Joint Design of Quasi-cyclic Low Density Parity Check Codes and Performance Analysis of Multi-source Multi-relay Coded Cooperative System
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摘要: 为解决多信源多中继低密度奇偶校验(LDPC)码编码协作系统编码复杂度高、编码时延长的问题，该文引入一种特殊结构的LDPC码—基于生成矩阵的准循环LDPC码(QC-LDPC)码。该类码结合了QC-LDPC码与基于生成矩阵LDPC (G-LDPC)码的特点，可直接实现完全并行编码，极大地降低了中继节点的编码时延及编码复杂度。在此基础上，推导出对应于信源节点和中继节点采用的QC-LDPC码的联合校验矩阵，并基于最大公约数(GCD)定理联合设计该矩阵以消除其所有围长为4, 6(girth-4, girth-6)的短环。理论分析和仿真结果表明，在同等条件下该系统的误码率(BER)性能优于相应的点对点系统。仿真结果还表明，与采用显式算法构造QC-LDPC码或一般构造QC-LDPC码的协作系统相比，采用联合设计QC-LDPC码的系统均可获得更高的编码增益。
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关键词:
- 准循环低密度奇偶校验码 /
- 编码协作 /
- 联合校验矩阵 /
- 最大公约数定理
Abstract: To solve the problems of high encoding complexity and long encoding delay in the multi-source multi-relay Low Density Parity Check (LDPC) coded cooperative system, a special kind of structured LDPC codes—Quasi-Cyclic LDPC (QC-LDPC) codes based on generator matrix is proposed, which combines the characteristics of QC-LDPC codes and Generator-matrix-based LDPC (G-LDPC) codes. It can perform completely parallel encoding, which greatly reduces the encoding complexity and delay at the relays. Based on this, a joint parity check matrix corresponding to the QC-LDPC codes adopted by the sources and relays is deduced, and the matrix is further jointly designed based on the Greatest Common Divisor (GCD) theorem to eliminate all cycles of girth-4 and girth-6. Theoretical analysis and simulation results show that under the same conditions, the Bit Error Rate (BER) performance of the proposed system is better than that of the corresponding point-to-point system. The simulation results also show that the cooperative system with jointly designed QC-LDPC codes can obtain a higher coding gain than the system with explicitly constructed QC-LDPC codes or generally constructed QC-LDPC codes. -
表 1 双信源双中继编码协作及对应点对点系统所采用的QC-LDPC码
信源节点所采用的QC-LDPC码 中继节点所采用的QC-LDPC码 双信源双中继系统 $ {{\text{H}}_{{S_{1}}}} = {{\text{H}}_{1(1100 \times 2200)}} $ $ {{\text{H}}_{{R_{1}}}} = [ {{{\text{A}}_{1(1100 \times 2200)}}}\quad {{{\text{B}}_{1(1100 \times 2200)}}}\quad {\text{I}}_{(1100 \times 1100)}]$ ${{\text{H}}_{{S_{2}}}} = {{\text{H}}_{2(1100 \times 2200)}}$ $ {{\text{H}}_{{R_{2}}}} = [ {{{\text{A}}_{2(1100 \times 2200)}}}\quad {{{\text{B}}_{2(1100 \times 2200)}}}\quad {\text{I}}_{(1100 \times 1100)}] $ Rate=1/2 Rate=4/5 点对点系统 ${{\text{H}}_S} = {{\text{H}}_{(2200 \times 6600)}}$
Rate=1/3\ 
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表 2 不同信源节点、中继节点数目情况下编码协作系统所采用的QC-LDPC码
信源节点所采用的QC-LDPC码 中继节点所采用的QC-LDPC码 双信源双中继 $ {{\text{H}}_{{S_{1}}}} = {{\text{H}}_{1(1100 \times 2200)}} $ $ { {\text{H} }_{ {R_{1} } } } = [\begin{array}{*{20}{c} } { { {\text{A} }_{1(1100 \times 2200)} } } & { { {\text{B} }_{1(1100 \times 2200)} } } & {\text{I} }_{(1100 \times 1100)} \end{array}] $ $ {{\text{H}}_{{S_{2}}}} = {{\text{H}}_{2(1100 \times 2200)}} $ $ { {\text{H} }_{ {R_{2} } } } = [\begin{array}{*{20}{c} } { { {\text{A} }_{2(1100 \times 2200)} } } & { { {\text{B} }_{2(1100 \times 2200)} } } & {\text{I}_{(1100 \times 1100)} } \end{array} ]$ Rate=1/2 Rate=4/5 双信源单中继 $ {{\text{H}}_{{S_{1}}}} = {{\text{H}}_{1(1100 \times 2200)}} $ ${ {\text{H} }_R} = [\begin{array}{*{20}{c} } { { {\text{A} }_{(1100 \times 2200)} } } & { { {\text{B} }_{(1100 \times 2200)} } } & {{\text{I}}_{(1100 \times 1100)} } \end{array} ]$ $ {{\text{H}}_{{S_{2}}}} = {{\text{H}}_{2(1100 \times 2200)}} $ Rate=1/2 Rate=4/5 单信源双中继 ${{\text{H}}_S} = {{\text{H}}_{1(1100 \times 2200)}}$ $\begin{gathered} {{\text{H}}_{{R_{1}}}} = [\begin{array}{*{20}{c}} {{{\text{A}}_{1(1100 \times 2200)}}}&{{{\text{I}}_{(1100 \times 1100)}}} \end{array}] \\ {{\text{H}}_{{R_{2}}}} = [\begin{array}{*{20}{c}} {{{\text{A}}_{2(1100 \times 2200)}}}&{{{\text{I}}_{(1100 \times 1100)}}} \end{array}] \\ \end{gathered} $ Rate=1/2 Rate=2/3
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表 3 采用一般构造QC-LDPC码的协作系统各节点所采用的码字
信源节点所采用的QC-LDPC码 中继节点所采用的QC-LDPC码 双信源双中继 ${d_{\rm v}} = 2$, ${d_{\rm c}} = 4$
B=550${d_{\rm v}} = 2$, ${d_{\rm c}} = 10$
B=550注：dv指每列“1”的个数，dc指每行“1”的个数
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表 4 采用显式构造QC-LDPC码的协作系统所采用的码字
信源节点所采用的QC-LDPC码 中继节点所采用的QC-LDPC码 双信源双中继 ${d_{\rm v}} = 2$, ${d_{\rm c}} = 3$
B=730${d_{\rm v}} = 2$, ${d_{\rm c}} = 8$
B=730
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