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MSc. Ing. Franco Martin Pessana
Tabla de Propiedades y algunas Transformadas de Fourier
f(t) ( )
∫
+∞
−∞
ω ω ω π
= F ·e d 2
(^1) j t F( ωωωω ) = (^) ( ) ∫
+∞
−∞
− ω f t·e dt
jt
1 a^1 f 1 (t)+^ a 2 f 2 (t) a^1 F 1 (ω)+a 2 F 2 (ω)
2 f^ (at) ,^ a≠^0
ω
a
F
a
3 f^ (t^ m^ t 0 ) e^ F( )
mj ωt (^0) ω
4 e^ f(t)
± jω 0 t F (ω m ω 0 )
5 f^ (t)⋅^ cos(^ ω 0 t) F(^ ) 2 F( 0 ) 1 (^20)
(^1) ω−ω + ω+ω
6 f^ (t)⋅^ sen(^ ω 0 t) F(^ ) 2 jF( 0 )
1 2 j 0
1 ω−ω − ω+ω
7 F^ (t)^2 πf(−ω)
8 n∈ℵ dt
d f(t ) n
n , (^ j )F( )
n ω ω
− ∞
t f (t)d t F( ) F(^0 )( ) j
ω +π δω ω
10 (^ −^ jt) f(t)n∈ℵ
n , n
n
d
dF( )
ω
ω
jt
f(t)
−
∫ ω′ ω′
ω
−∞
F( ) d
12 f^1 (t)*f 2 (t) F 1 ( ω)⋅F 2 (ω)
13 f^1 ( t)⋅^ f 2 (t) 2 F^1 ( )*F 2 ( ) (^1) ω ω π
14 ∫f^1 (t)f 2 (t)dt
∞
−∞
∫ ω ⋅ ω^ ω
∞
−∞
2 π F^1 ( )F^2 ( )d
1
15 e^ u(t)
−at a+ j ω
at e
− 2 2 a
2 a
(^17) e a 0 at 2 ≠
− ,^4 a
2
a e
ω − π
18 A^ ⋅^ p 2 T(t)^2 ATsinc(ωT)
( )
0 t T
A 1 t T ( t)
T
t
ω
2
T
ATsinc
2
e u(t) n 1!
t (^) at n 1 −
−
− (^ )
n j a
ω +
21 e^ sen 0 t u(t)
at ω ⋅ −
2 0
2
0 a + jω +ω
ω
22 e cos 0 tu(t) at ω ⋅ −
2 0
2 a j
a j
23 kδ(^ t) k
24 k 2 πkδ(ω)
25 sgn(^ t) jω
26 u(^ t) ( ) j
+πδω ω
27 cos^ ω 0 t π[δ^ (^ ω−ω 0 )^ +δ(ω^ +ω 0 )]
28 sen^ ω 0 t^ j^ π[δ^ (^ ω+ω 0 )^ −δ(^ ω−ω 0 )]
j 0 t e ± ω 2 ( ) πδ ωm ω 0
30 ( )^
T
ft·e dt, T
C e ,C 0
T 2
T 2
jn t n n
jn t n
(^0) = 0 ω = π
−
− ω
∞
=−∞
ω
∞
n= −∞
2 Cn n 0
(^31) ∑ (^ )
∞
=−∞
n
δ t nTS ( )
S
S n
S S T
n
∑ − ,^ =
∞
=−∞
32 t n∈ℵ n ,
n
n n d
d 2 j ω
δω π
MSc. Ing. Franco Martin Pessana
Tabla de Propiedades y algunas Transformadas de Laplace
∫^ ( )
σ+ ∞
σ−∞
=σ+ ω π
j
j
st Fs·e ds,s j 2 j
f(t) ( )
+∞ − = 0
st F(s) ft·e dt
1 a^1 f 1 (t)+^ a 2 f 2 (t) a^1 F 1 (s)+a 2 F 2 (s)
2 f^ (at) ,^ a≠^0
a
s F a
3 f^ (t−^ τ)u(t−τ) e^ F(s)
−τs
4 e f(t)
± at F (sma)
5 f^1 (t)*f 2 (t) F 1 ( s)⋅F 2 (s)
6 f^1 (t)⋅^ f 2 (t) ∫ τ −τ τ
−∞
c j
cj
F 1 ( )F 2 (s )d
7 n ,t 0 dt
d f(t) n
n , ∈ℵ ≥ s F(s) s f( 0 ) s f( 0 ) f ( 0 )
n n− 1 n− 2 (n− 1 ) − − ′ −L−
− ∞
t f(t)d t
s
f(t) dt
s
F(s )
0 ∫
−∞
9 ( − t) f(t)n∈ℵ
n , n
n
ds
dF(s )
t
f (t ) ∫
∞
s
F(u) du
11 f^ (t)=^ f(t+T) ∫ −
− −
T
0
st sT f(t)e dt 1 e
12 f^ (^0 ) lím s →∞sF(s)
13 límf(t) t →∞
límsF(s^ ) s→ 0
14 lím^ f (t)
(n 1 ) t
− →∞
lím sF(s^ )
n s→ 0
e , /Q( ) 0
Q
P
k k
n
k 1
t
k
k k (^) α α = ′α
α
=
α
gr( P) gr( Q) n
Q(s)
P( s) , < =
16 e u(t)
±at
s a
m
± t e u(t)n n at ,
n 1 s a
n!
m
18 u(^ t) s
at e
− 2 2 a s
2 a
−
, con ( )
+∞
−∞
− = f t·e dt st Fb (s)
20 ( 1 e ) u(t)
at − ⋅
−
s ( s a)
a
21 e sen 0 t u(t)
at ω ⋅
−
2 0
2
0 s + a +ω
ω
22 e^ cos 0 tu(t)
at ω ⋅ −
2 0
2 s a
s a
23 cos^ ω 0 t ⋅u(t) 2 0
2 s
s
24 sen^ ω 0 t⋅u(t) 2 0
2
0 s +ω
ω
25 kδ(^ t) k
(^26) n
n
dt
d δ( t ) n s
27 ∑ δ(^ − )
∞
n=−∞
t nT sT 1 e
− −
28 cosh^ ω 0 t⋅u(t) 2 0
2 s
s
− ω
29 senh^ ω 0 t ⋅u(t) 2 0
2
0 s −ω
ω
MSc. Ing. Franco Martin Pessana
Siendo:
[ ] [ ] [ ]
[ ] [ ] [ ]
−
n n 1 n
n n n 1
f f f
f f f
Identidad de Parseval
[ ] [ ] ∫^ (^ ) ω ω
ω
ω⋅ π
γ
∞ −
=−∞
d
F G
2 j
f g 1 n n n
[ ] ∫^ (^ ) ω ω
ω
ω⋅ π
γ
∞ −
=−∞
d
F F
2 j
f
1 n
2 n
Pares de Transformadas Z Unilateral
# [ ] ( )·^ { }^0
∫
− Fz z dz,n π j C
n f (^) n
∑ [ ]
∞
=
−
n 0
n F(z) fn z
Región de
Convergencia
1 δ[ n^ ]^1 ∀z∈^ Ζ
2 δ[ n^ −m] z−^ m ∀z∈^ Ζ−{^0 }
3 u[ n]
z 1
z
−
z> 1
4 n· u[ n]
2 z 1
z
−
z> 1
5 n·u[ n]
2
3 z 1
zz 1
−
z> 1
6 a·u[n ]
n z a
z
−
z >a
7 n· a ·u[ n]
n− 1
2 z a
z
−
z >a
8 ( n 1 )· a·u[n ]
n
2
2
z a
z
−
z >a
a·u[ ]n
m!
n + 1 n+ 2 Ln+ m n
m 1
m 1
z a
z
z >a
10 cos(^ Ω^0 n)·^ u[n^ ]
z 2 zcos 1
zz cos
0
2
0 − Ω +
z > 1
11 sen(^ Ω^0 n)·^ u[n^ ]
z 2 zcos 1
zsen
0
2
0 − Ω +
z > 1
12 a ·cos( 0 n)· u[ n]
n Ω
2 0
2
0 z 2 azcos a
zz acos
− Ω +
z >a
13 a ·sen( 0 n)· u[ n]
n Ω (^2) 0
2
0 z 2 azcos a
azsen
− Ω +
z >a
14 e u[n ]
−anT aT z e
z − −
aT z e − >
15 e ·cos( n 0 T)· u[ n]
anT ω
− (^ )^ 2 aT 0
2 aT
0
aT
z 2 ze cos T e
z z e cos T − −
−
− ω +
− ω aT z e − >
16 e ·sen( n 0 T)· u[ n]
anT ω − 2 aT 0
2 aT
0
aT
z 2 ze cos T e
ze sen T − −
−
− ω +
ω (^) aT z e
− >
17 senh(^ Ω 0 n)·^ u[n^ ]
z 2 zcosh 1
zsenh
0
2
0 − Ω +
z > e−Ω^0
18 cosh(^ Ω^0 n)·^ u[^ n]
z 2 zcosh 1
zz cosh
0
2
0 − Ω +
z > e−Ω^0
Expresiones útiles
2 2
senθ +cos θ=
2 2 cos −sen =cos
2 cos = +cos
2 sen = −cos
sen ( α ±β)=senα·cosβ±cosα·sen β
cos (α ±β)=cosα·cosβmsenα·sen β
senα ·senβ= 21 cos (α −β) − 21 cos(α + β)
cos α ·cosβ= 21 cos (α −β) + 21 cos(α + β)
senα ·cosβ= 12 sen (α −β) + 21 sen(α + β)
MSc. Ing. Franco Martin Pessana
Tabla de Propiedades Transformada de Fourier de una Secuencia (TFS)
# [ ]^ (^ )^ ω
ω
π
π
ω Fe e d π
j jn
−
f n [ ]
∞
=−∞
− = ⋅ n
jn fn e ω ω F(e ) j
1 a^1 f (^1) [ ]n +^ a 2 f (^2) [ ]n 11 ( ) 2 2 ( )
jω j ω a F e +aF e
2 f^ [^ nm^ n 0 ] ( )
jω jn 0 ω F e e m ⋅
3 [ ] 0
jn ω f n e
± ⋅
( )
jω m ω F e
4 f [n ] ( )
j ω F e −
5 f^ [−^ n] ( )
j ω F e
−
6 f [− n] ( )
j ω F e
7 Sobremuestrestreador:^ ( )[ ]^
[ ]
k
k
n kL
n kL
si
fnL si f (^) L n ,
jL F e
ω
8 Submuestreador:^ g[^ n^ ]^ =^ f[nM^ ] (^ )^
(( ) )
−
=
−
1
0
M
l
j j l M Fe M
Ge ω ω π
9 x[^ n]^ *^ h[n^ ] ( ) ( )
jω j ω X e ⋅He
10 x[^ n^ ]^ ⋅^ h[^ n] (^ )^
( )
−
− ⋅
π
π
θ ωθ
X e He d
j j
2
11 f^ [n^ ]^ −^ f[^ n−^1 ] ( ) ( )
jω j ω e Fe − 1 −
12 ∑ [ ]
=−∞
n
k
f k ( e jω^ ) F(e j^ ω)
1 1 − − −
13 nf^ [^ n]
ω
d
dF e j
j
14 f[^ n]^ =^ δ[^ n] ( ) = 1
j ω F e
15 f^ [^ n]^ =^ δ^ [^ n−n 0 ] ( )
jω jn 0 ω F e e − =
16 [ ]^ ∑ [^ ]
∞
=−∞
k
fn δ n kN ( ) ∑
∞
=−∞
k
j
N
k
N
Fe
ω 2 π^2
17 [ ]
j n f n e^0
ω
= (^ )^ ∑ (^ )
∞
=−∞
k
j
Fe π δω ω πk
ω 2 0 2
18 Serie Discreta de Fourier: [ ] ∑
−
=
1
0
N^2
k
N
nk j fn ak e
π
∞
=−∞
k
k
j N
k Fe a
ω 2 2
19 Relación de Parseval para señales aperiódicas:^ [ ]^ (^ )
−
∞
=−∞
π
π
ω
E fn Fe d j
n
2 2
2
Tabla de Propiedades de Transformada Discreta de Fourier (TDF)
# [ ] ∑ [ ]
−
=
⋅⋅
= ⋅
1 2 1
N
ko
N
nk j Fk e N
π
f n ; n = 0 , 1 , 2 , 3 ,L ,N− 1 [ ] ∑ [ ]
−
=
⋅⋅ − = ⋅
N 1 2
no
N
nk j fn e
π
F k ; k = 0 , 1 , 2 , 3 ,L,N− 1
1 a 1 f 1 [ n] +^ a 2 f 2 [ n] a 1 F 1 [^ k]^ +a 2 F 2 [^ k]
2 f^ (^ [^ nm^ n 0 ])N
[ ]
nk F k WN ⋅ ±^0 con N
j WN e
2 π − =
3 [ ]
kn f n WN ⋅ ±^0
([ ])
N F k± k 0 con N
j WN e
2 π − =
4 [ ]^ [ ]^ [ ]^ ([^ ])N
N
l
x n⊗ hn=∑xl⋅h n− l
−
=
1
0
, con n = 0 , 1 , 2 , 3 ,L, N− 1 X [ k] ⋅H[ k]
5 x[ n] ⋅ h[n ] [ ] ([^ ])N
N
l
Xl H k l N
−
=
1
0
, con k = 0 , 1 , 2 , 3 ,L,N− 1
6 f [ n] F ( [ −k])N
7 f^ ([^ −^ n])N F[ k]
