Degrees of Freedom

In Section 7.5.1, we have used the concept of effective degrees of freedom for a study with finite data rates. For some of the discussions in this chapter, we instead need the conventional notion ofdegrees of freedom, which describe the behavior of a system when the SNR and the achievable rates tend to infinity. The achievable (sum) DoF in an IFC are given by (see, e.g., [30])

D= lim

Q→∞

r(Q)

log₂(Q) (8.5)

wherer(Q)is the achievable sum rate with sum powerQ. For a givenD≥0, this means that the sum rater(Q)scales linearly withlog₂Qin the high-SNR regime (i.e., for high values ofQ), andDis the slope of this linear relationship. It is easy to verify1that we obtainD=µ in a single-user SISO system (e.g., [184]), and this is clearly an achievable value in theK-user SISO IFC (e.g., by shutting off all users but one).

Recall that allowing for RTS does not bring any gains for sum rate maximization (see Section 3.2.2). Moreover, the high-SNR approximationr(Q)≈Dlog₂Q+R₀, whereR₀is a constant, is concave inQ. Therefore, averaging over rates achieved with different values of Qcannot bring benefits in the high-SNR regime, i.e., TS cannot improve the achievable DoF.

Therefore, results based on DoF hold independently of whether or not (R)TS is allowed.

8.2 Benefits of Reduced-Entropy Transmission

In the literature, we find several results on, e.g., coding across carriers (e.g., [40–42, 186]), symbol extensions (e.g., [30, 184, 185]), and improper signaling (e.g., [30–34, 187–193]) in

1This could be calculated from (4.4) by plugging in the special caseMD=MS= 1withCx=Q.

interference channels. When interpreting them in terms of the nomenclature that we use here, these results can be summarized in the following general theorem.

Theorem 8.2.1. Let(PS_k)_∀kand(PD_k)_∀kbe power shaping spaces fulfilling the compatibility assumption (Definition 3.1.3). Even if the noise vectorsη_kare maximum-entropy signals with respect to(PDk)_∀k, it can happen that reduced-entropy transmit signalsx_k with respect to (PS_k)_∀kare needed to achieve the whole capacity region of theK-user RoP MIMO IFC or the whole rate region of theK-user RoP MIMO IFC-TIN. This statement holds for systems with and without (R)TS.

Proof. Examples where this happens are given below.

Just like in Chapter 7, the examples to establish the theorem can be special cases with relatively small system dimensions. Some examples from the literature are revisited below, and references to further examples are provided.

Example 8.2.1. In [40], the three-user two-carrier SISO IFC was considered. For a particular channel realization, it was shown that joint coding across carriers achieves a sum rate that is higher than the sum of the individual per-carrier sum capacities. In the achievable scheme that was used for joint coding, each user transmits exactly the same signal on both carriers, and the signals are combined by a linear filter at the receiver, while the interference coming from the two other users aligns in the nullspace of the receive filter. This is clearly an application of CC transmission, i.e., an example for a gain by reduced-entropy transmission, where the combined real transmit covariance matrices lie outside of the power shaping spaces(PS^CN_k )_∀k. It is remarkable that this TIN strategy outperforms the best separate coding scheme that utilizes optimal multiuser detection [40]. Therefore, this example suffices as a proof both in terms of the capacity region and in terms of the TIN rate region.

In [41], it was shown that such aninseparabilitycan occur in more complicated single-antenna interference networks, where transmitters may have messages for multiple receivers, and receivers may be interested in messages from multiple transmitters. Indeed, it was shown that only a very special class of interference networks with a so-calledMAC-Z-BCstructure is always separable in the sense that the sum capacity can be achieved without joint coding.

However, when the aim is to achieve not only the sum capacity, but the whole capacity region, coding across carriers can be necessary even in this kind of network. This was shown in [42,186] along with some further results on cases in which coding across carriers is necessary to achieve the capacity of interference networks.

Example 8.2.2. A technique called interference alignment was studied in [184] for theK-user (MIMO) IFC. The idea behind this method is to design the transmit signals such that several interfering signals arriving at a receiver fall into the same lower-dimensional subspace and can easily be canceled by linear zero-forcing as the remaining space stays interference-free (see [184] for more details). Note that a similar effect was exploited in Example 8.2.1. To achieve the sum capacity at high SNR, i.e., the maximum DoF, in theK-user SISO IFC with time-varying channels, the authors of [184] apply interference alignment in combination with symbol extensions. As stated above (see Section 8.1.1), the latter can be considered as a form of CC transmission.

8.2. Benefits of Reduced-Entropy Transmission 145

Interference alignment and signal extensions have also been studied in many further publications, e.g., for a more general interference network called MIMO X channel in [185], and the concept of interference alignment is particularly promising in theK-user MIMO IFC with multiple antennas at all terminals (see, e.g., [184]). Moreover, interference alignment and symbol extensions form the basis of the following example.

Example 8.2.3. In [30], it was proposed to combine improper transmit signals and symbol extensions in order to achieve interference alignment in the three-user SISO IFC. The resulting transmit scheme is a TIN strategy with widely linear zero-forcing and achievesD= 1.2DoF, while proper Gaussian transmit signals can achieve onlyD= 1DoF in this setting [30]. This combination of CC transmission (due to symbol extensions, see Section 8.1.1) and improper signaling is clearly an example for a gain by reduced-entropy transmission. The combined real transmit covariance matrices lie neither in(PS^CN_k )_∀knor in( `PSk)_∀k. As the study is based on DoF, the result holds irrespective of whether (R)TS is allowed (see Section 8.1.2).

This result was extended to the four-user SISO IFC in [187], whereD= ⁴₃DoF were shown to be achievable with improper signaling. Moreover, we can find several examples of improper signaling at finite SNR in the literature.

Example 8.2.4. The complex two-user SISO IFC-TIN was studied in [189], and the results were generalized to a SISO system withK >2users in [33] and to the complexK-user MISO IFC-TIN in [34,190]. In these papers, suboptimal algorithms for rate balancing with improper signaling were proposed, and it was demonstrated that the achieved solutions enlarge the rate region compared to optimized TIN strategies with proper signals, both in the case without (R)TS and in the case with RTS. However, the application of TS was not considered in these papers.

In two earlier papers [31, 188], the complex two-user SISO IFC-TIN was studied from a game theoretic perspective and the cooperative solutions based on improper signaling were shown to outperform the noncooperative Nash equilibrium with proper signaling. Moreover, a parametrization of the Pareto boundary of the achievable rate region with maximally improper signals (corresponding to one-dimensional real-valued signals) was derived. Further algorithms that can be applied for transceiver design without (R)TS in the SISO IFC-TIN with improper signaling were presented in [32, 191]. In [192], it was demonstrated that gains by improper signaling are possible for a large range of channel realizations.

The complexK-user MIMO IFC-TIN, i.e., the case of multiple antennas at transmitters and receivers, was studied in [35, 193], and a suboptimal algorithm for widely linear transceiver design was derived based on the weighted MSE method [180]. Note that it was discussed in [26] how the algorithm from [35, 193], which is based on the complex formulation, can be related to the application of a previously existing algorithm for linear transceivers [194] in the composite real representation. In numerical studies in [35,193], it was shown that the solutions obtained with widely linear transceivers and improper signals lead to higher achievable rates than the solutions with linear transceivers and proper signals. However, as this is a numerical comparison between two suboptimal schemes, a more rigorous or even analytical comparison of the achievable rate regions of the complexK-user MIMO IFC-TIN with proper signals and with improper signals is still an open problem.

Finally, benefits of improper signaling have recently been studied in several closely related system models, e.g., in interfering BCs [195, 196], in interfering MACs with TIN strategies

S₁ D₁

D₂ S₂

H₁

H

H₂

Figure 8.2: Illustration of the MIMO ZIFC, where one of the unintended channels is negligible due to the positioning of the terminals.

[197], in theK-user IFC with practical coding and modulation schemes [198], in cognitive radio systems [199–205], and in heterogeneous cellular networks [36].

All these examples show that reduced-entropy transmission can bring gains in the complex K-user multicarrier MIMO IFC and in special cases thereof, as well as in other interference networks. However, the overview also makes clear that there is a large variety of interference networks that can be studied and, accordingly, a large variety of open research questions, especially in MIMO settings with multiple antennas at all transmitters and receivers.