802. 16m Physical Layer (Part II) Maximum Sum Capacity Capacity Region Near-Far Capacity Region Capacity of User 1 FIGURE 10-33 An example illustration of capacity region signal vector, kECNikXN, denotes the flat-fading channel matrix and n is the independent and identically distributed additive white Gaussian noise n = (n1, N2, nNr), nk~N(0,1) vector at the BS. We assume that the receiver k has perfect and instantaneous knowledge of the channel matrix In the downlink as illustrated in Figure 10-32, the received signal at the kth receiver can be written as Yk = Hkx + nk Ak = 1, 2, .,Nu where Hke CNrk NN is the downlink channel and nke CNrk NN is the complex-valued - additive Gaussian noise at the kth receiver. We assume that each receiver also has perfect and instantaneous knowledge of its own channel matrix Hk The transmitted signal X is a function of the multiple users' information data, i.e., X = where Xk is the signal carrying kth user's message with covariance matrix Ok = E{X&x}}. The power allocated to the kth user is given by tr{Qk}. Under a sum power constraint at the BS, the power allocation needs to maintain < Ptotal. Assuming a unit variance for the noise, it is now known that the capacity region for a given matrix channel realization can be written as: (10-46) where R+N1 is the Nu-dimensional set of positive real numbers. The above equation may be optimized over each possible user ordering. Although difficult to realize in practice, the computation of the capacity region can be simplified using the assumption that the downlink capacity region can be calculated through the union of regions of the dual multiple access channel with all uplink power allocation vectors meeting the sum power constraint. The fundamental effect of the use of multiple antennas at either the BS or the user terminals in increasing the channel capacity is best understood by examining how the sum capacity, i.e., the point obtained by the maximum Rk in the capacity region, scales with the number of active users. 10.4 Multi-antenna 