CN110768677B - Optimization method of multi-edge type low-density parity check code - Google Patents

Optimization method of multi-edge type low-density parity check code Download PDF

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CN110768677B
CN110768677B CN201910887440.6A CN201910887440A CN110768677B CN 110768677 B CN110768677 B CN 110768677B CN 201910887440 A CN201910887440 A CN 201910887440A CN 110768677 B CN110768677 B CN 110768677B
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游莹
陈平平
林捷
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Fujian Polytechnic Normal University
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
    • H03M13/11—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits using multiple parity bits
    • H03M13/1102—Codes on graphs and decoding on graphs, e.g. low-density parity check [LDPC] codes
    • H03M13/1105—Decoding
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
    • H03M13/11—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits using multiple parity bits
    • H03M13/1102—Codes on graphs and decoding on graphs, e.g. low-density parity check [LDPC] codes
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Abstract

The invention belongs to the technical field of digital communication, and particularly relates to an optimization method of a multilateral low-density parity check code. The method of the invention comprises the following steps: setting a threshold value of an SPC codec; setting a linear programming objective function for optimizing an SPC codec according to a code rate formula; setting a coefficient vector of the linear programming objective function; setting linear inequality constraints according to the data flow of the EXIT graph of the SPC codec; setting a linear equation constraint condition; setting an upper bound, a lower bound and an initial value of the optimal degree distribution vector lambda of the matrix to be optimized; setting selection parameters of a linear programming function; and (5) solving the optimal solution of lambda by adopting a linear programming function to obtain the optimized degree distribution condition of the SPC coder. The method of the application calculates the optimal degree distribution of the MET-LDPC code by combining an external information transfer graph, a linear programming method and an interpolation method, not only reduces the calculation complexity, but also ensures that the decoding performance of the optimized code word is obviously improved.

Description

Optimization method of multilateral low-density parity check code
Technical Field
The invention belongs to the technical field of digital communication, and particularly relates to an optimization method of a multilateral low-density parity check code.
Background
With the continuous development of modern communication technology, the channel error correction coding theory has also gained vigorous development. Of most interest is the Low Density Parity Check (LDPC) code proposed by Gallager in 1963. Although the LDPC code has excellent error correction performance, its development is limited by the disadvantages of high coding and decoding complexity. To solve this problem, richardson proposed a new class of code patterns, namely the polygonal-type low-density parity-check codes (MET-LDPC), in 2004. (T.J.Richardson and R.L.Urbanke.Multi-Edge Type LDPC codes.accessed: jan.2002.[ Online ]. Available: https:// www.research. Net/publication/37439748. Multi-Edge _ Type _ LDPC codes). The multilateral low-density parity check (MET-LDPC) code covers the structural characteristics of most of the existing code words and has excellent coding and decoding performance. It is worth noting that the MET-LDPC code introduces variable nodes with degree 1, namely a single check coder (SPC coder), which has a large influence on the convergence behavior during decoding. The degree distribution of an SPC coder in the MET-LDPC code is reasonably designed, and the degree distribution has positive influence on the performance of code words. Therefore, it is necessary to design a good degree distribution of SPC codec in MET-LDPC code, but there is currently a lack of research in this direction.
Density evolution algorithms can be used to design the degree distribution of codewords, but the computational complexity is very high. Therefore, stephan ten Brink in 1999 proposed that the convergence behavior of iterative decoding of concatenated codewords can be effectively analyzed also with extrinsic information transfer (EXIT) maps. The EXIT graph not only can ensure the correctness of the decoding threshold, but also greatly reduces the calculation complexity compared with a density evolution algorithm. At present, the EXIT graph has been successfully used in the traditional irregular LDPC code and E 2 RC codes, etc. However, there is currently no research on a degree distribution optimization method for MET-LDPC codes having unique structures, excellent decoding performance, and low encoding and decoding complexity.
Disclosure of Invention
One of the objectives of the present invention is to overcome the above disadvantages and provide an optimization method for a multi-edge low density parity check code, which utilizes the multi-edge structure characteristics of the parity check matrix of MET-LDPC code, and uses EXIT diagram, linear programming method and interpolation method to obtain the threshold value σ at a given value n The optimal degree distribution under the condition of (2), and the decoding performance of the code word after the degree distribution optimization is obviously improved compared with the code word which is not optimized.
In order to solve the technical problem, the invention provides an optimization method of a multilateral low-density parity check code, which comprises the following steps:
step 1, setting the threshold value of the SPC codec as σ to obtain the optimal distribution under the given threshold value n ；
Step 2, setting and optimizing a linear programming objective function of the SPC codec according to a code rate formula;
and 3, setting a coefficient vector f of the linear programming objective function as:
Figure BDA0002207743800000021
dv is the maximum value of the node degree of the information bit; />
Step 4, setting according to the data flow of the EXIT graph of the SPC codecLinear inequality constraint: a is more than b, wherein A is a linear inequality constraint coefficient matrix, b is a constraint vector of a linear inequality, and lambda is an optimal degree distribution vector of a matrix to be optimized, namely lambda = [ lambda ] 2 ，λ 3 ，......，λ dv ] T ；
Step 5, setting a linear equation constraint condition: aeq × λ = beq, where Aeq is a full 1 matrix of 1 row and dv column, and beq =1;
step 6, setting lambda in the optimal degree distribution vector lambda of the matrix to be optimized i Has an upper boundary ub of 1 and a lower boundary lb of 0, and is set to lambda i The initial value start of (1) is 0, wherein i is more than or equal to 2 and less than or equal to dv;
step 7, setting a selection parameter option of the linear programming function;
and 8, solving the optimal solution of lambda by adopting a linear programming function routine (f, A, b, aeq, beq, lb, ub, start and option), and obtaining the degree distribution condition of the optimized SPC coder.
Further, the code rate formula is:
Figure BDA0002207743800000022
the linear programming objective function is:
Figure BDA0002207743800000023
wherein λ is i H being a low density parity check code of the multilateral type 3 The proportion of the number of edges of the information bit nodes with the moderate degree of i in the matrix to the total number of edges of the information bit nodes, dv is the maximum value of the degrees of the information bit nodes, and M is the total number of edges of the information bit nodes.
Further, the "setting the linear inequality constraint according to the data flow of the EXIT graph of the SPC codec: a x lambda < b ", comprising the steps of:
step 4.1, according to the set threshold value sigma of SPC codec n And calculating the variance of the channel likelihood information, wherein the calculation formula is as follows:
Figure BDA0002207743800000024
step 4.2, initializing degree distribution k =2 of the current information bit node;
step 4.3, calculating the information bit variable node V with the medium value of k in the EXIT graph Inf To check node C Spc (V Inf →C Spc ) Likelihood information of (1) and (V) Inf Average mutual information IEV between k The calculation formula is as follows:
Figure BDA0002207743800000031
wherein, IAV StoI For check node C in EXIT graph Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of (1) and (V) Inf The prior mutual information between the two is that the value range is more than 0 and less than IAV StoI ＜1，/>
Figure BDA0002207743800000032
The variance of the channel likelihood information is expressed, and the formula of the J function is as follows:
Figure BDA0002207743800000033
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，C 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608，
J -1 the function is the inverse of the J function;
4.4, judging whether k is smaller than the maximum value dc of the node degree of the information bit, if k is smaller than dv, calculating k = k +1, and then skipping to the step 4.3, otherwise skipping to the step 4.5;
step 4.5, updating coefficient matrix A = [ -IEV) of linear inequality constraint 2 ，-IEV 3 ，...-IEV k ，...，-IEV dv ]Where k is an integer satisfying 2 ≦ k ≦ dv, which represents a maximum value of the information bit node degree.
Step 4.6, meterCalculating single check code V in EXIT graph Spc To check node C Spc (V Spc →C Spc ) Likelihood information of and V Spc Average mutual information IEV between StoS The calculation formula is as follows:
Figure BDA0002207743800000034
wherein +>
Figure BDA0002207743800000035
The variance of the channel likelihood information is expressed, and the formula of the J function is as follows:
Figure BDA0002207743800000036
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608；
step 4.7, calculating the secondary check code V in the EXIT graph Spc For check node C Spc (V Spc →C Spc ) Likelihood information of (1) and (V) Spc Prior mutual information IAC between StoS The calculation formula is as follows: IAC StoS ＝IEV StoS ；
Step 4.8, calculating check node C in the EXIT graph Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of and V Inf Average mutual information IEC between StoI The calculation formula is as follows:
Figure BDA0002207743800000041
wherein, IAC ItoS For varying the node V from the information bit in the EXIT diagram Inf For check node C Spc (V Inf →C Spc ) Likelihood information of and V Inf A priori mutual information between them, the values of whichRanging between 0 and 1, dc 2 Degree of connection between check node Cspc and information bit variable node, dc 3 For the degree of connection between the check node Cspc and the single check code variable node, the J function formula is as follows:
Figure BDA0002207743800000042
wherein, the first and the second end of the pipe are connected with each other,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608，
the J-1 function is the inverse of the J function;
step 4.9, solving the current IEC by using a linear interpolation method StoI When the known IAC is known (expressed by Iec, namely O < Iec < 1), the IAC corresponding to the known IAC ItoS The value situation (expressed by Iac) is calculated by the following formula: iac = interp1 (IED) StoI ，IAC ItoS Iec, 'linear'), where interp1 is a linear interpolation function.
Step 4.10, calculating the value of the inequality constraint vector b, wherein the calculation formula is as follows: b = [ -Iac T ]。
In summary, the technical scheme of the invention has the following beneficial effects:
1. according to the structure characteristics of multiple edges of a parity check matrix of the MET-LDPC code, the degree distribution of a single check coder (SPC coder) which has great influence on convergence during decoding is reasonably designed, and the optimal effect of performance optimization of code words is ensured.
2. By combining an external information transfer (EXIT) graph, a linear programming method and an interpolation method, the optimal degree distribution under the condition of a given threshold value is obtained, so that the calculation complexity is greatly reduced compared with the existing density evolution algorithm, and the decoding performance of the code words with optimized degree distribution is obviously improved compared with the code words without optimization.
Drawings
Fig. 1 is a flowchart of the steps of an optimization method for a multi-edge low density parity check code according to the present invention.
FIG. 2 is a flow chart of the inventive step of setting linear inequality constraints based on the data flow of the SPC codec EXIT graph.
FIG. 3 is a EXIT data flow diagram of an SPC codec in the MET-LDPC code of the present invention.
FIG. 4 is a graph comparing the BER performance of MET-LDPC code before and after optimization according to the present invention.
FIG. 5 is a graph of the present invention for different matrices (H) 2 Or H 3 Matrix) to optimize the BER performance of the MET-LDPC code.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The multi-edge type low-density parity check code (MET-LDPC code) is firstly proposed by Tom Richardson and the like in 2004, the code type has better performance and wider application range, the MET-LDPC code comprises a wider coding framework, and the edge class of the MET-LDPC code covers the structural characteristics of CT codes, RA codes and LDPC codes. The MET-LDPC code has the structural characteristics that various types of edge classes are introduced into a factor graph, variable nodes with the degree of 1 and variable nodes with the degree of 2 are also introduced, and therefore, a parity check matrix of the MET-LDPC code can be divided into 4 types of sub-matrices as follows:
Figure BDA0002207743800000051
wherein H 1 Is a dual diagonal matrix, e.g.
Figure BDA0002207743800000061
H 4 Is a single diagonal matrix, e.g.
Figure BDA0002207743800000062
H 2 Is a matrix with row weights and column weights all 3, H 3 Is the matrix to be optimized. In addition, according to the code word structure definition of MET-LDPC code, the matrix H 1 And H 2 Accumulated code (ACC) codec, matrix H, forming MET-LDPC codes 3 And H 4 A single check code (SPC) codec of the MET-LDPC code is composed. In the MET-LDPC code, the variable node with the degree of 1, that is, the SPC codec has a significant influence on the decoding performance, and therefore, the MET-LDPC code needs to be optimized, mainly for the SPC codec, so that a better effect can be achieved. From the sub-matrix description of the parity-check matrix described above, it can be seen that H in the SPC codec 4 The structure of the matrix is fixed, so the SPC codec is optimized, i.e. for H 3 The degree distribution of the matrix is optimized. Therefore, the optimization of the MET-LDPC code described in the present application is to find the H in the MET-LDPC code under the condition of a given threshold value 3 Optimal degree distribution of matrix λ = [ λ ] 2 ，λ 3 ，......，λ dv ] T 。
Fig. 1 is a flowchart illustrating steps of an optimization method for a multilateral low density parity check code according to the present invention, which includes the following steps:
step 1, setting the threshold value of the SPC codec as σ to obtain the optimal distribution under the given threshold value n ；
Step 2, setting and optimizing a linear programming objective function of the SPC codec according to a code rate formula;
the present application employs a linear programming function to compute H 3 The optimal degree distribution of the matrix firstly has to set a linear programming objective function, and the H of the MET-LDPC code is assumed 3 In the matrix, the ratio of the number of edges of the information bit node with the degree i to the total number of edges of the information bit node is lambda i Wherein i is more than or equal to 2 and less than or equal to dv, dv represents the maximum value of the node degree of the information bit, and the total number of edges of the information bit node is M, the code rate R of the SPC codec in the MET-LDPC code can be calculated as:
Figure BDA0002207743800000063
in this case, to obtain the optimal degree distribution, it is required to ensure that C-R is the minimum, where C represents the shannon limit, is a fixed value, and thus requires the code rate R to be the maximum. From the above formula of the code rate R, it is necessary to maximize R
Figure BDA0002207743800000071
Minimum, and therefore linear programmed, objective function Φ (λ) of SPC codec i ) Is set to->
Figure BDA0002207743800000072
And 3, setting a coefficient vector f of the linear programming objective function, wherein the coefficient vector f is fixedly set in the application
Figure BDA0002207743800000073
Dv is the maximum value of the node degree of the information bit;
and 4, setting linear inequality constraint conditions according to the data flow of the EXIT graph of the SPC codec: a is multiplied by lambda and is less than b, the constraint condition of the linear inequality is used as a parameter for solving the subsequent linear programming function, wherein A is a linear inequality constraint coefficient matrix, b is a constraint vector of the linear inequality, and lambda is an optimal degree distribution vector of a matrix to be optimized, namely lambda = [ lambda ] 2 ，λ 3 ，......，λ dv ] T ；
An external Information Transfer (EXIT) diagram is a new method for analyzing the performance of a code word proposed by Stephan ten Brink in "conversion of iterative decoding" of its article. The EXIT graph describes the relationship between input information and output information in the iterative decoding process, and the decoding track shows the external information exchange between the decoders in the EXIT transition graph. The EXIT graph has been successfully used in parallel concatenated codes (PC codes), repeat accumulate codes (RA codes), low density parity check codes (LDPC codes), and protograph LDPC codes as a new analysis tool.
Fig. 2 is a flowchart of the step of setting linear inequality constraints according to the data flow of the EXIT graph of the SPC codec, that is, a specific setting process of the parameters a and b, according to the present invention, including the following steps:
step 4.1, according to the set threshold value sigma of SPC codec n And calculating the variance of the channel likelihood information, wherein the calculation formula is as follows:
Figure BDA0002207743800000074
step 4.2, initializing a degree distribution value k =2 of the current information bit node;
step 4.3, calculating the information bit variable node V with the medium value of k in the EXIT graph Inf To check node C Spc (V Inf →C Spc ) Likelihood information of and V Inf Average mutual information IEV between k The calculation formula is as follows:
Figure BDA0002207743800000075
wherein, IAV StoI For check node C in EXIT graph Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of and V Inf The prior mutual information between the two has the value range of O < IAV StoI ＜1，/>
Figure BDA0002207743800000076
The variance of the channel likelihood information is expressed, and the formula of the J function is as follows:
Figure BDA0002207743800000081
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608，
J -1 the function is the inverse of the J function;
4.4, judging whether the degree distribution value k of the current information bit node is smaller than the maximum value dv of the degree of the information bit node, if so, adding 1 to the current k value and then skipping to the step 4.3 to continue execution, otherwise, skipping to the step 4.5 to continue execution;
step 4.5, when the degree distribution value k of the information bit node is equal to the maximum value dv of the information bit node degree, all the calculated average mutual information IEV are calculated k Coefficient matrix A = [ -IEV) updated to linear inequality constraint 2 ，-IEV 3 ，...-IEV k ，...，-IEV dv ]Where k is an integer satisfying 2 or more and dv or less, dv represents the maximum value of the node degree of the information bit.
Step 4.6, calculating single check code V in EXIT graph Spc To check node C Spc (V Spc →C Spc ) Likelihood information of and V Spc Average mutual information IEV between StoS The calculation formula is as follows:
Figure BDA0002207743800000082
wherein it is present>
Figure BDA0002207743800000083
The variance of the channel likelihood information is expressed, and the formula of the J function is as follows:
Figure BDA0002207743800000084
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608。
step 4.7, calculating the secondary check code V in the EXIT graph Spc For check node C Spc (V Spc →C Spc ) Likelihood information of and V Spc Prior mutual information IAC between StoS The calculation formula is as follows: IAC StoS ＝IEV StoS ；
Step 4.8, calculating check node C in the EXIT graph Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of and V Inf Average mutual information IEC between StoI The calculation formula is as follows:
Figure BDA0002207743800000085
wherein, IAC IToS For varying the node V from the information bit in the EXIT diagram Inf For check node C Spc (V Inf →C Spc ) Likelihood information of and V Inf The prior mutual information between the two has the value range of 0 to 1, dc 2 Degree of connection between check node Cspc and information bit variable node, dc 3 For the degree of connection between the check node Cspc and the single check code variable node, the J function formula is as follows:
Figure BDA0002207743800000091
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608，
the J-1 function is the inverse of the J function;
step 4.9, in step 4.8, the current IAC is found ItoS Is a known quantity and takes a value between 0 and 1, the unknown quantity IEC StoI And IAC ItoS And taking values in a one-to-one correspondence manner. When calculating the inequality constraint vector b, the IEC is found StoI The unknown quantity IAC is a known quantity and has a value ranging from 0 to 1 ItoS And IEC StoI One-to-one correspondence of fetchingThe value is obtained. According to the IAC obtained in the step 4.8 ItoS And IEC StoI In the application, linear interpolation method is adopted to obtain current IEC StoI When the known (for distinguishing from 4.8 steps, represented by Iec) O < Iec < 1, the IAC corresponding to the known ItoS (for distinguishing from the 4.8 step, here denoted by Iac) the calculation formula is: iac = interp1 (IEC) StoI ，IAC ItoS Iec, 'linear'), where interp1 is a linear interpolation function.
Step 4.10, calculating the value of inequality constraint vector b according to the Iac result obtained by the linear interpolation function, wherein the calculation formula is as follows: b = [ -Iac T ]。
To more clearly illustrate the above steps, fig. 3 shows an EXIT data flow diagram of an SPC codec in MET-LDPC code of the present invention, and the direction of the arrow indicates the data flow, wherein V Inf Representing information bit variable nodes, C Spc Represents a check node, V Spc Representing a single check code. IEV ItoS Representing variable nodes V by information bits Inf To check node C Spc (V Inf →C Spc ) Likelihood information of and V Inf Average mutual information between; IAC ItoS Representing slave information bit variable node V Inf For check node C Spc (V Inf →C Spc ) Likelihood information of and V Inf Prior mutual information between; IEC StoI Representation by check node C Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of (1) and (V) Inf Average mutual information between; IAV StoI Representation by check node C Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of and V Inf Prior mutual information between; IEV StoS Indicating by a single check code V Spc To check node C Spc (V Spc →C Spc ) Likelihood information of and V Spc Average mutual information between; IAC StoS Indicating a secondary check code V Spc For check node C Spc (V Spc →C Spc ) Likelihood information of and V Spc Prior mutual information between; IEC StoS Representation by check node C Spc Passes to a single check code V Spc (C Spc →V Spc ) Likelihood information of and V Spc Average mutual information between; IAV StoS Representation by check node C Spc Passes to single check code V Spc (C Spc →V Spc ) Likelihood information of and V Spc A priori mutual information between
Step 5, setting a linear equation constraint condition: aeq x λ = beq, where Aeq is an equality constraint coefficient matrix and beq is an equality constraint vector due to λ i The ratio of the number of edges of the information bit node with the representation degree i to the total number of edges of the information bit node must satisfy
Figure BDA0002207743800000101
In the method, an Aeq is set to be a full 1 matrix with 1 row and dv column, and beq =1; />
Step 6, setting lambda in the optimal degree distribution vector lambda of the matrix to be optimized i Has an upper limit ub (i.e., upper limit value) of 1 and a lower limit lb (lower limit value) of 0, while setting all λ i The initial value start of (1) is 0, wherein i is more than or equal to 1 and less than or equal to dv;
step 7, setting a selection parameter option of the linear programming function; the option represents the optimization parameter options of the linear programming function, and in a specific application scene, the optimization parameter options can be specifically modified by calling an optimest function in the matlab function library according to needs.
And 8, solving the optimal solution of lambda by adopting a linear programming function (f, A, b, aeq, beq, lb, ub, start and option) to obtain the optimized degree distribution condition of the SPC coder. In a specific embodiment, the solution may be performed by calling a linear programming function in a matlab function library, where in the parameters of the function, f is a coefficient vector of a set linear programming target function, a is a linear inequality constraint coefficient matrix, b is a linear inequality constraint vector, aeq is an equality constraint coefficient matrix, beq is an equality constraint vector, ub is an upper bound value of λ, lb is a lower bound value of λ, and start is an upper bound value of all λ i The option represents a linear programming functionOptimization parameter options of numbers.
The above steps clearly illustrate the optimization process of the MET-LDPC code, and in the following, a specific embodiment is used to illustrate the optimization method of the MET-LDPC code of the present invention, where the embodiment takes a MET-LDPC code as an example, and a parity check matrix before the codeword optimization is:
Figure BDA0002207743800000102
wherein H 1 Is a dual diagonal matrix
Figure BDA0002207743800000103
H 4 Is a single diagonal matrix->
Figure BDA0002207743800000104
H 2 Is a matrix with row weights and column weights all 3, H 3 And are matrices with row weights and column weights all 3.
The MET-LDPC code described above is optimized, i.e. at a given threshold value σ n In the case of (1), H is obtained by using the linear programming objective function of the SPC codec 3 Optimal degree distribution of matrix λ = [ λ ] 2 ，λ 3 ，......，λ dv ] T . The method comprises the following specific steps:
step 1, threshold value sigma of SPC coder in given MET-LDPC code n ＝0.397；
Step 2, setting and optimizing a linear programming objective function of the SPC codec according to a code rate formula;
setting SPC codec optimization, H 3 The maximum value dv =6 of the node degree of the middle information bit. Assuming SPC codec optimization, H 3 The ratio of the number of edges of the information bit nodes with the medium degree of i to the total number of edges of the information bit nodes is lambda i Wherein i is more than or equal to 2 and less than or equal to 6, and the total number of edges of the information bit nodes is M, then the code rate R of the SPC coder in the MET-LDPC code can be calculated as:
Figure BDA0002207743800000111
in this case, to obtain the optimal degree distribution, C-R should be guaranteed to be minimum, where C represents shannon limit, and is a fixed value, and thus, the required code rate R is maximum. From the formula of the code rate R, to maximize R, it is necessary to make R the maximum
Figure BDA0002207743800000112
Minimum, and therefore the objective function of the linear programming of the SPC codec is set to @>
Figure BDA0002207743800000113
/>
And 3, setting a coefficient vector f of the SPC coder as follows according to the target function of the SPC coder:
Figure BDA0002207743800000114
and 4, setting a linear inequality constraint condition A multiplied by lambda < b according to the EXIT graph of the SPC detector, wherein A represents a coefficient matrix of the linear inequality constraint, b represents a constraint vector of the linear inequality, and lambda = [ lambda ] 2 ，λ 3 ，λ 4 ，λ 5 ，λ 6 ] T The specific setting steps of the parameters A and b are as follows:
step 4.1, according to the given threshold value as sigma n =0.397, calculate variance of channel likelihood information by:
Figure BDA0002207743800000115
step 4.2, setting the degree distribution value k =2 of the current information bit node;
step 4.3, information bit variable node V with k degree of calculation Inf To check node C Spc (V Inf →C Spc ) Likelihood information of and V Inf Average mutual information IEV between k The calculation formula is as follows:
Figure BDA0002207743800000116
wherein the content of the first and second substances,IAV StoI representation by check node C Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of and V Inf The prior mutual information between the two is a known value, and the value range is more than 0 and less than IAV StoI ＜1，/>
Figure BDA0002207743800000117
A variance representing channel likelihood information; the formula of the J function is:
Figure BDA0002207743800000121
wherein, the first and the second end of the pipe are connected with each other,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608，
the J-1 function is the inverse of the J function;
step 4.4, judging whether the degree distribution value k of the information bit node is smaller than the maximum value 6 of the degree of the information bit node, if the k is smaller than 6, increasing the value k by 1 and jumping to the step 4.3 to continue the execution, otherwise, jumping to the step 4.5 to continue the execution;
step 4.5, when the degree distribution value k of the information bit node is equal to 6, the calculated IEV k Updating the value to a coefficient matrix A constrained by a linear inequality to obtain A = [ -IEV 2 ，-IEV 3 ，-IEV 4 ，-IEV 5 ，-IEV 6 ]；
Step 4.6, updating the single check code V Spc To check node C Spc (V Spc →C Spc ) Likelihood information of (1) and (V) Spc Average mutual information IEV between StoS The calculation formula is as follows:
Figure BDA0002207743800000122
wherein +>
Figure BDA0002207743800000123
The variance of the channel likelihood information is expressed, and the formula of the J function is as follows:
Figure BDA0002207743800000124
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608。
step 4.7, updating the subordinate bill check code V Spc For check node C Spc (V Spc →C Spc ) Likelihood information of and V Spc Prior mutual information IAC between StoS The calculation formula is as follows: IAC StoS ＝IEV StoS ；
Step 4.8, updating the check node C Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of and V Inf Average mutual information IEC between StoI The calculation method is as follows:
Figure BDA0002207743800000125
wherein, IAC ItoS Representing slave information bit variable node V Inf For check node C Spc (V Inf →C Spc ) Likelihood information of and V Inf The prior mutual information between the two is a known quantity, and the value range is between 0 and 1. dc 2 Represents check node C spc Degree of connection to information bit variable node, dc 2 ＝3；dc 3 Representing check node C spc Degree of connection to a single check code variable node, dc 3 =3; the formula of the J function is:
Figure BDA0002207743800000131
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝-0.142675，c 2 ＝-0.0822054，d 2 ＝0.0549608，
the J-1 function is the inverse of the J function;
step 4.9, in the above step, IAC ItoS Is a known quantity, and IEC StoI Is unknown, is to find the current IEC StoI When the quantity is a known quantity (the value range is between 0 and 1), the IAC corresponding to the known quantity ItoS The linear interpolation method is introduced in the step to calculate the current IEC StoI When the value is known (expressed by Iec), namely O < Iec < 1, the corresponding IAC ItoS (in Iac), iac = interp1 (IEC) StoI ，IAC ItoS Iec, 'linear'), where interp1 is a linear interpolation function;
step 4.10, calculating the value of an inequality constraint vector b according to the Iac value obtained by the linear interpolation method, wherein the formula is as follows: b = [ -Iac T ]；
Step 5, setting linear equality constraint condition Aeq multiplied by lambda = beq, wherein Aeq is an equality constraint coefficient matrix, and beq is an equality constraint vector, and lambda is caused by i The ratio of the number of edges of the information bit node with the representation degree i to the total number of edges of the information bit node must satisfy
Figure BDA0002207743800000132
Thus, set Aeq to be a full 1 matrix with row weight of 1 and column weight of 6, beq to be 1;
step 6, setting lambda in the optimal degree distribution vector lambda of the matrix to be optimized i The upper boundary ub is 1, the lower boundary lb is 0, wherein i is not less than 2 and not more than 6, and lambda is set in lambda i The initial value start of (2) is 0;
step 7, setting selection parameters of the linear programming function by calling an optiset function in the matlab function library;
step 8, calling a linear programming function (f, A, b, aeq, beq, lb, ub, start, option) in the matlab function library to obtain an optimal solution of lambda, and obtaining H in the optimized SPC codec 3 Degree distribution of the matrix, wherein f is a coefficient vector of the set linear programming objective function, A is a linear inequality constraint coefficient matrix, b is a linear inequality constraint vector, aeq is an equality constraint coefficient matrix, beq is an equality constraint vector, ub is an upper boundary value of lambda, lb is a lower boundary value of lambda, and start is all lambda i Represents the optimization parameter options of the linear programming function. In this example, H 3 The comparison before and after matrix degree distribution optimization is shown in the following table:
λ i before optimization After optimization
λ
2 0 0.0849
λ 3 1 0.9151
The above table gives H 3 Degree distribution before and after matrix optimization, it can be seen that H is before optimization 3 The ratio lambda of the number of edges of the information bit node with the matrix middle degree of 2 to the total number of edges of the information bit node 2 λ, ratio of number of edges of information bit node of degree 3 to total number of edges of information bit node of =0 3 =1, after the optimization method of the multilateral low-density parity check code of the application, the proportion of the number of edges of the information bit node with the middle degree of 2 in the H3 matrix to the total number of edges of the information bit node is changed into lambda 2 The ratio of the number of edges of the information bit node having a degree of 3 to the total number of edges of the information bit node is =0.0849, and λ 3 =0.9151. At this time, from H 1 、H 2 、H 4 And optimized H 3 And a new H matrix formed by the matrix is the parity check matrix of the optimized MET-LDPC code. As shown in fig. 4, a comparison graph of the decoding BER performance of the MET-LDPC code before and after the optimization of the embodiment is shown, and it can be seen that the BER =10 -6 And the optimized MET-LDPC code has about 1.6dB gain compared with the MET-LDPC code before optimization.
In addition, FIG. 5 shows the MET-LDPC code pair different matrices (H) described above 2 Or H 3 Matrix) is optimized, the BER performance of the MET-LDPC code is compared, and the BER =10 -6 Then, H is respectively subjected to the optimization method of the multilateral type low-density parity check code 2 And H 3 After the matrix is optimized, compared with MET-LDPC code before optimization, the matrix has obvious gain, and the H is measured 3 The gain effect of matrix optimization is more remarkable.
The above embodiments are merely illustrative of the technical solutions of the present invention, and the present invention is not limited to the above embodiments, and any modifications or alterations according to the principles of the present invention should be within the protection scope of the present invention.

Claims (1)

1. A method for optimizing a multilateral low-density parity-check code is characterized by comprising the following steps:
step 1, in order to obtain the optimal distribution under the given threshold, the threshold of SPC codec is set as σ n ；
Step 2, setting and optimizing a linear programming objective function of the SPC codec according to a code rate formula,
the code rate formula is as follows:
Figure FDA0004044636780000011
the linear programming objective function is: />
Figure FDA0004044636780000012
Wherein λ is i H being a low density parity check code of the multilateral type 3 The proportion of the number of edges of the information bit nodes with the matrix moderate degree i to the total number of edges of the information bit nodes, dv is the maximum value of the information bit node degree, and M is the total number of edges of the information bit nodes;
and 3, setting a coefficient vector f of the linear programming objective function as:
Figure FDA0004044636780000013
dv is the maximum value of the node degree of the information bit;
and 4, setting linear inequality constraint conditions according to the data flow of the EXIT graph of the SPC codec: a is more than b, wherein A is a linear inequality constraint coefficient matrix, b is a constraint vector of a linear inequality, and lambda is an optimal degree distribution vector of a matrix to be optimized, namely lambda = [ lambda ] 2 ，λ 3 ，......,λ dv ] T The method comprises the following steps:
step 4.1, according to the set threshold value sigma of SPC codec n And calculating the variance of the channel likelihood information, wherein the calculation formula is as follows:
Figure FDA0004044636780000014
step 4.2, initializing degree distribution k =2 of the current information bit node;
step 4.3, calculating the information bit variable node V with the medium value of k in the EXIT graph Inf To check node C Spc (V Inf →C Spc ) Likelihood information of (1) and (V) Inf Average mutual information IEV between k The calculation formula is as follows:
Figure FDA0004044636780000015
wherein, IAV StoI For check node C in EXIT graph Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of and V Inf The prior mutual information between the two has the value range of 0 < IAV StoI ＜1，/>
Figure FDA0004044636780000016
The variance of the channel likelihood information is expressed, and the formula of the J function is as follows:
Figure FDA0004044636780000017
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491,b 2 ＝-0.142675,c 2 ＝-0.0822054,d 2 ＝0.0549608，
J -1 the function is the inverse of the J function;
step 4.4, judging whether k is smaller than the maximum value dv of the node degree of the information bit, if k is smaller than dv, calculating k = k +1, and then jumping to step 4.3, otherwise jumping to step 4.5;
step 4.5, updating coefficient matrix A = [ -IEV) of linear inequality constraint 2 ,-IEV 3 ,...-IEV k ,...,-IEV dv ]Wherein k is an integer satisfying 2 ≦ k ≦ dv, which represents a maximum value of the information bit node degree;
step 4.6, calculating single check code V in EXIT graph Spc To check node C Spc (V Spc →C Spc ) Likelihood information of (1) and (V) Spc Average mutual information IEV between StoS The calculation formula is as follows:
Figure FDA0004044636780000021
wherein +>
Figure FDA0004044636780000022
The variance of the channel likelihood information is expressed, and the formula of the J function is as follows:
Figure FDA0004044636780000023
wherein the content of the first and second substances,
ω＝1.6363,
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491，b 2 ＝0.142675，c 2 ＝0.0822054，d 2 ＝0.0549608；
step 4.7, calculating the subordinate single check code V in the EXIT graph Spc For check node C Spc (V Spc →C Spc ) Likelihood information of (1) and (V) Spc Prior mutual information IAC between StoS The calculation formula is as follows: IAC StoS ＝IEV StoS ；
Step 4.8, calculating check node C in the EXIT graph Spc To information bit variable node V Inf (C Spc →V Inf ) Likelihood information of and V Inf Average mutual information IEC between StoI The calculation formula is as follows:
Figure FDA0004044636780000024
wherein, IAC ItoS For varying the node V from the information bit in the EXIT diagram Inf For check node C Spc (V Inf →C Spc ) Likelihood information of (1) and (V) Inf The prior mutual information between the two has the value range of 0 to 1, dc 2 Degree of connection between check node Cspc and information bit variable node, dc 3 For the degree of connection between the check node Cspc and the single check code variable node, the J function formula is as follows:
Figure FDA0004044636780000031
wherein the content of the first and second substances,
ω＝1.6363，
a 1 ＝-0.0421061，b 1 ＝0.209252，c 1 ＝-0.00640081，
a 2 ＝0.00181491,b 2 ＝-0.142675,c 2 ＝-0.0822054,d 2 ＝0.0549608，
J -1 the function is the inverse of the J function;
step 4.9, linear interpolation method is used for obtaining current IEC StoI Is an IAC corresponding to the known time ItoS The calculation formula is as follows: iac = interp1 (IEC) StoI ,IAC ItoS Iec, 'linear'), where interp1 is a linear interpolation function, iec represents the known IEC StoI And satisfies 0 < Iec < 1, iac represents IAC corresponding to Iec ItoS The value of (1);
step 4.10, calculating the value of the inequality constraint vector b, wherein the calculation formula is as follows: b = [ -Iac T ]；
Step 5, setting a linear equation constraint condition: aeq × λ = beq, where Aeq is a full 1 matrix of 1 row and dv column, beq =1;
step 6, setting lambda in the optimal degree distribution vector lambda of the matrix to be optimized i Has an upper boundary ub of 1 and a lower boundary lb of 0, and sets λ i The initial value start of (1) is 0, wherein i is more than or equal to 2 and less than or equal to dv;
step 7, setting a selection parameter option of the linear programming function;
and 8, solving the optimal solution of lambda by adopting a linear programming function routine (f, A, b, aeq, beq, lb, ub, start and option), and obtaining the degree distribution condition of the optimized SPC coder.
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