which is obtained from (7.13) with µ = 12. If the noise is proper and CN,HUL,k has the structure that we expect from a combined real channel matrix, i.e.,HUL,k = `HUL,kis aBSC2
matrix with block-diagonal submatrices. The corresponding complex channel matrix on carrier ccan then be expressed as
HUL,k(c) =Hk(c)C−12
η(c)k (7.44)
where we have used thatCηˇk = 12<`(Cηk)due to (2.34).
If the downlink noise is CC, the transmission on the various carriers of the dual uplink is no longer orthogonal. Analogously, the uplink is a widely linear system instead of a linear one if the downlink noise is improper. In these cases, we cannot describe the dual uplink by matrices HUL,k(c) . However, this does not lead to any problems as long as we perform our derivations in the combined real representation. The only thing that has to be kept in mind is thatHUL,kis then not necessarily compatible with the power shaping spaces under consideration.
From the two duality theorems shown above, we obtain the following corollaries for various types of entropy reduction. The first of them was used implicitly in [1], and the second one was proven and exploited in [16].
Corollary 7.2.1(Pk= `PUCNk, ∀kandPB = `PBCN). In the complex multicarrier MIMO BC-TIN with proper CN noise, a rate vector is achievable with sum powerQusing proper CN input signals if and only if it is achievable with the same sum powerQin the dual MIMO MAC-TIN using proper CN transmit signals. The same is true in case of zero-forcing constraints.
Corollary 7.2.2(Pk = `PUk, ∀kandPB= `PB). In the complex (multicarrier) MIMO BC-TIN with proper noise, a rate vector is achievable with sum powerQusing proper input signals if and only if it is achievable with the same sum powerQin the dual MIMO MAC-TIN using proper transmit signals. The same is true in case of zero-forcing constraints.
Corollary 7.2.3(Pk=PUCNk, ∀kandPB =PBCN). In the complex multicarrier MIMO BC-TIN with CN noise, a rate vector is achievable with sum powerQusing CN input signals if and only if it is achievable with the same sum powerQin the dual MIMO MAC-TIN using CN transmit signals. The same is true in case of zero-forcing constraints.
7.3 Optimization of the Transmit Covariance Matrices
When trying to find Pareto-optimal transmit strategies for the MIMO BC-TIN, we are facing the problem that the rate equations (7.1) are not concave in the input covariance matrices.
Unfortunately, going via the dual uplink, as we did in Chapter 6 for the MIMO BC with DPC, does not help either since the uplink rates (7.14) are nonconcave functions as well if all interference is treated as noise. Filter-based formulations as in (7.5) and (7.17) do not lead to more tractable expressions either. Therefore, we cannot express transceiver optimizations for the MIMO BC-TIN as convex programs, meaning that there is no simple way to find globally optimal solutions to such optimizations.
Several suboptimal approaches can be found in the literature, e.g., for weighted sum rate maximization (3.12) in [145, 151, 153–155, 158–160], for weighted sum rate maximization with additional constraints on the per-user rates in [91, 93, 96], for rate balancing (3.13) in [9,93,161], and for minimization of the total transmit power under constraints on the per-user
rates in [2,88,89,91–93,156,161,162]. We discuss some of these suboptimal algorithms in Section 7.6, but as an ingredient for the proofs in the following sections, we need a method that is able to find globally optimal transmit strategies instead.
7.3.1 Globally Optimal Rate Balancing in the RoP MISO BC-TIN
To simplify the problem, we restrict ourselves to the special case of single-antenna users, i.e., to the RoP multiple-input single-output (MISO) BC. While the weighted sum rate maximization is still a difficult nonconvex problem in this setting (see, e.g., [163]), the abovementioned power minimization and the so-called SINR balancing problem can be solved efficiently in this case [94,162,164] as long as we do not allow (R)TS. These approaches can be easily extended to the rate balancing problem (3.13) as shown, e.g., in [9], [97, Sec. 4.1.3].
However, when applying the dual approach from Sections 3.2.3.1 and 3.2.3.2 in order to solve the rate balancing problem with TS or RTS, we need to find a solution to the inner problem (3.19) or (3.24), respectively, which is difficult due to the nonconcave weighted sum rate in the objective functions of these problems. In this section, we propose an algorithm for the rate balancing problem with TS based on the branch-and-bound method (e.g., [165, Sec. 6.2]). We briefly comment on the case with RTS afterwards.
7.3.1.1 Branch-And-Bound Method
The branch-and-bound method is summarized below in Algorithm 7.3.1. It needs to be initialized with a collectionBof disjoint setsBsuch that the optimizer is contained in one of the sets B ∈B. To implement the method, we have to decide for a subdivision rule that divides a setB into disjoint setsB1andB2such thatB=B1∪ B2. Moreover, we need a method to calculate an upper boundU(B)≥f(p), ∀p∈ Band an achievable valueL(B) =f(p)for somep∈ B, wheref is the function to be maximized.
Algorithm 7.3.1Branch-and-Bound Method
1. Find the set with the highest upper bound, i.e.,Bˆ= argmaxB∈BU(B).
2. ReplaceBby(B\ {B}ˆ )∪ {B1,B2}whereB1andB2form a subdivision ofB.ˆ 3. Repeat Steps 1. and 2. untilmaxB∈BU(B)−maxB∈BL(B)≤.
4. Return the vectorpthat achievesmaxB∈BL(B).
Convergence of the method can be shown (see [165, Sec. 6.2]) if the subdivision rule is exhaustive, i.e., the setsB finally converge to singletons, and if the utopian boundU(B) is consistent, i.e., it converges to an achievable value when the setBconverges to a singleton. The stopping criterion in Step 3 leads to an-optimal solution, i.e., the obtained valuef(p)is at mostaway from the global optimum.
7.3. Optimization of the Transmit Covariance Matrices 101
7.3.1.2 Application to the Rate Balancing Problem with TS
To solve the rate balancing problem with TS in the RoP MISO BC-TIN, we can apply the dual approach from Section 3.2.3.1 if we have a solver for the inner problem (3.19). Accordingly, with a solver for (3.33), the dual decomposition from Section 3.3.2 can be applied in the RoP CN MISO BC-TIN. Therefore, we now discuss how the branch-and-bound method can be applied to solve problem (3.19) or (3.33).
To this end, we exploit that the transmit covariance matrices degenerate to scalar variances Cxk in the dual uplink of the RoP MISO BC-TIN, i.e., the transmit strategy is completely determined by the per-user transmit powers(pk=Cxk)∀k. As the analogous statement holds in a carrier-wise manner for each carrier of the RoP CN MISO BC-TIN, the following derivation can be easily extended to solve (3.33).
We letf(p)denote the objective function of (3.19) withrkfrom (7.14), and we introduce the function
It can be easily verified (e.g., by calculating the partial derivatives) that the functionF is nondecreasing inpand nonincreasing inp. Consequently, if the considered sets are boxes¯ B= [a, b] ={p|a≤p≤b}, it holds that
f(p) =F(p,p)≤F(b,a)
| {z }
U([a,b])
, ∀p∈[a, b]. (7.46)
Even though there is usually nop∈[a, b]fulfilling (7.46) with equality, this bound becomes tight asb−a→ 0, i.e., this utopian bound is consistent. On the other hand, an achievable value off(p)inside a box[a, b]is, e.g., given byL([a, b]) =F(a,a) =f(a).
An exhaustive subdivision rule that is appropriate for boxesB = [a, b]is the adaptive bisection [165, Sec. 6.2]
This can be interpreted as cutting theBbox along its longest edge into two subboxes.
To obtain an initial setB, we introduce the functions fˆk(pk) =eTkν1µlog2
maxpk≥0fˆk(pk)for allk. Moreover, we can apply simple root finding to find a valuep0,k such thatfˆk(pk) +P
j6=kfˆmax,j ≤0, ∀pk≥p0,k. As choosingpk ≥p0,kfor anykthen implies thatfˆ(p)≤0, all positive values offˆ(p)lie in[0, p0], wherep0= [p0,1, . . . , p0,K]T. Since f(0) = 0is an achievable value andf(p)≤fˆ(p), the maximum off is surely contained in [0, p0], and we can use the initializationB={[0, p0]}.
These choices for the initialization, the upper bound, and the subdivision rule fulfill the requirements stated in Section 7.3.1.1. It is thus guaranteed that the proposed application of the branch-and-bound method converges to an-optimal solution of (3.19). By choosingsmall enough, an arbitrarily precise solution to the overall problem (3.15) can be obtained if enough computation time is invested (see Section 7.3.1.5).
7.3.1.3 Application to the Rate Balancing Problem with RTS
When applying the method to solve problem (3.24) for the case of RTS in the dual uplink of the RoP MISO BC-TIN, the difference is that we have a compact constraint set in (3.24). The initialization can then be done by finding a boxBsuch that all feasible power vectorspare contained inB. As the rectangular shape of the boxes does not fit to the shape of the upper boundary of the constraint set, e.g., in case of a sum power constraintPK
k=1pk≤Q, a so-called reduction step as described in [166] should be incorporated into the algorithm. As an alternative, we can add a step that drops boxes which do not contain any feasible point.
7.3.1.4 Related Approaches
A problem equivalent to (3.19) arises as a subproblem of the power minimization with TS performed in [3]. To solve this problem, a similar monotonic optimization method as above was applied, but with the per-user rates as optimization variables instead of the per-user powers.
The advantage of the above formulation is that the rates as a function of the powers can be calculated explicitly while it is necessary to apply an iterative numerical algorithm to calculate the powers as a function of the rates. This means that an inner loop is executed wheneverU(B) andL(B)are calculated in the method used in [3].
Other examples of monotonic optimization approaches for solving nonconvex transceiver optimization problems using the branch-and-bound method can be found, e.g., in [10,14,167].
In [97,163,168–171], the so-called polyblock method (e.g., [165, Sec. 11.2]) was used, which can be considered as a modified branch-and-bound method.
7.3.1.5 Computational Complexity
The striking disadvantage of all these methods including the one proposed above is that their worst-case complexity order is c1cN
2, wherec1andc2are constants that depend on properties of the objective function, andN is the number of optimization variables. This can be concluded from [172, Theorem 4], and it means that the complexity of the above method is exponential in the number of usersK. However, for the considered nonconvex problem, no globally optimal solution with lower complexity order is known. Therefore, suboptimal methods for transceiver design as discussed in Section 7.6 are more appropriate for practical implementations, but whenever we need a globally optimal solution as a benchmark in offline simulations or as an