Ortografía editándolo
Introduccion y Conceptos [ editar ] En las modulaciones angulares, como su propio nombre indica, es la fase ( o frecuencia) de la portadora la que es modulada, y no la amplitud, que es lo que ocurría en las anteriores lineales.
c
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cos
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{\displaystyle c(t)=A_{c}\cos \left(\omega _{c}t\right)}
Los tipos de modulaciones angulares son:
FM (Frequency Modulation)
y PM (Phase Modulation)
Podemos representar ambas modulaciones como:
x
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P
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{\displaystyle {\begin{aligned}&x_{\begin{smallmatrix}FM\\PM\end{smallmatrix}}(t)=A_{c}\cos \left(\theta _{i}\left(t\right)\right)\\&\theta _{i}\left(t\right)=\omega _{c}t+\varphi _{i}(t)\\\end{aligned}}}
θ
i
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{\displaystyle \theta _{i}\left(t\right)}
φ
i
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{\displaystyle \varphi _{i}(t)}
Como se relacionan la frecuencia y la fase de una señal?
La frecuencia es la derivada de la fase
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∂
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{\displaystyle {\begin{aligned}&\omega _{i}\left(t\right)={\frac {\partial \theta _{i}\left(t\right)}{\partial t}}\\&f_{i}\left(t\right)={\frac {1}{2\pi }}{\frac {\partial \theta _{i}\left(t\right)}{\partial t}}\to \\&\left\{\theta _{i}\left(t\right)=\omega _{c}t+\varphi _{i}(t)\right\}\to f_{i}\left(t\right)={\frac {1}{2\pi }}{\frac {\partial \left\{\omega _{c}t+\varphi _{i}(t)\right\}}{\partial t}}=f_{c}+{\frac {1}{2\pi }}{\frac {\partial \varphi _{i}\left(t\right)}{\partial t}}\\\end{aligned}}}
ω
i
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t
)
{\displaystyle \omega _{i}\left(t\right)}
f
i
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t
)
{\displaystyle f_{i}\left(t\right)}
Igualmente, podemos decir que la fase, es la integral de la frecuencia
ω
i
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∂
θ
i
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∂
t
⇔
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∫
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∞
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f
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⇔
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{\displaystyle {\begin{aligned}&\omega _{i}\left(t\right)={\frac {\partial \theta _{i}\left(t\right)}{\partial t}}\Leftrightarrow \theta _{i}\left(t\right)=\int _{-\infty }^{t}{\omega _{i}\left(\lambda \right)\partial \lambda }\\&f_{i}\left(t\right)={\frac {1}{2\pi }}{\frac {\partial \theta _{i}\left(t\right)}{\partial t}}\Leftrightarrow \theta _{i}\left(t\right)=2\pi \int _{-\infty }^{t}{f_{i}\left(\lambda \right)\partial \lambda }\\\end{aligned}}}
Debido a que solo estamos modulando la fase, sin cambiar la amplitud de la señal, la potencia de cualquier modulación angular será:
P
F
M
P
M
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c
2
2
{\displaystyle P_{\begin{smallmatrix}FM\\PM\end{smallmatrix}}={\frac {A_{c}^{2}}{2}}}
Representación en el tiempo de modulación PM y FM [ editar ]
x
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PM
→
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Δ
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cos
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FM
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⋅
∂
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i
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∂
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∫
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∞
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∂
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→
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F
M
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cos
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∞
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∂
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{\displaystyle {\begin{aligned}&x(t)=A_{c}\cos \left(\theta _{i}\left(t\right)\right)=A_{c}\cos \left(\omega _{c}t+\varphi _{i}\left(t\right)\right)\\&{\text{PM}}\to \varphi _{i}(t)=\phi _{\Delta }x(t)\to \\&x_{PM}(t)=A_{c}\cos \left(\omega _{c}t+\phi _{\Delta }x(t)\right)\\&{\text{FM}}\\&f_{i}(t)=f_{c}+f_{\Delta }x(t)\to \\&f_{i}(t)={\frac {1}{2\pi }}\cdot {\frac {\partial \theta _{i}\left(t\right)}{\partial t}}=f_{c}+{\frac {1}{2\pi }}{\frac {\partial \varphi _{i}\left(t\right)}{\partial t}}\\&f_{\Delta }x(t)={\frac {1}{2\pi }}{\frac {\partial \varphi _{i}\left(t\right)}{\partial t}}\to \varphi _{i}\left(t\right)=2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda \to }\\&x_{FM}(t)=A_{c}\cos \left(\omega _{c}t+2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda }\right)\\\end{aligned}}}
Características PM [ editar ]
Ejemplo con moduladora sinusoidal:
En este ultimo caso la relacion no es tan clara.
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ϕ
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x
min
≤
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max
E
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:
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{\displaystyle {\begin{aligned}&\varphi _{i}(t)=\phi _{\Delta }x(t)\to \\&x_{PM}(t)=A_{c}\cos \left(\omega _{c}t+\phi _{\Delta }x(t)\right)\\&\phi _{\Delta }x_{\min }\leq \varphi _{i}(t)\leq \phi _{\Delta }x_{\max }\\&Ej:\\&x(t)=A_{m}\cos \left(\omega _{m}t\right)\\&\varphi _{i\max }(t)=\phi _{\Delta }A_{m}\\&\left|x(t)\right|\leq 1\to \varphi _{i\max }(t)=\phi _{\Delta }\\&\varphi _{i}(t)\leq \pi \to \phi _{\Delta }\leq \pi \\\end{aligned}}}
El motivo de esta restricción es debido a que el coseno ( o el seno) es periodica cada
2
π
{\displaystyle 2\pi }
Esto es debido a que:
cos
(
π
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=
cos
(
3
π
)
=
cos
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5
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=
cos
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{\displaystyle \cos(\pi )=\cos(3\pi )=\cos(5\pi )=\cos(7\pi )}
Características FM [ editar ]
Ejemplo con moduladora sinusoidal:
f
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+
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x
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∫
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∞
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∂
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F
M
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∫
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∂
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f
d
≤
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c
{\displaystyle {\begin{aligned}&f_{i}(t)=f_{c}+f_{\Delta }x(t)\to \\&\varphi _{i}\left(t\right)=2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda \to }\\&x_{FM}(t)=A_{c}\cos \left(\omega _{c}t+2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda }\right)\\&f_{d}\leq f_{c}\\\end{aligned}}}
La desviación de frecuencia tiene que ser menor que la frecuencia portadora para poder demodular correctamente la señal, a pesar de ello, no es una restricción especialmente limitadora.
Se define también la desviación máxima de frecuencia, esto es, la máxima frecuencia que puede alcanzar nuestra señal teniendo como referencia la frecuencia central de la portadora.
f
i
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E
j
:
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Δ
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max
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m
{\displaystyle {\begin{aligned}&f_{i}(t)=f_{c}+f_{\Delta }x(t)\\&Ej:\\&x(t)=A_{m}\cos \left(\omega _{m}t\right)\\&\Delta f_{\max }=f_{\Delta }A_{m}\\\end{aligned}}}
Propiedades [ editar ]
x
P
M
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=
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cos
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x
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∫
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∂
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x
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cos
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→
{
cos
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=
cos
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cos
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−
sin
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sin
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}
→
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cos
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cos
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−
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sin
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sin
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{
PM
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cos
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FM
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cos
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x
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sin
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sin
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e
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{\displaystyle {\begin{aligned}&x_{PM}(t)=A_{c}\cos \left(\omega _{c}t+\phi _{\Delta }x(t)\right)\\&x_{FM}(t)=A_{c}\cos \left(\omega _{c}t+2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda }\right)\\&x(t)=A_{c}\cos \left(\omega _{c}t+\varphi _{i}\left(t\right)\right)\to \left\{\cos \left(a+b\right)=\cos a\cos b-\sin a\sin b\right\}\to \\&A_{c}\cos \left(\omega _{c}t+\varphi _{i}\left(t\right)\right)=A_{c}\cos \left(\omega _{c}t\right)\cos \left(\varphi _{i}\left(t\right)\right)-A_{c}\sin \left(\omega _{c}t\right)\sin \left(\varphi _{i}\left(t\right)\right)\\&x(t)=x_{I}(t)\cos \left(\omega _{c}t\right)-x_{Q}(t)\sin \left(\omega _{c}t\right)=e(t)\cos \left(\omega _{c}t+\varphi (t)\right)\to \\&x_{I}(t)=A_{c}\cos \left(\varphi _{i}\left(t\right)\right)=\left\{{\begin{aligned}&{\text{PM}}\to A_{c}\cos \left(\phi _{\Delta }x(t)\right)\\&{\text{FM}}\to A_{c}\cos \left(2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda }\right)\\\end{aligned}}\right.\\&x_{Q}(t)=A_{c}\sin \left(\varphi _{i}\left(t\right)\right)=\left\{{\begin{aligned}&{\text{PM}}\to A_{c}\sin \left(\phi _{\Delta }x(t)\right)\\&{\text{FM}}\to A_{c}\sin \left(2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda }\right)\\\end{aligned}}\right.\\&e(t)=A_{c}\\\end{aligned}}}
NarrowBand modulation [ editar ]
x
P
M
F
M
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cos
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cos
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cos
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−
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sin
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sin
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{\displaystyle {\begin{aligned}&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t+\varphi _{i}\left(t\right)\right)\\&A_{c}\cos \left(\omega _{c}t+\varphi _{i}\left(t\right)\right)=A_{c}\cos \left(\omega _{c}t\right)\cos \left(\varphi _{i}\left(t\right)\right)-A_{c}\sin \left(\omega _{c}t\right)\sin \left(\varphi _{i}\left(t\right)\right)\\\end{aligned}}}
Ahora, si consideramos que la fase modulada es muy pequeña
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≪
1
lim
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→
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sin
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≈
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lim
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→
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cos
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≈
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x
P
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cos
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cos
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sin
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sin
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≈
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Δ
X
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∂
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→
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Δ
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=
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⋅
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)
{\displaystyle {\begin{aligned}&\varphi _{i}\left(t\right)\ll 1\to \left\{{\begin{aligned}&PM\to \phi _{\Delta }x(t)\ll 1\\&FM\to 2\pi f_{\Delta }\int _{-\infty }^{t}{x(\lambda )\partial \lambda \ll 1}\\\end{aligned}}\right.\\&{\underset {x\to 0}{\mathop {\lim } }}\,{\text{ }}\sin x\approx x\\&{\underset {x\to 0}{\mathop {\lim } }}\,{\text{ }}\cos x\approx 1\\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t+\varphi _{i}\left(t\right)\right)=A_{c}\cos \left(\omega _{c}t\right)\underbrace {\cos \left(\varphi _{i}\left(t\right)\right)} _{\approx 1}-A_{c}\sin \left(\omega _{c}t\right)\underbrace {\sin \left(\varphi _{i}\left(t\right)\right)} _{\approx \varphi _{i}\left(t\right)}\to \\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)\approx A_{c}\cos \left(\omega _{c}t\right)-A_{c}\sin \left(\omega _{c}t\right)\varphi _{i}\left(t\right)\\&X_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(f)=A_{c}\left({\frac {\delta \left(f-f_{c}\right)+\delta \left(f+f_{c}\right)}{2}}\right)-A_{c}\left({\frac {\varphi \left(f-f_{c}\right)-\varphi \left(f+f_{c}\right)}{2j}}\right)\\&\\&PM\to \phi _{\Delta }x(t){\xrightarrow[{\mathbb {F} }]{}}\phi _{\Delta }X(f)\\&FM\to 2\pi f_{\Delta }\int _{-\infty }^{t}{x(\lambda )\partial \lambda {\xrightarrow[{\mathbb {F} }]{}}}2\pi f_{\Delta }{\frac {X(f)}{j2\pi f}}={\frac {f_{\Delta }X(f)}{j\cdot f}}\\&X_{PM}(f)=A_{c}\left({\frac {\delta \left(f-f_{c}\right)+\delta \left(f+f_{c}\right)}{2}}\right)-{\frac {A_{c}\phi _{\Delta }}{2j}}\left(X\left(f-f_{c}\right)-X\left(f+f_{c}\right)\right)\\&X_{FM}(f)=A_{c}\left({\frac {\delta \left(f-f_{c}\right)+\delta \left(f+f_{c}\right)}{2}}\right)+{\frac {A_{c}f_{\Delta }}{2}}\left({\frac {X(f-f_{c})}{f-f_{c}}}-{\frac {X(f-f_{c})}{f-f_{c}}}\right)\\\end{aligned}}}
Modulación de un tono [ editar ] Apliquemos este caso para una moduladora senoidal.
x
P
M
F
M
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t
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≈
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c
cos
(
ω
c
t
)
−
A
c
sin
(
ω
c
t
)
φ
i
(
t
)
P
M
→
x
(
t
)
=
A
m
sin
(
ω
m
t
)
F
M
→
x
(
t
)
=
A
m
cos
(
ω
m
t
)
}
→
{
P
M
→
φ
i
(
t
)
=
ϕ
Δ
A
m
sin
(
ω
m
t
)
≪
1
→
β
P
M
=
ϕ
Δ
A
m
F
M
→
φ
i
(
t
)
=
f
Δ
f
m
A
m
sin
(
ω
m
t
)
≪
1
→
β
F
M
=
f
Δ
f
m
A
m
x
P
M
F
M
(
t
)
=
A
c
cos
(
ω
c
t
)
−
A
c
sin
(
ω
c
t
)
β
P
M
F
M
sin
(
ω
m
t
)
→
{
sin
a
sin
b
=
cos
(
a
−
b
)
−
cos
(
a
+
b
)
2
}
x
P
M
F
M
(
t
)
=
A
c
cos
(
ω
c
t
)
−
A
c
β
(
cos
(
(
ω
c
−
ω
m
)
t
)
−
cos
(
(
ω
c
+
ω
m
)
t
)
2
)
=
A
c
cos
(
ω
c
t
)
−
A
c
β
2
cos
(
(
ω
c
−
ω
m
)
t
)
+
A
c
β
2
cos
(
(
ω
c
+
ω
m
)
t
)
{\displaystyle {\begin{aligned}&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)\approx A_{c}\cos \left(\omega _{c}t\right)-A_{c}\sin \left(\omega _{c}t\right)\varphi _{i}\left(t\right)\\&\left.{\begin{aligned}&PM\to x(t)=A_{m}\sin(\omega _{m}t)\\&FM\to x(t)=A_{m}\cos(\omega _{m}t)\\\end{aligned}}\right\}\to \left\{{\begin{aligned}&PM\to \varphi _{i}\left(t\right)=\phi _{\Delta }A_{m}\sin(\omega _{m}t)\ll 1\to \beta _{PM}=\phi _{\Delta }A_{m}\\&FM\to \varphi _{i}\left(t\right)={\frac {f_{\Delta }}{f_{m}}}A_{m}\sin(\omega _{m}t)\ll 1\to \beta _{FM}={\frac {f_{\Delta }}{f_{m}}}A_{m}\\\end{aligned}}\right.\\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t\right)-A_{c}\sin \left(\omega _{c}t\right)\beta _{\begin{smallmatrix}PM\\FM\end{smallmatrix}}\sin(\omega _{m}t)\to \\&\left\{\sin a\sin b={\frac {\cos(a-b)-\cos(a+b)}{2}}\right\}\\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t\right)-A_{c}\beta \left({\frac {\cos \left(\left(\omega _{c}-\omega _{m}\right)t\right)-\cos \left(\left(\omega _{c}+\omega _{m}\right)t\right)}{2}}\right)=\\&A_{c}\cos \left(\omega _{c}t\right)-{\frac {A_{c}\beta }{2}}\cos \left(\left(\omega _{c}-\omega _{m}\right)t\right)+{\frac {A_{c}\beta }{2}}\cos \left(\left(\omega _{c}+\omega _{m}\right)t\right)\\\end{aligned}}}
Para el dibujo vectorial, lo mejor es usar la envolvente compleja:
x
~
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M
F
M
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t
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=
A
c
+
A
c
β
2
(
e
+
j
ω
m
t
−
e
−
j
ω
m
t
)
{\displaystyle {\tilde {x}}_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}+{\frac {A_{c}\beta }{2}}\left(e^{+j\omega _{m}t}-e^{-j\omega _{m}t}\right)}
X
P
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F
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(
f
)
=
A
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2
δ
(
f
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f
c
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(
δ
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m
)
)
−
δ
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f
−
(
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c
+
f
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)
)
;
f
>
0
{\displaystyle X_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(f)={\frac {A_{c}}{2}}\delta \left(f-f_{c}\right)-{\frac {A_{c}\beta }{4}}\left(\delta \left(f-\left(f_{c}-f_{m}\right)\right)-\delta \left(f-\left(f_{c}+f_{m}\right)\right)\right);f>0}
FullBand modulation (Modulación de un tono) [ editar ]
x
P
M
F
M
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t
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=
A
c
cos
(
ω
c
t
+
φ
i
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→
x
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t
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=
A
m
sin
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ω
m
t
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F
M
→
x
(
t
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=
A
m
cos
(
ω
m
t
)
x
P
M
(
t
)
=
A
c
cos
(
ω
c
t
+
ϕ
Δ
x
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t
)
)
=
A
c
cos
(
ω
c
t
+
ϕ
Δ
A
m
sin
(
ω
m
t
)
)
x
F
M
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t
)
=
A
c
cos
(
ω
c
t
+
2
π
f
Δ
∫
−
∞
t
x
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λ
)
∂
λ
)
→
∫
−
∞
t
A
m
cos
(
ω
m
t
)
∂
t
=
A
m
ω
m
sin
(
ω
m
t
)
ω
m
=
2
π
f
m
x
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
2
π
f
Δ
A
m
2
π
f
m
sin
(
ω
m
t
)
)
=
A
c
cos
(
ω
c
t
+
A
m
f
Δ
f
m
sin
(
ω
m
t
)
)
{\displaystyle {\begin{aligned}&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t+\varphi _{i}\left(t\right)\right)\\&PM\to x(t)=A_{m}\sin \left(\omega _{m}t\right)\\&FM\to x(t)=A_{m}\cos \left(\omega _{m}t\right)\\&x_{PM}(t)=A_{c}\cos \left(\omega _{c}t+\phi _{\Delta }x(t)\right)=A_{c}\cos \left(\omega _{c}t+\phi _{\Delta }A_{m}\sin \left(\omega _{m}t\right)\right)\\&x_{FM}(t)=A_{c}\cos \left(\omega _{c}t+2\pi f_{\Delta }\int _{-\infty }^{t}{x\left(\lambda \right)\partial \lambda }\right)\to \int _{-\infty }^{t}{A_{m}\cos \left(\omega _{m}t\right)\partial t={\frac {A_{m}}{\omega _{m}}}\sin \left(\omega _{m}t\right)}\\&\omega _{m}=2\pi f_{m}\\&x_{FM}(t)=A_{c}\cos \left(\omega _{c}t+2\pi f_{\Delta }{\frac {A_{m}}{2\pi f_{m}}}\sin \left(\omega _{m}t\right)\right)=A_{c}\cos \left(\omega _{c}t+{\frac {A_{m}f_{\Delta }}{f_{m}}}\sin \left(\omega _{m}t\right)\right)\\\end{aligned}}}
Ahora, llamemos
β
{\displaystyle \beta }
β
P
M
=
ϕ
Δ
A
m
β
F
M
=
A
m
f
Δ
f
m
s
(
t
)
=
x
P
M
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
β
P
M
F
M
sin
(
ω
m
t
)
)
{\displaystyle {\begin{aligned}&\beta _{PM}=\phi _{\Delta }A_{m}\\&\beta _{FM}={\frac {A_{m}f_{\Delta }}{f_{m}}}\\&s(t)=x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t+\beta _{\begin{smallmatrix}PM\\FM\end{smallmatrix}}\sin \left(\omega _{m}t\right)\right)\\\end{aligned}}}
Ahora, queremos encontrar la representación en frecuencia de esta modulación angular, para ello nos será útil encontrar primero su envolvente compleja:
s
+
(
t
)
=
A
c
e
j
(
ω
c
t
+
β
sin
(
ω
m
t
)
)
s
~
(
t
)
=
A
c
e
j
β
sin
(
ω
m
t
)
{\displaystyle {\begin{aligned}&s^{+}(t)=A_{c}e^{j\left(\omega _{c}t+\beta \sin \left(\omega _{m}t\right)\right)}\\&{\tilde {s}}(t)=A_{c}e^{j\beta \sin \left(\omega _{m}t\right)}\\\end{aligned}}}
Esta señal es periódica, por lo que, por definición, toda señal periódica puede representarse como una serie de senos/cosenos, esto es, una serie de Fourier Pares clasicos de la transformada de Fourier
f
(
t
)
=
∑
k
=
−
∞
∞
C
k
⋅
e
j
2
π
T
k
t
→
s
~
(
t
)
=
∑
k
=
−
∞
∞
C
k
⋅
e
j
ω
m
k
t
C
k
=
1
T
∫
−
T
╱
2
T
╱
2
f
(
t
)
⋅
e
−
j
2
π
T
k
t
∂
t
→
C
k
=
1
T
m
∫
−
T
m
╱
2
T
m
╱
2
s
~
(
t
)
e
−
j
ω
m
k
t
∂
t
=
?
{\displaystyle {\begin{aligned}&f(t)=\sum \limits _{k=-\infty }^{\infty }{C_{k}\cdot e^{j{\frac {2\pi }{T}}kt}}\to {\tilde {s}}(t)=\sum \limits _{k=-\infty }^{\infty }{C_{k}\cdot e^{j\omega _{m}kt}}\\&C_{k}={\frac {1}{T}}\int \limits _{{}^{-T}\!\!\diagup \!\!{}_{2}\;}^{{}^{T}\!\!\diagup \!\!{}_{2}\;}{f(t)\cdot e^{-j{\frac {2\pi }{T}}kt}\partial t}\to C_{k}={\frac {1}{T_{m}}}\int \limits _{{}^{-T_{m}}\!\!\diagup \!\!{}_{2}\;}^{{}^{T_{m}}\!\!\diagup \!\!{}_{2}\;}{{\tilde {s}}(t)e^{-j\omega _{m}kt}\partial t}=?\\\end{aligned}}}
Para sacar los coeficientes, tenemos una integral irresoluble, pero afortunadamente, este tipo de integral esta tabulada y estudiada, por lo que sabemos que:
s
~
(
t
)
=
A
c
e
j
β
sin
(
ω
m
t
)
C
k
=
1
T
m
∫
−
T
m
╱
2
T
m
╱
2
s
~
(
t
)
e
−
j
ω
m
k
t
∂
t
J
n
(
x
)
=
1
2
π
∫
−
π
π
e
j
(
x
sin
λ
−
n
λ
)
∂
λ
=
1
2
π
∫
−
π
π
e
j
x
sin
λ
e
−
j
n
λ
∂
λ
{
λ
=
ω
m
t
↔
t
=
λ
/
ω
m
=
λ
T
m
2
π
∂
λ
=
ω
m
∂
t
↔
∂
t
=
T
m
2
π
∂
λ
}
t
=
T
m
╱
2
=
λ
T
m
2
π
→
λ
=
π
t
=
−
T
m
╱
2
→
λ
=
−
π
C
k
=
1
T
m
∫
t
=
−
T
m
╱
2
t
=
T
m
╱
2
A
c
e
j
β
sin
(
ω
m
t
)
e
−
j
ω
m
k
t
∂
t
=
A
c
1
T
m
∫
−
π
π
e
j
β
sin
λ
e
−
j
k
λ
T
m
2
π
∂
λ
=
A
c
1
2
π
∫
−
π
π
e
j
β
sin
λ
e
−
j
k
λ
∂
λ
→
C
k
=
A
c
J
k
(
β
)
{\displaystyle {\begin{aligned}&{\tilde {s}}(t)=A_{c}e^{j\beta \sin \left(\omega _{m}t\right)}\\&C_{k}={\frac {1}{T_{m}}}\int \limits _{{}^{-T_{m}}\!\!\diagup \!\!{}_{2}\;}^{{}^{T_{m}}\!\!\diagup \!\!{}_{2}\;}{{\tilde {s}}(t)e^{-j\omega _{m}kt}\partial t}\\&J_{n}(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }{e^{j\left(x\sin \lambda -n\lambda \right)}\partial \lambda }={\frac {1}{2\pi }}\int _{-\pi }^{\pi }{e^{jx\sin \lambda }e^{-jn\lambda }\partial \lambda }\\&\left\{{\begin{aligned}&\lambda =\omega _{m}t\leftrightarrow t=\lambda /\omega _{m}={\frac {\lambda T_{m}}{2\pi }}\\&\partial \lambda =\omega _{m}\partial t\leftrightarrow \partial t={\frac {T_{m}}{2\pi }}\partial \lambda \\\end{aligned}}\right\}\\&t={}^{T_{m}}\!\!\diagup \!\!{}_{2}\;={\frac {\lambda T_{m}}{2\pi }}\to \lambda =\pi \\&t=-{}^{T_{m}}\!\!\diagup \!\!{}_{2}\;\to \lambda =-\pi \\&C_{k}={\frac {1}{T_{m}}}\int \limits _{t={}^{-T_{m}}\!\!\diagup \!\!{}_{2}\;}^{t={}^{T_{m}}\!\!\diagup \!\!{}_{2}\;}{A_{c}e^{j\beta \sin \left(\omega _{m}t\right)}e^{-j\omega _{m}kt}\partial t}=A_{c}{\frac {1}{T_{m}}}\int \limits _{-\pi }^{\pi }{e^{j\beta \sin \lambda }e^{-jk\lambda }{\frac {T_{m}}{2\pi }}\partial \lambda }=A_{c}{\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }{e^{j\beta \sin \lambda }e^{-jk\lambda }\partial \lambda }\to \\&C_{k}=A_{c}J_{k}\left(\beta \right)\\\end{aligned}}}
Propiedades de funciones de Bessel [ editar ] 1º Propiedad
J
n
(
x
)
=
(
−
1
)
n
J
−
n
(
x
)
n
par
→
J
n
(
x
)
=
J
−
n
(
x
)
n
impar
→
J
n
(
x
)
=
−
J
−
n
(
x
)
{\displaystyle {\begin{aligned}&J_{n}\left(x\right)=\left(-1\right)^{n}J_{-n}\left(x\right)\\&n{\text{ par }}\to J_{n}\left(x\right)=J_{-n}\left(x\right)\\&n{\text{ impar }}\to J_{n}\left(x\right)=-J_{-n}\left(x\right)\\\end{aligned}}}
2º Propiedad
J
0
(
0
)
=
1
J
n
(
0
)
=
0
,
n
≠
0
,
1
,
−
1
J
1
(
β
)
≃
β
╱
2
J
−
1
(
β
)
≃
−
β
╱
2
{\displaystyle {\begin{aligned}&J_{0}\left(0\right)=1\\&J_{n}\left(0\right)=0,n\neq 0,1,-1\\&J_{1}\left(\beta \right)\simeq {}^{\beta }\!\!\diagup \!\!{}_{2}\;\\&J_{-1}\left(\beta \right)\simeq {}^{-\beta }\!\!\diagup \!\!{}_{2}\;\\\end{aligned}}}
3º Propiedad
En algunos puntos vale cero
4º Propiedad
∑
n
=
−
∞
∞
J
n
2
(
x
)
=
1
{\displaystyle \sum \limits _{n=-\infty }^{\infty }{J_{n}^{2}\left(x\right)}=1}
En ciertos valores la funcion de bessel vale 0, por lo que no transmitimos potencia en esos casos.
Por lo que, ahora tenemos:
s
~
(
t
)
=
∑
k
=
−
∞
∞
C
k
⋅
e
j
ω
m
k
t
=
∑
k
=
−
∞
∞
A
c
J
k
(
β
)
⋅
e
j
ω
m
k
t
s
+
(
t
)
=
s
~
(
t
)
e
j
ω
c
t
=
A
c
∑
k
=
−
∞
∞
J
k
(
β
)
⋅
e
j
(
ω
c
t
+
ω
m
k
t
)
x
P
M
F
M
(
t
)
=
s
(
t
)
=
ℜ
{
s
+
(
t
)
}
=
ℜ
{
A
c
∑
k
=
−
∞
∞
J
k
(
β
)
⋅
e
j
(
ω
c
t
+
ω
m
k
t
)
}
x
P
M
F
M
(
t
)
=
s
(
t
)
=
A
c
∑
k
=
−
∞
∞
J
k
(
β
)
cos
(
ω
c
t
+
k
ω
m
t
)
{\displaystyle {\begin{aligned}&{\tilde {s}}(t)=\sum \limits _{k=-\infty }^{\infty }{C_{k}\cdot e^{j\omega _{m}kt}}=\sum \limits _{k=-\infty }^{\infty }{A_{c}J_{k}\left(\beta \right)\cdot e^{j\omega _{m}kt}}\\&s^{+}(t)={\tilde {s}}(t)e^{j\omega _{c}t}=A_{c}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\cdot e^{j\left(\omega _{c}t+\omega _{m}kt\right)}}\\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=s(t)=\Re \left\{s^{+}(t)\right\}=\Re \left\{A_{c}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\cdot e^{j\left(\omega _{c}t+\omega _{m}kt\right)}}\right\}\\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=s(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\cos \left(\omega _{c}t+k\omega _{m}t\right)}\\\end{aligned}}}
Ahora si podemos sacar la representación en frecuencia de una señal con modulación angular.
x
P
M
F
M
(
t
)
=
s
(
t
)
=
A
c
∑
k
=
−
∞
∞
J
k
(
β
)
cos
(
(
ω
c
+
k
ω
m
)
t
)
=
A
c
[
J
0
(
β
)
cos
(
ω
c
t
)
+
J
1
(
β
)
cos
(
(
ω
c
+
ω
m
)
t
)
+
J
−
1
(
β
)
cos
(
(
ω
c
−
ω
m
)
t
)
+
J
2
(
β
)
cos
(
(
ω
c
+
2
ω
m
)
t
)
]
+
.
.
.
X
P
M
F
M
(
f
)
=
A
c
2
∑
k
=
−
∞
∞
J
k
(
β
)
[
δ
(
f
−
(
f
c
+
k
f
m
)
)
+
δ
(
f
+
(
f
c
+
k
f
m
)
)
]
{\displaystyle {\begin{aligned}&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=s(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\cos \left((\omega _{c}+k\omega _{m})t\right)}=\\&A_{c}\left[J_{0}\left(\beta \right)\cos \left(\omega _{c}t\right)+J_{1}\left(\beta \right)\cos \left((\omega _{c}+\omega _{m})t\right)+J_{-1}\left(\beta \right)\cos \left((\omega _{c}-\omega _{m})t\right)+J_{2}\left(\beta \right)\cos \left((\omega _{c}+2\omega _{m})t\right)\right]+...\\&X_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(f)={\frac {A_{c}}{2}}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\left[\delta \left(f-\left(f_{c}+kf_{m}\right)\right)+\delta \left(f+\left(f_{c}+kf_{m}\right)\right)\right]}\\\end{aligned}}}
Representacion frecuencial de la envolvente compleja de PM/FM (centrada en el origen)
Ancho de banda segun Carson [ editar ] Viendo ahora la representación en frecuencia, podemos preguntarnos, cuanto es el ancho de banda necesario para una señal PM/FM?
Para una señal AM o DSB
β
T
=
2
W
{\displaystyle \beta _{T}=2W}
β
T
=
W
{\displaystyle \beta _{T}=W}
β
T
=
∞
{\displaystyle \beta _{T}=\infty }
x
P
M
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
φ
i
(
t
)
)
P
M
→
φ
i
(
t
)
=
ϕ
Δ
x
(
t
)
F
M
→
f
i
(
t
)
=
f
c
+
f
Δ
x
(
t
)
s
(
t
)
=
x
P
M
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
β
P
M
F
M
sin
(
ω
m
t
)
)
β
P
M
=
ϕ
Δ
A
m
β
F
M
=
A
m
f
Δ
f
m
β
T
c
a
r
s
o
n
=
2
(
β
P
M
F
M
+
1
)
W
;
W
=
f
m
β
T
P
M
=
2
(
ϕ
Δ
A
m
+
1
)
f
m
=
2
ϕ
Δ
A
m
f
m
+
2
f
m
β
T
F
M
=
2
(
A
m
f
Δ
f
m
+
1
)
f
m
=
2
A
m
f
Δ
+
2
f
m
=
2
(
A
m
f
Δ
+
f
m
)
{\displaystyle {\begin{aligned}&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t+\varphi _{i}(t)\right)\\&PM\to \varphi _{i}(t)=\phi _{\Delta }x(t)\\&FM\to f_{i}(t)=f_{c}+f_{\Delta }x(t)\\&s(t)=x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t+\beta _{\begin{smallmatrix}PM\\FM\end{smallmatrix}}\sin \left(\omega _{m}t\right)\right)\\&\beta _{PM}=\phi _{\Delta }A_{m}\\&\beta _{FM}={\frac {A_{m}f_{\Delta }}{f_{m}}}\\&\beta _{T_{carson}}=2\left(\beta _{\begin{smallmatrix}PM\\FM\end{smallmatrix}}+1\right)W;\\&W=f_{m}\\&\beta _{T_{PM}}=2\left(\phi _{\Delta }A_{m}+1\right)f_{m}=2\phi _{\Delta }A_{m}f_{m}+2f_{m}\\&\beta _{T_{FM}}=2\left({\frac {A_{m}f_{\Delta }}{f_{m}}}+1\right)f_{m}=2A_{m}f_{\Delta }+2f_{m}=2\left(A_{m}f_{\Delta }+f_{m}\right)\\\end{aligned}}}
DIBUJO
Relación señal a ruido de una señal PM/FM, detección por envolvente [ editar ] A relación señal a ruido en recepción será, como en todos los casos:
(
S
╱
N
)
R
=
S
R
N
R
=
S
R
η
β
T
=
A
R
2
╱
2
η
2
(
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{\displaystyle \left({}^{S}\!\!\diagup \!\!{}_{N}\;\right)_{R}={\frac {S_{R}}{N_{R}}}={\frac {S_{R}}{\eta \beta _{T}}}={\frac {{}^{A_{R}^{2}}\!\!\diagup \!\!{}_{2}\;}{\eta 2\left(\beta _{\begin{smallmatrix}PM\\FM\end{smallmatrix}}+1\right)W}}}
Para detección, la demostración se detalla en un vinculo aparte dada su complejidad. La relación señal a ruido en detección de las señales con modulación angular serán respectivamente:
Demostracion de SNR en deteccion para PM y FM
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{\displaystyle {\begin{aligned}&\left({}^{S}\!\!\diagup \!\!{}_{N}\;\right)_{D}^{PM}=\phi _{\Delta }^{2}S_{x}\gamma =\phi _{\Delta }^{2}S_{x}{\frac {S_{R}}{\eta W}}\\&\left({}^{S}\!\!\diagup \!\!{}_{N}\;\right)_{D}^{FM}=3\left({\frac {f_{\Delta }}{W}}\right)^{2}S_{x}\gamma =3\left({\frac {f_{\Delta }}{W}}\right)^{2}S_{x}{\frac {S_{R}}{\eta W}}\to \left\{D={\frac {f_{\Delta }}{W}}\right\}\to 3D^{2}S_{x}\gamma \\\end{aligned}}}
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![{\displaystyle {\begin{aligned}&\varphi _{i}\left(t\right)\ll 1\to \left\{{\begin{aligned}&PM\to \phi _{\Delta }x(t)\ll 1\\&FM\to 2\pi f_{\Delta }\int _{-\infty }^{t}{x(\lambda )\partial \lambda \ll 1}\\\end{aligned}}\right.\\&{\underset {x\to 0}{\mathop {\lim } }}\,{\text{ }}\sin x\approx x\\&{\underset {x\to 0}{\mathop {\lim } }}\,{\text{ }}\cos x\approx 1\\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=A_{c}\cos \left(\omega _{c}t+\varphi _{i}\left(t\right)\right)=A_{c}\cos \left(\omega _{c}t\right)\underbrace {\cos \left(\varphi _{i}\left(t\right)\right)} _{\approx 1}-A_{c}\sin \left(\omega _{c}t\right)\underbrace {\sin \left(\varphi _{i}\left(t\right)\right)} _{\approx \varphi _{i}\left(t\right)}\to \\&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)\approx A_{c}\cos \left(\omega _{c}t\right)-A_{c}\sin \left(\omega _{c}t\right)\varphi _{i}\left(t\right)\\&X_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(f)=A_{c}\left({\frac {\delta \left(f-f_{c}\right)+\delta \left(f+f_{c}\right)}{2}}\right)-A_{c}\left({\frac {\varphi \left(f-f_{c}\right)-\varphi \left(f+f_{c}\right)}{2j}}\right)\\&\\&PM\to \phi _{\Delta }x(t){\xrightarrow[{\mathbb {F} }]{}}\phi _{\Delta }X(f)\\&FM\to 2\pi f_{\Delta }\int _{-\infty }^{t}{x(\lambda )\partial \lambda {\xrightarrow[{\mathbb {F} }]{}}}2\pi f_{\Delta }{\frac {X(f)}{j2\pi f}}={\frac {f_{\Delta }X(f)}{j\cdot f}}\\&X_{PM}(f)=A_{c}\left({\frac {\delta \left(f-f_{c}\right)+\delta \left(f+f_{c}\right)}{2}}\right)-{\frac {A_{c}\phi _{\Delta }}{2j}}\left(X\left(f-f_{c}\right)-X\left(f+f_{c}\right)\right)\\&X_{FM}(f)=A_{c}\left({\frac {\delta \left(f-f_{c}\right)+\delta \left(f+f_{c}\right)}{2}}\right)+{\frac {A_{c}f_{\Delta }}{2}}\left({\frac {X(f-f_{c})}{f-f_{c}}}-{\frac {X(f-f_{c})}{f-f_{c}}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f776371f51c92d6f4ef215dc54875fd769f586ce)
![{\displaystyle {\begin{aligned}&x_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(t)=s(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\cos \left((\omega _{c}+k\omega _{m})t\right)}=\\&A_{c}\left[J_{0}\left(\beta \right)\cos \left(\omega _{c}t\right)+J_{1}\left(\beta \right)\cos \left((\omega _{c}+\omega _{m})t\right)+J_{-1}\left(\beta \right)\cos \left((\omega _{c}-\omega _{m})t\right)+J_{2}\left(\beta \right)\cos \left((\omega _{c}+2\omega _{m})t\right)\right]+...\\&X_{\begin{smallmatrix}PM\\FM\end{smallmatrix}}(f)={\frac {A_{c}}{2}}\sum \limits _{k=-\infty }^{\infty }{J_{k}\left(\beta \right)\left[\delta \left(f-\left(f_{c}+kf_{m}\right)\right)+\delta \left(f+\left(f_{c}+kf_{m}\right)\right)\right]}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7caff6560ff77718ae355e933c7eff8cc25080d4)
