Joint ZF-IN feasibility conditions

We consider MISO IBRC with MIMO FD relay with direct links (Figure6.3).

So, BS and users communicate both directly and via relay. h^{U B}_c,c,k and h^{U R}_c,k are

Figure 6.3: Relay and direct links

now vectors of dimensions 1×MBS and 1×MRS respectively. In this section we do not use system-wide numbering of the users. Let us suppose that we haveK users per cell. The received signal can be written as:

y_c,k=h^{U B}_c,c,kg_c,kx_c,k+

(C,K)

X

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

h^{U B}_c,j,kg_j,ix_j,i+v_c,k

| {z }

direct signal +h^{U R}_c,kF{

(C,KX) (j,i)=(1,1)

H^RB_j g_j,ix_j,i+n_c,k

| {z }

link via relay

} (6.29)

where the conditions for joint ZF-IN on the BF vectorsg_j,i and the AF matrix Fare indicated. The noise-free received signal can be rewritten as:

y_c,k= (h^{U B}_c,c,k+h^{U R}_c,kFH^RB_c )g_c,k

| {z }

6

= 0

x_c,k+

(C,K)X

(j,i)=(1,1),6=(c,k)

(h^{U B}_c,j,k+h^{U R}_c,kFH^RB_j )g_j,i

| {z }

=0

x_j,i

(6.30)

These conditions can perhaps be more easily interpreted in a dual UL in which we have an Interfering Multiple Access Channel (IMAC) plus Relay:

g^H_j,i(h^{U B,H}_c,j,k +H^RB,H_j F^Hh^{U R,H}_c,k ) = 0∀(j, i)6= (c, k) (6.31) in which the BFg^H_j,i now plays the role of ZF Rx. HavingMBS antennas, the BS Rx can zero force M_BS −1 interfering streams while still receiving the stream of interest. For user (j, i), let S_j,i denote the set of M_BS−1 users that will be suppressed byg_j,i. Then, the conditions (6.31) become IN conditions for the AF matrixFfor the interfering users (c, k)6∈ {{(j, i)}, S_j,i}.The number of such conditions is KC(KC−1)(M_BS −1)KC =KC(KC−M_BS). Note that the ZF conditions for theg_j,i and the IN conditions forFinvolve different (and hence independent) user channelsh_c,j,k. Hence, even though the ZF and IN conditions are coupled, the BF can be considered as independent ofFin the IN conditions. For the same reason also, the direct overall channel gains appearing in (6.30) (for (c, j, k) = (c, c, k)) will be non-zero, in spite of the conditions (6.31).

By introducing the vec(:) operator, which stacks consecutive columns of a matrix in a supervector, with the property vec(AXB) = (B^T ⊗A)vec(X) where⊗denotes the Kronecker product, and taking Hermitian transpose of the scalars in (6.31), we can rewrite the IN conditions from (6.31) as:

vec^H(F^H)(h^{U R,T}_c,k ⊗H^RBg_j,i) =−h^{U B}_c,j,kg_j,i (6.32) which need to hold for ∀(c;k) ∈ {{/ (j;i)}, S_j,i}. There are many ways of selecting the setsS_j,i, leading to many solutions for joint ZF-IN. Each solution will correspond to a local optimum for utility optimization designs. Let us consider one specific choice for the S_j,i in which the M_BS −1 users to be ZF’d comprise in any case the K −1 other users in cell j and such that Sj = {{(j, i)}, Sj,i} is independent of i. Then let H^{U R}_j = [h^{U R,T}_c,k ,(c, k) ∈/ Sj] which is a matrix of sizeM_RS×(CK−M_BS). IntroduceG_j = [g_j,1· · ·g_j,k] of sizeM_BS×K andh^{U B}_j = [h_c,j,kG_j,(c, k)∈/ S_j], then we can rewrite (6.32) as vec^H(F^H)[H^{U R}₁ ⊗H^RB₁ G₁· · ·H^{U R}_C ⊗H^RB_C G_C] =−[h^{U B}₁ · · ·h^{U B}_C ] (6.33) This system of equations can be solved forvec^H(F^H) if the matrix of coefficients has full column rank. To investigate this, we can use the following Lemma.

Lemma 4.1: Full column rank conditions of Khatri-Rao product. We consider the block matricesA= [A₁· · ·A_n],B= [B₁· · ·B_n] with compatible column block structure, their Khatri-Rao productA·B= [A₁⊗B₁· · ·A_n⊗B_n] has full column rank if and only if (iff)

1. allA_i and B_i have column rank

2. at least one ofA orB has full column rank.

Proof. Sufficiency is fairly straightforward. For necessity, (1) is a result of rank(A_i⊗B_i) =rank(A_i)rank(B_i). (2) for the case n=2, by contradiction:

given that theA_i andB_i have full column rank, but if bothA andB didn’t have full column rank, then vectors ai,bi exist so that A1a1 = A2a2 and B₁b₁=B₂b₂. Then A₁a₁b^T₁B₁ =A₂a₂b^T₂B₂ and

vec(B_ib_ia^T_i A^T_i) = (A_i⊗B_i)vec(b_ia^T_i ) = (A_i⊗B_i)(a_i⊗b_i) (6.34) Hence, (A·B)[(a1⊗b1)^T −(a2⊗b2)^T]^T = 0, which means that A·B would not have full column rank. Applying Lemma 4.1 to (6.33) leads to the following.

Theorem 4.1: Interference Neutralization Feasibility IN the MISO IRBC with MIMO relay with the dimensions considered above, IN is feasible iff

M_RS ≥max(K, CK−M_BS, Cmin(K, CK−M_BS)), K ≤M_BS (6.35) This leads to the following evolution for the number of relay antennas:

M_RS =











0, 1≤K ≤ ^M_C^BS

C²(K−^M_C^BS), ^M_C^BS ≤K≤ ^M_C−^BS₁ CK, ^M_C^BS₋₁ ≤K≤M_BS

where in the first regime only ZF BF is needed. The following are two variations on the basic scenario.

Intracell BF. In this case, the BF is non-cooperative between cells and only considers the intracell users (the BF is multicell oblivious). All intercell interference needs to be canceled by IN. Hence, N = C(CK−M_BS) gets replaced byM_RS =C(C−1)K.

BF-independent AF The IN equations will not depend on the BF G_j (though the BF will still depend on the AFF) if interference is not neutralized starting from the BF inputs but starting from the BS antennas. Then, the factorsGj disappear from the equation in (6.33). This leads to IN conditions:

MRS ≥ max(MBS, CK −MBS, Cmin(MBS, CK −MBS). The ZF and IN conditions can be solved iteratively as follows. Start e.g. withF= 0.

1. The BFs g_j,i can be solved by ZF the direct links in (6.30) w.r.t. the effective channels in (6.31) of the other users in S_j.

2. The AF matrixFcan then be determined from the equations (6.33).

Iterate (1) and (2) until convergence. Whereas joint ZF-IN can have many solutions, fixing the setsS_j forces convergence to one particular solution (apart from underdeterminacy issues of course if N is larger than necessary). For the case ofC= 2 cells, M_RS = 4(K−^M₂^BS)⁺ which evolves from 0 to 2M_BS asK evolves from ^M₂^BS toM. For the Intracell BF case, we getM_RS = 2K, whereas the BF-independent AF case (typically) also leads toM_RS = 4(K− ^M₂^BS)⁺.

6.7 Conclusion

In this chapter, we treat the problem of communications with relays. In the first study, we treat the case where the BSs and the users can’t communicate directly but via a relay. Again, the problem of interest is to design jointly all the BFs and the relay matrix. We suppose perfect CSIT. A WSMSE-type solution exists. We propose a WSMSE-DC based solution where the relay matrix is always given by the WSMSE approach, but the BFs at the Tx side are given by a DC approach. Briefly, the BFs are given by linearizing the objective function corresponding to the sum rate of all the users except for the one of interest. We show that this solution is better. In the second study, we suppose that the BSs and the users can communicate via two ways, directly or via the relay. Two interference management techniques are at stake: IA and IN.

IA is the interference management technique highlighted in all of the chapters of this thesis. IN appears here for the first time, where artificial multipath is introduced to provoke destructive interference superposition at Rxs. We derive the DoFs of such scenario. The difficulty of IN is that the relay must know the channels from BSs to users which is hard to be achieved in practice.

Conclusions and Future