DELEGACIÓN DE HACIENDA
BASES I.- NORMAS GENERALES
In this subsection, we analyze the network proportional-fair metric expression in high and low SNR regimes and consider some special cases of efi(Pi; ¯Pi).
1) τi= 0.
If τi= 0, then we can rewrite the optimization problem (3.12) as follow
maximize
Pi∈K
wilog2det(INr+ CiPi) − Tr (EiPi) (3.14)
where Ei, ∑j6=iwjMj|Pi= ¯Pi. Note that (3.14) essentially maximizes the same objective function
as (3.4) with respect to Pi, except that the proportional-fair metric of the other link is approximated
to the first order at the point ¯Pi. The trace term in (3.14) plays the role of interference tax, dis-
couraging selfish behavior of link i, which would otherwise just want to maximize its own rate (if the trace term be equal to zero, each link selfishly maximizes its own rate). We can reconsider it as follows: for a given LTE-Advanced CoMP-CB system, in each iteration, i-th MS announces an
interference price to all BSs, which is the marginal decrease in utility for an increase in received interference. The transmitters update their power to maximize their own utility minus the cost of interference they produce.
Theorem 5. In an LTE-Advanced CoMP-CB system in which the proportional-fair metric of all the interference links are approximated to the first order (at the point ¯Pi), the closed-form solution
of power allocation for CoMP transmissions is available and can be expressed as
Pi(k, k) =
wiCi(k, k) − Gi(k, k)
Ci(k, k)Gi(k, k)
+
(3.15)
whereCi, VHi HHiiR−1i HiiVi,Gi, VHi ( ˜Ei+ µiI)Vi, and ˜Ei, ∑j6=iwjHHjiR−1j χjZ−1j R −1
j Hjieval-
uated atPi= ¯Pi. The remaining elements ofPiare zero.
Proof. See Appendix A.0.5.
Theorem 6. An optimal solution of downlink precoding matrices for the aforementioned system is
the generalized eigenmatrix ofHHiiR−1i Hiiand ˜Ei+ µiI, with HHiiR−1i HiiV˜i= ( ˜Ei+ µiI) ˜ViΛi, where
˜
Vi= ViP1/2i is an unnormalized transmit precoder of i-th BS and the elements of the diagonal
matrix Λiare the generalized eigenvalues ofHHiiRi−1Hii and ˜Ei+ µiI.
Proof. See Appendix A.0.6.
2) High SNR.
In the high SNR regime, the achievable rate Ri(Pi; ¯Pi) can be approximated by ˜Ri(Pi; ¯Pi) =
log2det(χi) − log2det(Ri). Then, one can formulate the high SNR proportional-fair maximization
problem as follows maximize Pi K
∑
i=1 wiR˜i(Pi; ¯Pi) subject to Pi 0, Tr(Pi) ≤ pi,max (3.16)the signal term in ˜Ri, we can bound it as follows ˜ Ri> − log det(Ri) (a) > n
∑
k=1 − log([Ri]kk) (3.17) (b) > − n∑
k=1 [Ri]kk= −Tr(Ri)where (a) comes from applying Hadamard’s inequality, i.e., det(M)6 ∏jMj j for M 0, (b)
follows from the fact that ∀x > 0, x > log(x), and n = min(Nr, Nt). It shows that minimizing the interference leakage at each user results in optimizing a lower bound on the user’s high SNR rate.
Back to the optimization problem at high SNR regimes, and by retaining only the linear term in the Taylor’s expansion of nonconvex part of the above objective function around ¯Pi, it is possible
to approximate (3.16) by a set of L per-link problems given for i ∈L by maximize
Pi
wilog2det(χi) − Tr(AiPi)
subject to Tr(Pi) ≤ pi,max, Pi 0,
(3.18)
where Ai, ∑j6=iwjSj evaluated at Pi= ¯Pi. Note that unlike (3.16), (3.18) is convex in Piand can
be efficiently solved by numerical iterative algorithms.
Theorem 7. In the high SNR regime, the power allocation of each BS in an LTE-Advanced CoMP- CB system can expressed as follows
Pi= UHdiag wi σk+ µi U (3.19)
whereU is a unitary matrix and can be achieved by eigenvector decomposition of Ai.
Proof. See Appendix A.0.7.
3) Low SNR.
In the low SNR regime, we can replace log2det(INr+ R −1/2
i χiR
−1/2
i ) with ∑i=1log(1 + λi),
where λiis the i−th eigenvalues of R −1/2
i χiR
−1/2
λiare small, so to first order we have log(1 + λi) ≈ λi. Using this first-order approximation in the
expression above, we get ∑i=1log(1 + λi) = ∑i=1λi= Tr(R−1/2i χiR−1/2i ) = Tr(R−1i χi). Thus, we have maximize Pi∈K K
∑
i=1 wiTr(R−1i χi) (3.20)which is a nonconvex optimization problem. By retaining only the linear term in the Taylor’s expansion of nonconvex part of the above objective function around ¯Pi, it is possible to approximate
(3.20) by a set of L per-link problems given for i ∈L by
maximize
Pi∈K
Tr ((wiCi− Di)Pi) (3.21)
where Di , ∑j6=iwjYHjχjYj evaluated at Pi = ¯Pi. Note that unlike (3.20), (3.21) is convex (a
Semidefinite Programming (SDP)) in Piand is therefore amenable to a wide variety of optimization
techniques.
Theorem 8. In the low SNR regime, the power allocation of each BS in an LTE-Advanced system can expressed as follows
Pi= VHdiag ε γk+ µi V (3.22)
whereV is a unitary matrix and can be achieved by eigenvector decomposition of matrix −wiCi+
Di.
Proof. See Appendix A.0.8.
Remark 5. It is worth mentioning that in the low SNR regime, the interference due to other BSs
is overwhelmed by the noise power seen at the MSs. The proportional-fair metric maximizing beamformers in this regime are simply the Ns dominant right singular vectors obtained from the singular value decomposition of the direct linkHiiof the i-th BS. The MSs are the corresponding Ns
left singular vectors. The power allocation strategy reduces to that of single-user MIMO scenario, i.e., water filling on the corresponding Ns dominant singular values. Which is not surprising
b Pi( ¯Pi, τi) , arg max Pi∈K wilog2det (INr+ CiPi) − Tr (EiPi) − τikPi− ¯Pik 2 F. (3.23)
because when the noise dominates the received signal, the benefit of self interference cancelation is marginal.
4) Hi( ¯Pi) = I.
If Hi( ¯Pi) = I, the quadratic term in (3.10) reduces to the standard proximal regularization
τikPi− ¯Pik2F, and then the best response matrix function of each BS is given by Eqn. (3.23), at the
top of the next page.
Theorem 9. If Hi( ¯Pi) = I, then the closed-form solution of the above-mentioned optimization
problem can be expressed as
Pi= ¯ Pi− 1 2τi (µ∗I + Z 0 0 0 ) + (3.24)
whereZ is the matrix of lagrangian multipliers associated to the linear constraints, [X]+ denotes the projection ofX onto the cone of positive semidefinite matrices, and µ∗is the multiplier which can be found by bisection.
Proof. See Appendix A.0.9.