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
(
t
)
=
A
c
cos
(
ω
c
t
)
{\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
F
M
P
M
(
t
)
=
A
c
cos
(
θ
i
(
t
)
)
θ
i
(
t
)
=
ω
c
t
+
φ
i
(
t
)
{\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
(
t
)
{\displaystyle \theta _{i}\left(t\right)}
φ
i
(
t
)
{\displaystyle \varphi _{i}(t)}
Como se relacionan la frecuencia y la fase de una señal?
La frecuencia es la derivada de la fase
ω
i
(
t
)
=
∂
θ
i
(
t
)
∂
t
f
i
(
t
)
=
1
2
π
∂
θ
i
(
t
)
∂
t
→
{
θ
i
(
t
)
=
ω
c
t
+
φ
i
(
t
)
}
→
f
i
(
t
)
=
1
2
π
∂
{
ω
c
t
+
φ
i
(
t
)
}
∂
t
=
f
c
+
1
2
π
∂
φ
i
(
t
)
∂
t
{\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
(
t
)
{\displaystyle \omega _{i}\left(t\right)}
f
i
(
t
)
{\displaystyle f_{i}\left(t\right)}
Igualmente, podemos decir que la fase, es la integral de la frecuencia
ω
i
(
t
)
=
∂
θ
i
(
t
)
∂
t
⇔
θ
i
(
t
)
=
∫
−
∞
t
ω
i
(
λ
)
∂
λ
f
i
(
t
)
=
1
2
π
∂
θ
i
(
t
)
∂
t
⇔
θ
i
(
t
)
=
2
π
∫
−
∞
t
f
i
(
λ
)
∂
λ
{\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
=
A
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
(
t
)
=
A
c
cos
(
θ
i
(
t
)
)
=
A
c
cos
(
ω
c
t
+
φ
i
(
t
)
)
PM
→
φ
i
(
t
)
=
ϕ
Δ
x
(
t
)
→
x
P
M
(
t
)
=
A
c
cos
(
ω
c
t
+
ϕ
Δ
x
(
t
)
)
FM
f
i
(
t
)
=
f
c
+
f
Δ
x
(
t
)
→
f
i
(
t
)
=
1
2
π
⋅
∂
θ
i
(
t
)
∂
t
=
f
c
+
1
2
π
∂
φ
i
(
t
)
∂
t
f
Δ
x
(
t
)
=
1
2
π
∂
φ
i
(
t
)
∂
t
→
φ
i
(
t
)
=
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
→
x
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
)
{\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.
φ
i
(
t
)
=
ϕ
Δ
x
(
t
)
→
x
P
M
(
t
)
=
A
c
cos
(
ω
c
t
+
ϕ
Δ
x
(
t
)
)
ϕ
Δ
x
min
≤
φ
i
(
t
)
≤
ϕ
Δ
x
max
E
j
:
x
(
t
)
=
A
m
cos
(
ω
m
t
)
φ
i
max
(
t
)
=
ϕ
Δ
A
m
|
x
(
t
)
|
≤
1
→
φ
i
max
(
t
)
=
ϕ
Δ
φ
i
(
t
)
≤
π
→
ϕ
Δ
≤
π
{\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
)
=
cos
(
3
π
)
=
cos
(
5
π
)
=
cos
(
7
π
)
{\displaystyle \cos(\pi )=\cos(3\pi )=\cos(5\pi )=\cos(7\pi )}
Características FM
editar
Ejemplo con moduladora sinusoidal:
f
i
(
t
)
=
f
c
+
f
Δ
x
(
t
)
→
φ
i
(
t
)
=
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
→
x
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
)
f
d
≤
f
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
(
t
)
=
f
c
+
f
Δ
x
(
t
)
E
j
:
x
(
t
)
=
A
m
cos
(
ω
m
t
)
Δ
f
max
=
f
Δ
A
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}}}
x
P
M
(
t
)
=
A
c
cos
(
ω
c
t
+
ϕ
Δ
x
(
t
)
)
x
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
)
x
(
t
)
=
A
c
cos
(
ω
c
t
+
φ
i
(
t
)
)
→
{
cos
(
a
+
b
)
=
cos
a
cos
b
−
sin
a
sin
b
}
→
A
c
cos
(
ω
c
t
+
φ
i
(
t
)
)
=
A
c
cos
(
ω
c
t
)
cos
(
φ
i
(
t
)
)
−
A
c
sin
(
ω
c
t
)
sin
(
φ
i
(
t
)
)
x
(
t
)
=
x
I
(
t
)
cos
(
ω
c
t
)
−
x
Q
(
t
)
sin
(
ω
c
t
)
=
e
(
t
)
cos
(
ω
c
t
+
φ
(
t
)
)
→
x
I
(
t
)
=
A
c
cos
(
φ
i
(
t
)
)
=
{
PM
→
A
c
cos
(
ϕ
Δ
x
(
t
)
)
FM
→
A
c
cos
(
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
)
x
Q
(
t
)
=
A
c
sin
(
φ
i
(
t
)
)
=
{
PM
→
A
c
sin
(
ϕ
Δ
x
(
t
)
)
FM
→
A
c
sin
(
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
)
e
(
t
)
=
A
c
{\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
(
t
)
=
A
c
cos
(
ω
c
t
+
φ
i
(
t
)
)
A
c
cos
(
ω
c
t
+
φ
i
(
t
)
)
=
A
c
cos
(
ω
c
t
)
cos
(
φ
i
(
t
)
)
−
A
c
sin
(
ω
c
t
)
sin
(
φ
i
(
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)\\&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
φ
i
(
t
)
≪
1
→
{
P
M
→
ϕ
Δ
x
(
t
)
≪
1
F
M
→
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
≪
1
lim
x
→
0
sin
x
≈
x
lim
x
→
0
cos
x
≈
1
x
P
M
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
φ
i
(
t
)
)
=
A
c
cos
(
ω
c
t
)
cos
(
φ
i
(
t
)
)
⏟
≈
1
−
A
c
sin
(
ω
c
t
)
sin
(
φ
i
(
t
)
)
⏟
≈
φ
i
(
t
)
→
x
P
M
F
M
(
t
)
≈
A
c
cos
(
ω
c
t
)
−
A
c
sin
(
ω
c
t
)
φ
i
(
t
)
X
P
M
F
M
(
f
)
=
A
c
(
δ
(
f
−
f
c
)
+
δ
(
f
+
f
c
)
2
)
−
A
c
(
φ
(
f
−
f
c
)
−
φ
(
f
+
f
c
)
2
j
)
P
M
→
ϕ
Δ
x
(
t
)
→
F
ϕ
Δ
X
(
f
)
F
M
→
2
π
f
Δ
∫
−
∞
t
x
(
λ
)
∂
λ
→
F
2
π
f
Δ
X
(
f
)
j
2
π
f
=
f
Δ
X
(
f
)
j
⋅
f
X
P
M
(
f
)
=
A
c
(
δ
(
f
−
f
c
)
+
δ
(
f
+
f
c
)
2
)
−
A
c
ϕ
Δ
2
j
(
X
(
f
−
f
c
)
−
X
(
f
+
f
c
)
)
X
F
M
(
f
)
=
A
c
(
δ
(
f
−
f
c
)
+
δ
(
f
+
f
c
)
2
)
+
A
c
f
Δ
2
(
X
(
f
−
f
c
)
f
−
f
c
−
X
(
f
−
f
c
)
f
−
f
c
)
{\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
(
t
)
≈
A
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
~
P
M
F
M
(
t
)
=
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
M
F
M
(
f
)
=
A
c
2
δ
(
f
−
f
c
)
−
A
c
β
4
(
δ
(
f
−
(
f
c
−
f
m
)
)
−
δ
(
f
−
(
f
c
+
f
m
)
)
)
;
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
(
t
)
=
A
c
cos
(
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
)
x
P
M
(
t
)
=
A
c
cos
(
ω
c
t
+
ϕ
Δ
x
(
t
)
)
=
A
c
cos
(
ω
c
t
+
ϕ
Δ
A
m
sin
(
ω
m
t
)
)
x
F
M
(
t
)
=
A
c
cos
(
ω
c
t
+
2
π
f
Δ
∫
−
∞
t
x
(
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)
