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VI INTERNATIONAL TELECOMMUNICATIONS SYMPOSIUM (ITS2006), SEPTEMBER 3-6, 2006, FORTALEZA-CE, BRAZILand ˜S i,u (b, n), respectively, we can rewrite (5) asy u (b, n) =U∑ω H ˜S i i,u (b, n) + ν(b, n) , (6)i=1] Twhere ω i =[w i (1) T w i (2) T · · · w i (L) Trepresents the i-th user’s [ precoder in vector formand ˜Si,u (b, n) = ˜s i,u,1 (b, n) T ˜s i,u,2 (b, n) T · · ·˜s i,u,L (b, n) T ] T. The precoders ωi are assumed to benormalized, i.e., ‖ω i ‖ = 1.The vector ˜S i,u (b, n) is related to the signals s i,l (b, n) bythe channel convolution matrix H u (b) as˜S i,u (b, n) = H u (b)s i (b, n) , (7)[] Twhere s i (b, n) = s i,1 (b, n) s i,2 (b, n) · · · s i,L (b, n)is the coded signal vector and the KL×L channel convolutionmatrix is given by⎡⎤h u (b) 0 0 · · · 00 h u (b) 0 · · · 0H u (b) = ⎢⎣.. . . ..⎥. ⎦ , (8)0 0 0 · · · h u (b)where 0 is a column vector composed by K zeros.Finally, we can write y u (b, n) asy u (b, n) = ω H u H u(b)s u (b, n) +} {{ }y u,u(b,n)U∑ω H i H u(b)s i (b, n) +ν(b, n) . (9)i=1i̸=u} {{ }y u,i(b,n)It is worth recalling that for each block b the channel h u (b)presents a different condition, i.e., it can be in a deep fade orin a good condition. This condition affects the received powerand the SINR of each user.Based on (9) we can write the useful received power foruser u at block b as{∣∣yu,u(b, n) ∣ }2P u (b) = E= p u ω H u R u (b)ω u , (10)where R u (b) = H u (b)H u (b) H is the instantaneous DCCM ofthe u-th user at block b, which has a block hermitian structure.We have used the fact that the coded symbols are i.i.d. andtheir power is p u , which came from the fact that an OSTBCis used, which is an orthogonal code, and the assumption thatthe “uncoded” symbols s u (b, n) are i.i.d. and have power p u .Remark that p u is the transmit power for user u, while thetotal transmit power is given by P TX = ∑ p u .uFollowing the same reasoning as for the useful power, wecan write the interference caused by all other users in user uasU∑I u (b) = p i ω H i R u(b)ω i . (11)i=1i̸=uFinally, the SINR of the u-th user at the block b is givenbyp u ω H u R u (b)ω uγ u (b) =, (12)U∑p i ω H i R u(b)ω i + σν2i=1i̸=uwhere σν 2 is the noise variance.Assuming the use of a linear modulation, the BER for blockb can be approximated [11], at relatively high SINR, by)BER u (b) ≈ N eN Q (√d2min2 γ u(b), (13)where N is the number of bits per symbol, d min is theminimum distance between 2 points for a unitary power constellation,N e is the mean number of neighbors at minimum∞∫distance and Q(x) = √ 12πx(exp−t 22)dt is the Q-function.III. MULTI-USER MINIMUM TRANSMIT POWER WITHBER CONSTRAINTSAs far as diversity is concerned, the optimum precoders aresuch that the virtual antennas are uncorrelated, which wouldprovide the maximum channel diversity for the OSTBC. Onthe other hand, from the point of view of multi-user downlinkbeamforming, it is desirable that each precoder layer w i (l)beamforms the signal s i,l (b, n) towards the desired user i andmitigates the interference radiated towards all the other users,called pollution here. These two goals are clearly contradictory,since uncorrelated antennas means omnidirectionaltransmission and beamforming requires correlated antennas.As we have mentioned earlier, we propose a design criterionto optimize the precoders ω i in order to find the best tradeoffbetween both wishes, i.e., maximum channel diversity andminimum pollution. These precoders lead to minimum BERfor a given transmit power. However, in a multi-user scenario,it makes more sense to minimize the total transmit power whilerespecting a target BER for each user within the consideredcell.From the BER expression in (13), we can write the proposedcriterion as∑min P TX = U p u ω H u ω uu=1{s.t.BER u EN eN Q ( √Nsγ u (b)) } = c u ∀u, (14)where c u is the target BER for user u and, for the sake oflegibility, the have defined N s = d2 min2 .By introducing the Lagrange multipliers λ u , the lagrangianassociated with (14) can be written as{U∑U∑J = p u ω H u ω N e(√ )u+ λ u(E} )N Q Ns γ u (b) −c u .u=1u=1(15)Let( us define f(γ u (b)) as minus the derivative of√Ns )N eN Q γ u (b) with respect to γ u (b), given by(N e N sf(γ u (b)) 2N √ 2πN s γ u (b) exp − N )sγ u (b). (16)2