软判决、迭代解码与航天发展中的纠错编码技术
1. 软判决与迭代解码基础
1.1 物理编码器分析
以物理编码器 (G’‘_1 = [1\frac{1 + D^2}{1 + D + D^2}]) 为例，通过 (s(i + 1) = x(i) + s(i) + s(i - 1)) 可求解 (c_2(i))：
[
\begin{align}
c_2(i) &= x(i) + s(i)\
&= x(i) + x(i - 1) + s(i - 1) + s(i - 2)\
&= x(i) + x(i - 1) + x(i - 2) + s(i - 2) + s(i - 3) + s(i - 2)\
&= x(i) + x(i - 1) + x(i - 2) + s(i - 3)\
&= x(i) + x(i - 1) + x(i - 2) + x(i - 4) + s(i - 4) + s(i - 5)\
\end{align}
]
继续推导可得：
[
c_2(i) = x(i) + x(i - 1) + x(i - 2) + x(i - 4) + x(i - 5)+ x(i - 7) + x(i - 8) + x(i - 10) + x(i - 11) + \cdots
]
通过练习 864 可知，(\frac{1 + D^2}{1 + D + D^2}) 的洛朗级数展开为 (1 + D + D^2 + D^4 + D^5 + D^7 + D^8 + D^{10} + D^{11} + D^{13} +
