5.5 Complex Multicarrier MIMO MAC
5.5.2 Optimal Transmit Covariance Matrices for the Case of Proper CN Noise . 75
When studying the capacity region of the complex multicarrier MIMO MAC (5.14) under the assumption that the noise is proper and CN as in Corollary 5.5.1, the combined real noise covariance matrixCηˇlies in the power shaping spaceP`BCN, and the optimal combined real transmit covariance matrices Cxˇk lie in the power shaping spacesP`UCNk for all k(see Proposition 3.4.2). This means that all these matrices are symmetric BSC2 matrices with block-diagonal submatrices (see Definition 3.4.1). In addition, the combined real channel matricesH`kareBSC2with block-diagonal submatrices as discussed in Section 3.4.4.
For the inequalities in (5.5), we thus obtain X
This means that we can decompose the problem of finding optimal covariance matrices into separate problems for each carrier which are only coupled by the first sum of (5.16) and by the covariance constraint. Indeed, the complex multicarrier MIMO MAC is a RoP CN MIMO system (see Definition 3.3.1) if the signals and the noise are proper and CN. Therefore, the dual decomposition framework presented in Section 3.3.2 can be applied to decouple the per-carrier problems.
For the case of per-user power constraints (3.39) and the case of a sum power constraint (3.42), such a dual decomposition was performed in [49]. The remaining task is then to solve the inner problem (3.33), i.e., to find optimal complex covariance matricesCx(c)
k on each carrier.
As we have a RoP MIMO MAC on each carrier, any weighted sum rate can be expressed as a concave function of these covariance matrices (see Section 5.3), so that (3.33) has a concave objective function and can be solved by convex programming methods.
An alternative decomposition was used in [50] by instead introducing per-carrier sum powers Q(c) as auxiliary variables. The proposed method then alternates between an optimization ofQ(c)for fixed covariance matrices and a covariance optimization under per-carrier power constraintsPK
k=1tr[Cx(c) k
]≤Q(c).
5.5.3 Optimal Transmit Covariance Matrices for the Case of Improper or CC Noise
If the noise is improper or CC or both, we can no longer apply Corollary 5.5.1, and at least one of the equalities in (5.15) becomes invalid. In particular, if the noise is both improper and CC, we need to apply the methods presented in Section 5.3 directly to the combined real representation in order to obtain an optimal transmit strategy.
In the case of improper CN noise, we can make use of the real-valued multicarrier formulation (3.36) in order to again be able to perform a decomposition into per-carrier problems. As pointed out in Section 3.4.1, the real-valued multicarrier formulation is a RoP CN MIMO system in this case.
On the other hand, if the noise is CC, but proper, the combined complex representation (3.35) can be used. In this case, it is no longer possible to perform a dual decomposition, but we can at least obtain a slight reduction of the computational complexity compared to the combined real representation (cf., e.g., the statements about computational complexity of complex formulations compared to composite real representations in [24]).
If we are interested in a universal method whose applicability does not depend on the noise properties, we can always apply a numerical solver directly to the combined real representation.
From the considerations in Section 5.3.1, we know that a special structure of the noise (e.g., CN or proper) automatically leads to a numerical solution where the transmit signals have the corresponding structure.
5.5.4 Worst-Case Noise
The following corollaries of Theorem 5.4.1 are obtained for the complex multicarrier MIMO MAC (5.14) with various choices of the considered power shaping spaces (see Proposition 3.4.2).
The termoptimizedis always understood in the sense of finding Pareto-optimal solutions under a transmit covariance constraint (3.38). Note that the statement of Corollary 5.5.5 was also shown in [28].
5.5. Complex Multicarrier MIMO MAC 77
Corollary 5.5.4(Pk= `PUCNk , ∀kandPB= `PBCN). In the complex multicarrier MIMO MAC (5.14)with constraints on the complex per-carrier noise covariance matrices, the worst-case noise is proper and CN if the transmit signals are either fixed to be proper and CN or optimized based on the noise properties.
Corollary 5.5.5(Pk= `PUk, ∀kandPB = `PB). In the complex (multicarrier) MIMO MAC (5.14)with constraints on the combined complex noise covariance matrix, the worst-case noise is proper if the transmit signals are either fixed to be proper or optimized based on the noise properties.
Corollary 5.5.6(Pk=PUCNk , ∀kandPB=PBCN). In the complex multicarrier MIMO MAC (5.14)with constraints on the per-carrier noise properties, the worst-case noise is CN if the transmit signals are either fixed to be CN or optimized based on the noise properties.
Chapter 6
The Gaussian MIMO Broadcast Channel
If we still consider a base stationBwith K user terminalsUk, but consider the downlink operation where information is transmitted from the base station to the users (see Figure 6.1), we have a so-called (Gaussian) MIMO broadcast channel (BC) [46, 121]. After introducing the system model and summarizing the capacity-achieving coding scheme, we revisit the uplink-downlink duality from [106], which enables us to study the MIMO BC via the dual MIMO MAC. By exploiting this duality, we can extend the results from Chapter 5 to the RoP MIMO BC and finally to the complex multicarrier MIMO BC.
6.1 System Model and Capacity Region
We introduce the abbreviationsyk =yUk,HkH=HUkB,x=xB,ηk=ηUk,M =MB, and Mk=MUk. The description of the RoP MIMO BC is obtained from the general model (3.1) as
yk=HkHx+ηk, ∀k (6.1)
with a covariance constraintCx ∈ Qas given by (3.3).
It was shown in [121] that the complete capacity region of the real-valued MIMO BC can be achieved by so-called dirty paper coding (DPC) with real-valued Gaussian input signals. As the proof is quite lengthy, we do not reproduce it here. For the complex MIMO BC, we can apply this result to the composite real representation to show that complex Gaussian signals are optimal in this case [121]. However, as pointed out in [21], this does not give a response to the question whether proper or improper signals should be used. Since the definition of a RoP MIMO system (Definition 3.1.1) anyway assumes that all signals in a complex system are proper, we come back to this question later when we study the complex multicarrier MIMO BC.
DPC exploits a result from [122] stating that interference that is noncausally known to the transmitter (but not to the receiver) can be pre-eliminated by means of coding without influencing the transmit covariance matrix. In the MIMO BC, this can be applied in a successive manner so that the interference from all users encoded earlier than userkis removed from the signal intended for userk(e.g., [46]). The interference caused by users encoded after userk cannot be removed in this manner and is treated as additional noise. Any point in the capacity region can be achieved by RTS between DPC strategies with different encoding orders [121].
79
B
U1 H1H
UK
HKH Uk
HkH
Figure 6.1: Illustration of the MIMO broadcast channel.
To describe this mathematically, we introduce auxiliary signals called per-user input signals ξk, which correspond to the messages intended for the various users. The transmit signalxthat is constructed by means of DPC then has the covariance matrixCx=PK
k=1Cξk.
With the encoding order specified by the permutation π¯ : k 7→ π(k), so that user¯ k: ¯π(k) = 1is encoded first, we can achieve the per-user rates [46,121]
rkBC=µlog2 detCyk
detCzk (6.2)
where
Cyk =Cηk+ X
j:¯π(j)≥¯π(k)
HkHCξjHk and Czk =Cyk −HkHCξkHk. (6.3) The resulting rate region
R= conv [
¯ π
[
(Cξk0)∀k
PK
k=1Cξk∈Q
ρ∈RK0,+
ρk≤rBCk , ∀k (6.4)
withrBCk from (6.2) is the capacity region of the RoP MIMO channel [112,113,121]. Note that the transmit strategy that was shown to be capacity-achieving in [121] only makes use of RTS, i.e., applying the more flexible TS would lead to the same rate regionR=R.
Just like in the MIMO MAC, we can find Pareto-optimal points of the capacity region C = Rby solving weighted sum rate maximizations (3.12). However, due to the different structure of the involved matrices in the rate equations (6.2) and (5.7), the MIMO BC does not feature the favorable property of the MIMO MAC that weighted sum rates can be written as concave functions of the covariance matrices of the per-user signals (e.g., [46]). A common approach to overcome this problem is the so-called uplink-downlink duality which allows us to solve optimization problems for the MIMO BC via a transformation to the dual MIMO MAC.