Desigualdad de Schwarz:
| ∫ f 1 ( x ) ⋅ f 2 ( x ) ∂ x | 2 ≤ ∫ | f 1 ( x ) | 2 ∂ x ⋅ ∫ | f 2 ( x ) | 2 ∂ x M a x → f 1 ( x ) = k ⋅ f 2 ∗ ( x ) {\displaystyle {\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}}}
y R ( t ) = s R ( t ) + n R ( t ) G n R ( f ) = η 2 ⋅ | H R ( f ) | 2 {\displaystyle {\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}}}
s R ( t ) = { ′ 1 ′ → s ′ 1 ′ ( t ) ′ 0 ′ → s ′ 0 ′ ( t ) ( S ╱ N ) R | T s = S R N R | T S = | s R ( t = T S ) | 2 N R → puede que no sea relacion se n ~ al a ruido {\displaystyle {\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}}}
S R ( f ) = S T ( f ) ⋅ H R ( f ) s R ( t ) = ∫ − ∞ ∞ S R ( f ) e + j 2 π f t ∂ f = ∫ − ∞ ∞ S T ( f ) H R ( f ) e + j 2 π f t ∂ f | s R ( t = T S ) | 2 N R = | ∫ − ∞ ∞ S T ( f ) H R ( f ) e + j 2 π f t ∂ f | t = T s 2 N R = | ∫ − ∞ ∞ S T ( f ) H R ( f ) e + j 2 π f T s ∂ f | 2 η 2 ∫ − ∞ ∞ | H R ( f ) | 2 ∂ f → | ∫ f 1 ( x ) ⋅ f 2 ( x ) ∂ x | 2 ≤ ∫ | f 1 ( x ) | 2 ∂ x ∫ | f 2 ( x ) | 2 ∂ x → f 1 ( x ) = H R ( f ) ; f 2 ( x ) = S T ( f ) e + j 2 π f T s M a x → f 1 ( x ) = k ⋅ f 2 ∗ ( x ) | s R ( t = T S ) | 2 N R ⇒ | ∫ f 1 ( x ) ⋅ f 2 ( x ) ∂ x | 2 η 2 ⋅ ∫ | f 2 ( x ) | 2 ∂ f ≤ ∫ | f 1 ( x ) | 2 ∂ x ∫ | f 2 ( x ) | 2 ∂ x η 2 ⋅ ∫ | f 2 ( x ) | 2 ∂ f → | ∫ − ∞ ∞ S T ( f ) H R ( f ) e + j 2 π f T s ∂ f | 2 η 2 ∫ − ∞ ∞ | H R ( f ) | 2 ∂ f ≤ ∫ − ∞ ∞ | H R ( f ) | 2 ∂ f ⋅ ∫ − ∞ ∞ | S T ( f ) e + j 2 π f T s | 2 ∂ f η 2 ∫ − ∞ ∞ | H R ( f ) | 2 ∂ f = ∫ − ∞ ∞ | S T ( f ) e + j 2 π f T s | 2 ∂ f η 2 | s R ( t = T S ) | 2 N R | max = ∫ − ∞ ∞ | S T ( f ) e + j 2 π f T s | 2 ∂ f η 2 {\displaystyle {\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}}}
f 1 ( x ) = H R ( f ) ; f 2 ( x ) = S T ( f ) e + j 2 π f T s M a x → f 1 ( x ) = k ⋅ f 2 ∗ ( x ) → H R ( f ) = k ⋅ 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 ] → h R ( t ) = F − 1 [ S T ∗ ( f ) e − j 2 π f T s ] { F [ x ∗ ( t ) ] = X ∗ ( − f ) F [ x ( t − t 0 ) ] = X ( f ) e − j 2 π f t 0 } G ( f ) = S T ∗ ( f ) → F − 1 [ S T ∗ ( f ) ⏞ = G ( f ) e − j 2 π f T s ] = g ( t − T s ) → g ( t ) = s T ∗ ( − t ) → g ( t − T s ) = s T ∗ ( − ( t − T s ) ) = s T ∗ ( T s − t ) M a x → f 1 ( x ) = k ⋅ f 2 ∗ ( x ) → h R ( t ) = s T ∗ ( T s − t ) {\displaystyle {\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}}}
y R ( t ) = [ s T ( t ) + n ( t ) ] ∗ h R ( t ) = ∫ − ∞ ∞ [ s T ( τ ) + n ( τ ) ] h R ( t − τ ) ∂ τ = h R ( t ) = k ⋅ p ∗ ( T s − t ) → ∫ − ∞ ∞ [ s T ( τ ) + n ( τ ) ] k p ∗ ( T s − ( t − τ ) ) ∂ τ → ∫ − ∞ ∞ [ s T ( τ ) + n ( τ ) ] k p ∗ ( T s − ( t − τ ) ) ∂ τ | t = T s = ∫ 0 T s [ s T ( τ ) + n ( τ ) ] k p ∗ ( τ ) ∂ τ = ∫ 0 T s [ s T ( t ) + n ( t ) ] k p ∗ ( t ) ∂ t {\displaystyle {\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}}}
Con filtro adaptado: h R ( t ) = p ∗ ( T s − t ) {\displaystyle h_{R}(t)=p^{*}\left(T_{s}-t\right)}
y P R ( t ) = ∫ 0 T s [ s T ( t ) + n ( t ) ] k p ∗ ( t ) ∂ t = ∫ 0 T s s T ( t ) k p ∗ ( t ) ∂ t + ∫ 0 T s n ( t ) k p ∗ ( t ) ∂ t m y P R ( t ) = E { y P R ( t ) } = E { ∫ 0 T s s T ( t ) k p ∗ ( t ) ∂ t } ⏟ Deterministico + E { ∫ 0 T s n ( t ) k p ∗ ( t ) ∂ t } ⏟ Aleatorio , E { n ( t ) } = 0 = ∫ 0 T s s T ( t ) k p ∗ ( t ) ∂ t σ y P R ( t ) 2 = E { y P R 2 ( t ) } − E 2 { y P R ( t ) } → E { y P R 2 ( t ) } = E { [ ∫ 0 T s [ s T ( t ) + n ( t ) ] k p ∗ ( t ) ∂ t ] 2 } = E { [ ∫ 0 T s s T ( t ) k p ∗ ( t ) ∂ t ] 2 } + E { [ ∫ 0 T s n ( t ) k p ∗ ( t ) ∂ t ] 2 } + 2 ⋅ E { ∫ 0 T s s T ( t ) k p ∗ ( t ) ∂ t ⋅ ∫ 0 T s n ( t ) k p ∗ ( t ) ∂ t ⏟ = 0 } E 2 { y P R ( t ) } = E { [ ∫ 0 T s s T ( t ) k p ∗ ( t ) ∂ t ] 2 } σ y P R ( t ) 2 = E { y P R 2 ( t ) } − E 2 { y P R ( t ) } = E { [ ∫ 0 T s n ( t ) k p ∗ ( t ) ∂ t ] 2 } E { ∫ 0 T s n ( t ) k p ∗ ( t ) ∂ t ⋅ ∫ 0 T s n ( λ ) k p ∗ ( λ ) ∂ λ } = E { ∫ 0 T s ∫ 0 T s n ( t ) n ( λ ) k 2 p ∗ ( t ) p ∗ ( λ ) ⏟ Determinista ∂ t ∂ λ } = ∫ 0 T s ∫ 0 T s E { n ( t ) n ( λ ) } ⏟ R n ( t − λ ) = η 2 δ ( t − λ ) k 2 p ∗ ( t ) p ∗ ( λ ) ∂ t ∂ λ = ∫ 0 T s ∫ 0 T s η 2 δ ( t − λ ) k 2 p ∗ ( t ) p ∗ ( λ ) ∂ t ∂ λ = ∫ 0 T s η 2 ∫ 0 T s δ ( t − λ ) p ∗ ( λ ) ∂ λ ⏟ = p ∗ ( t ) ∗ δ ( t ) = p ∗ ( t ) k 2 p ∗ ( t ) ∂ t = η 2 k 2 ∫ 0 T s p 2 ( t ) ∂ t ⏟ E p ( t ) σ y P R ( t ) 2 = η 2 ⋅ E p ( t ) {\displaystyle {\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:
m y P R ( t ) = ∫ 0 T s s T ( t ) k p ∗ ( t ) ∂ t → s T ( t ) = a k ⋅ p ( t ) , a k = ′ 0 ′ , ′ 1 ′ , . . . σ y P R ( t ) 2 = η 2 ⋅ E p ( t ) {\displaystyle {\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}}}
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42a4a3b687105c074bd10733b638c13901307aa1)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb0e256fffc28acee3b69a70ff1cdad478b1857)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5204bfc25d590ba96121589f83d65ed55e74076)
