Belief propagation-based multiuser receivers in optical code-division multiple access systems

In this study, the authors investigate the performance of optical code-division multiple access (OCDMA) systems with belief propagation (BP)-based receivers. They propose three receivers for the optical fibre channel that provide a trade-off between detecting complexity and system performance. The first proposed receiver achieves a performance very close to the so-called known interference lower bound. The second receiver exhibits a considerably less complexity at the expense of a slight degradation in performance. They show that the third BP-based receiver, which is a simplified version of the second receiver, is surprisingly the same as the so-called multistage detector in OCDMA systems. They then study the problem of finding proper spreading codes for the proposed receivers. BP-based receivers perform well if the graph corresponding to the spreading matrix has no short cycles. The probability of existence of short cycles directly depends on the sparsity of the spreading matrix. Therefore they look for sparse spreading matrices that are also uniquely detectable, that is, the corresponding input data vectors and the output spread vectors are in one-to-one correspondence. The existence of random uniquely detectable matrices (for which the elements are binary with equal probability) has already been proved by Edrös and Rényi when the dimensions of matrix tend to infinity. In this study, they prove the existence of sparse uniquely detectable spreading matrices in the large system limit, when the number of users and the number of chips approach infinity and their ratio is kept constant. For finite length systems, they propose to use optical codes with one chip interference between codes and show that they exhibit a better performance than random sparse codes.