• Aucun résultat trouvé

Alternative (sub-optimal) approach

Dans le document The DART-Europe E-theses Portal (Page 90-96)

Robust Beamformers for Partial CSIT

4.8 Alternative (sub-optimal) approach

We propose another precoder to optimize the EWSR under partial CSIT. We reconsider as the KG algorithm in1.4.2 but this time we apply results from a slightly different type of large system analysis which considers that the number of transmit antennas at the Tx and the number of receive antennas at the users are jointly going to infinity with bounded ratio whereas until now we have supposed that the number of transmit antennas and the number of users go jointly to infinity. Indeed, this particular regime of large systems is not in line with what future network will look like. In sub-6 GHz bands receivers will be merely equipped with 2 or 4 antennas. However, our study would still very beneficial, since all the results that we obtain are accurate even when the number of users’ antennas is finite or small. A recent paper in [43] used this assumption (regime), however, the authors used the large system to provide accurate estimates of the average achievable rates using some precoder and did not use the large system approach to design precoders as we do.

In what follows we shall go one step further in the separable channel correlation model and assumeCr,k,bi =Cr,k,∀bi. From (1.5) and using the notation in

(4.7),Rc,k andRc,k are given respectively by

Rc,k=I+Hc,kQHHc,k (4.56) and

Rc,k =I+Hc,kQc,kHHc,k (4.57) The large MIMO asymptotics from [44],[45], in which both M, N → ∞ at constant ratio, tend to give more precise approximations whenM is not so large. For the general case of Gaussian CSIT with separable (Kronecker) covariance structure, [44],[45] lead to asymptotic expressions of the form

EHlog det(I+HQHH)

where the maximization over zandw should be carried out alternately (and not jointly: the joint optimization may correspond to a global maximum or a saddle point; the cost function is concave however inz orw separately). We shall assume the same fully separable correlation Gaussian channel model. The EWSR of (1.24) can be written as

EW SR= EH

using (4.58), the EWSR now becomes

EW SR=

where Tc,k plays the role of some kind of total Tx side channel correlation matrix. Note that the weighting coefficients z, w depend on the BFs also though. The EWSR expression in (4.60) can be maximized alternatingly over the{gk}, the {zc,k, wc,k}and the{zc,k, wc,k}. For the optimization of the BFs gc,k, for givenz,w, introduce

Rbc,k,c,k = I+Qbc,kTk(zc,k, wc,k) Rbc,k = I+Qbc,kTc,k(zc,k, wc,k) Rbc,k = I+Qbc,kTk(zc,k, wc,k).

(4.63)

Inspired by section1.4.2, we get B˘c,k=IHc Tk(zc,k, wc,k)Rbc,k,c,k1 Ic

c,k= X

(j,i)6=(c,k)

uiIHj

Tj,i(zj,i, wj,i)Rbj,i1−Tj,i(zj,i, wj,i)Rbj,i1

Ij.

(4.64)

Note that in spite of their appearance, matrices of the formTR1are symmetric.

Indeed, if e.g. T is invertible then T(I +QT)1 = (T1 + Q)1. For the optimization max

z0,w0{log detS(Q, z, w)−zw}, we get from the extremum conditions

w=f(Q, z, w) = tr{QCt[I+QT(z, w)]1}

z=g(Q, z, w) = tr{Cr[I+wCrH(I+zQCt)1QHH]1} (4.65) which can be iterated until a fixed point. To get the normalized precoder we use

Gc,k=eigenmatrix( ˘Bc,k,A˘c,kcIM) (4.66) with eigenvalues Σc,k=eigenvalues( ˘Bc,k,A˘c,kcIM).

Let Σ(1)c,k =Gc,kHc,kGc,k(2)c,k =Gc,kHc,kGc,k. Powers Pc,k ≥0 are defined as in (1.19). λc is determined also as described in section 1.4.2. The algorithm can then be summarized as in Table 4.3. The performance of the proposed precoder is evaluated through numerical simulations. The algorithm in Table 4.3 is repeated and averaged for 100 realizations of the channels Hc,k for allc, kand their estimates Hc,k.

Figure4.12shows the EWSR versus transmit SNR for a cellular network having C= 6 cells and one BS at the center of each cell. Then, we assume that each

Table 4.3: The Iterative Algorithm

For (c, k) = (1,1). . .(C, K), initialize Qc,k,zc,k,zc,k,wc,k,wc,k, Tk(zc,k, wc,k),Tk(zc,k, wc,k)

Repeat until convergence Forj = 1 . . . C

Setλc = 0,λcmax For k

Compute ˘Ac,k using (4.64) Nextk

Repeat until convergence λc = 12cc)

Fork

Compute ˘Bc,k using (4.64)

Compute the generalized eigenmatrixGc,k of ˘Bc,k and ˆAc,kcIM

Normalize the generalized eigenmatrix so as to haveGc,k Compute Σ(1)c,k =Gc,kHc,kGc,k(2)c,k=Gc,kHc,kGc,k Compute Pc,k as in (1.19)

Nextk

ComputeP=P

kPc,k

iftr(P)≤Pc, setλcc , otherwise set λcc For allk, set Qc,k=Gc,kPc,kGc,k,H

For allk, compute zc,k,zc,k,wk,wc,k

and then computeTc,k(zc,k, wc,k) and Tc,k(zc,k, wc,k) Nextj

BS is endowed with a numberM = 6 Transmit antennas and serving one Rx equipped withN = 6 Receive antennas. We assumeCr=IN and we compare the performance of the algorithm explained in Table 4.3 to the performance of the naive approach ENAIVEKG of4.1. The Txs design their BFs according to their partial CSIT. Then, the BFs designed with the partial CSIT need to be evaluated with the real channel in order to get the actual resulting WSR.

We suppose that the error covariances matrices Θp,i,c,k for all i, c andk are identity matrices multiplied by a scalar α2; α2 = 101 in Figure 4.12. From (1.22), we construct the channelsHi,c,k and the channel estimatesHi,c,k, for

alli,cand k :

Hi,c,k =He(1)i,c,kΘ1/2t,i,c,k +He(2)i,c,kΘ1/2p,i,c,k (4.67) Hk,bk =He(1)i,c,kΘ1/2t,i,c,k. (4.68) Moreover, we have:

Θp,i,c,k2IM (4.69)

and

Θt,i,c,k= (1−α2)IM (4.70)

Figure 4.12: Sum rate comparisons for C=6, M=6 and N=6

We do not show simulations that compare the approach proposed in this section to the ESEI-WSR; however we can assure that the ESEI-WSR outperforms easily this approach.

4.9 Conclusion

In the previous chapters, we have talked about BFs. But a requirement to practically design the BFs is to know perfectly the channels. As explained in Chapter 1, the procedure to acquire the channels differ from FDD to TDD.

We have concluded as well that TDD is more suitable to Massive MIMO. Then

we have said that in TDD, DL channels which are estimated from UL training suffer from estimation noise. In order to design beamformers robust to that, a new optimization problem must be dealt with, the EWSR maximization problem. The objective function is the expected value of the weighted sum rate.

The constraints are always the same, a power budget limit per cell. The new problem of interest is a stochastic problem. Two solutions exist already, this first one is ENAIVEKG, which is the same as KG but it is proposed to replace the true channels but their estimates without considering the knowledge of channel estimation error covariance. The second one is EWSMSE, it proposes to reformulate the EWSR maximization problem into a minimization of the expected value of the MSE. Then comes our proposal. We propose to solve the EWSR as a DC approach. We divide the objective function into two parts corresponding to the expected value of the rate of the user of interest and to the expected sum rate of the others. A new objective function appears.

The solution is given by the eigenmatrix of some matrices. Then, we propose to calculate the expected values of these matrices as shown in Appendix.

Simulations show for different configurations that our approach is the best one. The gain over the other approaches comes from exploiting the channel covariance information not only in the interference terms, but also in the signal power. As in Chapter 2, we have derived deterministic expression for the rate for MISO. Moreover, we proposed another robust precoder for partial CSIT.

However, this proposed precoder is less beneficial and achieve less gains than its counterpart, the ESEI-WSR precoder.

Non-Linear Precoding

Dans le document The DART-Europe E-theses Portal (Page 90-96)