Ver el código de «Detección de señales digitales con ruido»

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Desigualdad de Schwarz:

<math>\begin{align}
  & \left| \int{f_{1}\left( x \right)\cdot f_{2}\left( x \right)\partial x} \right|^{2}\le \int{\left| f_{1}\left( x \right) \right|^{2}\partial x}\cdot \int{\left| f_{2}\left( x \right) \right|^{2}\partial x} \\ 
 & Max\to f_{1}\left( x \right)=k\cdot f_{2}^{*}\left( x \right) \\ 
\end{align}</math>

<math>\begin{align}
  & y_{R}(t)=s_{R}(t)+n_{R}(t) \\ 
 & G_{n_{R}}(f)=\frac{\eta }{2}\cdot \left| H_{R}(f) \right|^{2} \\ 
\end{align}</math>

<math>\begin{align}
  & s_{R}(t)=\left\{ \begin{align}
  & '1'\to s_{'1'}(t) \\ 
 & '0'\to s_{'0'}(t) \\ 
\end{align} \right. \\ 
 & \left. \left( {}^{S}\!\!\diagup\!\!{}_{N}\; \right)_{R} \right|_{T_{s}}=\left. \frac{S_{R}}{N_{R}} \right|_{T_{S}}=\frac{\left| s_{R}\left( t=T_{S} \right) \right|^{2}}{N_{R}}\to \text{puede que no sea relacion se }\!\!\tilde{\mathrm{n}}\!\!\text{ al a ruido} \\ 
\end{align}</math>

<math>\begin{align}
  & S_{R}(f)=S_{T}(f)\cdot H_{R}(f) \\ 
 & s_{R}(t)=\int_{-\infty }^{\infty }{S_{R}(f)e^{+j2\pi ft}\partial f}=\int_{-\infty }^{\infty }{S_{T}(f)H_{R}(f)e^{+j2\pi ft}\partial f} \\ 
 & \frac{\left| s_{R}\left( t=T_{S} \right) \right|^{2}}{N_{R}}=\frac{\left| \int_{-\infty }^{\infty }{S_{T}(f)H_{R}(f)e^{+j2\pi ft}\partial f} \right|_{t=T_{s}}^{2}}{N_{R}}=\frac{\left| \int_{-\infty }^{\infty }{S_{T}(f)H_{R}(f)e^{+j2\pi fT_{s}}\partial f} \right|^{2}}{\frac{\eta }{2}\int_{-\infty }^{\infty }{\left| H_{R}(f) \right|^{2}\partial f}}\to  \\ 
 & \left| \int{f_{1}\left( x \right)\cdot f_{2}\left( x \right)\partial x} \right|^{2}\le \int{\left| f_{1}\left( x \right) \right|^{2}\partial x}\int{\left| f_{2}\left( x \right) \right|^{2}\partial x}\to f_{1}\left( x \right)=H_{R}(f);f_{2}\left( x \right)=S_{T}(f)e^{+j2\pi fT_{s}} \\ 
 & Max\to f_{1}\left( x \right)=k\cdot f_{2}^{*}\left( x \right) \\ 
 & \frac{\left| s_{R}\left( t=T_{S} \right) \right|^{2}}{N_{R}}\Rightarrow \frac{\left| \int{f_{1}\left( x \right)\cdot f_{2}\left( x \right)\partial x} \right|^{2}}{\frac{\eta }{2}\cdot \int{\left| f_{2}\left( x \right) \right|^{2}\partial f}}\le \frac{\int{\left| f_{1}\left( x \right) \right|^{2}\partial x}\int{\left| f_{2}\left( x \right) \right|^{2}\partial x}}{\frac{\eta }{2}\cdot \int{\left| f_{2}\left( x \right) \right|^{2}\partial f}}\to  \\ 
 & \frac{\left| \int_{-\infty }^{\infty }{S_{T}(f)H_{R}(f)e^{+j2\pi fT_{s}}\partial f} \right|^{2}}{\frac{\eta }{2}\int_{-\infty }^{\infty }{\left| H_{R}(f) \right|^{2}\partial f}}\le \frac{\int_{-\infty }^{\infty }{\left| H_{R}(f) \right|^{2}\partial f}\cdot \int_{-\infty }^{\infty }{\left| S_{T}(f)e^{+j2\pi fT_{s}} \right|^{2}\partial f}}{\frac{\eta }{2}\int_{-\infty }^{\infty }{\left| H_{R}(f) \right|^{2}\partial f}}=\frac{\int_{-\infty }^{\infty }{\left| S_{T}(f)e^{+j2\pi fT_{s}} \right|^{2}\partial f}}{\frac{\eta }{2}} \\ 
 & \left. \frac{\left| s_{R}\left( t=T_{S} \right) \right|^{2}}{N_{R}} \right|_{\max }=\frac{\int_{-\infty }^{\infty }{\left| S_{T}(f)e^{+j2\pi fT_{s}} \right|^{2}\partial f}}{\frac{\eta }{2}} \\ 
\end{align}</math>


<math>\begin{align}
  & f_{1}\left( x \right)=H_{R}(f);f_{2}\left( x \right)=S_{T}(f)e^{+j2\pi fT_{s}} \\ 
 & Max\to f_{1}\left( x \right)=k\cdot f_{2}^{*}\left( x \right)\to H_{R}(f)=k\cdot S_{T}^{*}(f)e^{-j2\pi fT_{s}} \\ 
 & \mathbb{F}^{-1}\left[ H_{R}(f) \right]=\mathbb{F}^{-1}\left[ S_{T}^{*}(f)e^{-j2\pi fT_{s}} \right]\to h_{R}(t)=\mathbb{F}^{-1}\left[ S_{T}^{*}(f)e^{-j2\pi fT_{s}} \right] \\ 
 & \left\{ \begin{align}
  & \mathbb{F}\left[ x^{*}(t) \right]=X^{*}(-f) \\ 
 & \mathbb{F}\left[ x(t-t_{0}) \right]=X(f)e^{-j2\pi ft_{0}} \\ 
\end{align} \right\} \\ 
 & G(f)=S_{T}^{*}(f)\to \mathbb{F}^{-1}\left[ \overbrace{S_{T}^{*}(f)}^{=G(f)}e^{-j2\pi fT_{s}} \right]=g\left( t-T_{s} \right)\to  \\ 
 & g(t)=s_{T}^{*}(-t)\to g\left( t-T_{s} \right)=s_{T}^{*}\left( -\left( t-T_{s} \right) \right)=s_{T}^{*}\left( T_{s}-t \right) \\ 
 & Max\to f_{1}\left( x \right)=k\cdot f_{2}^{*}\left( x \right)\to h_{R}(t)=s_{T}^{*}\left( T_{s}-t \right) \\ 
\end{align}</math>

<math>\begin{align}
  & y_{R}(t)=\left[ s_{T}(t)+n(t) \right]*h_{R}(t)=\int_{-\infty }^{\infty }{\left[ s_{T}(\tau )+n(\tau ) \right]h_{R}(t-\tau )\partial \tau }= \\ 
 & h_{R}(t)=k\cdot p^{*}\left( T_{s}-t \right)\to  \\ 
 & \int_{-\infty }^{\infty }{\left[ s_{T}(\tau )+n(\tau ) \right]kp^{*}\left( T_{s}-\left( t-\tau  \right) \right)\partial \tau }\to  \\ 
 & \left. \int_{-\infty }^{\infty }{\left[ s_{T}(\tau )+n(\tau ) \right]kp^{*}\left( T_{s}-\left( t-\tau  \right) \right)\partial \tau } \right|_{t=T_{s}}=\int_{0}^{T_{s}}{\left[ s_{T}(\tau )+n(\tau ) \right]kp^{*}\left( \tau  \right)\partial \tau }=\int_{0}^{T_{s}}{\left[ s_{T}(t)+n(t) \right]kp^{*}\left( t \right)\partial t} \\ 
\end{align}</math>

== Filtro adaptado ==

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Con filtro adaptado:  <math>h_{R}(t)=p^{*}\left( T_{s}-t \right)</math>

[[Image:DeteccionDeSeñalDigitalFiltroAdaptado.png|left]]
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<math>\begin{align}
  & y_{PR}(t)=\int_{0}^{T_{s}}{\left[ s_{T}(t)+n(t) \right]kp^{*}\left( t \right)\partial t}=\int_{0}^{T_{s}}{s_{T}(t)kp^{*}\left( t \right)\partial t}+\int_{0}^{T_{s}}{n(t)kp^{*}\left( t \right)\partial t} \\ 
 & m_{y_{PR}(t)}=E\left\{ y_{PR}(t) \right\}=\underbrace{E\left\{ \int_{0}^{T_{s}}{s_{T}(t)kp^{*}\left( t \right)\partial t} \right\}}_{\text{Deterministico}}+\underbrace{E\left\{ \int_{0}^{T_{s}}{n(t)kp^{*}\left( t \right)\partial t} \right\}}_{\text{Aleatorio}\text{,}E\left\{ n(t) \right\}=0}=\int_{0}^{T_{s}}{s_{T}(t)kp^{*}\left( t \right)\partial t} \\ 
 & \sigma _{y_{PR}(t)}^{2}=E\left\{ y_{PR}^{2}(t) \right\}-E^{2}\left\{ y_{PR}(t) \right\}\to  \\ 
 & E\left\{ y_{PR}^{2}(t) \right\}=E\left\{ \left[ \int_{0}^{T_{s}}{\left[ s_{T}(t)+n(t) \right]kp^{*}\left( t \right)\partial t} \right]^{2} \right\}= \\ 
 & E\left\{ \left[ \int_{0}^{T_{s}}{s_{T}(t)kp^{*}\left( t \right)\partial t} \right]^{2} \right\}+E\left\{ \left[ \int_{0}^{T_{s}}{n(t)kp^{*}\left( t \right)\partial t} \right]^{2} \right\}+2\cdot E\left\{ \int_{0}^{T_{s}}{s_{T}(t)kp^{*}\left( t \right)\partial t}\cdot \underbrace{\int_{0}^{T_{s}}{n(t)kp^{*}\left( t \right)\partial t}}_{=0} \right\} \\ 
 & E^{2}\left\{ y_{PR}(t) \right\}=E\left\{ \left[ \int_{0}^{T_{s}}{s_{T}(t)kp^{*}\left( t \right)\partial t} \right]^{2} \right\} \\ 
 & \sigma _{y_{PR}(t)}^{2}=E\left\{ y_{PR}^{2}(t) \right\}-E^{2}\left\{ y_{PR}(t) \right\}=E\left\{ \left[ \int_{0}^{T_{s}}{n(t)kp^{*}\left( t \right)\partial t} \right]^{2} \right\} \\ 
 & E\left\{ \int_{0}^{T_{s}}{n(t)kp^{*}\left( t \right)\partial t}\cdot \int_{0}^{T_{s}}{n(\lambda )kp^{*}\left( \lambda  \right)\partial \lambda } \right\}=E\left\{ \int_{0}^{T_{s}}{\int_{0}^{T_{s}}{n(t)n(\lambda )\underbrace{k^{2}p^{*}\left( t \right)p^{*}\left( \lambda  \right)}_{\text{Determinista}}\partial t}\partial \lambda } \right\}= \\ 
 & \int_{0}^{T_{s}}{\int_{0}^{T_{s}}{\underbrace{E\left\{ n(t)n(\lambda ) \right\}}_{R_{n}\left( t-\lambda  \right)=\frac{\eta }{2}\delta \left( t-\lambda  \right)}k^{2}p^{*}\left( t \right)p^{*}\left( \lambda  \right)\partial t}\partial \lambda }=\int_{0}^{T_{s}}{\int_{0}^{T_{s}}{\frac{\eta }{2}\delta \left( t-\lambda  \right)k^{2}p^{*}\left( t \right)p^{*}\left( \lambda  \right)\partial t}\partial \lambda }= \\ 
 & \int_{0}^{T_{s}}{\frac{\eta }{2}\underbrace{\int_{0}^{T_{s}}{\delta \left( t-\lambda  \right)p^{*}\left( \lambda  \right)\partial \lambda }}_{=p^{*}\left( t \right)*\delta (t)=p^{*}\left( t \right)}k^{2}p^{*}\left( t \right)\partial t}=\frac{\eta }{2}\underbrace{k^{2}\int_{0}^{T_{s}}{p^{2}\left( t \right)\partial t}}_{E_{p(t)}} \\ 
 & \sigma _{y_{PR}(t)}^{2}=\frac{\eta }{2}\cdot E_{p(t)} \\ 
\end{align}</math>

Por lo que tenemos: 

<math>\begin{align}
  & m_{y_{PR}(t)}=\int_{0}^{T_{s}}{s_{T}(t)kp^{*}\left( t \right)\partial t}\to s_{T}(t)=a_{k}\cdot p(t),a_{k}='0','1',... \\ 
 & \sigma _{y_{PR}(t)}^{2}=\frac{\eta }{2}\cdot E_{p(t)} \\ 
\end{align}</math>

{{navegación|[[Departamento de Teoría de la Señal y Comunicaciones]]|[[Modulaciones digitales]]|[[Probabilidad de error de una señal aleatoria gaussiana]]}}

[[Categoría:Electrónica]]