一类QC-LDPC码的构造设计

【作者】 陈艳；

【导师】 唐元生；

【作者基本信息】 扬州大学 ， 数学， 2018， 硕士

【摘要】 近二十年来,低密度奇偶校验码(LDPCcodes)由于其逼近香农限的性能逐渐成为人们关注的焦点。研究发现,LDPC码的校验矩阵的稀疏性使其在译码复杂度、错误平层等各方面都有良好的表现。因此,它在光纤通信、声频广播、移动通信等各个领域都得到了广泛的应用。本文提出了拟循环LDPC(QC-LDPC)码的两种构造方法,给出了所构造出的QC-LDPC码的码率计算公式,并利用计算机模拟仿真分析了它们的性能。首先,本文研究了 QC-LDPC码的一种代数构造方法。L.Chen等人在1994年提出了QC-LDPC码的一种构造方法,其构造步骤为先构造满足RD约束的基矩阵,再通过矩阵弥散扩张成围长至少为6的低密度奇偶校验矩阵。关于基矩阵的构造他们给出了分别基于加法子群和乘法子群的两种构造方法,本文对这两种方法进行了改进和推广,提出了两种新的构造方法。第一种为基于乘法子群陪集的基矩阵构造法,以乘法子群及其陪集为基础集构造满足RD约束的基矩阵,其元素皆为陪集代表元的线性组合。第二种为基于Ω条件的基矩阵构造法,这里我们称有限域Fq上的集合U,V满足Ω条件是指它们满足:对任意(u’,v’),(u,v)∈U×V,若uv = uv,则u = u’,v = v’.接着,本文改进了 L.Zhang等人对基于加法子群构造的QC-LDPC码的码率的计算方法,并应用于计算我们提出的两种构造方法所得到的QC-LDPC码的码率。本文将对校验矩阵的秩的计算转化为对与其等价的分块对角矩阵的秩的计算,先将分块对角矩阵进行行列转换,后对转换后的分块对角矩阵分块求秩。我们还利用杨辉三角形分析了排列组合数的奇偶特性,进而得到了我们提出的两种QC-LDPC码的校验矩阵的秩的表达式。最后,本文采用比特翻转译码方法,用MATLAB对基于乘法子群,基于乘法子群陪集以及基于Ω条件等三种构造方法构造出的QC-LDPC码的纠错性能进行了仿真分析。计算结果表明,若取门限为3迭代次数为20,当误码率低于10-5时,基于乘法子群构造的码的误码平层更低,而基于Ω条件构造的码的误码率较优于基于乘法陪集构造的码的误码率。对译码门限研究时发现三种代数构造方法都较适用于门限为4的比特翻转译码算法,且仿真结果与比特翻转译码方法的瀑布特性相一致,因此我们提出的两种构造方法构造的QC-LDPC码和L.Chen提出的基于乘法子群的构造方法构造的码一样,在信噪比较高时均有较好的译码性能。更多还原

【Abstract】 In the past two decades,LDPC codes have gradually become the focus of attention due to its close proximity to the Shannon limit.It was found that the sparsity of the calibration matrix gives a good advantage in decoding complexity and error level.Therefore,it has been widely used in optical fiber communication,audio broadcasting,mobile communication and other fields.In this paper,two construction methods of the quasi-cyclic LDPC(QC-LDPC)codes are presented,the code rate calculation formula of the QC-LDPC code is given,and their performance is analyzed by computer simulation.Firstly,in this paper,we study an algebraic construction method of QC-LDPC code.L.Chen proposed a construction method of QC-LDPC code in 1994.The construction step is to construct the basis matrix which satisfies the RD constraint,and then expand the basis matrix into the low density parity matrix that the girth is at least 6.In terms of the structure of the basis matrix,they give two constructional methods based on the addition method group and the multiplication method group.We improve and popularize the construction methods based on these two methods,and put forward two new basic matrix construction methods.The first construction method is based on the structure of multiplicative group accompanying set.The basis matrix is based on the method group and its accompanying set,and it satisfies the RD constraint.Its elements are linear combinations of the representative elements of the accompanying set.The second is based on the condition Ω We call set U,V in finite field satisfy condition Ω that means,for the set U,V,(u’,v’),(u,v)∈U×X,if uv = u’v’,then u =u,v＝v.Then,we improve the calculation method of the code rate of the QC-LDPC code based on additive group structure,and apply it to the QC-LDPC code constructed by the two new construction methods.In this paper,the rank of the parity matrix is converted into the rank of the equivalent block diagonal matrix.First of all,we transform the parity check matrix into the equivalent block diagonal matrix.Secondly,we consider the block rank of the block diagonal matrix.By using the parity characteristics of the number of combinations in the Yang Hui triangle,we get the expression of the rank of each check matrix respectively.Finally,we adopt the bit flipping decoding method,and simulate the QC-LDPC codes by MATLAB.The result shows that,on the basis of the threshold of 3 and with 20 iteration,when the error rate is lower 10’5,the construction method based on the multiplicative set is lower,and the error rate of the constructed method based on Ω is better than that based of multiplication coset.These three methods are applicable to the four-bit inversion method.And the simulation results are consistent with the cascade characteristics of the bit flip decoding method,so both the QC-LDPC codes constructed by the method proposed by us and by L.Chen all have better decoding performance when the SNR is high.更多还原