Answer-Polar Code-定义错误概率

《Polar Code(4)编码之极化信道可靠性估计》提到,事件\({ {A}_{i}}\)表示序号为\(i\)的极化信道\(W_{N}^{\left( i \right)}\)所承载的比特经过传输后接收发生错误,即

\[ { {A}_{i}}=\left\{ u_{1}^{N},y_{1}^{N}:W_{N}^{\left( i \right)}\left( y_{1}^{N},u_{1}^{i-1}|{ {u}_{i}} \right)<W_{N}^{\left( i \right)}\left( y_{1}^{N},u_{1}^{i-1}|{ {u}_{i}}\oplus 1 \right) \right\} \]

则极化信道\(W_{N}^{\left( i \right)}\)的错误概率为\(P\left( { {A}_{i}} \right)\)

那么事件\({ {A}_{i}}\)怎么理解?

mark

举个例子,假设N=2。编码器输入是\(\left( \begin{matrix} { {u}_{1}} \\ { {u}_{2}} \\ \end{matrix} \right)\),编码器输出是\(\left( \begin{matrix} { {x}_{1}}={ {u}_{1}}\oplus { {u}_{2}} \\ { {x}_{2}}={ {u}_{2}} \\ \end{matrix} \right)\)。经过无线信道,假设接收端接收到的信号为\(y_{1}^{2}\)

对接收端来说,\(y_{1}^{2}\)是已知量,而\({ {u}_{i}}\)是未知量。转移概率\(W_{2}^{i}\left( y_{1}^{2}|{ {u}_{i}} \right)\)表示通过序号为\(i\)的极化信道\(W_{N}^{\left( i \right)}\),发送为\({ {u}_{i}}\)接收为\(y_{1}^{2}\)的概率。

\({ {u}_{i}}\)和$ ( { {u}_{i}} ) $这种表达形式使得这两个值正好是互斥的:

  • \({ {u}_{i}}=0\)时,有$ { {u}_{i}}=1 $
  • \({ {u}_{i}}=1\)时,有$ { {u}_{i}}=0 $

二进制比特\({ {u}_{i}}\)只有两种取值,无论发送“0”或“1”,由于无线信道的影响,接收端都有可能得到相同的\(y_{1}^{2}\)。那么到底是发送“0”使得接收为\(y_{1}^{2}\),还是发送“1”使得接收为\(y_{1}^{2}\)?可以通过比较这两种情况的转移概率得到答案。

给定\({ {u}_{1}}=0\)的情况

比较$ W_{N}^{( i )}( y_{1}{N},u_{1}{i-1}|{ {u}{i}} ) \(和\) W{N}^{( i )}( y_{1}{N},u_{1}{i-1}|{ {u}{i}} ) \(,也即比较\) W{2}^{( 1 )}( y_{1}^{2}|0 ) \(和\) W_{2}^{( 1 )}( y_{1}^{2}|1 ) \(。只有当\) W_{2}^{( 1 )}( y_{1}^{2}|0 )>W_{2}^{( 1 )}( y_{1}^{2}|1 ) \(时,才是正确接收。左边的式子和**给定\){ {u}_{1}}=0$相一致**。

给定\({ {u}_{1}}=1\)的情况

比较$ W_{N}^{( i )}( y_{1}{N},u_{1}{i-1}|{ {u}{i}} ) \(和\) W{N}^{( i )}( y_{1}{N},u_{1}{i-1}|{ {u}{i}} ) \(,也即比较\) W{2}^{( 1 )}( y_{1}^{2}|1 ) \(和\) W_{2}^{( 1 )}( y_{1}^{2}|0 ) \(。只有当\) W_{2}^{( 1 )}( y_{1}^{2}|1 )>W_{2}^{( 1 )}( y_{1}^{2}|0 ) \(时,才是正确接收。左边的式子和**给定\){ {u}_{1}}=1$相一致**。

这两种情况是等价的。即$ W_{N}^{( i )}( y_{1}{N},u_{1}{i-1}|{ {u}{i}} )>W{N}^{( i )}( y_{1}{N},u_{1}{i-1}|{ {u}{i}} ) \(时才是正确接收。反过来,当\) W{N}^{( i )}( y_{1}{N},u_{1}{i-1}|{ {u}{i}} )<W{N}^{( i )}( y_{1}{N},u_{1}{i-1}|{ {u}{i}} ) \(时意味着接收错误,这就是事件\){ {A}{i}}\(。事件\){ {A}{i}}\(发生的概率就是错误概率\)P( { {A}{i}} )$。