Joint LDPC Code and Spreading Optimization for Multi-User Communications

Closed-form expressions of the degree distribution of regular low-density parity-check codes and spreading length are presented over a Gaussian multiple access channel. Specifically, given a specific variable node degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d_{v}$ </tex-math></inline-formula>, the theoretical expressions of check node degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d_{c}$ </tex-math></inline-formula> and spreading length m are derived by fixed point equation array. Then, a reliable region where our multi-user system can decode successfully is obtained by solving uncertainty of the inverse problem. Finally, the optimal parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(d_{c}^{o}, d_{v}^{o},m^{o})$ </tex-math></inline-formula> of LDPC code degree distribution and spreading length in the reliable region are selected to maximize the transmission rate. Numerical results show that our derivation is effective and efficient, and exhibits much lower complexity than EXIT chart.