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Figure 6.5. Orthogonal transmit beamforming (OTBF) strategy with multiple antennas.
X = W PU = pk U k w k
k =1
where pk 0 is the average transmit power during the current scheduling instance for user k and e[|Uk|2] = 1. Since the encoding frame is short burst with quasistatic fading, no power adaptation within an encoding frame is required. At any scheduling slot, individual user(s) could be turned off by assigning pk = 0. An admitted user set, A = {k [1, K]: pk > 0}, is de ned as a set of selected users (users with nonzero allocated power) at any scheduling slot. The total transmit power out of the base station at any scheduling slot is constrained by P0:
Calculation of OTBF Weights. Given an admitted user set, the transmit power {p1, . . . , pK} and a realization of the channel fading { L1h1, . . . , LKhK}, the received signal of user k is given by
Yk = 144 h k w kU k + pkLk244 3
p jL j h k w jU j + Z k j A, j k 1444424444 3 4 4
Multibeam interference
where the rst term contains the desired signal and the middle term represents the multibeam interference due to simultaneous transmission of independent information streams. The following considerations apply: Spatial Multiplexing. The level of spatial multiplexing is controlled by the cardinality of the admitted user set |A|. The OTBF weight wk is selected to satisfy w *w k = 1 "k k and the orthogonality conditions h j w k = 0 " j A, j k (6.17) (6.16)
where the subscript * denotes complex conjugate transpose. Note that when pk = 0, the information stream for user k is turned off. In other words, the number of simultaneous transmissions is given by the cardinality of the admitted user set A. Intuitively, we would like the transmit beam of user k to be orthogonal to the rest of the selected users in A. Hence, the signal received by user k would consist of the desired signal only without the interference due to downlink signals to other users. Observe that there are 2nT degrees of freedom in wk and there are 2|A| - 1 equations from the constraints in Equations (6.16) and (6.17). Hence, we have A nT (6.18)
This means that with nT transmit antennas, the base station could support at most nT spatial channels. Spatial Diversity. The remaining degrees of freedom, 2(nT - |A|) + 1, are utilized to realize the diversity gain to maximize the received SNR, w*(h* hk)wk. k k Hence, given a certain admitted user set A, the overall weight determination problem is given by Problem 6.1. Problem 6.1 (Determination of OTBF Weights) wk = arg max w * h *h k w subject to w * w = 1 and h j w = 0 h
"j A , j k (6.19)
Please refer to Appendix 6B for solution of the weights.
Capacity Region of the OTBF. With the optimal beamforming weights {wk}, the multibeam interference becomes zero and there are |A| independent spatial channels. The received signal for mobile user k is given by Yk = pk Lk h k w kU k + Zk (6.20)
Hence, the maximumachievable date rate of the kth spatial channel during the fading block is given by the maximum mutual information between Uk and Yk as
2 pk Lk h k wk rk log 2 1 + 2 sz
"k A
In other words, the conditional instantaneous capacity region COTBF(h1, . . . , hK; A) of the multiuser systems with OTBF processing given the admitted user set A is given by Equation (6.21). For example, suppose K = 3, A = {1, 2} and nT = 2. The conditional capacity region COTBF(h1, . . . , hK, A) is given by COTBF (h 1 , . . . , h K , A ) =
2 2 pL h w pL h w (r1 , r2 , r3 ) : r1 log 2 1 + 1 1 21 1 , r2 log 2 1 + 2 2 22 2 , sz sz r3 = 0, p1 + p2 P0
To evaluate the unconditional instantaneous capacity region, we note the following convexity property of the region. Given any two feasible rate vectors rA and rB in the capacity region, we have r = lArA + lBrB belonging to the capacity region as well for any lA, lB satisfying lA + lB = 1. This is because the rate vector r can be achieved by timesharing between the two points rA and rB. Hence, the instantaneous capacity region COTBF(h1, . . . , hK) is given by the convex hull of the union of COTBF(h1, . . . , hK; A) over all possible admitted user set combinations A COTBF ( h 1 , . . . , h K ) =