TH`ESE - Library of Ph.D. Theses | EURASIP

VI INTERNATIONAL TELECOMMUNICATIONS SYMPOSIUM (ITS2006), SEPTEMBER 3-6, 2006, FORTALEZA-CE, BRAZILLet us further define the following variablesN u (b) ω H u R u (b)ω uU∑D u (b) p j ω H j R u(b)ω j + σ 2. (17)j=1j≠ux u (b) N u(b)D u (b)The optimum solution for the precoders ω u can be found bysetting to zero the derivatives of the lagrangian J with respectto ω i and p i . Firstly, we derive with respect to ω i to obtain{ (∂Jf γi (b) ) }= 2p i ω i − 2p i λ i E∂ω i D i (b) R i(b) ω i{ U (∑ f γk (b) ) }+ 2p i λ k ED k (b) γ k(b)R k (b) ω i . (18)k=1k≠iBy posing ∂J∂ω i= 0, the factor 2p i can be simplified andwe obtain that ω i is eigenvector of the following generalizedeigendecomposition⎛ ⎞⎜U∑ ⎟⎝ λ k ˜Rk + I⎠ω i − λ i R i ω i = 0 , (19)k=1k≠i{ ( ) }f γ k (b)where ˜Rk = ED k (b)γ k (b)R k (b) and{ ( ) }f γ i(b)R i = ED i(b)R i (b) . Let us further defineR INTi= U ∑k=1k≠iλ k ˜Rk + I.The eigendecomposition (19) can be used to find the precodersω i for a given set of λ k , with k ≠ i. We observethat (19) is similar to the eigendecomposition for the criterionof minimum transmit power with SINR constraints, see [3,eq. (14)], but here the matrices R i (b) and R k (b) are weightedinside the expectation. Since these weights are all positives,R i and ˜R i are homogenous to channel covariance matrices.We still need to determine which eigenvector to takefrom (19). To do so, we observe that if we had SINRconstraints instead of BER constraints, it is clear that wehave to take the eigenvector corresponding to the maximumeigenvalue, as proved in [3] for the DBPC algorithm.Remark that in order to obtain ω i , it is necessary to knowall the Lagrange multipliers λ i . Thus, to find the optimal λ i ’swe derive J with respect to p i , obtaining∂J∂p i= ω H i ω i − λ i E+ ∑ k≠iλ k E{f ( γ i (b) ) x i (b){f(γk (b) )}D k (b) γ k(b)ω H i R k(b)ω i}(20)and then we make ∂J∂p i= 0, which give us the linear systembelowAλ = b , (21)where λ = [ λ 1 λ 2] T. . . λ U and⎧b i = ω H i ω i = 1⎧ {⎪⎨ E f ( γ ⎪⎨ i (b) ) }x i (b)A i,k = { ( )}f γ k (b)⎪⎩⎪⎩ − ED k (b)γ k (b)ω H i R k(b)ω ii = ki ≠ k(22)where b i corresponds to the i-th element of the column vectorb and A i,k corresponds to the element in the row i andcolumn k of the matrix A. Remark that, in order to computeλ, we have to known all the ω i ’s and vice-versa, leading toan iterative procedure.Finally, we still need to determine the transmit powers p u .To do so, we can derive J with respect to λ u and make it equalzero. However, by doing so, we obtain the BER constraints.Thus, given a set of λ u , we can compute the precoder ω u andthe transmit powers are obtained such that the constraints aresatisfied.It is worth to note, however, that the BER is given by theexpectation of a non-linear function (the Q-function) of theSINR. Moreover, one transmit power p u affects the BER of allusers. Then, compute p u analytically is not a straightforwardtask. We propose to consider only the influence of the transmitpower p u on the BER u and linearize the Q-function in orderto find the approximate value of p u that satisfy the constraints.This is done by using the Newton method [12] to find the rootof the u-th constraint g u (p u ), expressed byg u (p u ) = E{N e(√ ) }N Q Ns γ u (b) −c u . (23)The derivative of g u (p u ) with respect to p u is{∂g u= − E∂p uf ( γ u (b) ) }x u (b) . (24)So, from a given value of p − u , one Newton iteration givesA. Proposed algorithmp + u = p− u − g u∂g u. (25)∂p uWe propose an iterative algorithm to find the optimumprecoders ω u and transmit powers p u , since each step describedbefore is connected to the others by the precoderω u , the transmit powers p u and the Lagrange multipliersλ u . This algorithm is described in Table I and is calledMulti-User constrained BER (MU-cBER). Note that, in apractical implementation of the algorithm, we must replacedthe expectations by averages over the blocks b.We start from a set of orthogonal layers w u (l) and unitarytransmit powers. Then, after computing A and b, we updateλ following (21). After computing R u and R INTu , we proposeto use a power iteration to find the maximum eigenvector. Thepower method [13] is used to find the maximum eigenvectorof an eigendecomposition, but since the matrices R u andR INTu depends on ω u , we don’t want to fully update ω u ,,