# Detección de señales digitales con ruido

Desigualdad de Schwarz:
${\begin{aligned}&\left|\int {f_{1}\left(x\right)\cdot f_{2}\left(x\right)\partial x}\right|^{2}\leq \int {\left|f_{1}\left(x\right)\right|^{2}\partial x}\cdot \int {\left|f_{2}\left(x\right)\right|^{2}\partial x}\\&Max\to f_{1}\left(x\right)=k\cdot f_{2}^{*}\left(x\right)\\\end{aligned}}$
${\begin{aligned}&y_{R}(t)=s_{R}(t)+n_{R}(t)\\&G_{n_{R}}(f)={\frac {\eta }{2}}\cdot \left|H_{R}(f)\right|^{2}\\\end{aligned}}$
${\begin{aligned}&s_{R}(t)=\left\{{\begin{aligned}&'1'\to s_{'1'}(t)\\&'0'\to s_{'0'}(t)\\\end{aligned}}\right.\\&\left.\left(^{S}\!\!\diagup \!\!_{N}\;\right)_{R}\right|_{T_{s}}=\left.{\frac {S_{R}}{N_{R}}}\right|_{T_{S}}={\frac {\left|s_{R}\left(t=T_{S}\right)\right|^{2}}{N_{R}}}\to {\text{puede que no sea relacion se }}\!\!{\tilde {\mathrm {n} }}\!\!{\text{ al a ruido}}\\\end{aligned}}$
${\begin{aligned}&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}\leq \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}}}\leq {\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}}}\leq {\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{aligned}}$
${\begin{aligned}&f_{1}\left(x\right)=H_{R}(f);f_{2}\left(x\right)=S_{T}(f)e^{+j2\pi fT_{s}}\\&Max\to f_{1}\left(x\right)=k\cdot f_{2}^{*}\left(x\right)\to H_{R}(f)=k\cdot S_{T}^{*}(f)e^{-j2\pi fT_{s}}\\&\mathbb {F} ^{-1}\left[H_{R}(f)\right]=\mathbb {F} ^{-1}\left[S_{T}^{*}(f)e^{-j2\pi fT_{s}}\right]\to h_{R}(t)=\mathbb {F} ^{-1}\left[S_{T}^{*}(f)e^{-j2\pi fT_{s}}\right]\\&\left\{{\begin{aligned}&\mathbb {F} \left[x^{*}(t)\right]=X^{*}(-f)\\&\mathbb {F} \left[x(t-t_{0})\right]=X(f)e^{-j2\pi ft_{0}}\\\end{aligned}}\right\}\\&G(f)=S_{T}^{*}(f)\to \mathbb {F} ^{-1}\left[\overbrace {S_{T}^{*}(f)} ^{=G(f)}e^{-j2\pi fT_{s}}\right]=g\left(t-T_{s}\right)\to \\&g(t)=s_{T}^{*}(-t)\to g\left(t-T_{s}\right)=s_{T}^{*}\left(-\left(t-T_{s}\right)\right)=s_{T}^{*}\left(T_{s}-t\right)\\&Max\to f_{1}\left(x\right)=k\cdot f_{2}^{*}\left(x\right)\to h_{R}(t)=s_{T}^{*}\left(T_{s}-t\right)\\\end{aligned}}$
${\begin{aligned}&y_{R}(t)=\left[s_{T}(t)+n(t)\right]*h_{R}(t)=\int _{-\infty }^{\infty }{\left[s_{T}(\tau )+n(\tau )\right]h_{R}(t-\tau )\partial \tau }=\\&h_{R}(t)=k\cdot p^{*}\left(T_{s}-t\right)\to \\&\int _{-\infty }^{\infty }{\left[s_{T}(\tau )+n(\tau )\right]kp^{*}\left(T_{s}-\left(t-\tau \right)\right)\partial \tau }\to \\&\left.\int _{-\infty }^{\infty }{\left[s_{T}(\tau )+n(\tau )\right]kp^{*}\left(T_{s}-\left(t-\tau \right)\right)\partial \tau }\right|_{t=T_{s}}=\int _{0}^{T_{s}}{\left[s_{T}(\tau )+n(\tau )\right]kp^{*}\left(\tau \right)\partial \tau }=\int _{0}^{T_{s}}{\left[s_{T}(t)+n(t)\right]kp^{*}\left(t\right)\partial t}\\\end{aligned}}$

## Filtro adaptado

Con filtro adaptado: $h_{R}(t)=p^{*}\left(T_{s}-t\right)$
${\begin{aligned}&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{aligned}}$
Por lo que tenemos:
${\begin{aligned}&m_{y_{PR}(t)}=\int _{0}^{T_{s}}{s_{T}(t)kp^{*}\left(t\right)\partial t}\to s_{T}(t)=a_{k}\cdot p(t),a_{k}='0','1',...\\&\sigma _{y_{PR}(t)}^{2}={\frac {\eta }{2}}\cdot E_{p(t)}\\\end{aligned}}$
* Proyecto: Departamento de Teoría de la Señal y Comunicaciones
* Anterior: Modulaciones digitales — Detección de señales digitales con ruido — Siguiente: Probabilidad de error de una señal aleatoria gaussiana