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ßÞÐ ßÞÐVI INTERNATIONAL TELECOMMUNICATIONS SYMPOSIUM (ITS2006), SEPTEMBER 3-6, 2006, FORTALEZA-CE, BRAZIL8TABLE I: MU-cBER Algorithm71) Initialize the precoders ω u and the transmit powers6ω u =Ö10 . . . 0 1 0 . . .0 1 · · ·T, pu = 1 ∀u5L1M zeros M zeros42) Compute N u(b), D u(b) and fγ u(b), as defined in (17) and (16)3) Compute A and b using (22)34) Update λ by solving the linear system (21)5) Compute R u and R INTu26) Update ω u by making one power iterationtransmit powerω u =R INTu−1Ru ω u (26)1user 1user 27) Normalize ω uω u = ωu‖ω u‖8) Update N u(b), D u(b) and fγ u(b), since the ω u’s were changed9) Compute g u and ∂gu∂p uas in (23) and (24)10) Update the transmit powers p u using (25)11) Go to 2 until convergence00 2 4 6 8 10 12iterationFig. 2: Evolution of the transmit powers p 1 and p 2 as afunction of the algorithm iterations.user 2user 110 0 iterationbut only adapt it to go in the direction of the maximumeigenvector. This is what is done in ((26). After ) normalizingω u , the values of N u (b), D u (b) and f γ u (b) are updated andused to compute g u and ∂gu∂p u. Finally the transmit powers p uare updated and the constraints values g u are tested. Iterationsare made until the constraints are within a given tolerance,when the algorithm stops.Although the analytical proof of convergence of this algorithmis a complex task, among all the simulations performed,we have not observed one single case of divergence of thealgorithm. Moreover, this algorithm is similar to the DBPC,whose convergence was proved in [3].TABLE II: 2-path scenario parametersUser #1 User #2path #1 path #2 path #1 path #2DOA −35 ◦ −5 ◦ +25 ◦ +55 ◦power −3 dB −3 dB −3 dB −3 dBIV. SIMULATION RESULTSWe consider the downlink of a wireless system, where theBS serves 2 co-channel users. The BS is equipped with a lineararray of K = 4 antennas and the inter-element distance is λc2 ,where λ c is the carrier wavelength. The transmit precoder isK × L (see Fig. 1), so that we have L = 2 virtual antennas,and the Alamouti scheme [4] is used as OSTBC. We assumethat the instantaneous DCCMs R u (b) for all users and forall blocks are perfectly known at the BS. Moreover, withoutloss of generality, we assume that the channel realization isindependent from one block to another.In order to assess the performance of the proposed technique,we have simulated N t = 10 4 training blocks thatwere used to obtain the optimum precoders and N d = 10 6data blocks were used to evaluate the performance of thissolution. The transmit powers p u were normalized with respectto the receiver noise variance σ 2 , so that 0 dB correspondsto p u = σ 2 (note that this is equivalent to say that 0 dBBER10 −110 −20 2 4 6 8 10 12Fig. 3: Evolution of the users’ BER as a function of thealgorithm iterations.corresponds to the transmit power necessary, when using anomnidirectional antenna at the BS, to have a SNR of 0 dB atthe mobile user).In the following, we compare the MU-cBER with the multiuserbeamforming-only DBPC solution of [3] in a 2-pathscenario and 4-QAM modulation 1 . This scenario correspondsto a flat-fading channel for each user. The DOAs and powersof each user are summarized in Table II. Although this is avery simple scenario, it shows the gain obtained by addingdiversity to multi-user downlink beamforming and allows usto physically interpret the results. Initially, the target rawBER was set to 10 −2 , which corresponds to a target SINRof 16.86 dB for the DBPC algorithm. This target SNR wascalculated by considering a Rayleigh channel 2 .Firstly, to illustrate the convergence of the MU-cBER technique,Fig. 2 shows the evolution of the users’ transmit powers,while Fig. 3 shows the evolution of the BER, for a target BERof 10 −2 . It can be seen that the MU-cBER converges in fewiterations.Fig. 4 shows the radiation pattern obtained with the DBPCtechnique for each user, where the vertical dashed lines1 For a 4-QAM modulation [11], we have NeN = 1 and Ns = d2 min= 1. 2 2 For a Rayleigh channel, we have [1]: BER = 21 1 −ÕSINR1+SINR.