SISO decoder for repetition (supplementary Chapter 4)

Preface ：

        stay LDPC,Polar Code will be involved repetion code, Here we will focus on the corresponding principle ,

Why can we add directly

---

One     Overall process

Repetition code(3,1)：

Sending process ：

news bit: m Range [0,1],    1bit

After the coding ： M=[m,m,m],3bit

BPSK:     1 Modulation into -1,

                0 Modulation into +1

Receiver decoding process

        beliefs that is 0

Here we mainly focus on the decoder ：

Practical application factor Fixed for 1（ Because it doesn't affect the final hardout This value is greater than 0, Namely 0, Less than 0 Namely 1）

  Example ：

For example, I received it r[3,2,4]

3:      beliefs about first bit       base on r1 alone

2:      beliefs about second bit base on r2 alone

4:      beliefs about third bit      base on r3 alone

as follows r[0.1,-2.-0.5]

---

Two   intrinsic LLR

      We also call it channel LLR perhaps input LLR

      Here's a detailed derivation of its principle

    Let's start with a priori probability （Prior）, front BEC,BSC It's all like this

     

     ,

     

be Using Bayesian principles ：

    

   

     

            among

            

            

              

    be

       

              

      Simplify writing

       

---

      3、 ... and   extrinsic LLR

          Now it's given   How to find

         

          Here we use For example ：

          

          

    

          

                    

         

           

    When

   

   

   

    because   Conditions are independent

  therefore   

 

  Empathy