7.6 Algorithmic Aspects
7.6.4 Summary and Discussion
The analysis in this section has shown that many procedures that are applied in algorithms for transceiver design tend to preserve matrix structures that correspond to maximum-entropy transmission. If the aim is to obtain a reduced-entropy strategy, such as improper signaling or CC transmission, by means of an iterative algorithm, it is thus necessary to account for this already in the initialization. Note that there is then still no guarantee that the obtained solutions always correspond to reduced-entropy transmission, but this is not a problem. We have to recall that reduced-entropy signals are not always beneficial, but it rather depends on
the scenario and on the channel realization. It is thus only important to avoid an inherent restriction to maximum-entropy transmission in order to obtain an algorithm that is able to converge to reduced-entropy or maximum-entropy strategies depending on the particular channel realizations. Numerical experiments reveal that such a convergence to reduced-entropy strategies is indeed possible (see, e.g., the simulation results in Section 7.5.5 and in [11,20,23,53]).
While we have identified several initializations of combined real transmit filters that correspond to improper signaling, finding initializations that correspond to CC transmission is less obvious. An initialization that is appropriate no matter which power shaping space is considered is a random initialization, but even then it is necessary to know about the possible pitfalls (e.g., the case whereTkTkH=IMk in Section 7.6.2.4).
In summary, we can say that the combined representations from Table 3.1 enable us to optimize improper signaling and CC transmission by means of algorithms that have originally been developed, e.g., for proper signaling in single-carrier systems. However, we usually do not obtain the intended reduced-entropy solutions when we apply such methods thoughtlessly.
Instead, it is necessary to adapt the initializations carefully.
A final remark is in order concerning algorithms that have been designed for SISO and MISO systems. The combined representations in Table 3.1 always correspond to a true MIMO system withMT>1for all terminalsT, no matter how many antennas the original complex multicarrier system has. Thus, the approach pursued in this section is possible only with algorithms that are able to handle such true MIMO systems. If the aim is, e.g., to extend an algorithm that was developed for proper signaling in a MISO system to the case of improper signaling, different approaches are necessary. An example of such an extension can be found in [34].
7.7 Worst-Case Noise
Just like in the previous chapters, let us conclude our study of the MIMO BC-TIN by analyzing the worst-case noise. Theorem 6.5.1 for the RoP MIMO BC with DPC can be easily extended to the case of TIN strategies.
Theorem 7.7.1. Let(Pk =PUk)∀kandPBbe power shaping spaces fulfilling the compatibility assumption (Definition 3.1.3), and let the constraint(Cηk)∀k ∈ Qnoise be compatible with NK
k=1Pk (in the sense of Definition 3.1.2). If it holds for all per-user input signals in the RoP MIMO BC-TIN that ξk is an arbitrary maximum-entropy signal with respect to PB, maximum-entropy noise with respect to(Pk)∀k minimizes the achievable rates of all users simultaneously.
Proof. We can considerCzkfrom (7.2) as effective noise covariance matrix and apply the same argumentation as in Theorem 6.5.1.
Even though this worst-case noise theorem, which applies for the case that the input signals are maximum-entropy signals, can be obtained using the same proof technique as before, the situation changes if we are interested in the case of optimized transmit strategies.
Unlike for the MIMO BC with DPC, optimal transmit strategies in the MIMO BC-TIN do not necessarily consist of maximum-entropy input signals even if we have maximum-entropy noise (Theorem 7.4.1). Therefore, we cannot easily obtain a corollary for the case of optimized
7.7. Worst-Case Noise 139
transmission. Some further comments on this aspect are given when we discuss the worst-case noise in interference channels in Section 8.4.
For the complex multicarrier MIMO BC-TIN, we have the following specializations based on the power shaping spaces from Proposition 3.4.2. Note that the case of optimized transmit strategies is missing in the corollaries as well.
Corollary 7.7.1(Pk= `PUCNk, ∀kandPB = `PBCN). In the complex multicarrier MIMO BC-TIN with constraints on the complex per-carrier noise covariance matrices, the worst-case noise is proper and CN if the per-user input signals are proper and CN.
Corollary 7.7.2(Pk = `PUk, ∀kandPB= `PB). In the complex (multicarrier) MIMO BC-TIN with constraints on the combined complex noise covariance matrices, the worst-case noise is proper if the per-user input signals are proper.
Corollary 7.7.3(Pk=PUCNk, ∀kandPB =PBCN). In the complex multicarrier MIMO BC-TIN with constraints on the per-carrier noise properties, the worst-case noise is CN if the per-user input signals are CN.
Chapter 8
Gaussian MIMO Interference Channels
Up to this point, we have considered systems with either only a single transmitter or only a single receiver. By instead considering configurations where both the number of transmitters and the number of receivers are larger than one, we arrive at so-called interference networks. We restrict the following considerations to the special case where each receiver is only interested in the data from a particular transmitter and each transmitter has data for only one receiver, so that the involved terminals can be grouped into transmitter-receiver pairs, which are also referred to as users (see Figure 8.1). This setting is called (Gaussian)K-user MIMO interference channel (IFC) [184].
After introducing the system model, we summarize results from the literature which show that reduced-entropy transmission can be beneficial in theK-user MIMO IFC. As a special case, we consider the so-called MIMO Z-interference channel (ZIFC), where only partial interference is present. In this setting, we study proper and improper signaling both for systems with and without TS. Finally, we compare the results obtained in theK-user MIMO IFC and the MIMO ZIFC to our observations from the previous chapters.
8.1 System Model and Achievable Rates
From the general model (3.1), we obtain the model for theK-user RoP MIMO IFC as
yk =
K
X
j=1
Hk,jxj+ηk, ∀k (8.1)
with a covariance constraint (Cxk)∀k ∈ Q as given by (3.3). We have introduced the abbreviationsyk=yDk,Hk,j =HDkSj,xk=xSk, andηk=ηDk. The system model for the complexK-user multicarrier MIMO IFC can be obtained in an analogous manner from the general model in (3.34).
The capacity region of theK-user MIMO IFC is still an open problem apart from some special cases. Therefore, it is assumed in many publications (e.g., [32, 34, 35, 170] and the references therein) that interference is treated as noise, i.e., TIN strategies are applied, and that all transmit signals are Gaussian, even though this is not necessarily the optimal input
141
S1
D1
D2 S2
H1,1
H1,2 H2,2
H2,1
Figure 8.1: Example of the MIMO interference channel withK= 2transmitter-receiver pairs. The unintended channels are drawn in gray.
distribution (see Chapter 10). We refer to the system model with these two restrictions as K-user MIMO IFC-TIN.
In theK-user RoP MIMO IFC-TIN, we can achieve the per-user rates rk=µlog2 detCyk
detCzk (8.2)
with
Cyk =Cηk+
K
X
j=1
Hk,jCxjHk,jH and Czk =Cyk −Hk,kCxkHk,kH . (8.3) The corresponding achievable rate region is
RTIN = [
(Cxk0)∀k
(Cxk)∀k∈Q
ρ∈RK0,+
ρk≤rk, ∀k (8.4)
withrkfrom (8.2). As this rate region is nonconvex in general (see, e.g., the plots in [33]), allowing (R)TS (see Section 3.2.1) can lead to enlarged rate regionsRTIN (with RTS) and RTIN (with TS).