosh ω0 t ⋅ u( t )

29

senh ω0 t ⋅ u( t )

ω0 s 2 − ω02 MSc. Ing. Franco Martin Pessana E-mail: [email protected]

Tabla de Propiedades de la Transformada Z Propiedades Comunes para TZB y TZU #

f [n ] =

1 F ( z )·z n−1dz , n ∈ Ζ 2πj C∫

1

a1 f 1 [n ] + a 2 f 2 [ n ]

2

f [n ]

Fb (z) =

∞

Im{ f [n ]}

5

n k f [n ]

6

a ± n f [n ] , a ≠ 0

n=0

a1 F1 ( z ) + a2 F2 ( z )

F (z )

1 2

4

F(z) = ∑ f [n ] z − n

n= −∞

Re{ f [n ]}

3

∞

∑ f [n ] z − n ;

1 2j

[F (z ) + F (z )] [F (z ) − F (z )] k

 d  ( −1 )k  z  F ( z ), k ∈ ℵ ∪ {0}  dz  F (a m1 z )

{f [ ]}* {g[ ]}

7

n

F ( z ) ⋅ G( z )

n

1 n −1 ∫ F ( z )z dz , n ∈ Ζ 2πj γ

8

f [n ] =

9

T f [nTS ] = S 2π

π TS −

∫ F (e π

j ωT S

F( z )

) ⋅ e jnωTS dω

F( z )

TS

1 1 z ∫ F (ω) ⋅ G dω 2πj γ ω  ω

10

f [ n ] ⋅ g [n ]

#

1 f [n ] = F ( z )·z n−1dz , n ∈ Ζ 2πj C∫

1

f [n + a ]

z a F( z )

2

f [n −a ]

z −a F( z )

3

f [− n ]

1 F  z

4

f [n ] n+a

− z a ∫ z −1−a F ( z )dz

5

f [−n ]

1 F  z

#

1 f [n ] = F (z )·z n−1dz , n ∈ ℵ ∪ {0} ∫ 2 πj C

F(z) = ∑ f [n ] z −n

1

f [n + a ]

z a F (z ) − f [0 ] − z −1 f [1] − L − z − a+1 f [a −1] , a ∈ ℵ

2

f [n −a ]

z − a F (z ) , a ∈ ℵ

3

f [n −a ]

4

f [0 ]

z − a F ( z ) + f [−a ] + z −1 f [−a +1] + z −2 f [− a+ 2 ] + L + z − a +1 f [−1] lim F (z )

5

lim f [n ] n→∞

lim 1 − z −1 ·F (z )

6

∇f [ n ]

z −1 F (z ) z

7

∇ m f [n ]

8

∆f [n ]

9

∆m f [n ]

10

∑ f [k ]

Propiedades Exclusivas de TZB Fb (z) =

∞

∑ f [n ] z − n

n= −∞

Propiedades Exclusivas de TZU ∞

n=0

[

z →∞

z →1

(

]

)

 z −1   F (z )  z  (z − 1)F (z ) − zf [0 ] m

(z − 1) F (z ) − z ∑ (z − 1) m

m − k −1

k =0

z F (z ) z −1 ∞ f F (z ′) dz ′ + lím [n ] ∫ n→∞ n z z′

n

11

m −1

k =0

f [n ] , n ∈ℵ n MSc. Ing. Franco Martin Pessana E-mail: [email protected]

∆k f [0 ]

Identidad de Parseval ∞ 1  1  −1 ∑ f [n ] g [n ] = ∫ F (ω) ⋅ G ω dω n = −∞ 2πj γ  ω

∇f [n ] = f [n ] − f [n−1]  ∆f [n ] = f [n+1] − f [n ]

Siendo:

∞

2

∑ f [n ] =

n = −∞

1  1  −1 ∫ F (ω) ⋅ F  ω dω 2πj γ  ω

Pares de Transformadas Z Unilateral #

f [n ] =

1 F (z )·z n −1dz , n ∈ ℵ ∪ {0} 2πj

∞

∫

C

F(z) = ∑ f [n ] z −n n=0

Región de Convergencia

1

∀z ∈ Ζ

δ[n ]

1 2

δ[n − m]

3

u[n ]

4

n·u[n ]

5

n 2 ·u[n]

6

a n ·u[n]

7

n·a n−1 ·u[n]

8

(n + 1)·a n ·u[n]

9

(n + 1)(n + 2 )L (n + m ) a n ·u[n]

10

cos(Ω0 n )·u[n]

11

sen(Ω 0 n )·u[n]

12

a n ·cos(Ω 0 n )·u[n]

z ( z − a cos Ω0 ) z 2 − 2az cos Ω 0 + a 2

z >a

13

a n ·sen(Ω 0 n )·u[n]

az sen Ω0 z − 2az cos Ω 0 + a 2

z >a

14

e − anT u[n]

15

e − anT ·cos(nω0T )·u[n]

z z − e −aT z z − e − aT cos ω0T z 2 − 2 ze −aT cos ω0T + e −2 aT

16

e − anT ·sen(nω0T )·u[n]

ze − aT sen ω0T z 2 − 2 ze −aT cos ω0T + e −2 aT

z > e − aT

17

senh(Ω 0 n )·u[n]

z senh Ω 0 z 2 − 2 z cosh Ω 0 + 1

z > e − Ω0

18

cosh(Ω 0 n )·u[n]

m!

z

∀z ∈ Ζ − {0}

−m

z z −1 z (z − 1)2 z (z + 1) (z − 1)3 z z−a z (z − a )2

z >1 z >1 z >1 z >a z >a

z2 (z − a )2

z >a

z m +1 (z − a )m+1 z (z − cos Ω0 ) 2 z − 2 z cos Ω0 + 1 z sen Ω0 z − 2 z cos Ω0 + 1 2

2

(

)

z (z − cosh Ω0 ) z − 2 z cosh Ω 0 + 1 2

z >a z >1 z >1

z > e − aT z > e − aT

z > e − Ω0

Expresiones útiles

sen 2 θ + cos 2 θ = 1 cos θ − sen θ = cos 2θ 2

2

(1 + cos 2θ ) 2 sen θ = 12 (1 − cos 2θ ) sen(α ± β ) = sen α ·cos β ± cos α ·sen β

cos 2 θ =

1 2

cos(α ± β ) = cos α ·cos β m sen α ·sen β

sen α ·sen β = 12 cos(α − β ) − 12 cos(α + β )

cos α ·cos β = 12 cos(α − β ) + 12 cos(α + β )

sen α ·cos β = 12 sen(α − β ) + 12 sen(α + β )

MSc. Ing. Franco Martin Pessana E-mail: [email protected]

Tabla de Propiedades Transformada de Fourier de una Secuencia (TFS) 1 f [n] = 2π

#

π

∫π ( )

a1F1 (e

f [n] ⋅ e

3

± jnω0

j

f [− n]

6

f [− n]

j

 f [n L ] si n = kL , k ∈ Ζ Sobremuestrestreador: f (L ) [n] =  , si n ≠ kL , k ∈ Ζ  0

8

Submuestreador: g [n] = f [nM ]

9

x[n ]* h[n]

10

x[n ] ⋅ h[n ]

11

f [n] − f [n − 1]

j L

( )

G e jω =

1 2π

14

f [n] = δ [n] ∞

∑

18

−

−1

−j

j

j

dω F e jω = 1

( ) F (e ω ) = e j

( )

F e jω =

∑

j

ak e

2π nk N

( )

∑δ (ω − ω

E=

∑

0

− 2π k )

∞

∑ a δ ω − k

k = −∞

k =0

Relación de Parseval para señales aperiódicas:

k = −∞

k = −∞

F e jω = 2π ∞

2πk   N 

∑ δ ω −

∞

F e jω = 2π N −1

− jn0ω

∞

2π N

( )

f [n] = e jω0n

19

j

(1 − e ω ) F (e ω ) dF (e ω ) j

δ [n − kN ]

Serie Discreta de Fourier: f [n] =

)

−

j

−j

k = −∞

17

− 2π l ) M )

l =0

j

f [n] = δ [n − n0 ] f [n] =

j

(ω θ ) θ ∫π X (e )⋅ H (e )dθ (1 − e ω )F (e ω )

k = −∞

nf [n]

∑ F (e ((ω

π

∑ f [k ]

13

M −1

1 M

X (e j ω ) ⋅ H (e j ω )

n

16

0

−j

7

15

m

−j

5

12

) + a2 F2 (e jω )

F e jω ⋅ e m jn 0ω

f [n]

4

jω

( ) F (e (ω ω ) ) F (e ω ) F (e ω ) F (e ω ) F (e ω )

f [n m n0 ]

2

− jnω

n

n = −∞

−

a1 f1[n ] + a 2 f 2 [n ]

1

∞

∑ f[ ] ⋅ e

F(e jω ) =

F e jω e jnω dω

f [n] = 2

n =−∞

1 2π

π

∫π ( ) F e jω

2

2π k   N  dω

−

Tabla de Propiedades de Transformada Discreta de Fourier (TDF) #

f [n] =

1 N

N −1

∑ k =o

F[k ] ⋅ e

j

2π ⋅ n ⋅ k N

; n = 0,1,2,3, L , N − 1

a1 f 1 [n ] + a 2 f 2 [ n ]

2

f ( [n m n0 ] )N

3

f [n] ⋅ W N ± k0n x[n] ⊗ h[n] =

N −1

∑ n =o

1

4

F [k ] =

N −1

∑ x[l ]⋅ h([n − l ])

N

f [n ] ⋅ e

−j

2π ⋅ n ⋅ k N

; k = 0,1,2,3, L , N − 1

a1 F1 [k ] + a2 F2 [k ]

F [k ] ⋅ W N ± n0k con WN = e

−j

2π N

F ( [k ± k 0 ])N con WN = e

−j

2π N

X [k ]⋅ H [k ]

, con n = 0,1,2,3, L , N − 1

l =0

5 6 7

x[n ] ⋅ h[n ]

f [n]

1 N

N −1

∑ X [l ] ⋅ H ([k − l ]) l =0

f ( [− n])N

N

, con k = 0,1,2,3, L , N − 1

F ( [− k ] )N F [k ]

MSc. Ing. Franco Martin Pessana E-mail: [email protected]