ords makes the near-optimal detection feasible through iterative MPA. Such low complexity of multi-layer detection allows excessive codeword overload- ing in which the dimension of multiplexed layers exceeds the dimension of codewords. Belief propagation, also known as sum-product message passing, is a message-passing algorithm for perform- ing inference on graphical models such as Bayesian networks and Markov random fields. It calculates the marginal distribution for each unobserved node conditional on any observed nodes. Belief propagation is commonly used in artificial intelligence and information theory and has demonstrated empirical success in numerous applications including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability. The algorithm was formulated as an exact inference algorithm on trees, which was later extended to polytrees. While it is not accurate for general graphs, it has been shown to be a useful approxi- mation algorithm. If X = {Xi} is a set of discrete random variables with a joint mass function p, the marginal distribution of a single Xi is the summation of p over all other variables Px,(Xj) = However, this would become computationally prohibitive, whereas by exploiting the polytree structure, belief propaga- tion would allow the marginals to be computed more efficiently [50-52]. 360 Chapter 3 Optimization of overloading factor along with modulation/coding levels of layers provides a more flexible and efficient link-adaptation mechanism. On the other hand, the signal spread- ing feature of SCMA can improve link-adaptation as a result of less colored interference. In SCMA, incoming bits are directly mapped to multi-dimensional complex codewords selected from predefined codebook sets. The co-transmitted spread data are carried over super-imposed layers. Since layers are not fully separated in a NOMA system, a nonlinear receiver is required to detect the intended layer of each user. Therefore, additional detection complexity is the cost of the nonorthogo