G ¯ x ( f ) = σ x 2 ⋅ R s | P ( f ) | 2 + m x 2 ⋅ R s 2 ∑ k = − ∞ ∞ | P ( k R s ) | 2 δ ( f − k R s ) {\displaystyle {\begin{aligned}&{\bar {G}}_{x}(f)=\sigma _{x}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{x}^{2}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\end{aligned}}}
H ( f ) = H β ( f ) ∗ 1 R ∏ ( f R ) = π 4 β cos ( π 2 β f ) ∏ ( f 2 β ) ∗ 1 R ∏ ( f R ) h ( t ) = h β ( t ) ⋅ sinc ( R t ) = cos ( 2 β π t ) 1 − ( 4 β t ) 2 ⋅ sinc ( R t ) {\displaystyle {\begin{aligned}&H(f)=H_{\beta }(f)*{\frac {1}{R}}\prod {\left({\frac {f}{R}}\right)}={\frac {\pi }{4\beta }}\cos \left({\frac {\pi }{2\beta }}f\right)\prod {\left({\frac {f}{2\beta }}\right)}*{\frac {1}{R}}\prod {\left({\frac {f}{R}}\right)}\\&h(t)=h_{\beta }(t)\cdot \operatorname {sinc} \left(Rt\right)={\frac {\cos \left(2\beta \pi t\right)}{1-\left(4\beta t\right)^{2}}}\cdot \operatorname {sinc} \left(Rt\right)\\\end{aligned}}}
Con filtro optimo: h R ( t ) = p ∗ ( T s − t ) {\displaystyle h_{R}(t)=p^{*}\left(T_{s}-t\right)}
∫ V T ∞ 1 2 π σ 2 ⋅ e − ( t − m ) 2 2 σ 2 ∂ t = Q ( | V T − m | σ 2 ) P e = p r ( ′ 0 ′ s e n t ) ⋅ p ( e r r o r ╱ ′ 0 ′ s e n t ) + p r ( ′ 1 ′ s e n t ) ⋅ p ( e r r o r ╱ ′ 1 ′ s e n t ) = p r ( ′ 0 ′ s e n t ) ⋅ Q ( | V T − m ′ 0 ′ | σ 2 ) + p r ( ′ 1 ′ s e n t ) ⋅ Q ( | V T − m ′ 1 ′ | σ 2 ) s T ( t ) = ∑ k = 0 ∞ a k p ( t − k T s ) m y P R ( t ) = ∫ 0 T s s T ( t ) k p ∗ ( t ) ∂ t σ y P R ( t ) 2 = η 2 ⋅ E p ( t ) = η 2 ∫ 0 T s p 2 ( t ) ∂ t {\displaystyle {\begin{aligned}&\int \limits _{V_{T}}^{\infty }{{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{\frac {-\left(t-m\right)^{2}}{2\sigma ^{2}}}\partial t}=Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)\\&P_{e}=pr\left('0'sent\right)\cdot p\left({}^{error}\!\!\diagup \!\!{}_{'0'sent}\;\right)+pr\left('1'sent\right)\cdot p\left({}^{error}\!\!\diagup \!\!{}_{'1'sent}\;\right)=\\&pr\left('0'sent\right)\cdot Q\left({\frac {\left|V_{T}-m_{'0'}\right|}{\sqrt {\sigma ^{2}}}}\right)+pr\left('1'sent\right)\cdot Q\left({\frac {\left|V_{T}-m_{'1'}\right|}{\sqrt {\sigma ^{2}}}}\right)\\&s_{T}(t)=\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}\\&m_{y_{PR}(t)}=\int _{0}^{T_{s}}{s_{T}(t)kp^{*}\left(t\right)\partial t}\\&\sigma _{y_{PR}(t)}^{2}={\frac {\eta }{2}}\cdot E_{p(t)}={\frac {\eta }{2}}\int _{0}^{T_{s}}{p^{2}(t)\partial t}\\\end{aligned}}}
∑ k = 0 ∞ a k p ( t − k T s ) {\displaystyle \sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}}
A c ∑ k = 0 ∞ a k p ( t − k T s ) cos ( ω c t ) {\displaystyle A_{c}\sum \limits _{k=0}^{\infty }{a_{k}p\left(t-kT_{s}\right)}\cos \left(\omega _{c}t\right)}