formulario fourier unitec

SERIES DE FOURIER 

TRIGONOMETRICA  EXPONENCIAL O COMPLEJA 

    ( ) ( ) ( )0 0 01

cos senn nn

f t a a n t b n tω ω∞

=

= + +⎡ ⎤⎣ ⎦∑

    ( ) ( )0

00

2 cost T

nt

a f t n t dtT

ω+

= ∫  

    ( ) ( )0

00

2 sent T

nt

b f t n t dtT

ω+

= ∫  

   ( )0

00

1 t T

t

a f t dtT

+

= ∫   ;             00

2Tπω =  

     ( ) 0e jn tn

nf t C ω

∞

=−∞

= ∑  

     ( )00

0

1 et T jn t

n tC f t dt

Tω+

= ∫  

    Módulo:   { } { }2 2Re Imn n nC C C= +  

    Fase:  { }{ }

Imarctan

Ren

n

CC

θ =  

TRANSFORMADA DE FOURIER D E F I N I C I O N 

DIRECTA:   ( ) ( )e j tF f t dtωω∞

−∞= ∫  

INVERSA:   ( ) ( )1 e2

j tf t F dωω ωπ

∞ −

−∞= ∫  

P R O P I E D A D E S 

Linealidad: Si                      ( ) ( )1 1f t F ω↔   Y   ( ) ( )2 2f t F ω↔  

⇒      ( ) ( ) ( ) ( )1 2 1 2f t f t F Fω ω+ ↔ +  

Desplazamiento en tiempo: 

Si                      ( ) ( )f t F ω↔  

⇒              ( ) ( )e jaf t a F ωω ±± ↔          siendo  a∈ . 

 Diferenciación en tiempo: 

Si                      ( ) ( )f t F ω↔  

⇒              ( ) ( ) ( )

nn

n

d f tj F

dtω ω↔  

Escalamiento: 

Si                      ( ) ( )f t F ω↔  

⇒                  ( ) 1f at Fa a

ω⎛ ⎞↔ ⎜ ⎟⎝ ⎠

           siendo  a∈ .  

Simetría: Si                      ( ) ( )f t F ω↔  

⇒                    ( ) ( )2F t fπ ω↔ −  

Desplazamiento en frecuencia: 

Si                      ( ) ( )f t F ω↔  

⇒            ( ) ( )e jatf t F aω↔ ±∓            siendo  a∈ . 

Diferenciación en frecuencia: 

Si                      ( ) ( )f t F ω↔  

⇒        ( ) ( ) ( )nn

n

d Fjt f t

dω

ω− ↔  

Modulación: 

Si                      ( ) ( )f t F ω↔  

⇒   ( ) ( ) ( ) ( )10 0 02cosf t t F Fω ω ω ω ω↔ + + −⎡ ⎤⎣ ⎦  

Y     ( ) ( ) ( ) ( )0 0 02sen jf t t F Fω ω ω ω ω↔ + − −⎡ ⎤⎣ ⎦  

IDENTIDADES TRIGONOMETRICAS: 

( ) ( ) ( )sen 2 2sen cosA A A=  

( ) ( ) ( )2 2cos 2 cos senA A A= −  

( ) ( ) ( ) ( ) ( )sen sen cos cos senA B A B A B± = ±  

( ) ( ) ( ) ( ) ( )cos cos cos sen senA B A B A B± = ∓  

( ) ( )2 12sen 1 cos 2A A= −⎡ ⎤⎣ ⎦  

( ) ( )2 12cos 1 cos 2A A= +⎡ ⎤⎣ ⎦  

( ) ( )e cos senjA A j A± = ±  

( ) ( ) ( ) ( )12sen sen cos cosA B A B A B= − − +⎡ ⎤⎣ ⎦

( ) ( ) ( ) ( )12cos cos cos cosA B A B A B= − + +⎡ ⎤⎣ ⎦

( ) ( ) ( ) ( )12sen cos sen senA B A B A B= − + +⎡ ⎤⎣ ⎦

( ) e ecos2

jA jA

A−+

=  

( ) e esen2

jA jA

Aj

−−=  

( )cos 2 1n nπ = ∀  

( )sen 0n nπ = ∀  

( ) ( )cos 1 nn nπ = − ∀  

TABLA DE INTEGRALES: 

a dt at=∫  

111

n nt dt tn

+=+∫  

 1e eat atadt =∫  

( ) ( )1sen cosaat dt at= −∫  

( ) ( )1cos senaat dt at=∫  

lndt tt

=∫  

( )ln lnt dt t t t= −∫  

( ) ( ) ( )( )

( )( )

sen sensen sen

2 2a b t a b t

at bt dta b a b− +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= −− +∫  

( ) ( ) ( )( )

( )( )

sen sencos cos

2 2a b t a b t

at bt dta b a b− +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= +− +∫  

( ) ( ) ( )( )

( )( )

cos cossen cos

2 2a b t a b t

at bt dta b a b− +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= − −− +∫

( ) ( ) ( )2 2

ee sen sen cosat

at bt dt a bt b bta b

= −⎡ ⎤⎣ ⎦+∫  

( ) ( ) ( )2 2

ee cos cos senat

at bt dt a bt b bta b

= +⎡ ⎤⎣ ⎦+∫  

( ) ( ) ( ) ( ) ( )2

1 1cos sen cost a bt dt t a bt btb b

± = ± +∫  

( ) ( ) ( ) ( ) ( )2

1 1sen cos sent a bt dt t a bt btb b

± = ± −∫  

( ) ( ) 2

1 1e e ebt bt btt a dt t ab b

± = ± −∫