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¢£¤¥¦¡¢¦¢¦ ¢£¤¥¦ ©¢¦©¢£¤¥¦¡¢£¤¥¦¡¢¦ ¢¦¡¢¦ ¢¦ ©¢¦¢¦ ©¢¦¨¡¢£¦ ¡¢£¤¥¦©¢£¤¥¦ ¨©¢£¦ ¨¢£¦ ¢£¤¥¦VI INTERNATIONAL TELECOMMUNICATIONS SYMPOSIUM (ITS2006), SEPTEMBER 3-6, 2006, FORTALEZA-CE, BRAZILOSTBC¢¦ ¢¦ Fig. 1: Proposed scheme: transmit precoder ω applied to the multiple antennas and OSTBC.§¢¦ §¢¦ Moreover, in [1], STBC was coupled with the proposedprecoder in order to exploit the channel diversity.Although the solution of [1] is optimal for Rayleigh flatfadingchannels, the mathematical formulation that leads to thederivation of this optimum solution is heavily based on thisassumption. So, in this work, we formulate a more general§¢¦criterion to design optimal transmitter precoders for any typeof channel (e.g., Rayleigh, Rician, Nakagami [9]). Our aim isto directly minimize the received BER for a given transmitpower.The rest of this paper is organized as follows. In the nextsection, we describe the system model used in this work. Theproposed criterion of minimum BER and an iterative algorithmto find the optimum solution is presented in Section III. SectionIV compares the performance of the proposed techniqueto other techniques presented in the literature. Conclusions aredrawn in Section V.The following notations are used throughout the paper. Boldlower letters denote column vectors, bold upper letters denotematrices; (·) T , (·) H and (·) ∗ denote transpose, conjugatetranspose, and conjugate, respectively. ‖x‖ is the 2-normof vector x, defined as ‖x‖ = √ x H x, and E{·} denotesmathematical expectation.II. SYSTEM MODELWe consider the downlink of one cell of a wireless communicationsystem, where the BS is equipped with K antennasand the MU has only one antenna. We only consider thetransmission towards one user, assuming that the multipleaccess technique used ensures that there is no interferencebetween users, i.e, co-channel interference.¨§¢£¦ The transmit processing is done by means of a precoder,as depicted in Fig. 1. This precoder is an extension of thepurely spatial downlink beamforming and can be seen asa transformation that transforms the K real antennas intoL virtual antennas, where L ≤ K. Transmit diversity isthus applied to these virtual antennas. Each precoder layerw(l) beamforms the signal s§¢£¤¥¦l (b, n) and scales its power. TheOSTBC block in Fig. 1 corresponds to the coding of thetransmitted signal by an Orthogonal STBC (OSTBC). Weassume also that the signal is transmitted in blocks of lengthN b , so that the channel variation during one block of data isnegligible. However, the channel changes from one block b toanother, characterizing a block-fading channel. Moreover, weassume that the channel is flat and that the DCCM is knownat the BS.The signal at the k-th antenna output for block b is givenbyx k (b, n) =L∑wk ∗ (l)s l(b, n) , (1)l=1where s l (b, n) are the coded symbols after the OSTBC andw k (l) are the coefficients of the precoder related to the realantenna k and virtual antenna l. We assume that within a block,the time index n varies from 0 to N b − 1.At a given block b, the received signal y(b, n) at the MUantenna can be expressed asK∑y(b, n) = h k (b)x k (b, n) + ν(b, n) (2)k=1where h k (b) are the coefficients of the flat channel that linksthe transmit antenna k and the receiver antenna, and ν(b, n)is the additive gaussian noise sample at the MU antenna.