Decentralized approach for large dimensions sys- sys-tem

The idea is to try to identify the quantities that require global knowledge of the channel vectors, the intercell interference Υ_inter,c,k andD in our case, and

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exchange them (or the quantities related to them) between the different BSs in such a way that the maximum WSR problem will decompose into parallel sub-problems (one per BS). However, it is necessary to limit as much as possible this exchange in order to be backhaul friendly (efficient). The solution in the last section can be reformulated as the following:

a_c,k=g^H_c,kh_c,c,k(σ²+Υ_intra,c,k+Υ_inter,c,k)⁻¹ (3.1)

e_c,k= (1 +γ_c,k)⁻¹ (3.2)

w_c,k=u_c,k(e_c,k)⁻¹ =u_c,k(1−a_c,kh^H_c,c,kg_c,k)⁻¹ (3.3) e

g_c,k = (H^H_c DH_c+trD_c

ρ_c I_M)⁻¹h_c,c,ka^H_c,kw_c,k (3.4) with

Υ_intra,c,k=X

n

h^H_c,c,kg_c,ng^H_c,nh_c,c,k (3.5) Υ_inter,c,k= X

m;m6=c

Υinter,m,c,k (3.6)

and

Υinter,m,c,k =X

n

h^H_m,c,kg_m,ng^H_m,nh_m,c,k (3.7) This solution can be initialized by a random precoder, e.g., a matched filter (MF) precoder. In general, it needs a central processing node to be imple-mented because of (3.6) which depends on global channels knowledge as shown in (3.7). In the case of absence of this central node, (3.6) can be detected by each receiver and then fed back using an over-the-air link as in [22]. However, this approach is spectral inefficient.

Another way to decentralize consists in that each BS m calculates the quantities in (3.7),Υinter,m,c,k considered as the interference leakage from BS mto user kof cell c6=m for all the users and sends them to the BScusing a backhaul link. This procedure is a bit heavy, so it is beneficial to limit as much as possible the number of iterations. However, at high SNR, the solution above requires a lot of iterations to converge, hence, requires an extensive exchange of information using the backhaul link which burdens this latter and makes it practically infeasible. For a limited number of iterations, the solution becomes very sub-optimal.

Thus, in the following we present a new initialization method which acceler-ates the convergence, hence, few iterations are no more sub-optimal and the

backhaul-based decentralization becomes realistic. In this following, perfor-mance analysis is conducted for the proposed precoder. The large-system limit is considered, where M and K go to infinity while keeping the ratioK/M finite such thatlimsup_MK/M <∞and liminf_MK/M >0.

Theorem 3.1: From 2.3.2, for a large MISO system, precoders ˜g_c,k can be written as the following:

e

g_c,keg^H_c,k−eg_c,keg^H_c,k −−−−→^M→∞ 0 (3.8) where

e

g_c,k= (H^H_c DHc+trDc

ρ_c IM)⁻¹h_c,c,ka^H_c,kw_c,k (3.9) We propose that (3.9) serves as an initialization for the iterative solution above. It also serves as a precoder itself, which is denoted as large system (LS)-precoder. Further details will be provided in the following sections. The fast-converging iterative algorithm behind (3.9) and the definitions of its terms are summarized in Algorithm 1. We give here the large system approximation of the intercell interference term as follows:

Υ_inter,c,k = lim

j→∞

ξ_c^2,(j)[ ˆΥ^(j)_c,k] d^(j)_c,k(1 +m^(j)_c,k)²

. (3.10)

3.2 Signaling

This section summarizes the iterative procedure to design transmit beamform-ers in a decentralized manner. The authors of [22] proposed a decentralized reasoning as well; so we will compare it to ours. They assumed that local channel information is available at each BS and for each user; we assume that as well. Moreover, they assume that each user has an additional channel to feedback information, which isd_i,k =|a_i,k|²w_i,k, to the BS; however we relax this assumption and we assume instead the existence of a backhaul link which is a way to save the wireless capacity consumption w.r.t an over-the-air link.

It is used as explained in the previous section.

We would then propose three different strategies: a) The intercell interference-free strategy, where at each iteration of the precoders design each BSccalculates

Algorithm 1 Large System Computation of Dual UL Scalars Step 2:Setj=j+ 1 and calculate the following quantities:

Υˆ^(j)_c,k= 1 Step 4:If converge stop and calculateeg^(j)_c,kas in Theorem 3.1, otherwise go to step 2.

*Note that allem,c,k=mm,c,k,e^′_c,kande^′_m,c,k,m,l

are obtained using the fixed-point iteration method as in Chapter2.

only the quantitiesΥ_intra,c,k using the local channel information and supposes the Υ_inter,c,k is null. b) The constant intercell interference strategy, where for every iteration of the precoders design each BSc calculates the quantities Υ_intra,c,k using the local channel information but utilizes the intercell interfer-ence given by Algorithm 1 using (3.10). c) The up-to-date intercell interferinterfer-ence strategy, where for every iteration of the precoders’ design, each BSccalculates the quantitiesΥ_intra,c,k using the local channel information, then calculates the intercell interferencesΥinter,m,c,k in (3.7) and sends them to the corresponding BS and finally this BS c collects the interference leakages corresponding to each user and sums them.

Clearly, the strategies (a) and (b) are sub-optimal but less demanding than (c) w.r.t to the backhaul capacity. Although the strategies (a) and (b) are sub-optimal, however they perform better than the approach in [22] by taking the MF initialization for a limited number of iterations. We recall that for each of the three strategies above, each BS calculates thed_c,k for all served users and then send them to all the neighbouring BSs via the backhaul link.

Furthermore, the fact that (c) requires that each BSmcalculates the quantities in (3.7) for all users not served bym and then sends them to the concerned BS consumes more backhaul capacity than (a) and (b). The maximum number of iterationsitermax is chosen to be very small, e.g.,itermax= 2 oritermax= 3.

The overall mechanism is described briefly in Algorithm 2.

Algorithm 2 The Decentralized Algorithm

Step 1:Set iter= 0. All BSs estimate local channel matrices (from BS to served users and to the users of the neighbouring cells). The BSs distribute the channel covariance matrices to neighbouring cells via the backhaul link only at slow fading rate. They apply Algorithm 1 and then calculate (3.9). They calculate (3.10) for strategy (b) and the intercell inteferenceΥinter,m,c,kwith (3.9) for strategy (c) and exchange them with the concerned BS.

Step 2:

All the BSs calculate Υ_intra,c,k, a_c,k, w_c,k and d_c,k = |a_c,k|²w_c,k using (3.5), (3.1) and (3.3) and send d_c,k to the neighbouring BSs, at fast fading rate.

Moreover, each BS calculates the interference leakages and collects the interference corresponding to its served users in strategy (c).

Step 3:All the BSs calculate their precoders using (3.4).

Step 4:iter=iter+ 1, if iter=iter_max stop, otherwise go to step 2.