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VI INTERNATIONAL TELECOMMUNICATIONS SYMPOSIUM (ITS2006), SEPTEMBER 3-6, 2006, FORTALEZA-CE, BRAZILBy inserting (1) into (2), we can write that(L∑ K)∑y(b, n) = wk ∗ (l)h k(b) s l (b, n) + ν(b, n) . (3)l=1k=1By defining the l-th precoder layer w(l) =[w 1 (l) w 2 (l)T· · · w K (l)], which maps the l-th virtual antenna into theK real antennas[(see Fig. 1), and the spatial channel vectorTh(b) = h 1 (b) h 2 (b) · · · h K (b)], which links themultiple antennas at the BS and the MU’s antenna, we canrewrite the summation over k in (3) as a dot product to getL∑y(b, n) = w(l) H h(b)s l (b, n) + ν(b, n) . (4)l=1We now define the signal s l (b, n) filtered by the spatialchannel h(b) as ˜s l (b, n) h(b)s l (b, n). By stacking theprecoder layers w(l) and the vectors ˜s l (b, n) for l = 1, . . .,Lin the vectors ω and ˜S(b, n), respectively, we can rewrite (4)asy(b, n) = ω H ˜S(b, n) + ν(b, n) , (5)[] Twhere ω = w(1) T w(2) T · · · w(L) T representsthe precoder in vector form and ˜S(b,[n) = ˜s 1 (b, n) T˜s 2 (b, n) T · · · ˜s L (b, n) T ] T. The precoder is assumed to benormalized, i.e., ‖ω‖ = 1.The vector ˜S(b, n) can be related to the coded symbolss l (b, n) by the channel convolution matrix H(b) as˜S(b, n) = H(b)s(b, n) , (6)[Twhere s(b, n) = s 1 (b, n) s 2 (b, n) · · · s L (b, n)]is thecoded signal vector and the KL × L channel convolutionmatrix is given by⎡⎤h(b) 0 0 · · · 00 h(b) 0 · · · 0H(b) = ⎢⎣.. . . ..⎥. ⎦ , (7)0 0 0 · · · h(b)where 0 is a column vector composed by K zeros.We can thus rewrite the received signal at the MU y(b, n)asy(b, n) = ω H H(b)s(b, n) +ν(b, n) , (8)} {{ }y u(b,n)It is worth recalling that for each block b the channel presentsa different condition, i.e., it can be in a deep fade or in a goodcondition. This condition affects the received signal power andthe SNR at the MU.The useful received power by the MU during block b isgiven by} ∣∣∣yu P(b) = E{(b, n) ∣ 2 (9)= ω H H(b)E{s(b, n)s H (b, n)}H(b) H ω .Assuming that the “uncoded” symbols s(b, n) are i.i.d. withpower P TX and using the fact that an OSTBC is used, whichis an orthogonal code, we obtain that the “coded” symbolsare { also i.i.d. with}the same power P TX . Thus, we have thatE s(b, n)s H (b, n) = P TX I and (9) becomesP(b) = P TX ω H R(b)ω , (10)where R(b) = H(b)H(b) H is the instantaneous DCCM forblock b. Further assuming that the noise power is σν 2 we obtainthe SNR for block b asω H R(b)ωγ b = P TXσν2 . (11)Assuming the use of a linear modulation, the BER for blockb can be approximated [10], at relatively high SNR, by)BER(b) ≈ N eN Q (√d2min2 γ b, (12)where N is the number of bits per symbol, d min isthe minimum distance between 2 points for a unitarypower constellation, N e is the mean number of neighborsat minimum distance and the Q-function is given by∞∫ ( )Q(x) = √ 12πexp dt.x−t 22III. MINIMUM BER CRITERIONAs mentioned before, downlink beamforming and transmitdiversity are both beneficial to the link quality, in terms ofreducing the BER for a given transmit power. However, theyhave opposite premises and can not be easily joined. In thisaim, we propose a design criterion to optimize the precoderω to directly minimize the BER of the received signal atthe MU, for a given transmit power at the BS. This can beseen as an extension of the purely spatial beamforming, whichaims to minimize the BER by maximizing the SNR of thereceived signal. We will show that this optimum precoder leadsto significantly better performances than the OSTBC applieddirectly to the real antennas.From the BER expression in (12), we can write the proposedcriterion as{(Nmin BER E eN Q √Ns ) }γ b, (13)s.t. ω H ω = 1where, for the sake of legibility, we have defined N s = d2 min2 .In order to find the optimum point of the criterion (13), wepropose to use an iterative algorithm based on the gradient (orsteepest descent) method [11]. The constraint will be treatedby applying the gradient method to the unconstrained functionand, after each iteration, projecting the unconstrained solutioninto the constrain set so that it satisfies the constraint. Thismethod is well suited for the type of constraint that we havehere, i.e., equally constrain of a simple form.Let us define f(γ b ) as the derivative of Q( √ N s γ b ) withrespect to γ b , given bydQ( √ N s γ b )dγ bN s= −2 √ exp2πN s γ b(− N sγ b2) −f(γ b ) .(14)