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The gradient { of the unconstrained cost function(NJ = E eN Q √Ns ) }γ b with respect to ω is givenby∇J ω = N eN= −2 P TXσ 2 νßÞÐ ßÞÐVI INTERNATIONAL TELECOMMUNICATIONS SYMPOSIUM (ITS2006), SEPTEMBER 3-6, 2006, FORTALEZA-CE, BRAZILTABLE I: mBER-TD-DB Algorithm1) Initializationω =Ö10 . . .0 1 0 . . .0 1 · · ·T(19)L1M zeros M zeros2) Compute f(γ b ) and R kdQ( √ N s γ b ) dγ bdγ b dωN eN E {f(γ b )R(b)}ω ,(15)where γ b is given by (11).{ By introducing } the iteration index k and by defining R k =E f(γ b )R(b) , we can write the normalized gradient updateasω k+1 = ω k − µ ′ ∇J ω‖∇J ω ‖(16)= ω k + µ ′ R k ω k‖R k ω k ‖ ,where µ ′ is the adaptation coefficient. This update is followedby the normalization of ω k+1ω k+1 = ω k+1‖ω k+1 ‖ . (17)Equations (16) and (17) compose the core of the proposedalgorithm. We highlight that, at the update step (16), ω k wasnormalized, and ω k+1 will be normalized at the end of theR kω k‖R k ω k ‖current iteration. Hence, the update term has only thefunction of changing the direction of the precoder vector ω kbut not its norm.It is worth mentioning that R k depends on ω k , sincef(γ b ) depends on ω k , but it is considered to be constant toperform one iteration of the algorithm and then its value isupdated to match the new transmit filter ω k+1 . So, we do notwant to change too much ω k in order to slowly converge tothe optimum solution. Thus, we propose to combine the oldR kω k‖R k ω k ‖precoder ω k with the update term in order to obtainthe new precoder ω k+1 . We obtain thus the following updateequation for ω kω k+1 = (1 − µ)ω k + µ R kω k‖R k ω k ‖ , (18)where µ is the adaptation coefficient.The proposed algorithm, called mBER-TD-DB (minimumBER for Transmit Diversity and Downlink Beamforming), issummarized in Table I, where the expectation was replaced bya temporal estimation over B blocks. The initialization givenby (19) corresponds to directly wiring each virtual antenna tothe corresponding real antenna. This initialization was chosensince it represents a neutral situation, where the precoder ωdoesn’t affect the transmission.For the simulations considered in this paper, we haveempirically determined that µ = 0.25 was a good choice for afast convergence. Higher values of µ are undesirable becauseR k depends on ω k and thus if we change too much the actualsolution, it can drastically change the matrix R k causing thenon-convergence of the algorithm. On the other side, lowervalues will only slow down the convergence.The analytical proof of convergence of the proposed algorithmis a complex task and have not yet been done, but weω H kγ b = P R(b)ω kN sTXσν2 , f(γ b ) =2 √ exp− Nsγ b2πN sγ b 2∀bR k = 1 B−1f(γ b )R(b)Bb=03) Update ω kω k+1 = (1 − µ) ω k + µ R kω k‖R k ω k ‖4) Normalization of ω k+1ω k+1 = ω k+1‖ω k+1 ‖5) Go to 2 until convergencehave not observed one single case of divergence among all thesimulations performed.IV. SIMULATION RESULTSWe consider the downlink of one cell of a wireless system.The data is transmitted in blocks of length N b and we assume,without loss of generality, that the channel realization isindependent from one block to another, i.e., path coefficientsvary from one block to another independently. The BS isequipped with a linear array of K = 4 antennas and the interelementdistance is λc2 , where λ c is the carrier wavelength. Thetransmit precoder is K×L (see Fig. 1) and we have consideredL = 2 virtual antennas. The Alamouti scheme [2] is usedas OSTBC. Furthermore, we assume that the instantaneousDCCMs R(b) for all blocks are perfectly known at the BS.In order to assess the performance of the proposed technique,we have simulated N t = 3 × 10 4 training blocks andN d = 10 5 data blocks. For each block, a different channelrealization was drawn according to the used channel model.The corresponding DCCMs R(b) of the training blocks wereused as input for the proposed algorithm, which computes theoptimum precoder ω opt , see Table I where we have B = N t .The transmit power P TX was normalized with respect to thereceiver noise variance σν 2, so that 0 dB corresponds to P TX =σν 2 (note that this normalization is equivalent to say that 0 dBcorresponds to the transmit power necessary, when using anomnidirectional antenna at the BS, to have a SNR of 0 dBat the MU). The optimum precoder is then used to evaluatethe raw BER of a 4-QAM modulation 1 at the MU for thesubsequent data blocks, using (12).In the following, we compare the performance of the proposedmBER-TD-DB, the 2D Eigen-Beamforming techniqueof [1], Downlink Beamforming (DB) [6] and the Alamoutischeme [2]. The Alamouti results were obtained by using theextreme real antennas to maximize the inter-antenna distanceand minimize the correlation between them. We will present1 For a 4-QAM modulation [10], we have N s = d2 min2 = 1 and NeN = 1.