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

Relación señal a ruido de una señal PM

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

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

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

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)

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

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

Como se aprecia en el dibujo:

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

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:

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

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

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

Ahora finalmente, en detección:

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

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

Relación señal a ruido de una señal FM

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

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

${\begin{aligned}&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{aligned}}$

Ahora, la relación señal a ruido en detección será:

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

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{aligned}&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{aligned}&x={\frac {f}{B_{de}}}\\&\partial x={\frac {\partial f}{B_{de}}}\\\end{aligned}}\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{aligned}}$

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

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

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

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

Proyecto: Departamento de Teoría de la Señal y Comunicaciones

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