X[k] - Instituto de Ingeniería Eléctrica

Serie de Fourier
x(t) funciōn periōdica, con perı̄odo T
1
X[k] =
T
Z
to +T
x(t)e−jk
2π
T t
dt
+∞
X
x(t) =
to
X(f ) =
X[k]e
jk 2π
T t
k=−∞
para cualquier to .
1
T
Z
to +T
x(t)y ∗ (t)dt =
to
∞
X
X[k]Y ∗ [k]
k=−∞
Z
∞
−∞
Z
x(−t) = x∗ (t) ⇐⇒ Im[X[k]] = 0
Re[x(t)] = 0 ⇐⇒ X[−k] = −X ∗ [k]
x(−t) = −x∗ (t) ⇐⇒ Re[X[k]] = 0
Funciōn
ax(t) + by(t)
f (t − τ )
ej2πmt/T x(t)
R
1 T
T 0 x(t − τ )y(τ )dτ
x(t)y(t)
P
t−nT
nΠ
τ
Transformada
aX[k] + bY [k]
e−j2πkτ /T X[k]
X[k − m]
X[k]Y [k]
X[k] ∗ Y [k]
τ
T sinc(kτ /T )
Identidades trigonomētricas
ejθ = cos θ + j sin θ
jθ
−jθ
cos θ = e +e
2
jθ
−jθ
sin θ = e −e
2j
cos θ = sin(θ + 90◦ )
sin θ = cos(θ − 90◦ )
sin2 θ + cos2 θ = 1
cos2 θ = 1/2(1 + cos 2θ)
cos3 θ = 1/4(3 cos θ + cos 3θ)
sin2 θ = 1/2(1 − cos 2θ)
sin3 θ = 1/4(3 sin θ − sin 3θ)
sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β
tan α±tan β
tan(α ± β) = 1∓tan
α tan β
sin α sin β = 1/2 cos(α − β) − 1/2 cos(α + β)
cos α cos β = 1/2 cos(α − β) + 1/2 cos(α + β)
sin α cos β = 1/2 sin(α − β) + 1/2 sin(α + β)
∞
−∞
x(t)y ∗ (t)dt =
−∞
Z
∞
X(f )Y ∗ (f )df
−∞
Im[x(t)] = 0 ⇐⇒ X(f ) = X ∗ (f )
x(−t) = x∗ (t) ⇐⇒ Im[X(f )] = 0
Re[x(t)] = 0 ⇐⇒ X(−f ) = −X ∗ (f )
x(−t) = −x∗ (t) ⇐⇒ Re[X(f )] = 0
∗
Im[x(t)] = 0 ⇐⇒ X[−k] = X [k]
Transformada de Fourier
Z ∞
X(f )ej2πf t df
x(t)e−j2πf t dt
x(t) =
Funciōn
ax(t) + by(t)
(x ∗ y)(t)
x(t)y(t)
x∗ (t)
X(t)
x(t − td )
x(t)ejωc t
x(t) cos(ωc t + φ)
x(αt)
dn x(t)
n
R dt
t
−∞ x(λ)dλ
n
t x(t)
δ(t − td )
ej(ωc t+φ)
2
e−π(at)
P∞
k=−∞ δ(t − kT )
sign t
u(t)
Π τt
Λ τt
Transformada
aX(f ) + bY (f )
X(f )Y (f )
(X ∗ Y )(f )
X ∗ (−f )
x(−f )
X(f )e−j2πf td
X(f − fc )
X(f −fc )ejφ +X(f +fc )e−jφ
2
f
1
X
|α|
α
n
(j2πf ) X(f )
X(0)
1
j2πf X(f ) + 2 δ(f )
n
)
(−j2π)−n d dfX(f
n
e−j2πf td
ejφ δ(f − fc )
1 −π(f /a)2
a eP
∞
k
1
k=−∞ δ f − T
T
1
jπf
1
j2πf
+ 21 δ(f )
τ sinc f τ
τ sinc2 f τ
Cola Gaussiana
Algunas funciones ūtiles
1
(x−m)2
p(x) = √
e− 2σ2
2
2πσZ
∞
2
1
Q(k) = √
e−λ /2 dλ
2π k
sin πt
sinc t =
πt
1
t>0
sign t =
−1 t < 0
1 t>0
u(t) =
0 t<0
1 |t| < τ2
t
Π τ =
τ
( 0 |t| > 2
|t|
1 − τ |t| < τ
Λ τt =
0
|t| > τ
Distribuciōn de Gauss
Cola Gaussiana
Sinc
Signo
Escalōn
Rectāngulo
Triāngulo
Cuando k > 3 la siguiente es una buena aproximaciōn
Q(k) ≈ √
2
1
e−k /2
2πk
c 2003 INSTITUTO DE INGENIER ĪA EL ĒCTRICA
Hoja de fōrmulas MPD/SISCOM ver 2.8 DTFT
jθ
X(e ) =
+∞
X
x[n]e
1
x[n] =
2π
−jnθ
n=−∞
+∞
X
DFT
1
x1 [n]x∗2 [n] =
2π
n=−∞
Z
Z
π
jθ
X(e )e
jθn
dθ
x[−n mod N ] = −x∗ [n] ⇐⇒ Re[X[k]] = 0
Transformada
a1 X(ejθ ) + a2 Y (ejθ )
X(ejθ )Y (ejθ )
Rπ
1
jλ
j(θ−λ
))dλ
2π −π X(e )Y (e
∗ −jθ
X (e )
X(ejθ )e−jno θ
X(ej(θ−θo ) )
n x[n]
x[−n]
δ[n]
δ[n − no ]
1
an u[n] (|a| < 1)
u[n]
(n + 1)an u[n] (|a| < 1)
)
j d X(e
dθ
X(e−jθ )
1
e−jno θ
P+∞
ejθo n
jθ
X(z) =
+ 2πk)
sin(θ(M +1)/2) −jθM/2
e
sin(θ/2)
Fōrmula de Poisson
e
k=−∞
+∞
2π X
2π
=
δ t−k
α
α
k=−∞
Desigualdad de Schwarz
Z
2 Z
Z b
b
b
∗
2
v(λ)w (λ)dλ ≤
|v(λ)| dλ
|w(λ)|2 dλ
a
a
a
La igualdad se da si v(λ) = K · w(λ) con K constante.
Transformada Z unilateral
X(z) =
+∞
X
+∞
X
x[n]z
−n
n=−∞
k=−∞ 2πδ(θ − θo + 2πk)
jkαt
Funciōn
ax[n] + by[n]
X[n]
x[(n − m) mod N ]
WN−ln x[n]
PN −1
m=0 x[m]y[(n − m) mod N ]
x[n]y[n]
x∗ [n]
x∗ [−n mod N ]
Transformada
aX[k] + bY [k]
N x[−k mod N ]
WNkm X[k]
X[(k − l) mod N ]
X[k]Y [k]
PN −1
1
l=0 X[l]Y [(k − l) mod N ]
N
X ∗ [−k mod N ]
X ∗ [k]
Transformada Z
P∞
+∞
X
k=0
Re[x[n]] = 0 ⇐⇒ X[−n mod N ] = −X ∗ [k]
Funciōn
a1 x[n] + a2 y[n]
(x ∗ y)[n]
x[n]y[n]
x∗ [n]
x[n − no ]
x[n]ejnθo
1 0≤n≤M
0 otro n
N −1
1 X
X[k]Y ∗ [k]
N
x[−n mod N ] = x∗ [n] ⇐⇒ Im[X[k]] = 0
x[−n] = −x∗ [n] ⇐⇒ Re[X(ejθ )] = 0
−j2π/N
Im[x[n]] = 0 ⇐⇒ X[−n mod N ] = X ∗ [k]
Re[x[n]] = 0 ⇐⇒ X(e−jθ ) = −X ∗ (ejθ )
x[n] =
x[n]y ∗ [n] =
n=0
k=−∞ 2πδ(θ + 2πk)
1
1−ae−jθ P
1
+ ∞
k=−∞ πδ(θ
1−e−jθ
1
(1−ae−jθ )2
1
1−2r cos
2 e−j2θ
θo e−jθ +r
P
θ+2πk
kΠ
2θo
N −1
1 X
X[k]WN−kn
N
k=0
N
−1
X
X1 (ejθ )X2∗ (ejθ )dθ
x[−n] = x∗ [n] ⇐⇒ Im[X(ejθ )] = 0
(|r| < 1)
x[n] =
donde WN = e
π
−π
x[n]WNkn
n=0
−π
Im[x[n]] = 0 ⇐⇒ X(e−jθ ) = X ∗ (ejθ )
r n sin θo (n+1)
u[n]
sin θo
sin θo n
πn
X[k] =
N
−1
X
1
x(n) =
2πj
I
X(z)z n−1 dz
C
donde C es una curva antihoraria en la regiōn de convergencia y
que envuelve al origen.
Secuencia
ax[n] + by[n]
x[n − no ]
zon x[n]
nk x[n]
x∗ [n]
x[−n]
(x ∗ y)[n]
δ[n]
an u[n]
−an u[−n − 1]
δ[n − no ]
cos(ωo n)u[n]
sin(ωo n)u[n]
Transformada Z
aX(z) + bY (z)
z −no X(z)
X(z/zo)
d k
X(z)
−z dz
X ∗ (z ∗ )
X(1/z)
X(z)Y (z)
1
1
1−az −1
1
1−az −1
−no
z
1−z −1 cos ωo
1−2z −1 cos ωo +z −2
z −1 sin ωo
1−2z −1 cos ωo +z −2
ROC
contiene Rx ∩ Ry
Rx , quizā ± 0 ō ∞
|zo |Rx
Rx , quizā ± 0 ō ∞
Rx
1/Rx
contiene Rx ∩ Ry
∀z
|z| > |a|
|z| < |a|
∀z excepto 0 ō ∞
|z| > 1
|z| > 1
Serie Geomētrica
N2
X
k=N1
αk =
αN1 − αN2 +1
1−α
si N2 > N1 y α 6= 1
x[n]z −n
n=0
x[n] ←→ Xu (z)
x[n + no ] ←→ z no Xu (z) − x[0]z no − · · · − x[no − 1]z
x[n − no ] ←→ z −no Xu (z) + x[−1]z −no +1 + · · · + x[−no ]
Fōrmula 1
ejπ + 1 = 0
c 2003 INSTITUTO DE INGENIER ĪA EL ĒCTRICA
Hoja de fōrmulas MPD/SISCOM ver 2.8