Analysis

remaining users. From an intuitive point of view, it makes sense to want to leverage between the benefit at a given cell’s own user and the harm caused at other cell’s users if the cells wish to cooperate. Thus:

γ_k^virtual = p_k|h_k,ku_k|² σ²+P

j6=kαkjpk|h_j,kuk|², (4.6) where α_kj ∈ R+, j, k = 1, . . . , K are a given set of weights. Recall that ku_kk= 1. This can be seen as the SINR achieved on the uplink of a system where at the kth base station, receive vector wk is used to process the received signal, mobile station k transmits its signal with power p_k, and mobile j,∀j 6= k transmit with power α_kjp_k: the ‘virtual uplink’ was first introduced in [23] in the context of downlink power control and beamforming in a multicell environment. Uplink-downlink duality is revisited in Chapter 7.

When transmitting at full power, Equation (4.6) becomes:

γ_k^virtual = |h_k,ku_k|²

1 ρ+P

j6=kαkj|h_j,kuk|², (4.7) where ρ= _σ^P2.

As the objective is to have a distributed algorithm which relies only on information local to each base station, we propose that each transmitter solve a VSINR maximization problem, which can be stated as follows:

w_k= arg max

kuk²=1

|h_k,kwk|²

1 ρ+P

j6=kα_kj|h_j,ku_k|². (4.8) This is justified in the following section.

4.4 Analysis

As first noted in [23], the same rate region may be achieved in the UL (for a reciprocal channel, in the virtual UL otherwise) and DL directions using the same set of vectors for receive and transmit beamforming respectively, but with different power levels in both directions that satisfy the same total power constraint. This is one form of what is referred to as uplink-downlink duality. In our later analysis, we will show that due to the power constraints at each transmitter a modified version of uplink-downlink duality needs to be considered, which can be related to Lagrangian duality (see [14, 51] for example).

84 Chapter 4 Coordinated MISO IC using the VSINR Framework

4.4.1 Virtual SINR Maximization as Pareto Boundary Achiev-ing Strategy

Transmission in the MISO IC may be viewed as a game, where each of the transmitters is a player trying to achieve a certain goal, wich may be a selfish one such as trying to maximize his own rate or an altruistic one such as trying to maximize his own rate without causing any interference at any other user. One can then define the Pareto boundary of the channel, as the set of Pareto-optimal rate-tuples: thus, a given tuple belongs to the Pareto boundary if it is not possible to increase any rate within that tuple without decreasing at least one of the others. As shown in [44, 49], rates on the Pareto boundary of the MISO interference channel are achieved by transmitting at full power (at least for the case whereN_t≥K).

We reproduce Proposition 1 and Corrolary 2 from [49]², which charac-terize the beamforming vectors along the Pareto boundary for the general case whenN_t ≥ K ≥2 and for the special case of K = 2, Nt ≥2, and re-late these characterizations to the solution of a virtual SINR maximization problem as follows.

Proposition 1 ( [49]). Let k be given and fixed. Suppose that hj,k are linearly independent for j= 1, . . . , K and that h^H_j,kh_j⁰_,k 6= 0 for allj, j⁰, j 6=

j⁰. Then if u_k is a beamforming vector on the Pareto boundary, there exist complex numbers {ξjk} such that

u_k=

K

X

j=1

ξjkh^H_j,k (4.9)

and

ku_kk²= 1, pk=P. (4.10)

Proposition 2. Maximizing (4.7) over u_k such that ku_kk = 1 for any se-lection of{α_kj}yields a beamforming vector choice that satisfies proposition 1.

Proof. Maximizing (4.7) can be reformulated as solving a generalized eigen-value problem. The unique solution such that the norm constraint is satisfied (up to a scalar rotation) is given by

uk= 1

ρI+P

j6=kαkjh^H_j,khj,k

−1

h^H_k,k k

1 ρI+P

j6=kαkjh^H_j,khj,k

−1

h^H_k,kk

(4.11)

2with appropriate changes to match our notation

4.4 Analysis 85

Such a uk belongs to the set specified in Proposition 1.

Note 1. The converse however is not true. The authors in [49] note that wlog one of theξ_jk may be constrained to be real-valued. A beamformer given by (4.11) imposes the following constraint on theξ_jk

ξkkkh_k,kk²+X

j6=k

ξjkh_k,kh^H_j,k∈R⁺. (4.12) For K > 2, one can easily find a set of ξ_jk which do not satisfy this con-straint, even if the constraint that one of them be real-valued is imposed. As shown below, this is not true when K = 2. However, as Theorem 3 states, any point on the Pareto boundary, as long asK ≤Nt, can indeed be achieved by maximizing a VSINR for appropriate choices of the α_jk’s.

Corollary 1 ( [49]). Any point on the Pareto boundary forK= 2 is achiev-able with the beamforming strategy:

u_k(λ_k) = λku^{N E}_k + (1−λk)u^ZF_k

kλ_ku^{N E}_k + (1−λ_k)u^ZF_k k, k= 1,2 (4.13) for some set of real-valuedparameters λk, 0≤λk ≤1, k= 1,2, where

u^{N E}_k = h^H_k,k

kh_k,kk and u^ZF_k = Π^⊥_¯kkh^H_k,k

kΠ^⊥_¯kkh^H_k,kk (4.14) are the Nash Equilibrium (NE) or Maximum Ratio Transmission (MRT) and ZF solutions, respectively. Π^⊥_kk¯ is the projection matrix onto the null space of h¯k,k, Π^⊥_¯kk=I_Nt−^{hk,k¯ h}

H¯ k,k

khk,k¯ k² .

Theorem 3. Any point on the Pareto boundary may be attained by solving the VSINR optimization problem, as given in (4.8), for an appropriate choice of α_kj ∈R+, j, k= 1, . . . , K, providedK ≤N_t.

Proof. Details are given in Appendix 4.A for the two-link case, N_t ≥2, by relating the solution to Corollary 1. Appendix 4.B generalizes this result for Nt≥K≥2.

4.4.2 Achieving a particular point on the Pareto Boundary for the Two Link Case

In general, a point on the Pareto boundary is obtained by maximizing a weighted sum rate for some set of weighting coefficients. Solving this in gen-eral requires either sharing the full CSIT or devising some iterative algorithm

86 Chapter 4 Coordinated MISO IC using the VSINR Framework whereby the transmitters exchange values of the Lagrange coefficients of the dual problem. For a specific choice of α_kj, in the two link case, attaining a point on the Pareto boundary is guaranteed in one shot.

Theorem 4. The rate pair obtained by beamforming using the solutions to problem (4.8) with α₁₂ = α₂₁ = 1 lies on the Pareto boundary of the rate region.

Proof. Appendix 4.C proves this.

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

R₁ (bits/sec/Hz) R2 (bits/sec/Hz)

Pareto boundary ZF solution NE solution Proposed algorithm

Figure 4.2: Pareto rate boundary, MRT, ZF andα12 =α21 = 1 points for a channel instance sampled from a channel with i.i.d. CN(0,1) coefficients, Nt= 3,K = 2.

This is illustrated in Figure 4.2, which also shows the rate pairs cor-responding to the NE (or MRT) and ZF solutions, which correspond to the most selfish and the most altruistic strategies, respectively, and whose beamforming vectors are given by (4.14) above.