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¡¤¥¦§¨¡¢£¤¨ ¡¢¤¥¦§¨ ¡¤¨ ¡¢¤¨¡¢£¤¥¦§¨ ¡¢¤¥¦§¨ ¡¤¨ ¡¤¨ ¡¢£¤¨ ¡¢£¤¨ ¢£¤¥¨ ¡¢£¤¥¦§¨¡¢¤¥¦§¨ ¢¤¥¨ ¢¤¥¨ ¡¢¤¥¦§¨VI INTERNATIONAL TELECOMMUNICATIONS SYMPOSIUM (ITS2006), SEPTEMBER 3-6, 2006, FORTALEZA-CE, BRAZILOSTBC ¡¢¤¨Fig. 1: Transmission of the i-th user signal s i (b, n) towards user u, using Orthogonal Space-Time Block Code (OSTBC) beforethe precoder ω i . ¡¢©¤¨ ¢©¤¥¨¡¢©¤öf the i-th user’s signal, received by user u. This precoder isan extension of the purely spatial downlink beamforming andcan be seen as a transformation that, for each user, transformsthe K real antennas into L virtual antennas, where L ≤ K.Transmit diversity is thus applied to these virtual antennas.¡¢©¤Ëach precoder layer w i (l) beamforms the signal s i,l (b, n) andalso scales its power. The OSTBC block in Fig. 1 correspondsto the coding of the transmitted signal by an Orthogonal Space-Time Block Code (OSTBC). We further assume that the signalis transmitted in blocks of length N b , so that the channelvariation during one block of data is negligible. However, thechannel changes from one block b to another, characterizing ablock-fading channel. Moreover, we assume that the channelis flat.Considering the transmission of the i-th user’s signal, thesignal at the k-th antenna output and block b is given byx i,k (b, n) =L∑wi,k(l)s ∗ i,l (b, n) , (1)l=1where s i,l (b, n) are the “coded” symbols of the i-th user (afterthe TD processing) and w i,k (l) are the coefficients of the i-thuser’s precoder related to real antenna k and virtual antenna l.We assume that within a block, the time index n varies from0 to N b − 1.The total multi-user signal at antenna k is then given byx k (b, n) =U∑x i,k (b, n) . (2)i=1So, at a given block b, the received signal y u (b, n) at the u-thuser’s antenna can be expressed by¡¢©¤¥¦§ÿ u (b, n) =U∑i=1 k=1K∑h u,k (b)x k (b, n) + ν(b, n) , (3)where h u,k (b) is the coefficient of the channel that linksantenna k and user u, and ν(b, n) is the additive gaussiannoise sample at the user’s antenna.By inserting (2) into (3) we can write(U∑L−1∑ K)∑y u (b, n) = wi,k(l)h ∗ u,k (b) s i,l (b, n)+ν(b, n) ,i=1 l=0k=1or rather, in vector formy u (b, n) =(4)U∑L−1∑w i (l) H h u (b)s i,l (b, n) + ν(b, n) , (5)i=1 l=0where the l-th precoder layer w i (l) =[w i,1 (l) w i,2 (l)T· · · w i,K (l)]is a vector that maps the i-th user’s l-thvirtual antenna[into the K real antennas (see Fig. 1) andTh u (b) = h u,1 (b) h u,2 (b) · · · h u,K (b)]is the spatialchannel vector that links the BS and user u.In order to obtain a full matrix notation, we now definethe signal s i,l (b, n) filtered by the spatial channel h u (b) as˜s i,u,l (b, n) h u (b)s i,l (b, n). By stacking the vectors w i (l)and the vectors ˜s i,u,l (b, n) for l = 1, . . .,L in the vectors ω i ¡¢¤¨ ¡¢¤¨ ¡¢¤¨ ¡¢¤¨
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
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THÈSEprésentéeau Conservatoire N
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ivRemerciementsJe remercie mes pare
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AbstractThis work deals with the us
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xTABLE DES MATIÈRES2.7.4 Contrôle
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Table des figures2.1 Schéma classi
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Liste des tableaux2.1 Algorithme DB
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Liste des acronymes et abréviation
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1IntroductionLe but d’un système
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1.1. Organisation et contributions
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6 Chapitre 2. Antenne multi-capteur
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10 Chapitre 2. Antenne multi-capteu
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2.7. Techniques de FV multi-utilisa
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2.7. Techniques de FV multi-utilisa
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3Techniques mono-utilisateur3.1 Int
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3.2. Précodeur et diversité de tr
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3.3. Diversité de transmission et
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3.3. Diversité de transmission et
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3.4. Allocation optimale de puissan
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3.4. Allocation optimale de puissan
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3.5. Minimisation de la variance 63
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3.6. Minimisation du TEB 85plat.On
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3.6. Minimisation du TEB 8710 0 RSB
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3.6. Minimisation du TEB 8910 2 γ
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4. Mise à jour de ω kω k = ω k
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3.6. Minimisation du TEB 933.6.4 R
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3.6. Minimisation du TEB 97351.1130
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Page 148 and 149: 128 Bibliographie[11] J. K. Cavers.
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Page 153 and 154: Index d’auteursAAlamouti, S. M.,
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