BER: Bit Error Rate
1 2 π σ 2 ⋅ e − ( t − m ) 2 2 σ 2 e r f ( x ) = 2 π ∫ λ = 0 λ = x e − λ 2 ∂ λ → e r f c ( x ) = e r f ( − x ) = 1 − e r f ( x ) , e r f ( ∞ ) = 1 ∫ − ∞ V T 1 2 π σ 2 ⋅ e − ( t − m ) 2 2 σ 2 ∂ t = ? , ∫ V T ∞ 1 2 π σ 2 ⋅ e − ( t − m ) 2 2 σ 2 ∂ t = ? , { − λ 2 = − ( t − m ) 2 2 σ 2 ⇒ λ = t − m 2 σ 2 ↔ t = 2 σ 2 λ + m ⇒ ∂ t = 2 σ 2 ∂ λ t = 2 σ 2 λ + m → { t = ∞ → λ = ∞ t = V T → λ = V T − m 2 σ 2 ∫ t = V T t = ∞ 1 2 π σ 2 ⋅ e − ( t − m ) 2 2 σ 2 ∂ t = ∫ t = V T t = ∞ 1 2 π σ 2 ⋅ e − λ 2 2 σ 2 ∂ λ = 1 π ∫ t = V T t = ∞ e − λ 2 ∂ λ = 1 π ∫ λ = V T − m 2 σ 2 λ = ∞ e − λ 2 ∂ λ e r f ( x ) 2 = 1 π ∫ λ = 0 λ = x e − λ 2 ∂ λ 1 π ∫ λ = V T − m 2 σ 2 λ = ∞ e − λ 2 ∂ λ = 1 π ∫ λ = 0 λ = ∞ e − λ 2 ∂ λ − 1 π ∫ λ = 0 λ = V T − m 2 σ 2 e − λ 2 ∂ λ = e r f ( ∞ ) = 1 2 − e r f ( V T − m 2 σ 2 ) 2 = 1 − e r f ( V T − m 2 σ 2 ) 2 = e r f c ( V T − m 2 σ 2 ) 2 → Q ( x ) = e r f c ( x 2 ) 2 → Q ( V T − m σ 2 ) ∫ V T ∞ 1 2 π σ 2 ⋅ e − ( t − m ) 2 2 σ 2 ∂ t = Q ( V T − m σ 2 ) ∫ − ∞ V T 1 2 π σ 2 ⋅ e − ( t − m ) 2 2 σ 2 ∂ t = 1 − ∫ V T ∞ 1 2 π σ 2 ⋅ e − ( t − m ) 2 2 σ 2 ∂ t = 1 − Q ( V T − m σ 2 ) = Q ( m − V T σ 2 ) Q ( | V T − m | σ 2 ) {\displaystyle {\begin{aligned}&{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{\frac {-\left(t-m\right)^{2}}{2\sigma ^{2}}}\\&erf(x)={\frac {2}{\sqrt {\pi }}}\int _{\lambda =0}^{\lambda =x}{e^{-\lambda ^{2}}\partial \lambda }\to erfc(x)=erf(-x)=1-erf(x),erf(\infty )=1\\&\int \limits _{-\infty }^{V_{T}}{{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{\frac {-\left(t-m\right)^{2}}{2\sigma ^{2}}}\partial t}=?,\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}=?,\\&\left\{{\begin{aligned}&-\lambda ^{2}={\frac {-\left(t-m\right)^{2}}{2\sigma ^{2}}}\Rightarrow \lambda ={\frac {t-m}{\sqrt {2\sigma ^{2}}}}\leftrightarrow t={\sqrt {2\sigma ^{2}}}\lambda +m\Rightarrow \partial t={\sqrt {2\sigma ^{2}}}\partial \lambda \\&t={\sqrt {2\sigma ^{2}}}\lambda +m\to \left\{{\begin{aligned}&t=\infty \to \lambda =\infty \\&t=V_{T}\to \lambda ={\frac {V_{T}-m}{\sqrt {2\sigma ^{2}}}}\\\end{aligned}}\right.\\\end{aligned}}\right.\\&\int \limits _{t=V_{T}}^{t=\infty }{{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{\frac {-\left(t-m\right)^{2}}{2\sigma ^{2}}}\partial t}=\int \limits _{t=V_{T}}^{t=\infty }{{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{-\lambda ^{2}}{\sqrt {2\sigma ^{2}}}\partial \lambda =}{\frac {1}{\sqrt {\pi }}}\int \limits _{t=V_{T}}^{t=\infty }{e^{-\lambda ^{2}}\partial \lambda }={\frac {1}{\sqrt {\pi }}}\int \limits _{\lambda ={\frac {V_{T}-m}{\sqrt {2\sigma ^{2}}}}}^{\lambda =\infty }{e^{-\lambda ^{2}}\partial \lambda }\\&{\frac {erf(x)}{2}}={\frac {1}{\sqrt {\pi }}}\int _{\lambda =0}^{\lambda =x}{e^{-\lambda ^{2}}\partial \lambda }\\&{\frac {1}{\sqrt {\pi }}}\int \limits _{\lambda ={\frac {V_{T}-m}{\sqrt {2\sigma ^{2}}}}}^{\lambda =\infty }{e^{-\lambda ^{2}}\partial \lambda }={\frac {1}{\sqrt {\pi }}}\int \limits _{\lambda =0}^{\lambda =\infty }{e^{-\lambda ^{2}}\partial \lambda }-{\frac {1}{\sqrt {\pi }}}\int \limits _{\lambda =0}^{\lambda ={\frac {V_{T}-m}{\sqrt {2\sigma ^{2}}}}}{e^{-\lambda ^{2}}\partial \lambda }={\frac {erf(\infty )=1}{2}}-{\frac {erf\left({\frac {V_{T}-m}{\sqrt {2\sigma ^{2}}}}\right)}{2}}=\\&{\frac {1-erf\left({\frac {V_{T}-m}{\sqrt {2\sigma ^{2}}}}\right)}{2}}={\frac {erfc\left({\frac {V_{T}-m}{\sqrt {2\sigma ^{2}}}}\right)}{2}}\to Q(x)={\frac {erfc\left({\frac {x}{\sqrt {2}}}\right)}{2}}\to Q\left({\frac {V_{T}-m}{\sqrt {\sigma ^{2}}}}\right)\\&\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 {V_{T}-m}{\sqrt {\sigma ^{2}}}}\right)\\&\int \limits _{-\infty }^{V_{T}}{{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{\frac {-\left(t-m\right)^{2}}{2\sigma ^{2}}}\partial t}=1-\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}=1-Q\left({\frac {V_{T}-m}{\sqrt {\sigma ^{2}}}}\right)=Q\left({\frac {m-V_{T}}{\sqrt {\sigma ^{2}}}}\right)\\&Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)\\\end{aligned}}}
∫ 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 ) {\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)\\\end{aligned}}}
