The MIMO ZIFC (or one-sided MIMO IFC) is a special case of the two-user MIMO IFC, where only one of the receivers is disturbed by interference from the other user. The system model of theK-user RoP MIMO IFC (8.1) then simplifies to
y1=H1x1+Hx2+η1 (8.6)
y2=H2x2+η2 (8.7)
with the abbreviationsHk=Hk,kandH=H1,2. For the achievable rates with TIN strategies (8.2), we obtain
r1=µlog2 det Cη1+H1Cx1H1H+HCx2HH
det (Cη1+HCx2HH) , (8.8) r2=µlog2 det Cη2+H2Cx2H2H
detCη2
. (8.9)
From a practical perspective, such a scenario could arise when the transmitting and receiving nodes are positioned in a way that one of the unintended links is negligibly weak (see, e.g., [36,206] and Figure 8.2). From a theoretical point of view, the ZIFC is an important starting point for analytical studies since it is easier to analyze due to the one-sided interference. The hope is then that fundamental insights gained in this simpler scenario also help to understand more complicated interference networks.
In this section, we discuss the application of improper signaling in the complex ZIFC. We restrict our considerations to TIN strategies with Gaussian transmit signals in a single-carrier single-antenna setting with individual power constraints. From the analysis of the resulting
8.3. Improper Signaling in the Complex MIMO Z-Interference Channel 147
SISO ZIFC-TIN, we obtain an insight on the interplay between TS and improper signaling, and we discuss possible implications for other systems at the end of this section.
8.3.1 Previous Results
From the literature on the capacity of the (MIMO) ZIFC (e.g., [42,186,207–210]), it is known that treating the interference as noise is in general not the optimal strategy in this type of system.
Nevertheless, TIN strategies have still attracted the interest of researchers due to their simplicity, and several recent publications have studied possible benefits of improper signaling in the (MIMO) ZIFC-TIN.
In [36,206], improper signaling was compared to proper signaling in the complex MIMO ZIFC-TIN. However, as the numerical analysis was based on heuristic transmit schemes, the observed gains do not allow conclusions about whether improper signaling can also outperform the (unknown) globally optimal proper transmit strategy. Despite the simplification due to one-sided interference, the MIMO case still does not seem to be easily tractable in an analytical study.
Analytical expressions for the achievable sum rate in the complex SISO ZIFC-TIN with improper signaling were derived in [37], and an extension to the whole Pareto-boundary of the rate region was presented in [38]. The obtained results show that the rate region with improper signaling is in general larger than with proper signaling, but the studies in [37,38] do not consider the possibility of TS (i.e., they apply only to systems where either RTS is used or (R)TS is avoided completely). We study the case with TS in the following.
8.3.2 Standard Form of the Complex SISO ZIFC As noted in [37,38], the SISO ZIFC
y1=|h1|ejθ1x1+|h|ejθx2+η1 (8.10)
y2=|h2|ejθ2x2+η2 (8.11)
with proper Gaussian noise and power constraintsCxk ≤Qk, ∀kcan always be transformed to a standard form
y10 =x10 +√
ax20 +η01 (8.12)
y20 =x20 +η02 (8.13)
where we have chosen
y10 = e−jθ1Cη−112y1, x10 =|h1|Cη−112x1, η01 = e−jθ1C−
1 2
η1 η1, (8.14) y20 = ej(θ−θ1−θ2)Cη−212y2, x20 =|h2|ej(θ−θ1)Cη−212x2, η02 = ej(θ−θ1−θ2)C−
1 2
η2 η2. (8.15) We can identify thata=|h|2|h2|−2Cη2Cη−11 ≥0and thatη0kis proper Gaussian noise with unit variance for allk. The transmit signalsxk0, ∀kare (possibly improper) complex Gaussian with power constraintsCxk0 ≤Q0k=|hk|2Cη−1k Qk, ∀k. In the following, we use the standard form, but we omit the prime symbols for the ease of notation.
8.3.3 Optimal Transmit Strategy with Time-Sharing in the Complex SISO ZIFC-TIN
When allowing TS as introduced in Section 3.2.1, we obtain the following result, which is also shown in [8].
Theorem 8.3.1. In the complex SISO ZIFC-TIN, the whole TS rate region is achieved by proper transmit signals.
Proof. We perform the proof in the standard form (8.12)–(8.13). When writing the transmit pseudovariances as˜Cxk =|˜Cxk|ejϕk, it can be shown that it is optimal to chooseϕ1 =ϕ2+π and thatϕ1 = 0can be chosen without loss of generality (see [38]). This means, we can assume
˜Cxkto be real-valued for allk. Note that this complies to our observations in Example 4.5.1 and in the proof of Proposition 7.4.2.
For the corresponding composite real transmit covariance matrices, we obtain from (2.34) Cxˇ0 optimal TS solution (which is equivalent to˜Cxk = 0).
The combined real representations (2.46) of hk = 1, ∀k and h = √
a are given by H`k=I2, ∀kandH` =√
aI2, respectively. For the combined real noise covariance matrices, we obtainCηˇk = 12I2 from (2.34) since the complex noise is proper with unit variance. As all these matrices are scaled identities, we can interpret (8.16) as CN transmission in a real-valued two-carrier system with equal channel conditions on both carriers. By applying (8.8)–(8.9) with µ= 12 on each carrier, we obtain the same rate equations
r1(c)= 1 on both carriers, so that the solution to the inner problem (3.33) is exactly the same on both carriers when we apply the dual decomposition from Section 3.3.2. Thus, the primal recovery (3.23) performs time-sharing between solutions for which it holds thatp(1)k =p(2)k , ∀k(see also the argumentation in the proof of Proposition 7.4.2).
8.3.4 Numerical Examples and Discussion
The above result might seem surprising since a gain by improper signaling was shown in terms of the sum rate in [37] and in terms the rate region in [38]. However, the difference comes from the fact that TS was not considered in these two publications. Instead, convex hulls of the rate regions were taken in [38], which corresponds to an application of RTS.
The more flexible TS not only renders improper signaling unnecessary, but even leads to a larger rate region than the one obtained in [38]. This can be seen from Figure 8.3, where we reproduce one of the examples from [38] along with an additional curve for proper signaling with TS.2To obtain this curve, we have adapted the branch-and-bound approach discussed in
2This simulation result was presented in a similar manner in [8].
8.3. Improper Signaling in the Complex MIMO Z-Interference Channel 149
0 1 2 3
0 1 2 3
r1 r2
a= 2,Q1 =Q2= 10
proper improper proper RTS improper RTS proper TS
0 1 2 3
0 1 2 3
r1 r2
a= 2,Q1= 10,Q2 = 5
SRmax improper SRmax proper TS
Figure 8.3: Achievable rate regions of the SISO ZIFC-TIN without (R)TS, with RTS and with TS. The solid gray lines are the tangents with slope−1at the sum rate points.
Section 7.3.1.2 to the considered scenario. In addition, we plot the rate region for a scenario with asymmetric power constraintsQ16=Q2, where it can be observed that proper signaling with TS can also improve the sum rate (SRmax) compared to the one obtained with improper signaling in [37,38].
The bad performance of proper signaling with RTS compared to improper signaling with RTS can be understood by considering the complex transmission of the second user as two real-valued data streams. When reducing the power of one of the real-valued streams, we do not only reduce the interference that this stream causes to the first user, but we can also increase the transmit power of the other real-valued stream without violating the power constraint.
Accordingly, we can redistribute power between the two real-valued streams of the first user in order to invest more power in the dimension where the interference is weaker (according to the waterfilling solution (4.10) with the noise covariance matrix replaced by the composite real interference-plus-noise covariance matrix).
When trying to achieve the same effect by considering two time slots in a transmit scheme with proper signals and RTS, we encounter the problem that reducing the power in one time slot does not allow us to increase the power in the other one. However, TS gives us this possibility (since the powers are averaged over the slots in case of TS, see Section 3.2.1), and it provides us with an even more flexible way of redistributing the power, namely by varying the length of the time slots in addition. As a result, proper signaling with TS can even outperform improper signaling with RTS.
It shall be noted that this argumentation is only possible due to the special structure of the combined real transmit covariance matrices in (8.16), where improper signaling boils down to a nonuniform power allocation between two real-valued streams. Thus, this justification of the optimality of proper signals is based on the fact that the system can be transformed to a standard model with purely real-valued channel coefficients. This is similar to Proposition 7.4.2, where the complex two-user MISO BC-TIN was transformed to a description with real-valued channel coefficients. On the other hand, such a transformation is not possible in the three-user examples in Proposition 7.4.1 and Example 8.2.3, where improper signaling was shown to be beneficial
even in case that TS is allowed.
It is therefore an interesting question whether or not Theorem 8.3.1 can be extended to the two-user IFC-TIN, where both users disturb each other. On the one hand, the improvements due to improper signaling reported for the two-user SISO IFC-TIN and the two-user MISO IFC-TIN in [31–34,188–192] (see Example 8.2.4) were all obtained in comparison to proper signaling with RTS (or without any form of (R)TS), and TS was not considered in these publications.
On the other hand, when transforming this scenario to a standard form, at least one complex coefficient remains, so that we cannot easily transfer the argumentation from the proof of Theorem 8.3.1 to these scenarios. A further study of this question is left open for future research.
Another aspect that is left open is the extension to the MIMO ZIFC-TIN and the two-user MIMO IFC-TIN (e.g., [35,36,193,206]). The step from scalar variances to covariance matrices makes it more difficult to obtain analytical results, and it does not seem to be possible to transform these systems to a standard form with real-valued coefficients, even if the considerations are restricted to one-sided interference. It is therefore again not clear whether Theorem 8.3.1 can be extended to these scenarios.
8.4 Worst-Case Noise
Just like in the MIMO BC-TIN, a difficulty when studying the worst-case noise in theK-user MIMO IFC is that the optimal transmit strategy is not known. In [66, 67], it was shown that Gaussian noise is the worst-case noise in arbitrary networks with multiple nodes, but this, e.g., does not allow direct conclusions about proper or improper noise since a real-valued setting was considered in [66, 67]. It might be possible to extend the proof technique from [66, 67], which does not require knowledge of the optimal transmit strategy, to complex systems in order to answer this question. At least from an intuitive point of view, it is to be expected that a disturbance with the highest differential entropy should always be the worst case. While we leave a proof of this aspect open for future research, we can at the current point at least state a worst-case noise theorem for theK-user MIMO IFC-TIN under the assumption that the transmit signals are fixed to be maximum-entropy signals.
Theorem 8.4.1. Let(PSk)∀kand(PDk)∀kbe power shaping spaces fulfilling the compatibility assumption (Definition 3.1.3), and let the constraint(Cηk)∀k ∈ Qnoise be compatible with NK
k=1PDk(in the sense of Definition 3.1.2). If it holds for all transmit signals in theK-user RoP MIMO IFC-TIN that xk is an arbitrary maximum-entropy signal with respect to PSk, maximum-entropy noise with respect to(PDk)∀kminimizes the achievable rates of all users simultaneously.
Proof. We can considerCzkfrom (8.3) as effective noise covariance matrix and apply the same argumentation as in Theorem 6.5.1.
Using the power shaping spaces from Proposition 3.4.2, we obtain the following corollaries for the complexK-user multicarrier MIMO IFC-TIN. The statement of Corollary 8.4.2 can also be found in [28].
Corollary 8.4.1(PSk = `PSCNk , ∀kand PDk = `PDCNk ). In the complex K-user multicarrier MIMO IFC-TIN with constraints on the complex per-carrier noise covariance matrices, the worst-case noise is proper and CN if the transmit signals are proper and CN.