can be treated as a constant.Eq.(4.4)can also be expressed as ±L(2 Pu:=圳=e4+e We define the soft bit()as the expectation of u.From the discussion above,we have A(4)=曰4] =(+)P(4=+)+(-1)P(4=-I) =tanhL(） 2 If P()is a random variable in the range of (1),then()is a random variable(r.v.)in the range (-1,+1).For two binary random variables u and uz,their GF(2)addition u transforms into uu Since Eluu]=E[u ]E[u]=()(u),the LLR of the sum L(4⊕42)equals tanh-tanh.tanh=L(t )L(u 2 abbreviated by the boxplus operation This relation is often used in the decoding of LDPC codes,and appeared in the following form. 定义(Box-PIus运算)：对于两个二元独立随机变量u和y,它们的box-plus运算定义 为 L0②Lm)=Lu田p=logP引=bg。+e4 * As well as the LLR L()based on the unconditional probabilities P(=),we are also interested in LLRs based on conditional probabilities.For example,we often use the conditional LLRs based on the probability that the receiver's matched filter output would be y&given that the corresponding transmitted bitx=+1 or-1.This conditional LLR,denoted by P(yx).is defined as w (4.5) If we assume that xa is transmitted over the(possibly fading)AWGN channel using BPSK 4-15 4-15 can be treated as a constant. Eq. (4.4) can also be expressed as ( )/2 ( )/2 ( )/2 ( 1) k k k L u k Lu Lu e P u e e      We define the soft bit ( ) k  u as the expectation of uk. From the discussion above, we have () [] k k  u u E ( 1) ( 1) ( 1) ( 1)         Pu Pu k k ( ) tanh 2   L uk      If P(uk) is a random variable in the range of (0, 1), then ( ) k  u is a random variable (r.v.) in the range (-1, +1). For two binary random variables u1 and u2, their GF(2) addition 1 2 u u  transforms into 1 2 u u . Since 12 1 2 1 2 Eu u Eu Eu u u [ ] [] [] () ()     , the LLR of the sum L  1 2 u u  equals 1 1 2 1 2 () () 2 tanh tanh tanh ( ) ( ) 2 2 Lu Lu Lu Lu                   abbreviated by the boxplus operation . This relation is often used in the decoding of LDPC codes, and appeared in the following form. 定义（Box-Plus 运算）：对于两个二元独立随机变量 u 和 v，它们的 box-plus 运算定义 为 () () () () ( 0) 1 ( ) ( ) ( ) log log ( 1) Lu Lv Lu Lv Pu v e Lu Lv Lu v Pu v e e           As well as the LLR L(uk) based on the unconditional probabilities ( 1) P uk   , we are also interested in LLRs based on conditional probabilities. For example, we often use the conditional LLRs based on the probability that the receiver’s matched filter output would be yk given that the corresponding transmitted bit 1 or 1 k x    . This conditional LLR, denoted by (|) Py x k k , is defined as ( | 1) ( | ) ln ( | 1) k k k k k k Py x Ly x Py x            (4.5) If we assume that xk is transmitted over the (possibly fading) AWGN channel using BPSK