# Polar Code Parameter Recognition Algorithm Based on Dual Space
## 3 Principle and Method of Polar Code Parameter Recognition
### Decision Threshold
, \tag{56}\]  
and the PDFs of Y are  
\[f_{Y}(y|R_{1}^{\prime})= \begin{cases}f(-y|R_{1}^{\prime})&,y>0\\ 0&,y\leq 0^{\prime}\end{cases}  \tag{57}\]  
\[f_{Y}(y|R_{2}^{\prime})= \begin{cases}f(y|R_{2}^{\prime})&,y>0\\ 0&,y\leq 0\end{cases} . \tag{58}\]  
It can be obtained from (55)-(58) that, when \(y\leq 0\), \(f_{Y}(y|R_{1}^{\prime}\cup R_{2}^{\prime})=0\), and, when \(y>0\), that  
\[f_{Y}(y|R_{1}^{\prime}\cup R_{2}^{\prime})=\frac{\sigma\cdot e^{-\sigma^{2} \cdot(y+2/\sigma^{2})^{2}/8}}{\sqrt{2\pi}(1-erf(1/(\sqrt{2}\sigma)))}. \tag{59}\]  
Assuming that there is an error in the element at position \(\{t_{v_{1}},t_{v_{2}},\cdots,t_{v_{s}}\}\) in \(l_{i,t_{1}},l_{i,t_{2}},\cdots,l_{i,t_{u}}\), it is easy to obtain the set of correct element positions as \(\big{\{}t_{j_{1}},t_{j_{2}},\cdots,t_{j_{u-s}}\big{\}}=\{t_{1},t_{2},\cdots,t_ {v}\}/\{t_{v_{1}},t_{v_{2}},\cdots,t_{v_{s}}\}\). Letting \(T_{1}^{s}=\min\Big{\{}|l_{i,t_{v_{1}}}|,|l_{i,t_{v_{2}}}|,\cdots,|l_{i,t_{v_{1 }}}|\Big{\}}\) and \(T_{2}^{s}=\min\Big{\{}|l_{i,t_{v_{1}}}|,|l_{i,t_{v_{2}}}|,\cdots,|l_{i,t_{v_{1 }}}|\Big{\}}\), it can be obtained from Inference 2-1 that, when \(y\leq 0\), \(f_{T_{1}^{s}}(y)=0\) and \(f_{T_{2}^{s}}(y)=0\), and, when \(y>0\), that  
\[f_{T_{1}^{s}}(y)=s\cdot f_{Y}(y|R_{1}^{\prime}\cup R_{2}^{\prime})(1-F_{Y}(y|R _{1}^{\prime}\cup R_{2}^{\prime}))^{s-1}, \tag{60}\]  
\[f_{T_{2}^{s}}(y)=(w-s)\cdot f_{Y}(y|R_{1}\cup R_{2})(1-F_{Y}(y|R_{1}\cup R_{2} ))^{w-s-1}, \tag{61}\]  
where \(F_{Y}(y|R_{1}^{\prime}\cup R_{2}^{\prime})\) and \(F_{Y}(y|R_{1}\cup R_{2})\) are the CFs of \(f_{Y}(y|R_{1}^{\prime}\cup R_{2}^{\prime})\) and \(f_{Y}(y|R_{1}\cup R_{2})\), respectively, so that, when \(y>0\),  
\[ \begin{array}{c}F_{Y}(y|R_{1}^{\prime}\cup R_{2}^{\prime})=\int_{0}^{y}f_{Y} (t|R_{1}^{\prime}\cup R_{2}^{\prime})dt=\\ \frac{1}{1-erf(1/(\sqrt{2}\sigma))}(erf(\frac{y0}{2\sqrt{2}}+\frac{1}{\sqrt{2 }\sigma})+erf(\frac{1}{\sqrt{2}\sigma}))\end{array} , \tag{62}\]  
\[ \begin{array}{c}F_{Y}(y|R_{1}\cup R_{2})=\int_{0}^{y}f_{Y}(t|R_{1}\cup R_{2} )dt=\\ \frac{1}{1+erf(1/(\sqrt{2}\sigma))}(erf(\frac{y0}{2\sqrt{2}}-\frac{1}{\sqrt{2 }\sigma})+erf(\frac{1}{\sqrt{2}\sigma}))\end{array} . \tag{63}\]  
Letting \(T^{s}=\min\{T_{1}^{s},T_{2}^{s}\}\), from Theorem 2 we can obtain  
\[f_{T^{s}}(y)=f_{T_{1}^{s}}(y)\cdot(1-F_{T_{2}^{s}}(y))+f_{T_{2}^{s}}(y)\cdot(1- F_{T_{1}^{s}}(y)), \tag{64}\]  
where \(F_{T_{1}^{s}}(y)\) and \(F_{T_{2}^{s}}(y)\) are the CFs of \(f_{T_{1}^{s}}(y)\) and \(f_{T_{2}^{s}}(y)\), respectively; that is,  
\[F_{T_{1}^{s}}(y)= \begin{cases}1-(1-F_{Y}(y|R_{1}^{\prime}\cup R_{2}^{\prime}))^ {s}&,y>0\\ 0&,y\leq 0\end{cases} , \tag{65}\]  
\[F_{T_{2}^{s}}(y)= \begin{cases}1-(1-F_{Y}(y|R_{1}\cup R_{2}))^{w-s}&,y>0\\ 0&,y\leq 0\end{cases} . \tag{66}\]When \(X_{i}=1\), the number of error symbols is even and the check relationship is still established, so  
\[E(X_{i}\cdot Y_{i},X_{i}=1|H_{1})=\sum\limits_{i=0}^{\lfloor w/2\rfloor}C_{w}^{2 i}p_{e}^{2i}(1-p_{e})^{w-2i}\int\limits_{0}^{+\infty}xf_{T^{2i}}(x)dx, \tag{67}\]  
where \(p_{e}\) is the theoretical bit error rate of the codewords modulated by BPSK, and the expression is  
\[p_{e}=0.5 erfc(\sqrt{1/(2\sigma^{2})}), \tag{68}\]  
where \(\sigma^{2}\) is noise variance, \(erfc(x)=\frac{2}{\sqrt{\pi}}\sum\limits_{-\infty}^{x}e^{-t^{2}}dt\).