Here, we first devise the optimal transmit scheme for each BS, and then derive a closed-form upper bound for the ergodic sum capacity using the matrix permanents. Based on the capacity bound, we develop low-complexity power allocation solutions using convex optimization techniques, followed by discussion of the beamforming optimality conditions for the BSs.
3.1 Optimal transmit scheme
We here assume that the mobile receiver has perfect instantaneous CSI, whereas the BSs only have the statistical CSI including Ut,i, U
r
, D
i
and M
i
(and thus Ω
i
) (i = 1, 2, ..., m), and this information can be exchanged among the BSs via the wired backbone. Under these assumptions, the ergodic sum capacity of the downlink system is achieved by selecting the transmitted signal vector x to follow a zero-mean proper Gaussian distribution [1]. Let , where
(11)
with . The power constraint (2) can be rewritten as , for i = 1, 2, ..., m, and the ergodic sum capacity is given by
(12)
where . Let , for i = 1, ..., m, with U
i
being the eigenvector matrix, and the diagonal matrix of the corresponding eigenvalues. The following theorem addresses the optimal transmit direction of each BS.
Theorem 1 The ergodic sum capacity is achieved if the BS transmit signals are all mutually independent (i.e., , for i ≠ j), and the eigenvector matrix offor the jointly-correlated channel (4) is given by U
i
= Ut,i. The ergodic sum capacity is then expressed as
(13)
Proof: Form (4) and (5), the channel matrix H can be expressed as
(15)
where
Defining and substituting (14) into (12) yields
(16)
where
(17)
Note that the optimization condition is met since . Now, define Π
l
for 1 ≤ l ≤ N
t
as diagonal matrices all of which have their diagonal entries being all 1s except for the (l,l)th entry as -1. As Π
l
is a unitary matrix, (17) can be written as
(18)
Note that is given by (15) and has the same distribution as , since D is a diagonal matrix plus the fact that the entries of M ʘ Hiid are independent and their distributions being symmetric, reversing the sign of some columns does not alter the distribution. Thus, we have
(19)
From Jensen's inequality, it follows that [37–39]
(20)
where the matrix has entries equal to those of except for the off-diagonals in the l th row and l th column, which are zero. In particular, its trace is identical to that of . As a result, nulling the off-diagonal entries of any column and the corresponding row of can only increase . Using the same process N
t
times, (17) is maximized with a diagonal , i.e., . As a result, we have , or
(21)
where Λ = diag {Λ1, Λ2, ..., Λ
m
}. As such, (16) can be rewritten as (13).
Theorem 1 reveals that the transmitted signals of all BSs should be mutually independent and the optimal signaling directions of the i-th BS align with the eigenvectors of the transmit-side correlation matrix of the MIMO channel of the i-th BS. This results extend the prior results in [29, 37, 40, 41] to the more general channel model given by (4).
3.2 Ergodic sum capacity upper bound
After knowing the optimal transmit directions of the BSs, the remaining challenge is to determine the eigenvalues of the capacity-achieving input covariance matrix Q
ii
for i = 1, ..., m. This is equivalent to optimally allocating the available transmit power over the optimized transmit eigen-directions that are determined by Theorem 1.
In the most general case, it is difficult to derive exact closed-form solutions for the power allocation problem. The main obstacle lies in the complexity in evaluating the expectation in (13) which is usually done by stochastic averaging over a large number of random samples. In this section, our approach is to derive a tight upper bound for the expectation in (13) which can serve as an approximation to the ergodic capacity. Based on this, we develop closed-form power allocation solutions which will be presented in Section 3.3.
Due to the concavity of the log(·) function, C is upper bounded by
(22)
where
(23)
with , in which λ
i
denotes an vector containing the eigenvalues for and i = 1, ..., m. The upper bound (22) can be rewritten as
(24)
The expectation derivation in (23) is based heavily on exploiting linear-algebraic concepts and the properties of matrix permanents. The permanent of a matrix is defined in a similar fashion to the determinant. The primary difference is that when taking the expansion over minors, all signs are positive. The permanents of M × N matrices have been investigated in [28, 42]. We introduce the definitions and properties of matrix permanents in Appendix 1.
From these definitions, we extend the results of [28] to the case of multiple BSs. We can derive a closed-form expression for the upper bound on the ergodic sum capacity.
Theorem 2 The ergodic sum capacity in (13) is upper bounded by
(25)
where.
Proof: We start by letting
(26)
where . The upper bound of mutual information (23) can be rewritten as
(27)
By using the known result [28, Theorem 2], E(λ) can be expressed as
(28)
Substituting (28) into (27) and using (24) completes the proof.
From Theorem 2, we can see that the upper bound of ergodic sum capacity depends on the average SNR and the eigenmode channel coupling matrices Ω
i
, for i = 1, 2, ..., m. Low-complexity algorithms about the computation of the matrix permanent were developed in [28].
3.3 Optimizing the power allocation policies
We now consider the transmitter power allocation optimization problem. Based on the upper bound in Theorem 2, we develop low-complexity power allocation solutions using convex optimization techniques and then propose a simple iterative water-filling algorithm (IWFA) for approaching the optimal power allocation policy.
From (25), the power allocation optimization problem can be formulated as
(29)
(30)
The above problem is a concave optimization problem [28] and the solution can be evaluated by employing standard convex optimization algorithms. In the following, we derive necessary and sufficient conditions for the optimal solution using the Karush-Kuhn-Tucker (KKT) conditions.
Theorem 3 The expected mutual information upper boundis concave with respect to λ, and the necessary and sufficient conditions for the optimal power allocation are given by
(31)
(32)
where
(33)
(34)
B = [Γ1Ω1Λ1, ..., Γ
m
Ω
m
Λ
m
], Bi(j)denotes the block matrix obtained by replacing the ith sub-matrix (i.e., Ω
i
Λ
i
) of B by Ωi(j)diag(λi(j)), Bi[j]denotes the block matrix obtained by replacing the ith submatrix of B by Ω
i
diag(λi[j]), Ωi(j)denotes the sub-matrix of Ω
i
obtained by deleting the jth column, λi(j)denotes thevector obtained by deleting the jth element of λ
i
, and λi[j]denotes thevector obtained by replacing the jth element of λ
i
by unity. In addition, (a)+ = max{0, a} andis chosen to satisfy the power constraints in (32).
Proof: See Appendix 2.
Since the right-hand-side of (31) is independent of , we propose a simple IWFA to evaluate the optimal power allocation policy which satisfies (31). Simulation results, to be given in Section 4, will demonstrate that this proposed approach works very well and is highly efficient; typically converging after only a few iterations, with the first iteration achieving near-optimal performance. The proposed algorithm includes the following steps:
Step 1 Initialize , and k = 0.
Step 2 Calculate and , for , and i = 1, ..., m.
Step 3 Calculate , for , and i = 1, ..., m, with the power constraints for i = 1, ..., m.
Step 4 Calculate .
Step 5 If set , and recalculate .
Step 6 Set k := k + 1 and return to Step 2 until the algorithm converges or the iteration number is equal to a predefined value.
In the above, the superscript k specifies the corresponding variable in the k th iteration so that λkstands for the value of λ in the k th iteration. In Step 1 of the first iteration, λ is initialized to 1, i.e., equal-power allocation. Note, however, that λ could also be initialized in a different way. For example, it is expected that the channel statistics change smoothly frame by frame, where a more appropriate initialization would be the optimal value of λ from the previous frame. In Step 3, the conventional water-filling algorithm is performed with the required variables p(λi(j)) and q(λi(j)) calculated in Step 2. Following the calculation of in Step 4, Step 5 is performed to guarantee convergence [28]. In Step 6, the convergence of the algorithm can be determined by checking whether is less than some predefined value for a given precision.
3.4 Optimality of beamforming
Here, we investigate the optimality of beamforming (i.e., rank-one transmission) [29–34] in the context of the multi-BS cooperation systems. We derive a necessary and sufficient condition for the optimality of beamforming in the multi-BS cooperation systems. For BS
i
, we assume that the transmit eigenmodes satisfy the following conditions
(35)
where and , for and i = 1, ..., m.
Theorem 4
For multi-BS cooperation systems, the transmit covariance matrices of all the BSs that achieve the sum capacity are of unit-rank (i.e., beamforming is optimal for all the BSs) if and only if the following inequality is fulfilled:
(36)
whereandare the first and jth columns of, respectively, withdefined as in (15), and d
ij
and m
ij
are the jth columns of D
i
and M
i
, respectively.
Proof: See Appendix 3.
Note that the proof is nontrivial generalization of the techniques in [33] to the jointly correlated MIMO multi-BS cooperation systems. We can make the following observations.
-
If for are i.i.d., the left-hand-side of (36) remains unchanged when j varies from 2 to , and the right-hand-side of (36) is maximized for i = 2. It means that if the condition for i = 2 holds, then it is also held for all other i. Thus, inserting j = 2 into (36) gives the following condition:
(37)
-
If the LOS D = 0 and , where a
l
and bi,jare square-roots of the eigenvalues of the receive correlation matrix Φ
r
and transmit correlation matrix Φt,i, respectively. In this case, the channel degenerates to the Kronecker channel. It can be proved that (37) reduces to the beamforming optimality condition in [33, Theorem 2].