XK] can be written as X max p(X|y). eral, the solution to the above problem requires a global search over the joint space of K Maximum-a-posteriori (MAP) estimate of random variable X given that we have observed Y = y, is given by the value of x that maximizes fx|y(x|y) if X is a continuous random variable, and Px|y(x|y) if X is a discrete random variable. The MAP estimate is shown by MAP = arg maxxfxy(x)y). New Radio Access Physical Layer Aspects (Part 1) 363 uplink users X1 X X2 X ... X . Due to the sparse nature of SCMA transmission scheme, the MPA detector can be applied to reduce the decoding complexity, which iteratively updates the belief associated with the underlying factor graph. Once X has been estimated, we can use the inverse mapping function (fk)-1 to recover the original user information bits Bk [50-52]. 3.3.3.2 Power-Domain Non-orthogonal Multiple Access Power-domain NOMA can serve multiple users in the same time slot, OFDMA subcarrier, or spreading code, and multiple-access is realized by allocating different power levels to different users depending on their relative position to the base station. Fig. 3.36 illustrates the concept of power-domain NOMA in the downlink with two UEs that utilize SIC receiver. For simplicity, we assume in this section the case of single trans- mit/receive antennas. The overall system transmission bandwidth is assumed to be 1 Hz. The base station transmits signal Xi to the ith UE with transmit power P where E{Ixil}} = 1 assuming . In power-domain NOMA, X1 and X2 are superposed as X = + 2; thus the received signal at the ith UE can be written as yi = hix+Wi, in which hi is the complex-valued channel coefficient between the ith UE and the base sta- tion, and Wi denotes a zero-mean AWGN plus inter-cell interference. The PSD of Wi is Noi. In the downlink NOMA, the SIC receiver is implemented at the UE receiver. The optimal order for decoding is in the order of decreasing channel gain normalized by noise and P2 (UE) Total P1 (UE1) SIC of UE2 signal Frequency Decode UE1 sig