信息处理 - 在 CCSDS 蓝皮书中验证 LDPC 矩阵 - 吾爱随笔录

我的理解是 LDPC 是一个线性块码，它的生成矩阵和奇偶校验矩阵满足。 $\mathbf{G}$ $\mathbf{H}$ $\mathbf{GH}^T=0$

我在这里使用 CCSDS 蓝皮书作为参考，特别是图 7-1 和 7-2。

图 7-1 和表 7-1 产生了一个奇偶校验矩阵，使用循环构造，列权重为 4，行权重为 32。除了给出常规权重之外，循环似乎有些随意；

流通者背后是否有原因/方法？

另外，我已经在 MATLAB 中实现了这两个矩阵，但是在将生成矩阵与奇偶校验矩阵的转置相乘时没有得到一个零矩阵，我也从GitHub - nicopi - LDPC Codes for Deep-Space Communications得到了这个结果（特别是 data.mat 中的和矩阵）。 $\mathbf{G}$ $\mathbf{H}$

有什么我在概念上遗漏的东西可以解释这个奇偶校验结果吗？

我编写了以下代码来创建矩阵：

在给定所有循环的第一行的情况下创建矩阵的辅助函数：

function [matrix,matrixHex] = auxiliary_expandCirculants(circulantSize,matrixSize,circulantFirstRow,circulantFirstRowType)
% Inputs:
%   circulantSize: size of the circulant
%   matrixSize: dimensions of the matrix in terms of circulants
%   circulantFirstRow: either 1) a matrix where each row corresponds to the
%       locations of 1s in the first row of each circulant, or 2) a cell
%       containing character arrays of the hex value of the first row of
%       each circulant; circulants are enumerated from left to right and
%       top to bottom
%   circulantFirstRowType: 'binary' if circulantFirstRow is a matrix or
%   'hex' if circulantFirstRow is a cell
% Outputs:
%   matrix: the matrix corresponding to the specified input circulants
%   matrixHex: a single character array containing the hex values of all
%       circulant first rows

% check for errors
if prod(matrixSize)~=length(circulantFirstRow)
    error('The number of circulants in the matrix and the number of entries in the input data file should be equal.');
end

if strcmp(circulantFirstRowType,'binary')
    matrixHex = '';
elseif strcmp(circulantFirstRowType,'hex')
    matrixHex = strjoin(circulantFirstRow);
    matrixHex = matrixHex(~isspace(matrixHex));
end

% fill matrix
matrix = zeros(circulantSize*matrixSize);
for ii=1:matrixSize(1)
    for jj=1:matrixSize(2)
        
        % used to index the input 'circulantFirstRow'
        idxMatrix = (ii-1)*matrixSize(2)+jj;
        
        % represent the first row of the circulant as a binary vector
        if strcmp(circulantFirstRowType,'binary')
            row = zeros(1,circulantSize);
            row(circulantFirstRow(idxMatrix,:)+1) = 1; % one-based indexing
        elseif strcmp(circulantFirstRowType,'hex')
            currentHex = circulantFirstRow{idxMatrix};
            row = [];
            for kk=1:ceil(circulantSize/4)
                row = [de2bi(hex2dec(currentHex(end-kk+1)),'left-msb',min(4,circulantSize-length(row))) row]; % generate from right to left
            end
        end
        
        circulant = toeplitz([row(1) fliplr(row(2:end))],row);
        idxMatrixRows = (ii-1)*circulantSize+1:ii*circulantSize;
        idxMatrixCols = (jj-1)*circulantSize+1:jj*circulantSize;
        matrix(idxMatrixRows,idxMatrixCols) = circulant;
        
        % generate hexadecimal value for circulant first rows
        if strcmp(circulantFirstRowType,'binary')
            circulantFirstRowHex = '';
            for kk=1:ceil(circulantSize/4)
                idxLeft = max(1,circulantSize-kk*4+1);
                idxRight = circulantSize-(kk-1)*4;
                circulantFirstRowHex = [dec2hex(bi2de(row(idxLeft:idxRight),'left-msb'),1) circulantFirstRowHex]; % generate each hex value from right to left
            end
            matrixHex = [matrixHex circulantFirstRowHex]; % concatenate all hex values from left to right
        end
    end
end

end

主文件：

circulantSize = 511;
baseMatrixSize = [2 16];
circulantFirstRow = readmatrix('data_8176-7156.csv'); % corresponds to Table 7-1 of the blue book
fileID = fopen('data_generatorParityColumns.txt','r'); % corresponds to Table C-1 of the blue book
formatSpec = '%s';
generatorParitySize = [baseMatrixSize(2)-baseMatrixSize(1) baseMatrixSize(1)];
generatorParityColumnsHex = textscan(fileID,formatSpec);
generatorParityColumnsHex = generatorParityColumnsHex{1};

% create base matrix as given by Figure 7-1 of the blue book
[baseMatrix,baseMatrixHex] = auxiliary_expandCirculants(circulantSize,baseMatrixSize,circulantFirstRow,'binary');

% create generator matrix as given by Figure 7-2 of the blue book
[generatorParityColumns,~] = auxiliary_expandCirculants(circulantSize,generatorParitySize,generatorParityColumnsHex,'hex');
generatorMatrix = [eye(size(generatorParityColumns,1)) generatorParityColumns];

generatorMatrix*baseMatrix.'

data_8176-7156.csv: 一个两列的csv文件，对应蓝皮书表7-1的中间列

'data_generatorParityColumns.txt'：蓝皮书表C-1对应的文本文件

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6855AE08698A50AA3051768793DC238544AF3FE987391021AAF6383A6503409C3CE971A80B3ECE12363EE809A01D91204F1811123EAB867D3E40E8C652585D28
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0008B4E899E5F7E692BDCE69CE3FAD997183CFAEB2785D0C3D9CAE510316D4BD65A2A06CBA7F4E4C4A80839ACA81012343648EEA8DBBA2464A68E115AB3F4034
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2CD8140C8A37DE0D0261259F63AA2A420A8F81FECB661DBA5C62DF6C817B4A61D2BC1F068A50DFD0EA8FE1BD387601062E2276A4987A19A70B460C54F215E184
06F1FF249192F2EAF063488E267EEE994E7760995C4FA6FFA0E4241825A7F5B65C74FB16AC4C891BC008D33AD4FF97523EE5BD14126916E0502FF2F8E4A07FC2
65287840D00243278F41CE1156D1868F24E02F91D3A1886ACE906CE741662B40B4EFDFB90F76C1ADD884D920AFA8B3427EEB84A759FA02E00635743F50B942F0
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