CC Transmission in the Complex Multicarrier MIMO BC-TIN

As a second example of reduced-entropy transmission, let us consider CC transmission in multicarrier systems.

7.4.2.1 Proper CN Transmission vs. Proper CC Transmission

In [1], it was shown that CN transmission not always achieves the complete rate region of the complex multicarrier MIMO BC-TIN with proper signaling, i.e., signals belonging to the power shaping spacesP_k⁰ = `PUk, ∀kandP<sub>B</sub><sup>0</sup> = `PB can be beneficial compared to staying in the power shaping spacesPk= `PU<sup>CN</sup><sub>k</sub> ⊂P`Uk, ∀kandPB= `PB<sup>CN</sup>⊂P`B. In [18], this result was extended to settings with zero-forcing constraints. Let us briefly summarize the example that was used in the proofs.

Example 7.4.1. Consider the complex three-user two-carrier MISO BC-TIN with proper noise andC

η^(c)_k = 1, ∀k,∀c. For allk, leth_k⁽¹⁾=hkandh_k⁽²⁾=h_k^∗withhkfrom(7.54).

Proper CN Signaling—Converse: According to [1], optimal proper CN transmission with TS can achieve the ratesρ_CN,k ≈1∀kunder a sum power constraint withQ= 3.5684. The optimal strategy consists of three equal-length time slots, where a different pair of two users is served over both carriers in each time slot. The optimality of this scheme can be verified by applying the method from Section 7.3.1.2 to the complex multicarrier formulation(3.34).

Proper CC Signaling—Achievability: In [1], a proper CC transmit strategy was constructed, which uses the combined complex transmit covariance matrices

Cxk = Q 6

"

1 e^jϕk e^−jϕk 1

#

(7.59) in the dual uplink. Withϕ1 = 0,ϕ2=−^5π₆ , andϕ3 = ^5π₆ , we can calculate by means of (7.14) withµ= 1that the ratesρ_CC,k ≈1.0986∀kare achievable using the same transmit power as before, i.e.,Q= 3.5684. From(7.23)withµ= 1, we obtain thatρ_ZF,CC,k ≈1.0648∀kis achievable with zero-forcing.

7.4.2.2 Comparison of Proper CC Transmission and Improper CN Transmission

We might wonder whether such a rate gain due to CC transmission is still possible in this scenario if improper signaling is allowed, but the answer is negative. If we apply the improper transmit strategy from the proof of Proposition 7.4.1 independently on each carrier using the per-carrier transmit power ^Q₂ = 1.7842, the per-carrier ratesρ^(c)_improper,k = 0.54928∀k,∀care achievable. Taking the sum over both carriers, we see that this improper CN scheme achieves exactly the same rate as the proper CC scheme from [1]. Similarly, we can show that the proper CC zero-forcing rates can instead be achieved with improper CN zero-forcing.

Does this mean that the two types of reduced-entropy transmission—improper signaling and CC transmission—can generally be substituted by each other? Obviously not, since CC

transmission is not a feasible approach in the single-carrier example from Proposition 7.4.1 or, e.g., in a multicarrier system where all carriers but one are negligibly weak. Thus, CC transmission is not a generally valid replacement of improper signaling. But what about the other direction? Improper signaling is always a feasible strategy in complex systems, and it is not obvious whether improper CN transmission can serve as a replacement of proper CC transmission in general.

7.4.2.3 Benefits of CC Transmission in Case of General Complex Signaling

When our aim is to demonstrate a gain by CC transmission that cannot be obtained by instead using improper CN transmission, the difficulty is that we need the globally optimal improper CN solution for comparison, but we are currently not aware of an algorithm to compute it. When optimizing such a strategy via the real-valued multicarrier formulation, we are always facing a true MIMO system withM_T >1for all terminalsTeven if the original complex system has single-antenna terminals. This means that we have to optimize covariance matrices, and we cannot switch to an optimization of scalar powers, as we do in Section 7.3.1.2. Therefore, we have to go a different path. To this end, let us first introduce the following lemma, which turns out to be helpful.

Lemma 7.4.1. In the RoP MIMO BC-TIN with a sum power constraint, letH_UL,k =H_UL,j for somej 6=k. Then, the rate region with (R)TS can be achieved without serving userskandj simultaneously in any time slot.

Proof. Consider a weighted sum rate maximization with weights(w_k)_∀k, where we assume w_k ≥w_j without loss of generality. For all other users, it does not matter how a given sum covariance matrixQ=C_xk+C_xj is split intoC_xkandC_xjsinceH_UL,k =H_UL,j. For any suchQ, we can use the raterQthat is achievable when decoding userskandjjointly (as in Section 5.1) as an upper bound, i.e.,

w_kr_k+wjrj ≤w_k(r_k+rj)≤w_krQ with rQ =µlog₂ detX

detX_k,j (7.60) whereX_k,j =X −H_UL,kQH_UL,k^H is the covariance of the noise plus the interference from the other usersk⁰ ∈ {/ k, j}. On the other hand,w_kr_k =w_krQ is obviously achievable with a TIN strategy by choosing C_xk = Q andC_xj = 0. By plugging inw_kr_k = w_kr_Q and wjrj = 0as optimal strategy in (3.19) and (3.24), we obtain the desired result for TS and for RTS, respectively.

We are now ready to demonstrate gains that cannot be obtained using improper signaling, but using CC transmission, i.e., reduced-entropy transmission with respect to the power shaping spacesPk=P_U^CN_k, ∀kandPB=P_B^CN.

Proposition 7.4.3. In the complex multicarrier MIMO BC-TIN with CN noise and a sum power constraint(3.4), CN per-user input signals do not always achieve the whole rate region. This statement holds for systems with and without (R)TS.

Proof. Consider the combined complex representation (3.35) of the complex three-user two-carrier MISO BC-TIN with noise covariance matricesCη_k =I₂, a sum power constraint with

7.4. Benefits of Reduced-Entropy Transmission 109

Q= 3, and the channel realization H₁^H=

CC Signaling—Achievability: Let the transmit signals in the dual uplink be proper with combined complex covariance matrices calculate that the ratesρ_CC,k ≈ 0.91083∀k are achievable. These rates are of course still achievable if (R)TS is allowed.

CN Signaling—Converse: To obtain an optimal solution with TS, we apply the dual decomposition from Section 3.3.2 to the real-valued multicarrier formulation (3.36). On each carrier, there are two users with exactly the same composite real per-carrier channel matrices, which means that no more than two users need to be considered simultaneously when solving (3.33) due to Lemma 7.4.1. We thus have to optimize a complex two-user MISO BC-TIN on each carrier in each time slot, so that Proposition 7.4.2 applies, i.e., improper signaling cannot bring any gains. Plugging in the assumption of proper signals, the complex multicarrier formulation (3.34) becomes a RoP CN MISO system, and we can optimize the transmit signals as described in Section 7.3.1.2. The obtained globally optimal proper CN rates areρ_CN,k ≈0.89117∀k.

This can be achieved with three equal-length time slots, where a pair of users is active on each carrier in each slot with per-user per-carrier transmit power ^Q₄. The rateρ_CN,kis then ⁴₃ times the rate that an active user achieves on a carrier in a slot, which can be calculated using (7.14) withµ= 1. Thus,ρ_CC,kfrom above is not achievable with CN transmission, and it of course stays unachievable if we consider only strategies without TS (where either RTS is used or (R)TS is avoided completely).

Note that the case of zero-forcing is not included in this proposition. However, this does not imply that improper signaling can always replace CC transmission if zero-forcing constraints are considered. It just means that we have not found a CC zero-forcing strategy that outperforms CN zero-forcing in this scenario. There might exist other examples in which gains that are only possible with CC transmission can be shown even though zero-forcing constraints are imposed.

This question is left open for future research.

7.4.2.4 Optimality of CN Transmission in Special Cases

We can also find examples for which CN transmission is provably optimal. A particularly interesting example is the following result from [15], which does not depend on a certain channel realization.

Example 7.4.2. Consider a complex two-user multicarrier MIMO BC-TIN with proper CN noise and proper signaling, and assume thatM ≥M1+M2, i.e., the number of base station antennas is at least as high as the total number of antennas at the user terminals. This case was called full multiplexingin [15, 150]. For this setting, CN transmission is optimal in the high-SNR regime, i.e., if the transmit power goes to infinity. This result was shown in [15] by applying the high-SNR rate balancing solution from [150] to the combined complex representation. The optimal combined complex transmit covariance matrices in the dual uplink then turn out to be block-diagonal, which corresponds to CN transmission.