Tabla de Transformadas

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

Tabla de Propiedades y algunas Transformadas de Fourier

f(t) ( )∫+∞

∞−

ω ωωπ

= de·F2

1 tj F(ωωωω)= ( )∫+∞

∞−

ω− dte·tf tj

1 )t(fa)t(fa 2211 + )(Fa)(Fa 2211 ω+ω

2 0a)at(f ≠ ,

ωa

Fa

1

3 )tt(f 0m )(Fe 0tj ωωm

4 )t(fe t0jω± )(F 0ωωm

5 )tcos()t(f 0ω⋅ )(F)(F 021

021 ω+ω+ω−ω

6 )tsen()t(f 0ω⋅ )(F)(F 0j21

0j21 ω+ω−ω−ω

7 )t(F )(f2 ω−π

8 ℵ∈ndt

)t(fdn

n

, )(F)j( n ωω

9 ∫ ′′∞−

t

td)t(f )()0(F)(Fj

1 ωδπ+ωω

10 ℵ∈− n)t(f)jt( n , n

n

d

)(Fd

ωω

11 jt

)t(f

− ∫ ω′ω′

ω

∞−d)(F

12 )t(f*)t(f 21 )(F)(F 21 ω⋅ω

13 )t(f)t(f 21 ⋅ )(F*)(F 2121 ωωπ

14 dt)t(f)t(f 21∫∞

∞− ω∫ ω⋅ω

∞

∞−π d)(F)(F 212

1

15 )t(ue at− ω+ ja

1

16 tae− 22a

a2

ω+

17 0ae2at ≠− , a4

2

a eω−

π

18 )t(pA T2⋅ )T(ATsinc2 ω

19 ( )

><−

=∆Tt0

Tt1A)t( T

t

,

,

ω2

TATsinc2

20 ( ) )t(ue!1n

t at1n

−−

− ( )n

aj

1

+ω

21 )t(utsene 0

at ⋅ω− ( ) 2

0

20

ja ω+ω+ω

22 )t(utcose 0

at ⋅ω− ( ) 2

0

2ja

ja

ω+ω+ω+

23 )t(kδ k

24 k )(k2 ωδπ

25 )tsgn( ωj2

26 )t(u )(j

1 ωπδ+ω

27 tcos 0ω ( ) ( )[ ]00 ω+ωδ+ω−ωδπ

28 tsen 0ω ( ) ( )[ ]00j ω−ωδ−ω+ωδπ

29 t0je ω± )(2 0ωωπδ m

30 ( )T

2,dte·tf

T

1C,eC 0

2T

2T

tjnn

n

tjnn

00π=ω= ∫∑

−

ω−∞

−∞=

ω ( )∑ ω−ωδπ∞

−∞=n0n nC2

31 ( )∑∞

−∞=

−n

SnTtδ ( )S

Sn

SS Tn

πωωωδω 2 , =−∑

∞

−∞=

32 ℵ∈nt n , ( )

n

nn

d

dj2

ωωδπ


Tabla de Propiedades y algunas Transformadas de Laplace

( )∫

∞+σ

∞−σ

ω+σ=π

=j

j

st js,dse·sFj2

1 f(t) ( )∫

+∞−=

0

stdte·tfF(s)

1 )t(fa)t(fa 2211 + )s(Fa)s(Fa 2211 +

2 0a)at(f ≠ ,

a

sF

a

1

3 )t(u)t(f τ−τ− )s(Fe sτ−

4 )t(fe at± )as(F m

5 )t(f*)t(f 21 )s(F)s(F 21 ⋅

6 )t(f)t(f 21 ⋅ ∫ ττ−τ∞+

∞−

jc

jc21 d)s(F)(F

7 0t,ndt

)t(fdn

n

≥ℵ∈ , )0(f)0(fs)0(fs)s(Fs )1n(2n1nn −−− −−′−− L

8 ∫ ′′∞−

t

td)t(f s

dt)t(f

s

)s(F

0

∫+ ∞−

9 ℵ∈− n)t(f)t( n , n

n

ds

)s(Fd

10 t

)t(f ∫

∞

s

du)u(F

11 )Tt(f)t(f += ∫−−

−

T

0

st

sTdte)t(f

e1

1

12 )0(f )s(sFlíms ∞→

13 )t(flímt ∞→

)s(sFlím0s→

14 )t(flím)1n(

t

−

∞→ )s(Fslím

n

0s→

15 ( )( ) ( ) 0Q/,e

Q

Pkk

n

1k

t

k

k k =ααα′α

∑=

α ( ) ( ) nQgrPgr)s(Q

)s(P =< ,

16 )t(ue at± as

1

m

17 ℵ∈± n)t(uet atn , ( ) 1nas

!n+

m

18 )t(u s

1

19 tae− 22 sa

a2

−, con ( )∫

+∞

∞−

−= dte·tf st(s)Fb

20 ( ) )t(ue1 at ⋅− − ( )ass

a

+

21 )t(utsene 0

at ⋅ω− ( ) 2

0

20

as ω++ω

22 )t(utcose 0

at ⋅ω− ( ) 2

0

2as

as

ω+++

23 )t(utcos 0 ⋅ω 2

0

2s

s

ω+

24 )t(utsen 0 ⋅ω 2

0

2

0

s ω+ω

25 )t(kδ k

26 n

n

dt

)t(d δ ns

27 ( )∑ −δ∞

−∞=nnTt

sTe1

1−−

28 )t(utcosh 0 ⋅ω 2

0

2s

s

ω−

29 )t(utsenh 0 ⋅ω 2

0

2

0

s ω−ω


Tabla de Propiedades de la Transformada Z

Propiedades Comunes para TZB y TZU

# [ ] ( ) Ζ∈= ∫− n ,dzz·zF

πj2

1

C

1n f n [ ]∑∞

−∞=

−=n

nn zf(z)Fb ; [ ]∑

∞

=

−=0n

nn zfF(z)

1 [ ] [ ]n22n11 fafa + )z(Fa)z(Fa 2211 +

2 [ ]nf ( )zF

3 [ ] nfRe ( ) ( )[ ]zFzF21 +

4 [ ] nfIm ( ) ( )[ ]zFzFj21 −

5 [ ]n

k fn 0k)z(Fdz

dz)1(

k

k ∪ℵ∈

− ,

6 [ ] 0afa n

n ≠± , ( )zaF 1m

7 [ ] [ ] nn g*f )z(G)z(F ⋅

8 [ ] ∫ Ζ∈π

=γ

− ndzz)z(Fj2

1f 1n

n , )z(F

9 [ ] ∫−

⋅=S

S

SS

S

T

T

TjnTjSnT deeF

Tf

π

π

ωω ωπ

)(2

)z(F

10 [ ] [ ]nn gf ⋅ ( )∫ ω

ω⋅ω

ωπ γd

zGF

1

j2

1

Propiedades Exclusivas de TZB

# [ ] ( ) Ζ∈= ∫− n ,dzz·zF

πj2

1

C

1n f n [ ]∑∞

−∞=

−=n

nn zf(z)Fb

1 [ ]anf + )z(Fza

2 [ ]anf − )z(Fz a−

3 [ ]nf −

z

1F

4 [ ]

an

f n

+ ∫− −− dz)z(Fzz a1a

5 [ ]nf −

z

1F

Propiedades Exclusivas de TZU

# [ ] ( ) 0n ,dzz·zFπj2

1

C

1n ∪ℵ∈= ∫− f n [ ]∑

∞

=

−=0n

nn zfF(z)

1 [ ]anf + ( ) [ ] [ ] [ ][ ] ℵ∈−−−− −+−− afzfzfzFz 1a

1a

1

1

0

a , L

2 [ ]anf − ( ) ℵ∈− azFz a ,

3 [ ]anf − ( ) [ ] [ ] [ ] [ ]11a

2a2

1a1

aa fzfzfzfzFz −

+−+−

−+−

−−

− +++++ L

4 [ ]0f ( )zFlimz ∞→

5 [ ]nn

flim∞→

( ) ( )zF·z1lim 1

1z

−

→−

6 [ ]nf∇ ( )zFz

1z −

7 [ ]n

m f∇ ( )zFz

1zm

−

8 [ ]nf∆ ( ) ( ) [ ]0zfzF1z −−

9 [ ]n

m f∆ ( ) ( ) ( ) [ ]∑ ∆−−−−

=

−−1m

0k0

k1kmm f1zzzF1z

10 [ ]∑=

n

0kkf ( )zF

1z

z

−

11 [ ] ℵ∈nn

f n , ( ) [ ]

∫ +′′′∞

∞→z

n

n n

flímzd

z

zF


Siendo: [ ] [ ] [ ]

[ ] [ ] [ ]

−=∆−=∇

+

−

n1nn

1nnn

fff

fff

Identidad de Parseval

[ ] [ ] ( )∫ ωω

ω⋅ω

π=∑

γ

−∞

−∞=d

1GF

j2

1gf 1

nnn

[ ] ( )∫ ωω

ω⋅ω

π=∑

γ

−∞

−∞=d

1FF

j2

1f 1

n

2

n

Pares de Transformadas Z Unilateral

# [ ] ( ) 0·2

1 1 ∪ℵ∈= ∫ − n ,dzzzFπj

C

n f n [ ]∑∞

=

−=0n

nn zfF(z) Región de

Convergencia 1 [ ]nδ 1 Ζ∈∀z

2 [ ]mn −δ mz− 0−∈∀ Ζz

3 [ ]nu 1z

z

− 1>z

4 [ ]nu·n ( )21z

z

− 1>z

5 [ ]nu·n2 ( )

( )31z

1zz

−+

1>z

6 [ ]nu·an az

z

− az >

7 [ ]nu·a·n 1n− ( )2az

z

− az >

8 ( ) [ ]nu·a·1n n+ ( )2

2

az

z

− az >

9 ( )( ) ( ) [ ]nu·a

!m

mn2n1n n+++ L ( ) 1m

1m

az

z+

+

− az >

10 ( ) [ ]nu·ncos 0Ω ( )

1cosz2z

coszz

02

0

+Ω−Ω−

1z >

11 ( ) [ ]nu·nsen 0Ω 1cosz2z

senz

02

0

+Ω−Ω

1z >

12 ( ) [ ]nu·n·cosa 0n Ω

( )2

02

0

acosaz2z

cosazz

+Ω−Ω−

az >

13 ( ) [ ]nu·n·sena 0n Ω 2

02

0

acosaz2z

senaz

+Ω−Ω

az >

14 [ ]nue anT− aTez

z−−

aTez −>

15 ( ) [ ]nu·Tn·cose 0anT ω−

( )aT2

0aT2

0aT

eTcosze2z

Tcosezz−−

−

+ω−ω−

aTez −>

16 ( ) [ ]nu·Tn·sene 0anT ω− aT2

0aT2

0aT

eTcosze2z

Tsenze−−

−

+ω−ω

aTez −>

17 ( ) [ ]nu·nsenh 0Ω 1coshz2z

senhz

02

0

+Ω−Ω

0ez Ω−>

18 ( ) [ ]nu·ncosh 0Ω ( )

1coshz2z

coshzz

02

0

+Ω−Ω−

0ez Ω−>

Expresiones útiles

122 =+ θθ cossen

θθθ 222 cossencos =−

( )θθ 21212 coscos +=

( )θθ 21212 cossen −=

( ) βαβαβα ·sencos·cossensen ±=±

( ) βαβαβα ·sensen·coscoscos m=±

( ) ( )βαβαβα +−−= coscos·sensen 21

21

( ) ( )βαβαβα ++−= coscos·coscos 21

21

( ) ( )βαβαβα ++−= sensen·cossen 21

21


Tabla de Propiedades Transformada de Fourier de una Secuencia (TFS)

# [ ] ( ) ωωπ

π

ω d eeFπ

jnj∫

−

=21

nf [ ]∑∞

−∞=

−⋅=n

jnn ef ωω )F(e j

1 [ ] [ ]nn fafa 2211 + )()( 2211ωω jj eFaeFa +

2 [ ]0nnf m ( ) ωω 0jnj eeF m⋅

3 [ ] 0ωjnenf ±⋅ ( )( )0ωωmjeF

4 [ ]nf ( )ωjeF −

5 [ ]nf − ( )ωjeF −

6 [ ]nf − ( )ωjeF

7 Sobremuestrestreador: ( )[ ] [ ]Ζ∈Ζ∈

≠=

=k

k

kLn

kLn

si

siLnfnf L ,

,,

0 ( )LjeF ω

8 Submuestreador: [ ] [ ]nMfng = ( ) ( )( )( )∑−

=

−=1

0

21 M

l

Mljj eFM

eG πωω

9 [ ] [ ]nhnx * )()( ωω jj eHeX ⋅

10 [ ] [ ]nhnx ⋅ ( ) ( )( )∫−

−⋅π

π

θωθ θπ

deHeX jj

21

11 [ ] [ ]1−− nfnf ( ) ( )ωω jj eFe−−1

12 [ ]∑−∞=

n

k

kf ( ) ( )ωω jj eFe1

1−−−

13 [ ]nnf ( )ω

ω

d

edFj

j

14 [ ] [ ]nnf δ= ( ) 1=ωjeF

15 [ ] [ ]0nnnf −= δ ( ) ωω 0jnj eeF −=

16 [ ] [ ]∑∞

−∞=

−=k

kNnnf δ ( ) ∑∞

−∞=

−=k

j

N

k

NeF

πωδπω 22

17 [ ] njenf 0ω= ( ) ( )∑∞

−∞=

−−=k

j keF πωωδπω 22 0

18 Serie Discreta de Fourier: [ ] ∑−

=

=1

0

2N

k

N

nkj

keanfπ

( ) ∑∞

−∞=

−=k

kj

N

kaeF

πωδπω 22

19 Relación de Parseval para señales aperiódicas: [ ] ( )∫∑−

∞

−∞=

==π

π

ω ωπ

deFnfE j

n

22

2

1

Tabla de Propiedades de Transformada Discreta de Fourier (TDF)

# [ ] [ ]∑−

=

⋅⋅

⋅=1 2

1 N

ok

N

knj

k eFN

π

nf ; 1,,3,2,1,0 −= Nn L [ ] [ ]∑−

=

⋅⋅−⋅=

1 2N

on

N

knj

n efπ

kF ; 1,,3,2,1,0 −= Nk L

1 [ ] [ ]n22n11 fafa + [ ] [ ]kFakFa 2211 +

2 [ ]( )Nnnf 0m [ ] knNWkF 0±⋅ con N

j

N eWπ2−

=

3 [ ] nkNWnf 0±⋅ [ ]( )NkkF 0± con N

j

N eWπ2−

=

4 [ ] [ ] [ ] [ ]( )N

N

l

lnhlxnhnx −⋅=⊗ ∑−

=

1

0

, con 1,,3,2,1,0 −= Nn L [ ] [ ]kHkX ⋅

5 [ ] [ ]nhnx ⋅ [ ] [ ]( )N

N

l

lkHlXN

−⋅∑−

=

1

0

1, con 1,,3,2,1,0 −= Nk L

6 [ ]nf [ ]( )NkF −

7 [ ]( )Nnf − [ ]kF

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Tabla de Transformadas