CN104734814A

CN104734814A - LDPC coding and decoding method applied in incoherent ultra-wideband communication system

Info

Publication number: CN104734814A
Application number: CN201510145176.0A
Authority: CN
Inventors: 梁中华; 臧俊杉; 刘瑾瑾; 李培培
Original assignee: Changan University
Current assignee: Changan University
Priority date: 2015-03-30
Filing date: 2015-03-30
Publication date: 2015-06-24
Anticipated expiration: 2035-03-30
Also published as: CN104734814B

Abstract

The invention discloses an LDPC encoding and decoding method applicable to a non-coherent ultra-wideband communication system, and relates to the field of non-coherent ultra-wideband (UWB) communication, in particular to a non-coherent UWB communication system based on a transmission reference pulse cluster (TRPC). The method of the invention uses two construction methods of LDPC codes in the system, and adopts BP algorithm to decode the LDPC codes, and adopts the algorithm to obtain prior probability according to channel characteristics. The method of the invention aims at the channel characteristics of the TRPC system, and uses the sample value to solve the channel noise variance in actual simulation. Simulation results show that the present invention can obtain better performance than RS codes, systematic long convolutional codes and non-systematic long convolutional codes currently used in non-coherent ultra-wideband communication systems.

Description

A LDPC encoding and decoding method suitable for non-coherent ultra-wideband communication systems

技术领域technical field

本发明属于非相干超宽带无线通信技术领域，涉及基于传输参考脉冲簇的非相干超宽带通信系统中的信道编码方法，具体涉及一种适用于非相干超宽带通信系统的LDPC编码及译码方法。The invention belongs to the technical field of non-coherent ultra-wideband wireless communication, and relates to a channel coding method in a non-coherent ultra-wideband communication system based on transmission reference pulse clusters, in particular to an LDPC encoding and decoding method applicable to an incoherent ultra-wideband communication system .

背景技术Background technique

随着无线通信与网络技术的发展、融合，超宽带(Ultra-wideband,UWB)技术以低成本、低功耗及良好的时域分辨能力等特点成为一种短距离无线通信解决方案。IEEE 802.15.4a标准采用基于脉冲无线电(Impulse radio,IR)制式的UWB方案。IR-UWB系统具备精确的定位性能，然而其仍然面临低成本、低功耗等现实要求的巨大挑战。因此，在低码率场合，更多地是采用复杂度较低、对采样速率要求不高且不需要信道估计的非相干接收机。其中，采用传输参考(Transmitter Reference,TR)技术的自相关接收机由于其结构简单、无需精准同步及信道估计、性能稳健等特点备受关注。然而TR-UWB系统在进行自相关接收时为了避免多径延时扩展造成的脉冲间干扰(Inter-pulse Interference,IPI)，需要引入过长的延迟线来隔离参考脉冲和数据脉冲，这以当前的技术水平来说是不可行的。With the development and integration of wireless communication and network technology, Ultra-wideband (UWB) technology has become a short-distance wireless communication solution with the characteristics of low cost, low power consumption and good time domain resolution. The IEEE 802.15.4a standard adopts the UWB scheme based on the Impulse radio (IR) system. The IR-UWB system has precise positioning performance, but it still faces huge challenges such as low cost and low power consumption. Therefore, in the case of low code rate, more non-coherent receivers with lower complexity, less requirement on sampling rate and no need for channel estimation are used. Among them, autocorrelation receivers using Transmitter Reference (TR) technology have attracted much attention due to their simple structure, no need for precise synchronization and channel estimation, and robust performance. However, in order to avoid inter-pulse interference (Inter-pulse Interference, IPI) caused by multipath delay extension in the TR-UWB system during autocorrelation reception, it is necessary to introduce an overly long delay line to isolate the reference pulse and the data pulse, which is currently unfeasible at the technical level.

传输参考脉冲簇(Transmitter Reference pulse cluster,TRPC)系统的提出就是为了解决TR系统中的长延迟线问题。TRPC信号的参考脉冲和数据脉冲具有更加统一、紧凑的结构。参考脉冲与数据脉冲之间以较小的延迟T_d顺序发送，最小延迟就是一个脉冲的宽度T_p，即T_d＝T_p。一个参考脉冲和一个数据脉冲组成一个脉冲对，每个脉冲对以2T_d秒的间隔重复发送。The transmission reference pulse cluster (Transmitter Reference pulse cluster, TRPC) system is proposed to solve the long delay line problem in the TR system. The reference pulse and data pulse of the TRPC signal have a more uniform and compact structure. The reference pulse and the data pulse are sent sequentially with a small delay T _d , and the minimum delay is the width T _p of one pulse, ie T _d =T _p . A reference pulse and a data pulse form a pulse pair, and each pulse pair is sent repeatedly at intervals of 2T _d seconds.

在TRPC系统中，信号若直接发送很难提供理想的误码率性能，因此在发送前进行信道编码来提高系统性能。IEEE 802.15.4a标准中规定的编码方式是RS码，但是实际中还存在一些比RS码性能更好的码，如系统长卷积码、非系统长卷积码等。In the TRPC system, if the signal is sent directly, it is difficult to provide ideal bit error rate performance, so channel coding is performed before sending to improve system performance. The coding method stipulated in the IEEE 802.15.4a standard is RS code, but in practice there are some codes with better performance than RS code, such as systematic long convolutional codes and non-systematic long convolutional codes.

近年来,LDPC码由于其接近香农限的良好性能而成为理论界研究的热点。在二十世纪六十年代，LDPC码由于受硬件条件的限制而未受到重视，如今LDPC码已得到广泛的应用，其中不乏在通信系统编译码和高码率UWB中的应用。然而，要想将LDPC码应用到非相干TRPC-UWB系统中，还存在如下问题：1、如何构造好的奇偶校验矩阵，避免短环并保证子矩阵的非奇异性，从而得到生成矩阵进行编译码；2、采用BP算法进行译码时，需已知信道的先验概率。3、要求信道的先验概率，还需已知多径信道噪声的方差。In recent years, LDPC codes have become a research hotspot in the theoretical field because of their good performance close to the Shannon limit. In the 1960s, LDPC codes were not paid much attention due to the limitation of hardware conditions. Today, LDPC codes have been widely used, including many applications in communication system coding and decoding and high bit rate UWB. However, in order to apply LDPC codes to non-coherent TRPC-UWB systems, there are still the following problems: 1. How to construct a good parity check matrix, avoid short loops and ensure the non-singularity of the sub-matrix, so as to obtain the generator matrix for Coding and decoding; 2. When using the BP algorithm for decoding, the prior probability of the channel needs to be known. 3. The prior probability of the channel is required, and the variance of the multipath channel noise needs to be known.

发明内容Contents of the invention

本发明的目的在于将性能更好的LDPC码用于非相干超宽带通信系统中，解决LDPC码在该系统编译码过程中的问题，提出了一种适用于非相干超宽带通信系统的LDPC编码及译码方法。The purpose of the present invention is to use the LDPC code with better performance in the non-coherent ultra-wideband communication system, solve the problem of the LDPC code in the system coding and decoding process, and propose a LDPC code suitable for the non-coherent ultra-wideband communication system and decoding method.

为达到上述目的，本发明采用了以下技术方案：To achieve the above object, the present invention adopts the following technical solutions:

包括以下步骤：Include the following steps:

1)、采用LDPC码对待发送信号进行信道编码，得到发送信号；1), using the LDPC code to perform channel coding on the signal to be transmitted to obtain the transmitted signal;

2)、将待发送信号通过UWB信道进行发送；2) Send the signal to be sent through the UWB channel;

3)、接收端接收待发送信号，并对接收到的待发送信号进行滤波，得到接收信号；3), the receiving end receives the signal to be sent, and filters the received signal to be sent to obtain the received signal;

4)、在接收端采用BP算法对接收信号进行译码，计算译码所需的各个参数，获得所需要的接收信号。4) At the receiving end, the BP algorithm is used to decode the received signal, and each parameter required for decoding is calculated to obtain the required received signal.

所述步骤(1)具体过程为：Described step (1) specific process is:

1.1)、设置待发送信号为N_f个紧密排列的脉冲对组成，且每个脉冲对包括一个参考脉冲和一个数据脉冲，按照延迟T_d顺序发送。在发射端生成的发送信号为：1.1), the signal to be sent is set to be composed of N _f closely arranged pulse pairs, and each pulse pair includes a reference pulse and a data pulse, and is sent in order with a delay of T _d . The transmitted signal generated at the transmitter is:

$\begin{matrix} {\overset{~ ~}{s the s}}_{n no} ((t t)) = = \sqrt{\frac{{E E.}_{b b}}{22 {N N}_{f f}}} {Σ Σ}_{i i = = 00}^{n no - - 11} {Σ Σ}_{m m = = 00}^{{N N}_{f f} - - 11} [[g g ((t t - - i i {T T}_{s the s} - - 22 m m {T T}_{d d})) + + {b b}_{n no} ((i i)) g g ((t t - - i i {T T}_{s the s} - - ((22 m m + + 11))))]] \\ = = \sqrt{\frac{{E E.}_{b b}}{22 {N N}_{f f}}} {Σ Σ}_{i i = = 00}^{N N - - 11} {s the s}_{i i} ((t t - - i i {T T}_{s the s})) \end{matrix} - - - - - - ((88))$

其中，E_b代表每比特的发射能量，N代表码字数据包的比特数，N_f代表每个符号中脉冲对重复的次数，g(t)代表能量归一化超宽带脉冲，b_n(i)∈{±1}为发射的二进制数据符号，T_S为符号周期，T_p为脉冲宽度，m为非负整数，t为连续时间变量，s_i(t)定义为：Wherein, E _b represents the transmission energy per bit, N represents the number of bits of the codeword data packet, N _f represents the number of repetitions of the pulse pair in each symbol, g(t) represents the energy normalized UWB pulse, b _n ( i) ∈ {±1} is the transmitted binary data symbol, T _S is the symbol period, T _p is the pulse width, m is a non-negative integer, t is a continuous time variable, s _i (t) is defined as:

${s the s}_{i i} ((t t)) \overset{Δ Δ}{= =} {Σ Σ}_{m m = = 00}^{{N N}_{f f} - - 11} [[g g ((t t - - 22 m m {T T}_{d d})) + + {b b}_{n no} ((i i)) g g ((t t - - ((22 m m + + 11)) {T T}_{d d}))]] - - - - - - ((99))$

1.2)、采用循环移位矩阵构造QC-LDPC码(N,j,k)，N代表码长，j代表列重，k代表行重。首先给出索引矩阵P：1.2), using a cyclic shift matrix to construct a QC-LDPC code (N, j, k), where N represents the code length, j represents the column weight, and k represents the row weight. First the index matrix P is given:

$P P = = [\begin{matrix} 11 & a a & . . . . . . & {a a}^{k k - - 11} \\ b b & ab ab & . . . . . . & {a a}^{k k - - 11} b b \\ . . . . . . \\ {b b}^{j j - - 11} & a a {b b}^{j j - - 11} & . . . . . . & {a a}^{k k - - 11} {b b}^{j j - - 11} \end{matrix}] - - - - - - ((33))$

其中，P是一个维度为j×k的索引矩阵，其中0≤s≤j-1，0≤t≤k-1且a，b是素数，j小于k；索引矩阵中的每个元素的值代表校验矩阵的子矩阵向右移的位数；Among them, P is an index matrix with dimension j×k, where 0≤s≤j-1, 0≤t≤k-1 and a, b are prime numbers, and j is less than k; the value of each element in the index matrix Represents the number of bits shifted to the right of the sub-matrix of the parity check matrix;

校验矩阵H为：Check matrix H is:

$H h = = [\begin{matrix} {I I}_{11} & {I I}_{a a} & . . . . . . & {I I}_{{a a}^{k k - - 11}} \\ {I I}_{b b} & {I I}_{ab ab} & . . . . . . & {I I}_{{a a}^{k k - - 11} b b} \\ . . . . . . \\ {I I}_{{b b}^{j j - - 11}} & {I I}_{{ab ab}^{j j - - 11}} & . . . . . . & {I I}_{{a a}^{k k - - 11} {b b}^{j j - - 11}} \end{matrix}] - - - - - - ((44))$

H的维度为jM×kM，M＝N/k代表子矩阵的维度；The dimension of H is jM×kM, and M=N/k represents the dimension of the sub-matrix;

1.3)将H表示为如下形式：1.3) Express H as the following form:

H＝[A B] (5)H＝[A B] (5)

其中A是维度为jM×jM的方阵，B的维度为jM×(k-j)M；则生成矩阵G：Among them, A is a square matrix with a dimension of jM×jM, and the dimension of B is jM×(k-j)M; then the matrix G is generated:

G＝[(A^-1·B)^T I] (6)G=[(A ^-1 B) ^T I] (6)

其中，方阵A是非奇异矩阵。Among them, the square matrix A is a non-singular matrix.

1.4)、根据生成矩阵G，对待发送信号进行编码，生成发送信号。1.4) According to the generator matrix G, encode the signal to be transmitted to generate the transmitted signal.

所述方阵A构造方法为：The construction method of the square matrix A is:

2.1)、改变索引矩阵P，得到新的索引矩阵P₁：2.1), change the index matrix P to get a new index matrix P ₁ :

${P P}_{11} = = [\begin{matrix} 00 & a a & {a a}^{22} & . . . . . . & {a a}^{k k - - 11} \\ z z & 00 & {a a}^{22} b b & . . . . . . & {a a}^{k k - - 11} b b \\ {b b}^{22} & z z & 00 & . . . . . . & {a a}^{k k - - 11} {b b}^{22} \\ . . . . . . \\ {b b}^{j j - - 11} & a a {b b}^{j j - - 11} & {a a}^{22} {b b}^{j j - - 11} & . . . . . . & {a a}^{k k - - 11} {b b}^{j j - - 11} \end{matrix}] - - - - - - ((77))$

2.2)、在该矩阵中，z代表的是零矩阵，相应的校验矩阵表示为：2.2), in the matrix, what z represents is a zero matrix, and the corresponding check matrix is expressed as:

${H h}_{11} = = [\begin{matrix} {I I}_{00} & {I I}_{a a} & {I I}_{{a a}^{22}} & . . . . . . & {I I}_{{a a}^{k k - - 11}} \\ 00 & {I I}_{00} & {I I}_{{a a}^{22} b b} & . . . . . . & {I I}_{{a a}^{k k - - 11} b b} \\ {I I}_{{b b}^{22}} & 00 & {I I}_{00} & . . . . . . & {I I}_{{a a}^{k k - - 11} {b b}^{22}} \\ . . . . . . \\ {I I}_{{b b}^{j j - - 11}} & {I I}_{a a {b b}^{j j - - 11}} & {I I}_{{a a}^{22} {b b}^{j j - - 11}} & . . . . . . & {I I}_{{a a}^{k k - - 11} {b b}^{j j - - 11}} \end{matrix}] - - - - - - ((88))$

从H₁中可以得到A₁ A ₁ can be obtained from H ₁

${A A}_{11} = = [\begin{matrix} {I I}_{00} & {I I}_{a a} & {I I}_{{a a}^{22}} & . . . . . . & {I I}_{{a a}^{j j - - 11}} \\ 00 & {I I}_{00} & {I I}_{{a a}^{22} b b} & . . . . . . & {I I}_{{a a}^{j j - - 11} b b} \\ {I I}_{{b b}^{22}} & 00 & {I I}_{{a a}^{22} {b b}^{22}} & . . . . . . & {I I}_{{a a}^{j j - - 11} {b b}^{j j - - 11}} \\ . . . . . . \\ {I I}_{{b b}^{j j - - 11}} & {I I}_{a a {b b}^{j j - - 11}} & {I I}_{{a a}^{22} {b b}^{j j - - 11}} & . . . . . . & {I I}_{00} \end{matrix}] - - - - - - ((99))$

通过这种改变，A₁是非奇异的。With this change, _A1 is non-singular.

所述方阵A构造方法为：The method for constructing the square matrix A is:

3.1)、首先交换公式(5)中H的各列，使A的对角线元素全部为1；3.1), first exchange the columns of H in the formula (5), so that all the diagonal elements of A are 1;

3.2)、当交换矩阵H的列时，矩阵的列重和环特性保持不变；3.2), when exchanging the columns of matrix H, the column weight and ring characteristics of matrix remain unchanged;

3.3)、对交换矩阵H进行变换：3.3), transform the exchange matrix H:

第一步：从第1行到第jM行，将方阵A对角线下方的元素1替换为0，同时检查方阵A是否非奇异；如果不是非奇异的，则重复进行该步骤，直到A非奇异；Step 1: From line 1 to line jM, replace the element 1 below the diagonal of square matrix A with 0, and check whether square matrix A is non-singular; if it is not non-singular, repeat this step until A is non-singular;

第二步：从方阵A得到生成矩阵。Step 2: Get the generating matrix from the square matrix A.

所述步骤3)的具体步骤为：The concrete steps of described step 3) are:

在TRPC系统中，信号经过编码且通过UWB信道及滤波器后，接收到的信号为：In the TRPC system, after the signal is encoded and passed through the UWB channel and filter, the received signal is:

${r r}_{n no} ((t t)) = = \overset{~ ~}{s the s} ((t t)) * * h h ((t t)) + + n no ((t t))$

$= = \sqrt{\frac{{E E.}_{b b}}{22 {N N}_{f f}}} {Σ Σ}_{k k = = 00}^{K K - - 11} {α α}_{k k} {Σ Σ}_{i i = = 00}^{N N - - 11} {s the s}_{i i} ((t t - - {τ τ}_{k k} - - i i {T T}_{s the s})) + + n no ((t t)) - - - - - - ((1010))$

其中，*代表线性卷积，n(t)代表加性高斯白噪声，在IEEE802.15.4a标准中，多径信道冲激响应表示为：Among them, * represents linear convolution, and n(t) represents additive white Gaussian noise. In the IEEE802.15.4a standard, the multipath channel impulse response is expressed as:

$h h ((t t)) = = {Σ Σ}_{k k = = 00}^{K K - - 11} {a a}_{k k} δ δ ((t t - - {τ τ}_{k k})) - - - - - - ((1111))$

其中，α_k和τ_k分别代表第K路多径分量的复衰落系数和到达延时；为了确保没有符号间干扰，假定T_S≥2N_fT_d+τ_max，其中τ_max代表信道的最大延时。Among them, α _k and τ _k represent the complex fading coefficient and arrival delay of the K-th multipath component respectively; in order to ensure that there is no inter-symbol interference, it is assumed that T _S ≥ 2N _f T _d +τ _max , where τ _max represents the maximum channel delay.

所述步骤4)的具体步骤为：The concrete steps of described step 4) are:

4.1)、接收端对接收到的信号以及自身的Td延迟进行自相关，得到的判决变量y(i)表示为：4.1), the receiving end performs autocorrelation on the received signal and its own Td delay, and the obtained decision variable y(i) is expressed as:

$y the y ((i i)) = = {&Integral; &Integral;}_{i i {T T}_{S S} + + {T T}_{11}}^{i i {T T}_{s the s} + + {T T}_{22}} {r r}_{n no} ((t t)) {r r}_{n no}^{* *} ((t t - - {T T}_{d d})) dt dt - - - - - - ((1010))$

其中，T₁和T₂分别表示自相关积分区间的起始点和终点，(·)^*表示复数的共轭运算；Among them, T ₁ and T ₂ represent the starting point and end point of the autocorrelation integral interval respectively, ( ) ^* represents the conjugate operation of complex numbers;

4.2)、获取先验概率，并根据先验概率对判决变量y(i)采用BP算法进行译码。4.2) Obtain the prior probability, and decode the decision variable y(i) using the BP algorithm according to the prior probability.

获取先验概率具体包括以下步骤：Obtaining the prior probability specifically includes the following steps:

5.1)、若x是等概率产生的随机变量，那么通过贝叶斯公式可得到：5.1), if x is a random variable generated with equal probability, then the Bayes formula can be obtained:

$\frac{p p ((x x = = {u u}_{00} / / y the y))}{p p ((x x = = {u u}_{11} / / y the y))} = = \frac{p p ((y the y / / x x = = {u u}_{00}))}{p p ((y the y / / x x = = {u u}_{11}))} - - - - - - ((1111))$

其中，u₀代表的发送为“0”时的比特，u₁代表的是发送为“1”时的比特，y代表的是接收到的信息比特。p(y/x＝u₁)代表的是当发送的信息为u₁时接收到y的概率，x是等概率产生的随机变量；Among them, u ₀ represents the bit when the transmission is “0”, u ₁ represents the bit when the transmission is “1”, and y represents the received information bit. p(y/x=u ₁ ) represents the probability of receiving y when the sent information is u ₁ , and x is a random variable generated with equal probability;

5.2)、设u₀代表当比特“0”通过去掉噪声的信道后的值，u₁代表的是比特“1”通过该信道后的值；y是实际系统中通过信道接收到的值；则公式(13)为：5.2), let u ₀ represent the value after the bit “0” passes through the noise-removed channel, u ₁ represents the value after the bit “1” passes through the channel; y is the value received through the channel in the actual system; then Formula (13) is:

$\frac{p p ((y the y / / x x = = {u u}_{00}))}{p p ((y the y / / x x = = {u u}_{11}))} = = \frac{f f ((y the y / / x x = = {u u}_{00}))}{f f ((y the y / / x x = = {u u}_{11}))} = = \frac{\frac{11}{\sqrt{22 π π} σ σ} {e e}^{- - \frac{{((y the y - - {u u}_{00}))}^{22}}{22 {δ δ}^{22}}}}{\frac{11}{\sqrt{22 π π} σ σ} {e e}^{- - \frac{{((y the y - - {u u}_{11}))}^{22}}{22 {δ δ}^{22}}}} - - - - - - ((1212))$

定义p(y/x＝u₁)＝p₁，通过化简(14)可以得到：Define p(y/x=u ₁ )=p ₁ , by simplifying (14), we can get:

$\frac{11 - - {p p}_{11}}{{p p}_{11}} = = {e e}^{\frac{{u u}_{11}^{22} - - {u u}_{00}^{22} + + 22 y the y {u u}_{00} - - 22 y the y {u u}_{11}}{22 {δ δ}^{22}}} - - - - - - ((1313))$

${p p}_{11} = = \frac{11}{11 + + {e e}^{\frac{{u u}_{11}^{22} - - {u u}_{00}^{22} + + 22 y the y {u u}_{00} - - 22 y the y {u u}_{11}}{22 {δ δ}^{22}}}} - - - - - - ((1414))$

5.3)、用先验概率进行初始化后，按照BP算法进行译码，得到u₀，u₁，方差δ²依赖于信道特性，可以通过样本值得到。5.3) After initialization with the prior probability, decode according to the BP algorithm to obtain u ₀ , u ₁ , and the variance δ ² depends on the channel characteristics and can be obtained from the sample value.

与现有技术比较，本发明具有以下有益效果：Compared with the prior art, the present invention has the following beneficial effects:

本发明针对TRPC-UWB系统的信道编码问题，提出一种适用于非相干超宽带通信系统的LDPC编码及译码方法。本发明方法首先将两种LDPC码的构造方法用于TRPC系统中，然后针对该信道特性求出先验概率，还求得信道噪声的方差。利用以上所求，采用BP算法进行译码。还将LDPC编码的TRPC系统与采用RS码、系统长卷积码以及非系统长卷积码的系统进行对比，得出使用LDPC码编码的系统比使用现有的其他编码方式的系统具有更好的误码性能。可见，本发明可以很好的提高TRPC系统的性能。Aiming at the channel coding problem of the TRPC-UWB system, the invention proposes an LDPC coding and decoding method suitable for a non-coherent ultra-wideband communication system. The method of the invention first uses two construction methods of LDPC codes in the TRPC system, and then obtains the prior probability and the variance of the channel noise according to the channel characteristics. Using the above requirements, the BP algorithm is used for decoding. The LDPC coded TRPC system is also compared with the systems using RS codes, systematic long convolutional codes and non-systematic long convolutional codes, and it is concluded that the system using LDPC codes has better error rate coding performance. It can be seen that the present invention can well improve the performance of the TRPC system.

附图说明Description of drawings

图1为本发明中所应用的系统模型；Fig. 1 is the system model applied among the present invention;

图2为本发明中使用两种LDPC码的TRPC系统与未编码的TRPC系统在信道1环境下的性能对比图；Fig. 2 is the performance comparison figure of using the TRPC system of two kinds of LDPC codes and the uncoded TRPC system under channel 1 environment in the present invention;

图3为本发明中使用两种LDPC码的TRPC系统与未编码的TRPC系统在信道8环境下的性能对比图；Fig. 3 uses the TRPC system of two kinds of LDPC yards among the present invention and the performance contrast figure of the unencoded TRPC system under channel 8 environment;

图4为本发明中使用LDPC码的TRPC系统与使用RS码、系统长卷积码以及非系统长卷积码的TRPC系统在信道1环境下的性能对比图；Fig. 4 is the TRPC system that uses LDPC code among the present invention and uses the TRPC system of RS code, systematic long convolutional code and non-systematic long convolutional code's performance contrast chart under channel 1 environment;

图5为本发明中使用LDPC码的TRPC系统与使用RS码、系统长卷积码以及非系统长卷积码的TRPC系统在信道8环境下的性能对比图；Fig. 5 is the TRPC system that uses LDPC code among the present invention and uses the TRPC system of RS code, systematic long convolutional code and non-systematic long convolutional code's performance contrast figure under channel 8 environment;

图6为本发明流程框图。Fig. 6 is a flow chart of the present invention.

具体实施方式Detailed ways

下面结合附图对本发明做进一步详细描述：The present invention is described in further detail below in conjunction with accompanying drawing:

如图1、6所示，示例性的图1中给出了应用场景，适用于非相干超宽带通信系统的LDPC编码及译码方法，具体包括以下步骤：As shown in Figures 1 and 6, an exemplary application scenario is given in Figure 1, which is applicable to an LDPC encoding and decoding method for a non-coherent ultra-wideband communication system, and specifically includes the following steps:

$\begin{matrix} {\overset{~ ~}{s the s}}_{n no} ((t t)) = = \sqrt{\frac{{E E.}_{b b}}{22 {N N}_{f f}}} {Σ Σ}_{i i = = 00}^{n no - - 11} {Σ Σ}_{m m = = 00}^{{N N}_{f f} - - 11} [[g g ((t t - - i i {T T}_{s the s} - - 22 m m {T T}_{d d})) + + {b b}_{n no} ((i i)) g g ((t t - - i i {T T}_{s the s} - - ((22 m m + + 11))))]] \\ = = \sqrt{\frac{{E E.}_{b b}}{22 {N N}_{f f}}} {Σ Σ}_{i i = = 00}^{N N - - 11} {s the s}_{i i} ((t t - - i i {T T}_{s the s})) \end{matrix} - - - - - - ((1515))$

${s the s}_{i i} ((t t)) \overset{Δ Δ}{= =} {Σ Σ}_{m m = = 00}^{{N N}_{f f} - - 11} [[g g ((t t - - 22 m m {T T}_{d d})) + + {b b}_{n no} ((i i)) g g ((t t - - ((22 m m + + 11)) {T T}_{d d}))]] - - - - - - ((1616))$

未编码的系统难以提供理想的误码率性能，采用信道编码可提高系统性能。虽然IEEE 802.15.4a标准规定了RS码、级联码等编码方式，但是仍然存在一些性能更好的码，如系统长卷积码、非系统长卷积码等。考虑到LDPC码是一种性能接近香农限的理想好码，近年来在超宽带通信的很多领域都得到应用。因此采用LDPC码作为TRPC系统的信道编码方式以获得更好的性能。It is difficult for an uncoded system to provide ideal bit error rate performance, and channel coding can improve system performance. Although the IEEE 802.15.4a standard specifies coding methods such as RS codes and concatenated codes, there are still some codes with better performance, such as systematic long convolutional codes and non-systematic long convolutional codes. Considering that LDPC code is an ideal code whose performance is close to the Shannon limit, it has been applied in many fields of ultra-wideband communication in recent years. Therefore, the LDPC code is adopted as the channel coding method of the TRPC system to obtain better performance.

1.2)、为提高LDPC码的性能减少校验矩阵因式图中的短环，采用循环移位矩阵构造QC-LDPC码(N,j,k)，N代表码长，j代表列重，k代表行重，首先给出索引矩阵P：1.2), in order to improve the performance of the LDPC code and reduce the short loop in the parity check matrix factor diagram, the cyclic shift matrix is used to construct the QC-LDPC code (N, j, k), N represents the code length, j represents the column weight, and k Representing the row weight, first give the index matrix P:

校验矩阵H为：Check matrix H is:

1.3)将H表示为如下形式：1.3) Express H as the following form:

H＝[A B] (5)H＝[A B] (5)

其中，A是维度为jM×jM的方阵，B的维度为jM×(k-j)M；则生成矩阵G：Among them, A is a square matrix with a dimension of jM×jM, and the dimension of B is jM×(k-j)M; then the matrix G is generated:

G＝[(A^-1·B)^T I] (6)G=[(A ^-1 B) ^T I] (6)

其中，方阵A是非奇异矩阵。这种方法构造的校验矩阵，既没有四环也没有六环。但是在实际中，这种方法构造的QC-LDPC码并不能保证是A非奇异的，生成矩阵也无法通过上述方法得到。Among them, the square matrix A is a non-singular matrix. The check matrix constructed by this method has neither four rings nor six rings. But in practice, the QC-LDPC code constructed by this method cannot be guaranteed to be A nonsingular, and the generator matrix cannot be obtained by the above method.

在TRPC系统中，n个信息数据包可以表示为：在该数据发送前采用LDPC码进行编码。编码需已知LDPC码的生成矩阵，而生成矩阵是由奇偶校验矩阵转化而来，在已知(4)的情况下，可采用(5)得到生成矩阵。LDPC码编码的主要步骤是构造好的奇偶校验矩阵，既能得到生成矩阵又能避免短环。In the TRPC system, n information packets can be expressed as: Before the data is sent, the LDPC code is used for encoding. Encoding needs to know the generator matrix of the LDPC code, and the generator matrix is transformed from the parity check matrix. If (4) is known, (5) can be used to obtain the generator matrix. The main step of LDPC code encoding is to construct a good parity check matrix, which can not only obtain the generator matrix but also avoid short loops.

采用索引矩阵(3)来构造奇偶校验矩阵如(4)所示。为确保(5)式中子矩阵A的非奇异性，又采用了两种方法对校验矩阵进行改造。第一种方法是改变索引矩阵如(7)式，相应的校验矩阵如(8)式来确保可得到生成矩阵。第二种方法是通过对(5)式进行列变换，然后按行将A的对角线以下的元素1替换为0，逐步判断A是否非奇异，从而得到生成矩阵进行编码。通过LDPC编码后相应的输出为： $c_{n} (i) &Element; {0,1}_{n = 0}^{N - 1} .$ The index matrix (3) is used to construct the parity check matrix as shown in (4). In order to ensure the non-singularity of the sub-matrix A in formula (5), two methods are used to transform the parity check matrix. The first method is to change the index matrix such as formula (7), and the corresponding parity check matrix such as formula (8) to ensure that the generator matrix can be obtained. The second method is to perform column transformation on formula (5), and then replace the elements 1 below the diagonal of A with 0 by row, and gradually judge whether A is non-singular, so as to obtain the generator matrix for encoding. The corresponding output after LDPC encoding is: $c_{no} (i) &Element; {0,1}_{no = 0}^{N - 1} .$

下面是对该校验矩阵进行改变的两种方法，目的是使A为非奇异，然后可以得到生成矩阵G进行编译码。The following are two methods for changing the parity check matrix, in order to make A non-singular, and then the generator matrix G can be obtained for encoding and decoding.

第一种方法：the first method:

从H₁中得到A₁ Get A ₁ from H ₁

通过这种改变，A₁是非奇异的，然后可以得到生成矩阵。这种方法设计的校验矩阵可以保证没有四环，但仍存在六环。With this change, _A1 is non-singular, and then the generator matrix can be obtained. The parity check matrix designed by this method can guarantee that there are no four rings, but there are still six rings.

第二种方法：The second method:

3.1)、首先交换公式(6)中H的各列，使A的对角线元素全部为1；3.1), first exchange the columns of H in the formula (6), so that all the diagonal elements of A are 1;

3.3)、对交换矩阵H进行变换：3.3), transform the exchange matrix H:

第二种方法中，由于只是交换矩阵的列且替换矩阵中的元素1，并不改变环特性，因此使用这种方法设计的校验矩阵不存在六环。In the second method, because only the columns of the matrix are exchanged and the element 1 in the matrix is replaced, the ring characteristics are not changed, so the check matrix designed by this method does not have six rings.

2)、将待发送信号通过UWB信道进行发送；据通过发射机发射后，变为具体表达式如公式(1)所示。2), send the signal to be sent through the UWB channel; after the data is transmitted through the transmitter, it becomes The specific expression is shown in formula (1).

3)、接收端接收待发送信号，并对接收到的待发射信号进行滤波，得到接收信号；3), the receiving end receives the signal to be transmitted, and filters the received signal to be transmitted to obtain the received signal;

$\begin{matrix} {r r}_{n no} ((t t)) = = \overset{~ ~}{s the s} ((t t)) * * h h ((t t)) + + n no ((t t)) \\ = = \sqrt{\frac{{E E.}_{b b}}{22 {N N}_{f f}}} {Σ Σ}_{k k = = 00}^{K K - - 11} {α α}_{k k} {Σ Σ}_{i i = = 00}^{N N - - 11} {s the s}_{i i} ((t t - - {τ τ}_{k k} - - i i {T T}_{s the s})) + + n no ((t t)) \end{matrix} - - - - - - ((1010))$

所述步骤4)的具体步骤为：The concrete steps of described step 4) are:

$y the y ((i i)) = = {&Integral; &Integral;}_{i i {T T}_{S S} + + {T T}_{11}}^{i i {T T}_{s the s} + + {T T}_{22}} {r r}_{n no} ((t t)) {r r}_{n no}^{* *} ((t t - - {T T}_{d d})) dt dt - - - - - - ((1717))$

4.2)、获取先验概率，并根据先验概率对判决变量y(i)采用BP算法进行译码。4.2) Obtain the prior probability, and use the BP algorithm to decode the decision variable y(i) according to the prior probability.

$\frac{p p ((x x = = {u u}_{00} / / y the y))}{p p ((x x = = {u u}_{11} / / y the y))} = = \frac{p p ((y the y / / x x = = {u u}_{00}))}{p p ((y the y / / x x = = {u u}_{11}))} - - - - - - ((1818))$

$\frac{p p ((y the y / / x x = = {u u}_{00}))}{p p ((y the y / / x x = = {u u}_{11}))} = = \frac{f f ((y the y / / x x = = {u u}_{00}))}{f f ((y the y / / x x = = {u u}_{11}))} = = \frac{\frac{11}{\sqrt{22 π π} σ σ} {e e}^{- - \frac{{((y the y - - {u u}_{00}))}^{22}}{22 {δ δ}^{22}}}}{\frac{11}{\sqrt{22 π π} σ σ} {e e}^{- - \frac{{((y the y - - {u u}_{11}))}^{22}}{22 {δ δ}^{22}}}} - - - - - - ((1919))$

定义p(y/x＝u₁)＝p₁，通过化简(14)可以得到Define p(y/x=u ₁ )=p ₁ , by simplifying (14), we can get

$\frac{11 - - {p p}_{11}}{{p p}_{11}} = = {e e}^{\frac{{u u}_{11}^{22} - - {u u}_{00}^{22} + + 22 y the y {u u}_{00} - - 22 y the y {u u}_{11}}{22 {δ δ}^{22}}} - - - - - - ((2020))$

该算法需已知信道的先验概率，先验概率公式推导过程见公式(13)-(16)。可见要得到先验概率需先求得信道噪声方差。由于实际推导过程需考虑随机信号的自相关与互相关作用，因此难以得出噪声方差的完整表达式，但是可以从样本值中得到。在实际的系统中，利用IEEE 802.15.4a信道模型标准程序建立一个有噪声的模板和一个去除噪声的模板，然后让一个训练符号分别通过这两个模板，所得到的值便可以用来计算噪声方差δ²。为使所求值具有一般性，分别采用不同的次数求取平均值，如100次、1000次或10000次，在仿真过程中发现1000次求平均时就接近最佳性能，因此选取1000次求平均值以达到良好的效果。This algorithm needs to know the prior probability of the channel, and the derivation process of the prior probability formula is shown in formulas (13)-(16). It can be seen that to obtain the prior probability, the channel noise variance must be obtained first. Since the actual derivation process needs to consider the autocorrelation and cross-correlation of random signals, it is difficult to obtain a complete expression of the noise variance, but it can be obtained from the sample value. In the actual system, use the IEEE 802.15.4a channel model standard program to establish a template with noise and a template to remove noise, and then let a training symbol pass through the two templates, and the obtained value can be used to calculate the noise Variance δ ² . In order to make the evaluated values general, different times are used to obtain the average value, such as 100 times, 1000 times or 10000 times. In the simulation process, it is found that the average performance of 1000 times is close to the best performance, so choose 1000 times to calculate the average value. average for good results.

本发明的仿真过程分别在信道1和信道8环境下进行。如图2所示为使用两种方法构造LDPC码的TRPC系统在信道1环境下的性能对比仿真图。如图3所示为使用两种方法构造LDPC码的TRPC系统在信道8环境下的性能对比仿真图。以上两图均与未编码系统进行了对比，由仿真曲线可见，采用LDPC码编码的系统比未编码系统有较大的性能改善，且采用第二种方法构造LDPC码的系统比采用第一种方法的系统具有更好的性能。图4和图5是使用第一种方法构造LDPC码的系统与使用RS码、系统长卷积码以及非系统长卷积码的系统在信道1与信道8的性能对比图。由图可见LDPC码编码的系统具有最好的性能。The simulation process of the present invention is carried out under the environment of channel 1 and channel 8 respectively. As shown in Figure 2, it is a performance comparison simulation diagram of the TRPC system using two methods to construct LDPC codes in the channel 1 environment. As shown in Fig. 3, it is a performance comparison simulation diagram of a TRPC system using two methods to construct LDPC codes under channel 8 environment. The above two figures are compared with the unencoded system. It can be seen from the simulation curve that the system using LDPC code encoding has a greater performance improvement than the unencoded system, and the system using the second method to construct LDPC codes is better than the first one. method of the system has better performance. Figure 4 and Figure 5 are performance comparison diagrams on channel 1 and channel 8 between the system using the first method to construct LDPC codes and the system using RS codes, systematic long convolutional codes and non-systematic long convolutional codes. It can be seen from the figure that the system encoded by LDPC code has the best performance.

以上所述TRPC-UWB系统仅为本发明非相干UWB通信系统的优选实施例而已，并不用于限制本发明，对于本领域的技术人员来说，本发明可以有各种更改和变化，该方法可用于其他非相干UWB通信系统中，如TR系统，NC-PPM系统等。凡在本发明的精神和原则之内，所作的任何修改、等同替换、改进等，均应包含在本发明的保护范围之内。The TRPC-UWB system described above is only a preferred embodiment of the non-coherent UWB communication system of the present invention, and is not used to limit the present invention. For those skilled in the art, the present invention can have various modifications and changes. The method It can be used in other non-coherent UWB communication systems, such as TR system, NC-PPM system, etc. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A kind of LDPC encoding and decoding method applicable to non-coherent ultra-wideband communication system, is characterized in that, comprises the following steps:

1), using the LDPC code to perform channel coding on the signal to be transmitted to obtain the transmitted signal;

2) Send the signal to be sent through the UWB channel;

3), the receiving end receives the signal to be sent, and filters the received signal to be sent to obtain the received signal;

4) At the receiving end, the BP algorithm is used to decode the received signal, and each parameter required for decoding is calculated to obtain the required received signal.

2. a kind of LDPC encoding and decoding method applicable to non-coherent ultra-wideband communication system according to claim 1, is characterized in that, described step 1) specific process is:

1.1), the signal to be sent is set to be composed of N _f closely arranged pulse pairs, and each pulse pair includes a reference pulse and a data pulse, which are sent in order of delay T _d , and the sent signal generated at the transmitter is:

\begin{matrix} {\overset{~ ~}{s the s}}_{n no} ((t t)) = = \sqrt{\frac{{E E.}_{b b}}{{22 N N}_{f f}}} {Σ Σ}_{i i = = 00}^{N N - - 11} {Σ Σ}_{m m = = 00}^{{N N}_{f f} - - 11} [[g g ((t t - - {iT i}_{s the s} - - {22 mT mT}_{d d})) + + {b b}_{n no} ((i i)) g g ((t t - - {iT i}_{s the s} - - ((22 m m + + 11))))]] \\ = = \sqrt{\frac{{E E.}_{b b}}{{22 N N}_{f f}}} {Σ Σ}_{i i = = 00}^{N N - - 11} {s the s}_{i i} ((t t - - {iT i}_{s the s})) \end{matrix} - - - - - - ((11))

Wherein, E _b represents the transmission energy per bit, N represents the number of bits of the codeword data packet, N _f represents the number of repetitions of the pulse pair in each symbol, g(t) represents the energy normalized UWB pulse, b _n ( i) ∈ {±1} is the transmitted binary data symbol, T _S is the symbol period, T _p is the pulse width, m is a non-negative integer, t is a continuous time variable, s _i (t) is defined as:

{s the s}_{i i} ((t t)) \overset{Δ Δ}{= =} {Σ Σ}_{m m = = 00}^{{N N}_{f f} - - 11} [[g g ((t t - - {22 mT mT}_{d d})) + + {b b}_{n no} ((i i)) g g ((t t - - ((22 m m + + 11)) {T T}_{d d}))]] - - - - - - ((22))

1.2), use the cyclic shift matrix to construct the QC-LDPC code (N, j, k), N represents the code length, j represents the column weight, k represents the row weight, first give the index matrix P:

P P = = [\begin{matrix} 11 & a a & . . . . . . & {a a}^{k k - - 11} \\ b b & ab ab & . . . . . . & {a a}^{k k - - 11} b b \\ . . . . . . \\ {b b}^{j j - - 11} & {ab ab}^{j j - - 11} & . . . . . . & {a a}^{k k - - 11} {b b}^{j j - - 11} \end{matrix}] - - - - - - ((33))

Among them, P is an index matrix with dimension j×k, where 0≤s≤j-1, 0≤t≤k-1 and a, b are prime numbers, and j is less than k; the value of each element in the index matrix Represents the number of bits shifted to the right of the sub-matrix of the parity check matrix;

Check matrix H is:

H h = = [\begin{matrix} {I I}_{11} & {I I}_{a a} & . . . . . . & {I I}_{{a a}^{k k - - 11}} \\ {I I}_{b b} & {I I}_{ab ab} & . . . . . . & {I I}_{{a a}^{k k - - 11} b b} \\ . . . . . . \\ {I I}_{{b b}^{j j - - 11}} & {I I}_{{ab ab}^{j j - - 11}} & . . . . . . & {I I}_{{a a}^{k k - - 11} {b b}^{j j - - 11}} \end{matrix}] - - - - - - ((44))

The dimension of H is jM×kM, and M=N/k represents the dimension of the sub-matrix;

1.3) Express H as the following form:

H＝[AB] (5)

Among them, A is a square matrix with a dimension of jM×jM, and the dimension of B is jM×(k-j)M; then the matrix G is generated:

G=[(A ^-1 B) ^T I] (6)

Among them, the square matrix A is a non-singular matrix;

1.4) According to the generator matrix G, encode the signal to be transmitted to generate the transmitted signal.

3. a kind of LDPC encoding and decoding method that is suitable for non-coherent ultra-wideband communication system according to claim 2, it is characterized in that, described square matrix A construction method is:

2.1), change the index matrix P to get a new index matrix P ₁ :

{P P}_{11} = = [\begin{matrix} 00 & a a & {a a}^{22} & . . . . . . & {a a}^{k k - - 11} \\ z z & 00 & {a a}^{22} b b & . . . . . . & {a a}^{k k - - 11} b b \\ {b b}^{22} & z z & 00 & . . . . . . & {a a}^{k k - - 11} {b b}^{22} \\ . . . . . . \\ {b b}^{j j - - 11} & {ab ab}^{j j - - 11} & {a a}^{22} {b b}^{j j - - 11} & . . . . . . & {a a}^{k k - - 11} {b b}^{j j - - 11} \end{matrix}] - - - - - - ((77))

2.2), in the matrix, what z represents is a zero matrix, and the corresponding check matrix is expressed as:

{H h}_{11} = = [\begin{matrix} {I I}_{00} & {I I}_{a a} & {I I}_{{a a}^{22}} & . . . . . . & {I I}_{{a a}^{k k - - 11}} \\ 00 & {I I}_{00} & {I I}_{{a a}^{22} b b} & . . . . . . & {I I}_{{a a}^{k k - - 11} b b} \\ {I I}_{{b b}^{22}} & 00 & {I I}_{00} & . . . . . . & {I I}_{{a a}^{k k - - 11} {b b}^{22}} \\ . . . . . . \\ {I I}_{{b b}^{j j - - 11}} & {I I}_{{ab ab}^{j j - - 11}} & {I I}_{{a a}^{22} {b b}^{j j - - 11}} & . . . . . . & {I I}_{{a a}^{k k - - 11} {b b}^{j j - - 11}} \end{matrix}] - - - - - - ((88))

A ₁ can be obtained from H ₁

{A A}_{11} = = [\begin{matrix} {I I}_{00} & {I I}_{a a} & {I I}_{{a a}^{22}} & . . . . . . & {I I}_{{a a}^{j j - - 11}} \\ 00 & {I I}_{00} & {I I}_{{a a}^{22} b b} & . . . . . . & {I I}_{{a a}^{j j - - 11} b b} \\ {I I}_{{b b}^{22}} & 00 & {I I}_{{a a}^{22} {b b}^{22}} & . . . . . . & {I I}_{{a a}^{j j - - 11} {b b}^{j j - - 11}} \\ . . . . . . \\ {I I}_{{b b}^{j j - - 11}} & {I I}_{{ab ab}^{j j - - 11}} & {I I}_{{a a}^{22} {b b}^{j j - - 11}} & . . . . . . & {I I}_{00} \end{matrix}] - - - - - - ((99))

With this change, _A1 is non-singular.

4. A kind of LDPC encoding and decoding method applicable to non-coherent ultra-wideband communication system according to claim 2, is characterized in that, described square matrix A construction method is:

3.1), first exchange the columns of H in the formula (5), so that all the diagonal elements of A are 1;

3.2), when exchanging the columns of matrix H, the column weight and ring characteristics of matrix remain unchanged;

3.3), transform the exchange matrix H:

Step 1: From line 1 to line jM, replace the element 1 below the diagonal of square matrix A with 0, and check whether square matrix A is non-singular; if it is not non-singular, repeat this step until A is non-singular;

Step 2: Get the generating matrix from the square matrix A.

5. a kind of LDPC encoding and decoding method that is suitable for non-coherent ultra wideband communication system according to claim 2, it is characterized in that, the concrete steps of described step 3) are:

In the TRPC system, after the signal is encoded and passed through the UWB channel and filter, the received signal is:

\begin{matrix} {r r}_{n no} ((t t)) = = \overset{~ ~}{s the s} ((t t)) * * h h ((t t)) + + n no ((t t)) \\ = = \sqrt{\frac{{E E.}_{b b}}{{22 N N}_{f f}}} {Σ Σ}_{k k = = 00}^{K K - - 11} {α α}_{k k} {Σ Σ}_{i i = = 00}^{N N - - 11} {s the s}_{i i} ((t t - - {τ τ}_{k k} - - {iT i}_{s the s})) + + n no ((t t)) \end{matrix} - - - - - - ((1010))

Among them, * represents linear convolution, and n(t) represents additive white Gaussian noise. In the IEEE 802.15.4a standard, the multipath channel impulse response is expressed as:

h h ((t t)) = = {Σ Σ}_{k k = = 00}^{K K - - 11} {a a}_{k k} δ δ ((t t - - {τ τ}_{k k})) - - - - - - ((1111))

Among them, α _k and τ _k represent the complex fading coefficient and arrival delay of the multipath component of the Kth path respectively; in order to ensure that there is no inter-symbol interference, set T _S ≥ 2N _f T _d +τ _max , where τ _max represents the maximum channel delay.

6. A kind of LDPC encoding and decoding method applicable to non-coherent ultra-wideband communication system according to claim 2, it is characterized in that, the concrete steps of described step 4) are:

4.1), the receiving end performs autocorrelation on the received signal and its own T _d delay, and the obtained decision variable y(i) is expressed as:

y the y ((i i)) = = {&Integral; &Integral;}_{{iT i}_{s the s} + + {T T}_{11}}^{{iT i}_{s the s} + + {T T}_{22}} {r r}_{n no} ((t t)) {r r}_{n no}^{* *} ((t t - - {T T}_{d d})) dt dt - - - - - - ((33))

Among them, T ₁ and T ₂ represent the starting point and end point of the autocorrelation integral interval respectively, ( ) ^* represents the conjugate operation of complex numbers;

4.2) Obtain the prior probability, and decode the decision variable y(i) using the BP algorithm according to the prior probability.

7. A kind of LDPC encoding and decoding method that is suitable for non-coherent ultra-wideband communication system according to claim 6, is characterized in that, obtaining prior probability specifically comprises the following steps:

5.1), if x is a random variable generated with equal probability, then the Bayes formula can be obtained:

\frac{p p ((x x = = {u u}_{00} / / y the y))}{p p ((x x = = {u u}_{11} / / y the y))} = = \frac{p p ((y the y / / x x = = {u u}_{00}))}{p p ((y the y / / x x = = {u u}_{11}))} - - - - - - ((44))

Among them, u ₀ represents the bit when the transmission is "0", u ₁ represents the bit when the transmission is "1", y represents the received information bit, and p(y/x=u ₁ ) represents the is the probability of receiving y when the sent information is u ₁ , and x is a random variable generated with equal probability;

5.2), let u ₀ represent the value after the bit “0” passes through the noise-removed channel, u ₁ represents the value after the bit “1” passes through the channel; y is the value received through the channel in the actual system; then Formula (13) is:

\frac{p p ((y the y / / x x = = {u u}_{00}))}{p p ((y the y / / x x = = {u u}_{11}))} = = \frac{f f ((y the y / / x x = = {u u}_{00}))}{f f ((y the y / / x x = = {u u}_{11}))} = = \frac{{\frac{11}{\sqrt{22 π π} σ σ} e e}^{- - \frac{{((y the y - - {u u}_{00}))}^{22}}{{22 δ δ}^{22}}}}{{\frac{11}{\sqrt{22 π π} σ σ} e e}^{- - \frac{{((y the y - - {u u}_{11}))}^{22}}{{22 δ δ}^{22}}}} - - - - - - ((55))

Define p(y/x=u ₁ )=p ₁ , by simplifying (14), we can get

\frac{11 - - {p p}_{11}}{{p p}_{11}} = = {e e}^{\frac{{u u}_{11}^{22} - - {u u}_{00}^{22} {+ + 22 yu you}_{00} - - {22 yu you}_{11}}{{22 δ δ}^{22}}} - - - - - - ((66))

{p p}_{11} = = \frac{11}{11 + + {e e}^{\frac{{u u}_{11}^{22} - - {u u}_{00}^{22} + + {22 yu you}_{00} - - {22 yu you}_{11}}{{22 δ δ}^{22}}}}

5.3) After initialization with the prior probability, decode according to the BP algorithm to obtain u ₀ , u ₁ , and the variance δ ² depends on the channel characteristics and can be obtained from the sample value.