7.5 Further Aspects in Systems without (Rate-)Time-Sharing
7.5.2 Feasibility of Zero-Forcing
In the presence of zero-forcing constraints, the question of feasibility becomes qualitatively different, but it turns out that the previously derived results are still helpful for the study of zero-forcing. In the RoP MISO BC-TIN, the number of users out of a groupKthat can be served simultaneously by a zero-forcing strategy is limited by the rank of the channel matrix HK(e.g., [124]). If this limitation is violated, the null space of the interference covariance matrix, which is necessary to construct the zero-forcing filters in (7.12) or (7.23), vanishes. We now formalize this for the RoP MIMO BC-TIN based of the previously stated feasibility results.
Theorem 7.5.3. In the RoP MIMO BC-TIN with zero-forcing constraints, the rate vector [ρ1, . . . , ρK]T ≥0is feasible without (R)TS if and only if
{k∈ K |ρk>0}
≤rank [HK] ∀K ⊆ {1, . . . , K}. (7.94) Proof. If a rate vectorρ= [ρ1, . . . , ρK]T ≥0is achievable with zero-forcing, then the rate vectorρ∞= limα→∞αρis also achievable if an infinite amount of transmit power is invested.
7.5. Further Aspects in Systems without (Rate-)Time-Sharing 121
The reason for this is that no interference is present and that the noise becomes negligible if the transmit power is sufficiently high. Conversely, the rate vectorρ∞is only achievable if zero-forcing is feasible because interference terms that grow together with the growing transmit powers prevent the SINRs from going to infinity (see the proof of [178, Theorem III.1]). Thus, ρis feasible with zero-forcing if and only if ρ∞lies in the QoS feasibility region given in Theorem 7.5.1, i.e., if and only if
X
k∈K
dZF,µ(ρk)≤µrank [HK] ∀K ⊆ {1, . . . , K} (7.95) with
dZF,µ(ρ) = lim
α→∞dµ(αρ) =
(µ ifρ >0
0 otherwise. (7.96)
The inequality in (7.95) is not strict since the bound in (7.68) can be reached with equality if the transmit powers tend to infinity (see [174,176]). This means that we only need to count the number of users with nonzero rate requirements, which is compactly written in (7.94).
Corollary 7.5.10. If the regularity condition is fulfilled in Theorem 7.5.3, the feasibility condition(7.94)reduces to
{k∈ {1, . . . , K} |ρk>0}
≤M. (7.97)
7.5.2.1 Feasibility of Zero-Forcing with Maximum-Entropy Signals
We are now interested in how this result changes if a restriction to maximum-entropy transmission is introduced.
Example 7.5.3. Let us first consider the effect observed in Example 7.5.2, where a restriction to the power shaping space of diagonal matrices was imposed. In this case, individual feasibility conditions(7.75)corresponding to each of the diagonal elements had to be considered instead of an overall feasibility condition(7.74). For unsuitable combinations of rate requirementsρk, it could then happen that there exists no choice of the transmit filters such that the individual conditions are satisfied, even though the overall criterion is fulfilled. However, such an effect is not possible for the case of zero-forcing because both sides of each inequality in(7.94)are integer multiples ofµ.
However, the situation is different when considering the effect observed in Example 7.5.1.
Example 7.5.4. If the restriction to a power shaping space requires the rank of the transmit covariance matrices to be a multiple ofβ >1, this means that at leastβstreams have to be transmitted ifρk>0. The other users have to eliminate the interference of these streams by means of zero-forcing filters. We obtain a sum EDoF requirement ofβdZF,µ(ρk) =µβfor each userkwithρk>0. This leads to the stricter conditionβ
{k∈ {1, . . . , K} |ρk>0} ≤M instead of (7.97)and reduces the set of rates that are feasible with zero-forcing.
Before turning our attention to the complex multicarrier MIMO BC-TIN, let us formalize this finding in a general theorem.
Theorem 7.5.4. The statement of Theorem 7.5.2 holds analogously if zero-forcing constraints are imposed.
Proof. Examples where this happens are given above and below.
7.5.2.2 Feasibility of Zero-Forcing in the Complex Multicarrier MIMO BC-TIN
From the proof of Theorem 7.5.3, we know an easy way to extend the results from Section 7.5.1.2 to the case of zero forcing constraints. We can simply replacedµby dZF,µ and <by≤in Corollaries 7.5.2 through 7.5.9. By doing so, we obtain the following results. Just like in Section 7.5.1.2, we consider only the case of proper CN noise, but results for other types of noise could be derived as well.
Corollary 7.5.11(see Corollary 7.5.2). Under the assumption of proper CN noise in the complex multicarrier MIMO BC-TIN with zero-forcing constraints, the rate vector[ρ1, . . . , ρK]T≥0is feasible without (R)TS if and only if
1 Corollary 7.5.12(see Corollary 7.5.3). If the regularity condition is fulfilled in Corollary 7.5.11, the feasibility condition(7.98)reduces to
{k∈ {1, . . . , K} |ρk>0}
≤2Cm. (7.99)
Under a restriction to proper signaling, we obtain the following stricter feasibility condition.
Corollary 7.5.13(see Corollary 7.5.4). Under the assumption of proper CN noise in the complex multicarrier MIMO BC-TIN with zero-forcing constraints, the rate vector[ρ1, . . . , ρK]T≥0is feasible using proper per-user input signals without (R)TS if and only if
Corollary 7.5.14(see Corollary 7.5.5). If the regularity condition is fulfilled in Corollary 7.5.13, the feasibility condition(7.100)reduces to
{k∈ {1, . . . , K} |ρk>0}
≤Cm. (7.101)
In comparison to proper signaling (Corollary 7.5.14), improper signaling doubles the number of users that can be served in the complex multicarrier MIMO BC-TIN under zero-forcing constraints (Corollary 7.5.12). This result was published in [5] and can be found in a similar manner in [173], where a study based on bit error rates was performed instead of based on achievable rates. Note that the number of users given for proper signaling in Corollary 7.5.14 can be achieved with linear zero-forcing (see Proposition 7.1.1) while widely linear zero-forcing is necessary to achieve any higher number [5,173].
Already in Section 7.5.1.2, we could notice a qualitative difference between the two kinds of reduced-entropy transmission that we consider, and we will see that this difference between CC transmission and improper signaling is even more pronounced in case of zero-forcing constraints.
Let us first note the following difference between QoS feasibility (see Example 7.5.2) and the feasibility of zero-forcing constraints.
Lemma 7.5.3. Under the assumption of proper CN noise in the complex multicarrier MIMO BC-TIN with zero-forcing constraints, the rate vector [ρ1, . . . , ρK]T ≥ 0 is feasible using (proper or general complex) CN per-user input signals without (R)TS if and only if it is feasible with single-stream transmission (of proper or real-valued streams, respectively).
7.5. Further Aspects in Systems without (Rate-)Time-Sharing 123
Proof. When adding another stream to reduce the rate requirementρk,1 of the first stream, this does not reduce the EDoF requirementdZF,µ(ρk,1).
Finding a feasible allocation of rate targets to carriers is still a combinatorial problem in the case of zero-forcing constraints, but due to Lemma 7.5.3, less possibilities than in the case of QoS feasibility (Corollaries 7.5.6 and 7.5.8) need to be considered.
Corollary 7.5.15(see Corollary 7.5.6 and Lemma 7.5.3). Under the assumption of proper CN noise in the complex multicarrier MIMO BC-TIN with zero-forcing constraints, the rate vector [ρ1, . . . , ρK]T≥0is feasible using CN per-user input signals without (R)TS if and only if we can find disjoint setsKc, ∀csuch thatSC
c=1Kc={k∈ {1, . . . , K} |ρk>0}and 1
2|K| ≤rankh HK(c)i
∀K ⊆ Kc, ∀c. (7.102) Corollary 7.5.16(see Corollary 7.5.8 and Lemma 7.5.3). Under the assumption of proper CN noise in the complex multicarrier MIMO BC-TIN with zero-forcing constraints, the rate vector [ρ1, . . . , ρK]T≥0is feasible using proper CN per-user input signals without (R)TS if and only if we can find disjoint setsKc, ∀csuch thatSC
c=1Kc={k∈ {1, . . . , K} |ρk>0}and
|K| ≤rankh HK(c)i
∀K ⊆ Kc, ∀c. (7.103) However, the combinatorial nature of the problem vanishes in case of regular channels, where we have the following result.
Proposition 7.5.1. Under the assumption of regular channels and proper CN noise in the complex multicarrier MIMO BC-TIN with zero-forcing constraints without (R)TS, a restriction to CN transmission does not impair the feasibility of a rate vectorρ.
Proof. In this case, the setsKccan be chosen arbitrarily as long as|Kc| ≤2min Corollary 7.5.15 and as long as|Kc| ≤min Corollary 7.5.16. The resulting feasibility conditions are equivalent to the ones in Corollary 7.5.11 and Corollary 7.5.13, respectively.
While improper signaling has the potential to double the number of users that can be served (Corollary 7.5.14 vs. Corollary 7.5.12) in scenarios with regular channels, CC transmission does not bring any gains in terms of the feasibility of zero-forcing constraints (Proposition 7.5.1).
From the reasoning in the proof of Theorem 7.5.3 (see also [178, Theorem III.1]), we can infer that these observations are also valid for QoS feasibility in systems without zero-forcing constraints that operate in the high-rate regime. This statement is supported by Figure 7.3, which shows that the EDoF requirementsdµ(ρ)converge quickly todZF,µ(ρ).
7.5.3 Reduced-Entropy Transmission in the Complex Two-User Multicarrier MIMO BC-TIN