Reduced-Entropy Transmission in the Complex Two-User Multicarrier

transmission can occur in a two-user BC-TIN. If (R)TS is not allowed, this question can be easily answered by means of the QoS feasibility results presented above.

Example 7.5.5. Consider the complex two-user single-carrier SISO BC with uplink chan-nels hUL,k = 1, ∀k. Due to Corollary 7.5.2, arbitrarily high rates are feasible since

P2 k=1d1

2(ρ_k)<1is satisfied for arbitraryρ_k >0, ∀k. However, if we restrict the users to apply proper signaling, the rates have to fulfillP2

k=11−2^−ρk <1sinceP2

k=1d1(ρ_k)<1is required in Corollary 7.5.4.

This is clearly a gain by reduced-entropy transmission in a two-user system without (R)TS, i.e., this example is sufficient to answer the question that was posed above. However, to get a better understanding of the effects that happen, it makes sense to also look at examples which not only show gains in terms of feasibility, but also for the case where QoS feasibility is not the limiting factor.

Example 7.5.6. Consider the system from Example 7.5.5. With proper signaling, the only freedom of choice that we have in the uplink is the power allocation. UsingCxk =p_k= ^√1

2 ∀k, we can achieve the ratesr_k= log₂(1 +_1+p^pk

j) = ¹₂ ∀k, j6=kdue to(7.14)withµ= 1. Using the framework from [179], it can be shown that the power allocation that is needed to achieve a particular rate vector in this setting is unique. Thus, the rate vectorρ= ¹₂1cannot be achieved with a sum transmit power smaller thanP2

k=1Cxk =√

2as long as we stick to proper signaling.

Using improper signaling with the combined real transmit covariance matricesCxˇ_k = ¹₂e_ke^T_k, we can calculate using(7.14)withµ= ¹₂ andH_UL,k = `H_UL,k =I₂, ∀kthat the same rate vectorρ= ¹₂1is achievable at a reduced sum transmit power ofPK

k=1tr[C_xˇk] = 1.

Note that we could apply zero-forcing receive filtersG^H_k,ZF = e^T_k without changing the achievable rates of the improper strategy. On the other hand, if we impose zero-forcing constraints for the case of proper signaling, only one user can be served at a time in this example.

This shows that gains by improper signaling occur as well if the same setting is considered under zero-forcing constraints.

The above example was constructed in a way that the (complex) degrees of freedomCm= 1 available at the base station are less than the number of users K = 2, i.e., the system is overloaded if we stick to proper signaling. The additional flexibility of improper signaling, which can allocate streams that correspond to half a degree of freedom of the complex system, can resolve this issue. However, the same result could be obtained by instead exploiting the flexibility offered by (R)TS, which we assume to not be allowed throughout this section.

Let us now turn our attention to systems where the degrees of freedom at the base station are not the limiting factor. An example for this can be found in [34] where gains by improper signaling were demonstrated in the complex two-user single-carrier MISO BC-TIN. However, just like in Example 7.5.6, these gains did not persist when allowing (R)TS.

As a further example, we consider improper signaling and CC transmission in a complex multicarrier setting with zero-forcing constraints.

Example 7.5.7. Consider the complex two-user two-carrier SISO BC with proper CN noise, where

H_k⁽¹⁾= 1, ∀k, H_k⁽²⁾= 0.1, ∀k, C

η^(c)_k = 1, ∀k, ∀c (7.104) i.e., both users have good channel conditions on the first carrier, but not on the second carrier.

Such a situation was described with the termspectrally similar channelsin [4,53]. When using proper CN signals, the setting can be studied in the complex multicarrier formulation, where

7.5. Further Aspects in Systems without (Rate-)Time-Sharing 125

the null space that is necessary to construct the zero-forcing filters vanishes as soon as the other user transmits. With zero-forcing constraints, we thus can only serve one user per carrier, i.e., we must haveC If we instead impose the zero-forcing constraints in the real-valued multicarrier formulation (3.36), we can allow for improper signaling. We have

H`<sub>k</sub><sup>(1),T</sup> =I<sub>2</sub>, ∀k, H`_k^(2),T= 0.1I₂, ∀k, C

ˇ η^(c)_k = 1

2I₂, ∀k, ∀c. (7.107) Using the transmit filtersB_k⁽¹⁾=

q3 This transmit strategy corresponds to improper CN transmission in the original complex system.

The relatively high transmit power in the case of proper CN transmission comes from the fact that only one user can be served via the strong carrier, so that a lot of transmit power is required to serve the second user via the weak carrier. Using improper signals, both users can share the strong carrier without violating the zero-forcing constraints. As pointed out in Section 7.5.2.2, this requires widely linear zero-forcing in the complex representation. An alternative method to avoid the competition for the strong carrier is using proper CC signals, as done in the following example from [4].

Example 7.5.8. The combined complex representation(3.35)of the system from Example 7.5.7 reads as the zero-forcing constraints R_k^HH_k^HBj = 0, j 6= k are fulfilled, and the achievable rates according to(7.12)withµ= 1are

using a sum transmit power of

QZF,proper,CC =

2

X

k=1

tr[BkB_k^H] = 206

3 < QZF,proper,CN. (7.112) In this transmit strategy, both users are served via a combination of the strong and the weak carrier, which leads to a moderate transmit power for both users. The resulting sum power is lower than in the case of proper CN transmission, where the users compete for the best carrier.

Note that an even lower transmit power can be achieved by optimizing the CC strategy as in [4]. Moreover, [4] established conditions on the channel coefficients that lead to the described competition for the strong carrier and, thus, to gains by CC transmission (as long as neither (R)TS nor improper signaling is used). Similar results were presented for systems with higher numbers of users, carriers, and antennas in [53]. In [18], the system from Example 7.5.8 was studied without zero-forcing constraints, and it was observed that proper CC transmission outperforms proper CN transmission in this case as well.

Note that we could also employ (R)TS in all these examples in order to avoid a competition for strong carriers. However, as long as (R)TS is not allowed, reduced-entropy transmission (in the form of CC transmission or improper signals) can apparently be necessary to achieve the whole rate region of a two-user BC-TIN. If (R)TS is allowed, it remains an open question whether we can find an example of a two-user BC-TIN where reduced-entropy transmission is beneficial.