4.5 Complex Multicarrier MIMO Channel
4.5.3 Optimal Transmit Covariance Matrix for the Case of Improper or CC Noise 65
ω(c)− 1 λ(c)i , 0
)
. (4.24)
For the sum power constraint (3.39), we can keep the modal matrixU(c)in (4.23), but we have to optimize the powersp(c)i jointly over all carriers. To do so, we replaceω(c)in (4.24) by a common water levelω, which we choose such that the sum power constraint (3.39) is fulfilled with equality.
Finally, if we assume per-carrier shaping constraints (3.41), we can use thatH(c)XH(c),H H(c)X0H(c),HforX X0 and thatdetX is nondecreasing inX 0[68, Sec. 7.7]. Thus, r(c) is nondecreasing inCx(c), and the optimal solution isCx(c) =Q(c)∀c, i.e., the shaping constraints are active.
Especially for per-carrier power constraints or per-carrier shaping constraints, solving the problem via the combined real representation would be more effort than the above derivations based on the corollaries from Section 4.5.1.
4.5.3 Optimal Transmit Covariance Matrix for the Case of Improper or CC Noise
The situation is different if the noise is improper (as considered, e.g., in [7,36] for the MIMO channel and in [107] for the continuous-time SISO channel) or CC or both. As Corollary 4.5.1 is no longer applicable in that case, the optimal transmit signal is a reduced-entropy signal in general. More precisely, the type of entropy reduction that we need for the optimal transmit signal is the same as the type of entropy reduction of the noise. That is, improper noise can require improper signaling, CC noise can require CC transmission, and noise that is both improper and CC can require a transmit signal that is both improper and CC.
In all these cases, at least one of the equalities in (4.21) is no longer valid. When using the complex multicarrier model, it would thus be necessary to derive a solution that accounts for the impropriety and/or the correlations between carriers when evaluating the mutual information I (x(1), . . . ,x(C)); (y(1), . . . ,y(C))
.
It seems to be much simpler to instead perform the optimization directly in the combined real representation in the form given in (4.5). As pointed out in Section 4.3, the solution can be
computed by a convex programming solver, and it can be obtained analytically from (4.10) in case of a sum power constraint. This approach was pursued, e.g., in [7,36].
To get a better intuition about the optimal transmit strategy in case of improper noise, we reconsider the minimal example from Section 4.3.2.
Example 4.5.1. Consider a complex single-carrier single-antenna channel, i.e.,C =mS= mD= 1, with the channelh = 1, and improper noise with varianceCη= 4and pseudovariance
˜Cη = 2. The combined real representation of this system corresponds exactly to the RoP MIMO channel considered in Example 4.3.1. We can identify the power shaping spaces considered therein asPS = `PS andPD = `PDsince any2×2symmetric BSC2 matrix is a scaled identity. The results obtained in Example 4.3.1 translate to˜Cx = −12˜Cη = −1 (obtained fromNx=−12Nηfor the constrainttr[Cxˇ]≤1⇔Cx ≤1) and˜Cx =−˜Cη=−2 (fromNx = −Nη fortr[Cxˇ]≤ 2 ⇔Cx ≤ 2). By parameterizing the pseudovariance as
˜Cx = |˜Cx|ejϕ, these results can be interpreted as follows. In the considered scenario, the impropriety of the intended signal should try to compensate the impropriety of the noise in the following sense. The direction of impropriety of the intended signal (parametrized byϕ) should be the exact opposite of the direction of impropriety of the noise, and the strength of impropriety of the intended signal (parametrized by|˜Cx|) should be the same as the strength of impropriety of the noise as long as this does not violate the constraints.
4.5.4 Scenarios that Violate the Compatibility Assumption
It shall be noted that there are situations that lead to a reduced-entropy signal as optimal input even if we have maximum-entropy noise. This can happen if the compatibility assumption from Definition 3.1.3 is violated for the power shaping spaces of interest, so that Theorem 4.2.1 is no longer applicable.
An important part of the compatibility assumption is that the channel matrix has to be compatible with the power shaping spaces(PD,PS). For the study of proper signaling, i.e., for the power shaping spaces( `PD,P`S), this assumption is violated if the original complex channel is widely linear (Definition 2.9.5) instead of linear, so that its real-valued representation (2.38) is no longer aBSC2matrix. In particular, this is the case if I/Q imbalance occurs (e.g., [108,109]) and is modeled as part of the channel (cf., e.g., [110] and the references therein). In a similar manner, if the assumption of orthogonal carriers is not fulfilled, the combined real channel matrix no longer consists of block-diagonal submatrices and is no longer compatible with(PDCN,PSCN), i.e., with the power shaping spaces that we use to study CN transmission.
We do not consider such scenarios in detail, but the solution that we obtain when applying (4.10) to the combined real representation is general enough to cover these cases as well. By plugging in a combined real channel matrix that is not compatible with(PD,PS), we obtain a solutionCxˇ that can have a nonzero entropy reduction componentprojNS(Cxˇ)even in the case of maximum-entropy noise withCηˇ ∈ PD.
The second part of the compatibility assumption requires that the constraintCxˇ ∈ Qacts only on the power shaping componentprojPS(Cxˇ)of the transmit covariance matrix. If this is violated, we can again obtain a reduced-entropy transmit signal as optimal solution even in the case of a compatible combined real channel matrix and maximum-entropy noise. However, we do not further discuss this case since the assumption is fulfilled by the most common transmit
4.5. Complex Multicarrier MIMO Channel 67
covariance constraints (see Section 3.4.2 and Proposition 3.4.4).
The aspects discussed above apply in a similar manner to the multiuser systems treated in the following chapters, but they will not be considered in detail any further.
4.5.5 Worst-Case Noise
By applying Theorem 4.4.1 and Corollary 4.4.1 to the combined real representation, we can obtain several specialized worst-case noise results for the complex multicarrier MIMO channel.
In all three corollaries, the termoptimizedhas to be understood as a rate maximization under a transmit covariance constraint (4.20). The various power shaping spaces from Proposition 3.4.2 that we use in the corollaries allow for different kinds of constraints on the statistical properties of the noise.
Corollary 4.5.4(PS = `PSCNandPD = `PDCN). In the complex multicarrier MIMO channel (4.19)with constraints on the complex per-carrier noise covariance matrices, the worst-case noise is proper and CN if the transmit signal is either fixed to be proper and CN or optimized based on the noise properties.
Corollary 4.5.5 (PS = `PS and PD = `PD). In the complex (multicarrier) MIMO channel (4.19)with constraints on the combined complex noise covariance matrix, the worst-case noise is proper if the transmit signal is either fixed to be proper or optimized based on the noise properties.
Corollary 4.5.6(PS =PSCNandPD =PDCN). In the complex multicarrier MIMO channel (4.19)with constraints on the per-carrier noise properties, the worst-case noise is CN if the transmit signal is either fixed to be CN or optimized based on the noise properties.
A theorem stating that proper noise is the worst-case noise was shown in [36] under the assumption that the transmit covariance matrix is either chosen optimally or at least with optimal modal matrix and a uniform power allocation. This result was generalized in [28] by showing that it holds even if the transmitter does not adapt its signal to the potential impropriety of the noise. Moreover, some restrictive assumptions that were made in [36] (e.g., on the rank of the channels) can be avoided by means of the proof technique used in [28] and for Theorem 4.4.1.
The result from [28], which includes the one from [36] as a special case, corresponds to Corollary 4.5.5, i.e., it can be considerd as a corollary of the more general Theorem 4.4.1. Note that this result holds under arbitrary constraints on the combined complex noise covariance matrixCη, including a limited total noise power and the case of a fixedCη (see discussion below Theorem 4.4.1).
Chapter 5
The Gaussian MIMO Multiple Access Channel
From now on, we turn our attention to systems with multiple users that are served in the same spectrum, so that inter-user interference occurs. The first example we consider is a MIMO uplink scenario, whereKuser terminalsUktransmit independent signalsxUk to a base stationB(see Figure 5.1). In the information theoretic literature, this scenario is referred to as a (Gaussian) MIMO multiple access channel (MIMO MAC, e.g., [46]) or Gaussian vector MAC (e.g., [57]).
Even though inter-user interference is present in this scenario, the MIMO MAC turns out to behave in many regards similar to an interference-free single-user MIMO channel as long as we eliminate part of the interference by means of coding. This can be done either by a joint decoding of all received signals or by a successive interference cancellation scheme (see Section 5.1.1). If we instead treat all interference as noise, we obtain a different situation, which is described at a later point in Section 7.2.
Just like in Chapter 4, we first consider a RoP MIMO system, for which we show that maximum-entropy transmission is optimal in case of maximum-entropy noise. The obtained result can then be used to study the complex multicarrier MIMO MAC afterwards, and it also forms the basis for our later derivations on the MIMO broadcast channel in Chapter 6 and on the MIMO relay channel in Chapter 9. In addition to the theoretical findings, we review several numerical algorithms from the existing literature and discuss how they can be applied in the cases of maximum-entropy noise and reduced-entropy noise.
5.1 System Model and Capacity Region
We introduce the abbreviationsy =yB, Hk =HBUk,xk =xUk, andη = ηB. From the general model in (3.1), we obtain the description of the RoP MIMO MAC as
y=
K
X
k=1
Hkxk+η. (5.1)
As we have multiple data rates of interest, namely the achievable ratesρkof all usersk, we
69
B U1
H1
UK
HK Uk
Hk
Figure 5.1: Illustration of the MIMO multiple access channel.
consider a rate region as introduced in Section 3.2. The following derivation can be obtained by applying a result for general memoryless multiple access channels (from [58,111] as cited in [48]) to the RoP MIMO MAC. Instead of focusing on power constraintstr[Cxk]≤Qk, ∀kas, e.g., in [57,112,113], we consider general covariance constraints of the form (3.3).
By decoding the signals of all users jointly, the ratesρ= [ρ1, . . . , ρK]Tare achievable if P
k∈Kρkfor any group of usersK ⊆ {1, . . . , K}fulfills [48]
X
k∈K
ρk≤I ({xk}k∈K;y|(xk)k /∈K) = h (yK)−h (yK| {xk}k∈K) = h (yK)−h (η) (5.2) where
yK=X
k∈K
Hkxk+η. (5.3)
Note thath (η)in (5.2) does not depend on the transmit strategy, and assume some fixed (Cxk)∀kso that
CyK =Cη+X
k∈K
HkCxkHkH. (5.4)
is fixed. Then,h (yK)is maximized ifyKfollows a Gaussian distribution (see [59, 62] for a real-valued system and [29,63] for a complex system). As the noiseηis assumed to be Gaussian, and asη,x1, . . . ,xKare independent, we can make allyKGaussian by choosing allxkto be Gaussian. The constraints on the transmit strategy affect only(Cxk)∀k, and the above reasoning holds for any(Cxk)∀k. Thus, Gaussian transmit signals are optimal for all inequalities in (5.2).
For a complex setting, note that Remark 4.1.1 applies accordingly to the current chapter.
As the optimal transmit signals are Gaussian, we can plug in the differential entropy from (2.7) to obtain the rate region
R= [
(Cxk0)∀k
(Cxk)∀k∈Q
(
ρ∈RK0,+
X
k∈K
ρk≤µlog2 detCyK detCη
, ∀K ⊆ {1, . . . , K} )
(5.5)
where the constraint setQis defined as in (3.3).
5.1. System Model and Capacity Region 71
As pointed out in [57], this rate region is convex, i.e., it cannot be enlarged by taking its convex hull, meaning that RTS does not bring any gains. To see that this is true and that more flexible TS does not help either, we express the rate region in (5.5) by means of (3.11) using
O= As the expressions on the right hand side of the rate inequalities are concave functions of (Cxk)∀k, the setOis convex and cannot be enlarged by taking its convex hull. As the constraint setQis convex, this implies thatR=R=R(see Section 3.2.1).
An alternative proof ofR=Rcan be obtained, e.g., for the case of a sum power constraint (3.4) by showing that the solution of the rate balancing problem (3.13) in the RoP MIMO MAC is a concave function of the available sum transmit power Q(see [17, Prop. 1]). This can be extended to other constraints, e.g., to a sum shaping constraint by following the lines of [114, Th. 1].
As Gaussian transmit signals are optimal and (R)TS is not necessary, the rate region given in (5.4) is the capacity region of the RoP MIMO MAC, i.e.,C=R.