Detección de señales digitales con ruido

Desigualdad de Schwarz:

$\begin{matrix} {| \int f_{1} (x) \cdot f_{2} (x) \partial x |}^{2} \leq \int {| f_{1} (x) |}^{2} \partial x \cdot \int {| f_{2} (x) |}^{2} \partial x \\ M a x \to f_{1} (x) = k \cdot f_{2}^{*} (x) \end{matrix}$

$\begin{matrix} y_{R} (t) = s_{R} (t) + n_{R} (t) \\ G_{n_{R}} (f) = \frac{η}{2} \cdot {| H_{R} (f) |}^{2} \end{matrix}$

$\begin{matrix} s_{R} (t) = {\begin{matrix} ' 1^{'} \to s_{' 1^{'}} (t) \\ ' 0^{'} \to s_{' 0^{'}} (t) \end{matrix} \\ {{(^{S} ╱_{N})}_{R} |}_{T_{s}} = {\frac{S_{R}}{N_{R}} |}_{T_{S}} = \frac{{| s_{R} (t = T_{S}) |}^{2}}{N_{R}} \to puede que no sea relacion se \tilde{n} al a ruido \end{matrix}$

$\begin{matrix} S_{R} (f) = S_{T} (f) \cdot H_{R} (f) \\ s_{R} (t) = \int_{- \infty}^{\infty} S_{R} (f) e^{+ j 2 π f t} \partial f = \int_{- \infty}^{\infty} S_{T} (f) H_{R} (f) e^{+ j 2 π f t} \partial f \\ \frac{{| s_{R} (t = T_{S}) |}^{2}}{N_{R}} = \frac{{| \int_{- \infty}^{\infty} S_{T} (f) H_{R} (f) e^{+ j 2 π f t} \partial f |}_{t = T_{s}}^{2}}{N_{R}} = \frac{{| \int_{- \infty}^{\infty} S_{T} (f) H_{R} (f) e^{+ j 2 π f T_{s}} \partial f |}^{2}}{\frac{η}{2} \int_{- \infty}^{\infty} {| H_{R} (f) |}^{2} \partial f} \to \\ {| \int f_{1} (x) \cdot f_{2} (x) \partial x |}^{2} \leq \int {| f_{1} (x) |}^{2} \partial x \int {| f_{2} (x) |}^{2} \partial x \to f_{1} (x) = H_{R} (f); f_{2} (x) = S_{T} (f) e^{+ j 2 π f T_{s}} \\ M a x \to f_{1} (x) = k \cdot f_{2}^{*} (x) \\ \frac{{| s_{R} (t = T_{S}) |}^{2}}{N_{R}} \Rightarrow \frac{{| \int f_{1} (x) \cdot f_{2} (x) \partial x |}^{2}}{\frac{η}{2} \cdot \int {| f_{2} (x) |}^{2} \partial f} \leq \frac{\int {| f_{1} (x) |}^{2} \partial x \int {| f_{2} (x) |}^{2} \partial x}{\frac{η}{2} \cdot \int {| f_{2} (x) |}^{2} \partial f} \to \\ \frac{{| \int_{- \infty}^{\infty} S_{T} (f) H_{R} (f) e^{+ j 2 π f T_{s}} \partial f |}^{2}}{\frac{η}{2} \int_{- \infty}^{\infty} {| H_{R} (f) |}^{2} \partial f} \leq \frac{\int_{- \infty}^{\infty} {| H_{R} (f) |}^{2} \partial f \cdot \int_{- \infty}^{\infty} {| S_{T} (f) e^{+ j 2 π f T_{s}} |}^{2} \partial f}{\frac{η}{2} \int_{- \infty}^{\infty} {| H_{R} (f) |}^{2} \partial f} = \frac{\int_{- \infty}^{\infty} {| S_{T} (f) e^{+ j 2 π f T_{s}} |}^{2} \partial f}{\frac{η}{2}} \\ {\frac{{| s_{R} (t = T_{S}) |}^{2}}{N_{R}} |}_{\max} = \frac{\int_{- \infty}^{\infty} {| S_{T} (f) e^{+ j 2 π f T_{s}} |}^{2} \partial f}{\frac{η}{2}} \end{matrix}$

$\begin{matrix} f_{1} (x) = H_{R} (f); f_{2} (x) = S_{T} (f) e^{+ j 2 π f T_{s}} \\ M a x \to f_{1} (x) = k \cdot f_{2}^{*} (x) \to H_{R} (f) = k \cdot S_{T}^{*} (f) e^{- j 2 π f T_{s}} \\ 𝔽^{- 1} [H_{R} (f)] = 𝔽^{- 1} [S_{T}^{*} (f) e^{- j 2 π f T_{s}}] \to h_{R} (t) = 𝔽^{- 1} [S_{T}^{*} (f) e^{- j 2 π f T_{s}}] \\ {\begin{matrix} 𝔽 [x^{*} (t)] = X^{*} (- f) \\ 𝔽 [x (t - t_{0})] = X (f) e^{- j 2 π f t_{0}} \end{matrix}} \\ G (f) = S_{T}^{*} (f) \to 𝔽^{- 1} [\overset{= G (f)}{\overset{⏞}{S_{T}^{*} (f)}} e^{- j 2 π f T_{s}}] = g (t - T_{s}) \to \\ g (t) = s_{T}^{*} (- t) \to g (t - T_{s}) = s_{T}^{*} (- (t - T_{s})) = s_{T}^{*} (T_{s} - t) \\ M a x \to f_{1} (x) = k \cdot f_{2}^{*} (x) \to h_{R} (t) = s_{T}^{*} (T_{s} - t) \end{matrix}$

$\begin{matrix} y_{R} (t) = [s_{T} (t) + n (t)] * h_{R} (t) = \int_{- \infty}^{\infty} [s_{T} (τ) + n (τ)] h_{R} (t - τ) \partial τ = \\ h_{R} (t) = k \cdot p^{*} (T_{s} - t) \to \\ \int_{- \infty}^{\infty} [s_{T} (τ) + n (τ)] k p^{*} (T_{s} - (t - τ)) \partial τ \to \\ {\int_{- \infty}^{\infty} [s_{T} (τ) + n (τ)] k p^{*} (T_{s} - (t - τ)) \partial τ |}_{t = T_{s}} = \int_{0}^{T_{s}} [s_{T} (τ) + n (τ)] k p^{*} (τ) \partial τ = \int_{0}^{T_{s}} [s_{T} (t) + n (t)] k p^{*} (t) \partial t \end{matrix}$

Filtro adaptado

Con filtro adaptado: $h_{R} (t) = p^{*} (T_{s} - t)$

$\begin{matrix} y_{P R} (t) = \int_{0}^{T_{s}} [s_{T} (t) + n (t)] k p^{*} (t) \partial t = \int_{0}^{T_{s}} s_{T} (t) k p^{*} (t) \partial t + \int_{0}^{T_{s}} n (t) k p^{*} (t) \partial t \\ m_{y_{P R} (t)} = E {y_{P R} (t)} = \underset{Deterministico}{\underset{⏟}{E {\int_{0}^{T_{s}} s_{T} (t) k p^{*} (t) \partial t}}} + \underset{Aleatorio, E {n (t)} = 0}{\underset{⏟}{E {\int_{0}^{T_{s}} n (t) k p^{*} (t) \partial t}}} = \int_{0}^{T_{s}} s_{T} (t) k p^{*} (t) \partial t \\ σ_{y_{P R} (t)}^{2} = E {y_{P R}^{2} (t)} - E^{2} {y_{P R} (t)} \to \\ E {y_{P R}^{2} (t)} = E {{[\int_{0}^{T_{s}} [s_{T} (t) + n (t)] k p^{*} (t) \partial t]}^{2}} = \\ E {{[\int_{0}^{T_{s}} s_{T} (t) k p^{*} (t) \partial t]}^{2}} + E {{[\int_{0}^{T_{s}} n (t) k p^{*} (t) \partial t]}^{2}} + 2 \cdot E {\int_{0}^{T_{s}} s_{T} (t) k p^{*} (t) \partial t \cdot \underset{= 0}{\underset{⏟}{\int_{0}^{T_{s}} n (t) k p^{*} (t) \partial t}}} \\ E^{2} {y_{P R} (t)} = E {{[\int_{0}^{T_{s}} s_{T} (t) k p^{*} (t) \partial t]}^{2}} \\ σ_{y_{P R} (t)}^{2} = E {y_{P R}^{2} (t)} - E^{2} {y_{P R} (t)} = E {{[\int_{0}^{T_{s}} n (t) k p^{*} (t) \partial t]}^{2}} \\ E {\int_{0}^{T_{s}} n (t) k p^{*} (t) \partial t \cdot \int_{0}^{T_{s}} n (λ) k p^{*} (λ) \partial λ} = E {\int_{0}^{T_{s}} \int_{0}^{T_{s}} n (t) n (λ) \underset{Determinista}{\underset{⏟}{k^{2} p^{*} (t) p^{*} (λ)}} \partial t \partial λ} = \\ \int_{0}^{T_{s}} \int_{0}^{T_{s}} \underset{R_{n} (t - λ) = \frac{η}{2} δ (t - λ)}{\underset{⏟}{E {n (t) n (λ)}}} k^{2} p^{*} (t) p^{*} (λ) \partial t \partial λ = \int_{0}^{T_{s}} \int_{0}^{T_{s}} \frac{η}{2} δ (t - λ) k^{2} p^{*} (t) p^{*} (λ) \partial t \partial λ = \\ \int_{0}^{T_{s}} \frac{η}{2} \underset{= p^{*} (t) * δ (t) = p^{*} (t)}{\underset{⏟}{\int_{0}^{T_{s}} δ (t - λ) p^{*} (λ) \partial λ}} k^{2} p^{*} (t) \partial t = \frac{η}{2} \underset{E_{p (t)}}{\underset{⏟}{k^{2} \int_{0}^{T_{s}} p^{2} (t) \partial t}} \\ σ_{y_{P R} (t)}^{2} = \frac{η}{2} \cdot E_{p (t)} \end{matrix}$

Por lo que tenemos:

$\begin{matrix} m_{y_{P R} (t)} = \int_{0}^{T_{s}} s_{T} (t) k p^{*} (t) \partial t \to s_{T} (t) = a_{k} \cdot p (t), a_{k} =^{'} 0^{'},^{'} 1^{'}, ... \\ σ_{y_{P R} (t)}^{2} = \frac{η}{2} \cdot E_{p (t)} \end{matrix}$

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Detección de señales digitales con ruido

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