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8/18/2019 Formulario Analisis de Señales
1/5
SERIE DE FOURIER CON PERIODO ENINTERVALO SIMETRICO
f ( t )=1
2a0+∑
n=1
∞
an cos( nπt L )+∑n=1∞
bn sen( nπt L )
a0=
1
L∫− L
L
f ( t ) dt
an=1
L∫− L
L
f ( t ) cos( nπt L )dt (n=1,2,3,…)
bn=1
L∫− L
L
f ( t ) sin( nπt L )dt (n=1,2,3,…)SERIE DE FOURIER CON PERIODO ARBITRARIO
f ( t )=12
a0+∑
n=1
∞
an cos (nωt )+∑n=1
∞
bn sen (nωt )
a0=
2
T ∫
d
d+T
f ( t )dt
an=2
T ∫
d
d+T
f ( t )cosnωtdt (n=1,2,3,…)
bn=2
T ∫d
d+T
f ( t )sinnωt dt (n=1,2,3,…)
SERIE DE FOURIER COMPLEJA
f ( t )=c0+∑−∞
∞
cn einωt
c0=
1
T ∫
d
d+T
f (t ) dt
cn=1
T ∫
d
d+T
f (t ) e−inωt dt (n=±1, ±2, ±3,…)
ESPECTRO DE AMPLITUD|Cn|
ESPECTRO DE FASEnω
CONVERSIÓN DE COEFICIENTES COMPLEJOS ATRIGONOMETRICOS (SERIE DE FOURIER)
an=2ℜ
bn=2iIm
CONVERSIÓN DE COEFICIENTESTRIGONOMETRICOS A COMPLEJOS (SERIE DEFOURIER)
c0=
1
2a
0
cn=an−i bn
2
TRANSFORMADA DE FOURIERf(t) F(ω)
e−at
u (t ) 1a+iω
−a∨t ∨¿e¿
2a
a2+ω2
e−at 2
√ π a e−ω2
4a
t e−at
u(t ) 1
( a+iω )2
t n−1
( n−1 ) ! e−at
u(t )
1
( a+iω )n
−a∨t ∨¿te
¿−4 aiω
( a2+ω2)2
−a∨t ∨¿¿ t ∨e¿
2(a2−ω2)
( a2+ω2 )2
1
a2+t 2
−a∨ω∨¿π
a e
¿
t
a2+t 2
−a∨ω∨¿−iπ 2a
ωe¿
cosbt
a2+t 2
−a∨ω+b∨¿−a∨ω−b∨¿+e¿
e¿
π
2a¿
sinbt
a2+t 2
−a∨ω+b∨¿−a∨ω−b∨¿−e¿
e¿
−iπ 2a
¿
A [u ( t +T )−u (t 2 AT Sa(ωT )
8/18/2019 Formulario Analisis de Señales
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Ga(t )T ={ A (¿ t 0 (¿ t 2 AT Sa(ωT )
δ (t ) 1
δ (t −t 0) e−iωt
0
δ ' (t ) iω
δ n(t ) (iω )n
u(t )πδ (ω )+
1
iω
u(t −t 0) πδ ( ω)+ 1
iω e−iω t
0
1 2πδ (ω )
t i2πδ ' (ω )
t n
in2π δ
n (ω )
eiω
0t
2πδ ( ω−ω0 )
cosω0
t π [δ (ω−ω0 )+δ ( ω+ω0 )]
sinω0
t −iπ [δ ( ω−ω0 )−δ (ω+ω0
u(t )cosω0 t iω
ω02
−ω2+
π
2
[δ (ω−ω0 )+δ (ω+
u (t ) sinω0
t ω0
ω0
2−ω2−
iπ
2 [δ (ω−ω0 )−δ (ω
1
t n
−iω ¿n−1
¿¿¿
sgn (t ) −i2ω
¿ t ∨¿ −2
ω2
TRANSFORMADA DE FOURIERf(t) F(ω)
t u(t )iπδ' ( ω)−
1
ω2
e−at
sin(bt )u(t ) b
( a+iω )2+b2
e−at
cos (bt )u(t ) a+iω
( a+iω )2+b2
sinat πt
Ga(ω)T
PROPIEDADES DE LA TRANSFORMADA DEFOURIER
f(t) F(ω)
af ( t )+bf (f ) aF ( ω)+b F (ω )
f ( ct )¿c∨¿ F ( ωc )
1
¿
f ( at −t 0 )¿a∨¿e
−iωt 0
a F ( ωa )1
¿
f (−t ) F (−ω)
f (t −t 0 ) F (ω )e−iωt
0
f (t )e−iω 0 t F (ω−ω0 )
f ( t )cosω0
t 1
2 F (ω−ω0 )+
1
2 F (ω+ω0 )
f (t )sinω0 t −i2
F ( ω−ω0 )+ i
2 F (ω+ω
F ( t ) 2πf (−ω)
f ' (t ) iωF (ω)
f n( t ) (iω )n F ( ω)
t n
f (t )(i )n
dn F (ω )
d ωn
(−it )n f ( t ) dn F (ω )
d ω
n
∫−∞
t
f ( x ) dx−iω
F (ω )+πF (0)δ (ω)
INTEGRAL TRIGONOMÉTRICA DE FOURIER
f ( t )=1
π ∫0
∞
( A ωcosωt +Bω sinωt ) dω
A ω=∫−∞
∞
f ( t )cosωtdt
B ω=∫−∞
∞
f ( t ) sinωt dt
INTEGRAL COMPLEJA DE FOURIER
8/18/2019 Formulario Analisis de Señales
3/5
f ( t )=∫−∞
∞1
2π e
−iωt ∫−∞
∞
f (τ )e−iωτ dτdω
TRANSFORMADA ZXk X() ROCδ k 1 Todo z
δ k!k " z−
0 Todo z,0 si k0>0
exp∝ si k01
-u[-k-1] z
z−1
|z||a|
!#k $%!k!&' z
z−a|z||a|
k#k az
( z−a )2|z|>|a|
!k#k %!k!&' az
( z−a )2
|z||a|
(k&)(k)#k
* z
3
( z−a )3|z|>|a|
(k&)(k)+(kn)#k
n*
zn
( z−a )n|z|>|a|
k z
( z−1 )2|z|>1
k
z ( z+1
)( z−1)3
|z|>1
(,&)#k!& z
( z−a )2|z|>a
-!kT z
z−e−T |z|>e-T
./0 k ωT z ( z−cosωT )
z2−2 z cosωT +
|z|>1
01n k ωT zsinωT
z2−2 z cosωT +
|z|>1
2k ./0 k ωT z ( z−!c"s ωT
z2−2 !z cosωT +
|z|>1
2k 01n k ωT !z sinωT
z2−2 !z cosωT +
|z|>1
-!kT ./0 k ωT
z ( z−e−T
cos
z2−2e−T zcosω
|z|>1
-!kT 01n k ωT z e−T
sinω
z2−2e−T zcosω
|z|>1
{a
0
0≤ ≤
"t"s $a%
z & −a &
z & −a z & −1
|z|>0
TRANSFORMADA Z UNILATERAL
( z )=∑ =0
∞ x
z
PROPIEDADES DE LA TRANSFORMADA ZI3 LINEALIDAD
Z {( x + ) * }=( ( z)+ )+ ( z)
II3 TRASLACIÓN#3 R-t2#0/
Z { x − 0}= 1
z 0
' ( z)
43 A5#n.-
Z { x + 0}= z 0 ( z )−∑
n=0
0−1
xn z 0−n
Z x +1 = z ( z )− z x0
Z { x +2 }= z2 ' ( z )− z2 x0− z x 1
III3 INVERSIÓN EN EL TIEMPO
Z { x− # }= ( 1, )IV3 MULTIPLICACIÓN #k
Z {a x }=
( z
a
)V3 MULTIPLICACIÓN k n (D1f-2-n.1#.16n 2-07-.t/ #
Z)
Z { n x }=(− z ddz )n
( z )
VI3 TEOREMA DEL VALOR INICIAL
lim z- ∞
x = x0
8/18/2019 Formulario Analisis de Señales
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VII3 TEOREMA DEL VALOR FINAL
lim -∞
x =lim, -1
(1−, −1 ) ( z)
VIII3 CONVOLUCIÓN
Z { x ∗. }= ( z ) / ( z )=+ ( z)
* =∑ 0=0
∞
. 0
0 x − 0
TRANSFORMADA DE FOURIER DISCRETA (TDF)
G =∑n=0
& −1
gn e−i2 π
& n
gn= 1
& ∑ =0
& −1
G ei2 π
& n
TRANSFORMADA DE FOURIER R8PIDA (TRF) (9:$-0t2#0)
[G
0
G1
G2
G3
]=[1
1
1
1
1
−i−1
i
1
−11
−1
1
i−1−i
][g0
g1
g2
g3
]=[ g
0+g
1+g
2+g
3
g0−i g
1−g
2+i g
3
g0−g
1+g
2−g
3
g0+i g
1−g
2−i g
3
]TRANSFORMADA DE LAPLACE
f(t) F(0)
δ (t ) 1
u(t) 1
s
c
(c=constante)c
s
t 1
s2
t n
(n=entero
positivo)
n!
sn+1
t a
(a=real) 1 (a+1)
sa+1
1
√ t √ π
seat
(a=constante)1
s−a
t eat 1
( s−a )2
t n eat n!
( s−a )n+1
sen at as2+a2
cos at s
s2+a2
ebt sen at a
(s−b)2+a2
ebt cos at s−b
(s−b)2+a2
senh at a
s2−a2
cosh at s
s2−a2
ebt senh at a
(s−b)2−a2
ebt cosh at s−b
(s−b)2−a2
ln(t) − ln ( s)+2 s
n√ t
s−n+1
n 1
(1+
1
n
)TRANSFORMADA DE LAPLACE
! [ f ( t )]=∫0
∞
e−st
f ( t ) dt = F (s)
PROPIEDADES DE LA TRANSFORMADA DELAPLACE
I3 LINEALIDAD
! {(f (t )+ )g(t )}=(F (s)+ )G(s)
II3 TRASLACIÓN
! {eat f (t )}= F (s−a)
! { f (t −a)u (t −a)}=e−as
F (s−a)
III3 ESCALAR
! { f (at )}=1
a F ( sa )
IV3 DERIVADA DE LA TRANSFORMADA
! {t n
f (t ) }=(−1)n dn F (s)
d sn
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V3 TRANSFORMADA DE LA DERIVADA
! { f ' (t )}=sF ( s )− f (0)!
{f n(t )}=sn F (s )−sn−1 f (0 )−sn−2 f 1 (0 )−3−f n−1
! {f n(t )}=sn F (s )−∑
i=1
n
sn−i
f i−1(0)
VI3 TRANSFORMADA DE INTEGRALES
! {∫4
t
f ( τ ) dτ }=1s F (s)VII3 TEOREMA DEL VALOR INICIAL
lims- ∞
F (s)=0
VIII3 TEOREMA DEL VALOR FINAL
t -0+¿
f (t )lims- ∞
sF (s)=lim¿¿
lims- 0
sF (s)=limt - ∞
f (t )
IX3 FUNCIONES PERIODICAS
! { f (t )}=∫0
T
e−st
f ( t ) dt
1−e−sT
X3 CONVOLUCIÓN
! f (t )∗g( t ) = F (s)0 G(s)