Relación señal a ruido de una señal PM/FM, detección por envolvente

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[editar] Relación señal a ruido de una señal PM

$\begin{align} & s_{R}(t)=x_{\begin{smallmatrix} PM \\ FM \end{smallmatrix}}(t)=A_{R}\cos \left( \omega _{c}t+\varphi _{i}(t) \right);PM\to \varphi _{i}(t)=\phi _{\Delta }x(t) \\ & y_{R}(t)=s_{R}(t)+n_{R}(t) \\ \end{align}$

En el bloque de recepción tenemos un detector de fase, esto es, “extraerá” la fase de la señal de entrada. Por lo tanto:

$y_{PD}(t)=\varphi _{T}(t)=\varphi _{y_{R}}(t)$

Para entenderlo mejor, supongamos que en nuestra señal recibida no hay ruido

$\begin{align} & y_{R}(t)=\underbrace{A_{R}\cos \left( \omega _{c}t+\phi _{\Delta }x(t) \right)}_{s_{R}(t)}+\underbrace{n_{R}(t)}_{0} \\ & y_{PD}(t)=\varphi _{y_{R}}(t)\to y_{PD}(t)=\varphi _{s_{R}}(t) \\ & y_{PD}(t)=\phi _{\Delta }x(t) \\ & G_{D}(f)=G_{Y_{PD}}(f)\cdot \left| H_{LPF}(f) \right|^{2}=\underbrace{G_{Y_{PD}}(f)}_{\phi _{_{\Delta }}^{2}\cdot S_{x}}\cdot \prod{\left( \frac{f}{2W} \right)}=\phi _{_{\Delta }}^{2}S_{x} \\ \end{align}$

Pero ahora, consideráramos el caso donde también tenemos ruido a la entrada del bloque de recepción. Haremos una representación vectorial para apreciarlo mejor. (Para vectores lo mejor es usar la envolvente compleja de la señal en cuestión)

$\begin{align} & s_{R}(t)=x_{\begin{smallmatrix} PM \\ FM \end{smallmatrix}}(t)=A_{R}\cos \left( \omega _{c}t+\varphi _{i}(t) \right) \\ & PM\to \varphi _{i}(t)=\phi _{\Delta }x(t) \\ & y_{R}(t)=s_{R}(t)+n_{R}(t) \\ & y_{PD}(t)=\varphi _{y_{R}}(t)=\varphi _{T}(t)? \\ & x(t)=x_{I}(t)\cos (\omega _{c}t)-x_{Q}(t)\sin (\omega _{c}t)=e(t)\cos (\omega _{c}t+\varphi (t)) \\ & n_{R}(t)=e_{n_{R}}(t)\cos (\omega _{c}t+\varphi _{n_{R}}(t)) \\ & y_{R}(t)=s_{R}(t)+n_{R}(t)=\underbrace{A_{T}(t)}_{=e_{y_{R}}(t)}\cos (\omega _{c}t+\varphi _{T}(t))=\underbrace{A_{R}}_{=e_{s_{R}}(t)}\cos \left( \omega _{c}t+\varphi _{i}(t) \right)+e_{n_{R}}(t)\cos (\omega _{c}t+\varphi _{n_{R}}(t)) \\ & \\ & \tilde{y}_{R}(t)=\tilde{s}_{R}(t)+\tilde{n}_{R}(t)=A_{T}(t)e^{j\varphi _{T}(t)}=A_{R}e^{j\varphi _{i}(t)}+e_{n_{R}}(t)e^{j\varphi _{n_{R}}(t)} \\ \end{align}$

$\varphi _{T}(t)=\varphi _{i}(t)+\varphi _{n_{TOTAL}}(t)$

Como se aprecia en el dibujo:

$\begin{align} & \arctan \left( \varphi _{n_{TOTAL}}(t) \right)=\frac{b(t)}{a(t)} \\ & \measuredangle \varphi _{n_{TOTAL}}(t)=\frac{b(t)}{a(t)}=\frac{b(t)}{A_{R}+c(t)}\to \\ & \cos \left( \varphi _{n_{R}}(t) \right)=\frac{c(t)}{e_{n_{R}}(t)}\to c(t)=e_{n_{R}}(t)\cos \left( \varphi _{n_{R}}(t) \right) \\ & \sin \left( \varphi _{n_{R}}(t) \right)=\frac{b(t)}{e_{n_{R}}(t)}\to b(t)=e_{n_{R}}(t)\sin \left( \varphi _{n_{R}}(t) \right) \\ & \left\{ \begin{align} & \tilde{x}(t)=x_{I}(t)+jx_{Q}(t)=e(t)e^{j\varphi (t)}\to \Re \left\{ \tilde{x}(t) \right\}=x_{I}(t)=\Re \left\{ e(t)e^{j\varphi (t)} \right\}=e(t)\cos (\varphi (t)) \\ & x_{Q}(t)=e(t)\sin (\varphi (t)) \\ \end{align} \right\} \\ & c(t)=e_{n_{R}}(t)\cos \left( \varphi _{n_{R}}(t) \right)=n_{R_{I}}(t) \\ & b(t)=e_{n_{R}}(t)\sin \left( \varphi _{n_{R}}(t) \right)=n_{R_{Q}}(t) \\ & \measuredangle \varphi _{n_{TOTAL}}(t)=\frac{b(t)}{A_{R}+c(t)}=\frac{n_{R_{Q}}(t)}{A_{R}+n_{R_{I}}(t)} \\ \end{align}$

Ahora, suponemos que $A_{R}\gg n_{R_{I}}(t)$ , esto es, que la amplitud de la señal es mas grande que la amplitud del ruido. Esta es una condición necesaria para que el detector funcione, igual que lo que ocurría en la detección AM. Entonces:

$\measuredangle \varphi _{n_{TOTAL}}(t)=\frac{n_{R_{Q}}(t)}{A_{R}+n_{R_{I}}(t)}\simeq \frac{n_{R_{Q}}(t)}{A_{R}}$

$\begin{align} & y_{PD}(t)\to PD:\text{Pre-Detection filter} \\ & y_{PD}(t)=\varphi _{T}(t)=\underbrace{\varphi _{i}(t)}_{\phi _{\Delta }x(t)}+\varphi _{n_{TOTAL}}(t)=\underbrace{\phi _{\Delta }x(t)}_{s_{PD}(t)}+\underbrace{\frac{n_{R_{Q}}(t)}{A_{R}}}_{n_{PD}(t)}\to \\ & G_{x}(f)=\underset{T\to \infty }{\mathop{\lim }}\,\frac{\left| X(f) \right|^{2}}{T} \\ & G_{Y_{PD}}(t)=G_{S_{PD}}(f)+G_{n_{R_{Q}}}(f)=\phi _{\Delta }^{2}G_{x}(f)+\frac{1}{A_{R}^{2}}G_{n_{R_{Q}}}(f) \\ & G_{n}(f)=\frac{\eta }{2} \\ & G_{n_{R}}(f)=\frac{\eta }{2}\left[ \prod{\left( \frac{f-f_{c}}{\beta _{T}} \right)+\prod{\left( \frac{f+f_{c}}{\beta _{T}} \right)}} \right] \\ & G_{n_{R_{Q}}}(f)=\frac{\eta }{2}\cdot 2\cdot \prod{\left( \frac{f}{\beta _{T}} \right)}=\eta \prod{\left( \frac{f}{\beta _{T}} \right)} \\ \end{align}$

(No olvidar el 2 en la DEP del ruido! )

Ahora finalmente, en detección:

$\begin{align} & y_{D}(t)=s_{D}(t)+n_{D}(t) \\ & G_{D}(f)=G_{PD}(f)\cdot \left| H_{LPF}(f) \right|^{2}=G_{PD}(f)\cdot \prod{\left( \frac{f}{2W} \right)} \\ & s_{D}(t)\to G_{S_{D}}(f)=G_{S_{PD}}(f)\cdot \prod{\left( \frac{f}{2W} \right)} \\ & G_{S_{PD}}(f)=\phi _{\Delta }^{2}G_{x}(f)\to G_{S_{D}}(f)=\phi _{\Delta }^{2}\underbrace{G_{x}(f)\prod{\left( \frac{f}{2W} \right)}}_{=G_{x}(f)} \\ & S_{D}=\int_{-\infty }^{\infty }{G_{S_{D}}(f)\partial f}=\phi _{\Delta }^{2}S_{x} \\ & \\ & n_{D}(t)\to G_{n_{D}}(f)=G_{n_{PD}}(f)\cdot \prod{\left( \frac{f}{2W} \right)} \\ & G_{n_{PD}}(f)=\frac{1}{A_{R}^{2}}G_{n_{R_{Q}}}(f)=\frac{1}{A_{R}^{2}}\eta \prod{\left( \frac{f}{\beta _{T}} \right)} \\ & G_{n_{D}}(f)=\frac{1}{A_{R}^{2}}\eta \prod{\left( \frac{f}{\beta _{T}} \right)}\cdot \prod{\left( \frac{f}{2W} \right)}\to \left\{ \beta _{T}>2W \right\}\to G_{n_{D}}(f)=\frac{1}{A_{R}^{2}}\eta \prod{\left( \frac{f}{2W} \right)} \\ & N_{D}=\int_{-\infty }^{\infty }{G_{n_{D}}(f)\partial f}=\frac{1}{A_{R}^{2}}\eta 2W \\ \end{align}$

$\begin{align} & \\ & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}=\frac{S_{D}}{N_{D}}=\frac{\phi _{\Delta }^{2}S_{x}}{\frac{1}{A_{R}^{2}}\eta 2W}=\frac{A_{R}^{2}}{2}\frac{\phi _{\Delta }^{2}S_{x}}{\eta W}\to \left\{ S_{\begin{smallmatrix} PM \\ FM \end{smallmatrix}}=\frac{A_{c}^{2}}{2},S_{R}=\frac{A_{R}^{2}}{2} \right\} \\ & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}=\frac{\phi _{\Delta }^{2}S_{x}S_{R}}{\eta W}\to \left\{ \gamma =\frac{S_{R}}{\eta W} \right\}\to \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}=\phi _{\Delta }^{2}S_{x}\gamma \\ \end{align}$

[editar] Relación señal a ruido de una señal FM

Para FM, en vez de la fase, detectamos la frecuencia:

$\begin{align} & s_{R}(t)=x_{\begin{smallmatrix} PM \\ FM \end{smallmatrix}}(t)=A_{R}\cos \left( \omega _{c}t+\varphi _{i}(t) \right) \\ & FM\to f_{i}(t)=f_{c}+f_{\Delta }x(t)\to \varphi _{i}(t)=2\pi f_{\Delta }\int_{-\infty }^{t}{x(\lambda )\partial \lambda } \\ & y_{R}(t)=s_{R}(t)+n_{R}(t) \\ & y_{PD}(t)=f_{T}(t)=\frac{1}{2\pi }\frac{\partial \varphi _{T}(t)}{\partial t}=\frac{1}{2\pi }\frac{\partial \varphi _{y_{R}}(t)}{\partial t}=? \\ & \\ & \varphi _{T}(t)=\varphi _{i}(t)+\varphi _{n_{TOTAL}}(t) \\ & f_{T}(t)=\frac{1}{2\pi }\frac{\partial \varphi _{T}(t)}{\partial t}=\frac{1}{2\pi }\frac{\partial \varphi _{i}(t)}{\partial t}+\frac{1}{2\pi }\frac{\partial \varphi _{n_{TOTAL}}(t)}{\partial t}= \\ & f_{T}(t)=f_{\Delta }x(t)+\frac{1}{2\pi }\frac{\partial \varphi _{n_{TOTAL}}(t)}{\partial t} \\ & \measuredangle \varphi _{n_{TOTAL}}(t)=\frac{n_{R_{Q}}(t)}{A_{R}+n_{R_{I}}(t)}\simeq \frac{n_{R_{Q}}(t)}{A_{R}} \\ & f_{T}(t)=f_{\Delta }x(t)+\frac{1}{2\pi }\frac{1}{A_{R}}\frac{\partial n_{R_{Q}}(t)}{\partial t} \\ \end{align}$

$\begin{align} & G_{x}(f)=\underset{T\to \infty }{\mathop{\lim }}\,\frac{\left| X(f) \right|^{2}}{T} \\ & \left\{ \mathbb{F}\left[ \frac{\partial f(t)}{\partial t} \right]=\left( j2\pi f \right)F(f) \right\} \\ & N_{PD}(f)=\mathbb{F}\left[ \frac{1}{2\pi }\frac{1}{A_{R}}\frac{\partial n_{R_{Q}}(t)}{\partial t} \right]=\frac{1}{2\pi }\frac{1}{A_{R}}\left( j2\pi f \right)N_{R_{Q}}(f)=\frac{j\cdot f}{A_{R}}N_{R_{Q}}(f) \\ & G_{Y_{PD}}(t)=G_{S_{PD}}(f)+G_{n_{PD}}(f)=f_{\Delta }^{2}G_{x}(f)+\frac{f^{2}}{A_{R}^{2}}G_{n_{R_{Q}}}(f) \\ & G_{n}(f)=\frac{\eta }{2} \\ & G_{n_{R}}(f)=\frac{\eta }{2}\left[ \prod{\left( \frac{f-f_{c}}{\beta _{T}} \right)+\prod{\left( \frac{f+f_{c}}{\beta _{T}} \right)}} \right] \\ & G_{n_{R_{Q}}}(f)=\frac{\eta }{2}\cdot 2\cdot \prod{\left( \frac{f}{\beta _{T}} \right)}=\eta \prod{\left( \frac{f}{\beta _{T}} \right)} \\ & y_{D}(t)=s_{D}(t)+n_{D}(t) \\ & G_{D}(f)=G_{PD}(f)\cdot \left| H_{LPF}(f) \right|^{2}=G_{PD}(f)\cdot \prod{\left( \frac{f}{2W} \right)} \\ & s_{D}(t)\to G_{S_{D}}(f)=G_{S_{PD}}(f)\cdot \prod{\left( \frac{f}{2W} \right)} \\ & G_{S_{PD}}(f)=f_{\Delta }^{2}G_{x}(f)\to G_{S_{D}}(f)=f_{\Delta }^{2}\underbrace{G_{x}(f)\prod{\left( \frac{f}{2W} \right)}}_{=G_{x}(f)} \\ & S_{D}=\int_{-\infty }^{\infty }{G_{S_{D}}(f)\partial f}=f_{\Delta }^{2}S_{x} \\ & \\ & n_{D}(t)\to G_{n_{D}}(f)=G_{n_{PD}}(f)\cdot \prod{\left( \frac{f}{2W} \right)} \\ & G_{n_{PD}}(f)=\frac{f^{2}}{A_{R}^{2}}G_{n_{R_{Q}}}(f)=\frac{f^{2}}{A_{R}^{2}}\eta \prod{\left( \frac{f}{\beta _{T}} \right)} \\ & G_{n_{D}}(f)=\frac{f^{2}}{A_{R}^{2}}\eta \prod{\left( \frac{f}{\beta _{T}} \right)}\cdot \prod{\left( \frac{f}{2W} \right)}\to \left\{ \beta _{T}>2W \right\}\to G_{n_{D}}(f)=\frac{f^{2}}{A_{R}^{2}}\eta \prod{\left( \frac{f}{2W} \right)} \\ & N_{D}=\int_{-\infty }^{\infty }{G_{n_{D}}(f)\partial f}=\int_{-W}^{W}{\frac{f^{2}}{A_{R}^{2}}\eta \partial f}=\frac{\eta }{A_{R}^{2}}\int_{-W}^{W}{f^{2}\partial f}=\frac{\eta }{A_{R}^{2}}\left. \frac{f^{3}}{3} \right|_{-W}^{W}= \\ & N_{D}=\frac{\eta }{A_{R}^{2}}\frac{2W^{3}}{3} \\ \end{align}$

Ahora, la relación señal a ruido en detección será:

$\begin{align} & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}=\frac{S_{D}}{N_{D}}=\frac{f_{\Delta }^{2}S_{x}}{\frac{\eta }{A_{R}^{2}}\frac{2W^{3}}{3}}=\frac{A_{R}^{2}}{2}\frac{3f_{\Delta }^{2}S_{x}}{\eta W^{3}}\to \left\{ S_{\begin{smallmatrix} PM \\ FM \end{smallmatrix}}=\frac{A_{c}^{2}}{2},S_{R}=\frac{A_{R}^{2}}{2} \right\} \\ & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}=S_{R}\frac{3f_{\Delta }^{2}S_{x}}{\eta W^{3}}=\frac{S_{R}}{\eta W}\frac{3f_{\Delta }^{2}S_{x}}{W^{2}}=3\left( \frac{f_{\Delta }}{W} \right)^{2}S_{x}\frac{S_{R}}{\eta W}\to \left\{ \gamma =\frac{S_{R}}{\eta W} \right\}\to \\ & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}=3\left( \frac{f_{\Delta }}{W} \right)^{2}S_{x}\gamma \\ \end{align}$

[editar] Filtro Pre-emphasis y De-emphasis

Para el caso de demodulación FM, En vez de usar un filtro paso-bajo ideal, ahora usaremos un:

Filtro De-emphasis:

$\begin{align} & H_{de}(f)=\frac{1}{1+\frac{j2\pi f}{f_{1}}}=\frac{1}{1+\frac{j\cdot f}{B_{de}}} \\ & f_{1}=\frac{1}{2\pi RC}\to B_{de}=\frac{1}{RC} \\ & \left| H_{de}(f) \right|^{2}=\frac{1}{1+\left( \frac{f}{B_{de}} \right)^{2}} \\ & \\ & S_{D}=\int_{-\infty }^{\infty }{G_{S_{D}}(f)\partial f}=f_{\Delta }^{2}S_{x} \\ & G_{n_{PD}}(f)=\frac{f^{2}}{A_{R}^{2}}G_{n_{R_{Q}}}(f)=\frac{f^{2}}{A_{R}^{2}}\eta \prod{\left( \frac{f}{\beta _{T}} \right)} \\ & n_{D}(t)\to G_{n_{D}}(f)=G_{n_{PD}}(f)\cdot \left| H_{LPF}(f) \right|^{2}\left| H_{DE}(f) \right|^{2}=\frac{f^{2}}{A_{R}^{2}}\eta \prod{\left( \frac{f}{\beta _{T}} \right)}\cdot \prod{\left( \frac{f}{2W} \right)}\left| H_{LPF}(f) \right|^{2}\to \left\{ \beta _{T}>2W \right\}\to \\ & G_{n_{D}}(f)=\frac{f^{2}}{A_{R}^{2}}\eta \prod{\left( \frac{f}{2W} \right)}\left| H_{DE}(f) \right|^{2}=\frac{f^{2}}{A_{R}^{2}}\eta \prod{\left( \frac{f}{2W} \right)}\frac{1}{1+\left( \frac{f}{B_{de}} \right)^{2}} \\ & N_{D}=\int_{-\infty }^{\infty }{G_{n_{D}}(f)\partial f}=\int_{-W}^{W}{\frac{f^{2}}{A_{R}^{2}}\eta \frac{1}{1+\left( \frac{f}{B_{de}} \right)^{2}}\partial f}=\frac{\eta }{A_{R}^{2}}\int_{-W}^{W}{\frac{f^{2}}{1+\left( \frac{f}{B_{de}} \right)^{2}}\partial f}\to \\ & \left\{ \int{\frac{1}{1+x^{2}}\partial x=\arctan x\to \int{1-\frac{1}{1+x^{2}}\partial x=\int{\frac{x^{2}}{1+x^{2}}\partial x=}x-\arctan x}} \right\}\to \left\{ \begin{align} & x=\frac{f}{B_{de}} \\ & \partial x=\frac{\partial f}{B_{de}} \\ \end{align} \right\}\to \\ & \int{\frac{x^{2}}{1+x^{2}}\partial x=}x-\arctan x=\int{\frac{\left( \frac{f}{B_{de}} \right)^{2}}{1+\left( \frac{f}{B_{de}} \right)^{2}}\frac{1}{B_{de}}\partial f=}\left( \frac{f}{B_{de}} \right)-\arctan \left( \frac{f}{B_{de}} \right) \\ & N_{D}=\frac{\eta }{A_{R}^{2}}\int_{-W}^{W}{\frac{f^{2}}{1+\left( \frac{f}{B_{de}} \right)^{2}}\partial f}=\frac{\eta }{A_{R}^{2}}\frac{\left( B_{de} \right)^{2}}{\left( B_{de} \right)^{2}}\frac{B_{de}}{B_{de}}\int_{-W}^{W}{\frac{f^{2}}{1+\left( \frac{f}{B_{de}} \right)^{2}}\partial f}=\frac{\eta }{A_{R}^{2}}\left( B_{de} \right)^{3}\int_{-W}^{W}{\frac{\left( \frac{f}{B_{de}} \right)^{2}}{1+\left( \frac{f}{B_{de}} \right)^{2}}\frac{1}{B_{de}}\partial f}= \\ & N_{D}=\frac{\eta }{A_{R}^{2}}B_{de}^{3}\left[ \left( \frac{f}{B_{de}} \right)-\underbrace{\arctan \left( \frac{f}{B_{de}} \right)}_{\approx \pi \ll {}^{W}\!\!\diagup\!\!{}_{B_{de}}\;} \right]_{-W}^{W}=\frac{\eta }{A_{R}^{2}}B_{de}^{3}\left( \frac{2W}{B_{de}} \right)=\frac{\eta }{A_{R}^{2}}B_{de}^{2}2W \\ \end{align}$

$\begin{align} & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}=? \\ & S_{D}=f_{\Delta }^{2}S_{x} \\ & N_{D}=\frac{\eta }{A_{R}^{2}}B_{de}^{2}2W \\ & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}=\frac{S_{D}}{N_{D}}=\frac{f_{\Delta }^{2}S_{x}}{\frac{\eta }{A_{R}^{2}}B_{de}^{2}2W}=\left( \frac{f_{\Delta }}{B_{de}} \right)^{2}S_{x}\frac{A_{R}^{2}}{2}\frac{1}{\eta W}=\left( \frac{f_{\Delta }}{B_{de}} \right)^{2}S_{x}\frac{S_{R}}{\eta W}=\left( \frac{f_{\Delta }}{B_{de}} \right)^{2}S_{x}\gamma \\ \end{align}$

Si ahora comparamos la relación señal a ruido en detección en FM con filtro deemphasis y si deemphasis:

$\begin{align} & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}^{FM}=3\left( \frac{f_{\Delta }}{W} \right)^{2}S_{x}\gamma \\ & \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}^{FM_{de}}=\left( \frac{f_{\Delta }}{B_{de}} \right)^{2}S_{x}\gamma \\ & \frac{\left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}^{FM_{de}}}{\left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{D}^{FM}}=\frac{\left( \frac{f_{\Delta }}{B_{de}} \right)^{2}S_{x}\gamma }{3\left( \frac{f_{\Delta }}{W} \right)^{2}S_{x}\gamma }=\frac{1}{3}\left( \frac{W}{B_{de}} \right)^{2}\Rightarrow \\ & >1?\to W>\sqrt{3}B_{de} \\ \end{align}$

Tiene sentido porque si nuestro filtro deemphasis es mas restrictivo (mas pequeño), menor será el ruido que deje pasar con lo que aumentará la relacion señal a ruido

Proyecto: Departamento de Teoría de la Señal y Comunicaciones

Anterior: Modulaciones angulares — Relación señal a ruido de una señal PM/FM, detección por envolvente — Siguiente: Resumen y tablas

Relación señal a ruido de una señal PM/FM, detección por envolvente

De Wikiversidad

[editar] Relación señal a ruido de una señal PM

[editar] Relación señal a ruido de una señal FM

[editar] Filtro Pre-emphasis y De-emphasis

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