consuming solution than the one obtained previously in chapter 2, since the addition of any new constraint to an optimization problem may only result in the degradation of the solution’s performance. Through comparison to the problem in chapter 2, having power constraints on some antennas results in a power transfer from the constrained antennas to the unconstrained ones, in such a way that the requested rates remain satisfied for each user. Hence, minimizing the total transmit power of the system under the user rate and antenna power limit constraints translates into searching for the best “power transfer”
scheme that minimizes the excess in power compared to the unconstrained DAS solutions in chapter 2. Note that the number of constrained antennas 𝐹 shall not reach 𝑅, that is to say that at least one antenna has to remain unconstrained in order to guarantee the satisfaction of the requested rate for all users. The global optimization problem of user-subcarrier-RRH assignment and PA, taking into account the rate requirements, power limits, and NOMA PMCs, can be formulated as:
OP1: {S𝑘, 𝑃𝑘 ,𝑛,𝑟}∗ =arg min As in the previous chapter, the problem at hand involves set selection as well as continuous variable optimization, hence its mixed-integer non-convex nature justifies the introduction of suboptimal schemes. Therefore, we follow the same approach for tackling the RA problem by first deriving the optimal PA for power minimization in the OMA context and for a fixed SA (section 3.3). Then the optimal PA properties lead to the elaboration of RA schemes for OMA (section 3.4) and NOMA (section 3.5).
3.3 Optimal Power Allocation for OMA HDAS
In the orthogonal scenario, every subcarrier𝑛 is allocated to one user and one antenna at most, referred to as 𝑘(𝑛) and 𝑟(𝑛) respectively. The optimal PA scheme for a predefined subcarrier allocation scheme is cast as the solution to the following problem:
OP2:{𝑃𝑘 ,𝑛,𝑟}∗ = min
Chapter 3. NOMA Mutual SIC for Power Minimization in Hybrid Distributed Antenna The problem in hand can be solved by means of standard convex optimization techniques, its Lagrangian is given by:
where 𝜆𝑘 and 𝛼𝑖 represent the Lagrangian multipliers relative to the rate and power constraints respectively. The corresponding KKT conditions are:
∇𝐿(𝑃∗
The expressions of the partial derivatives of 𝐿 with respect to the power variables change according to whether the powering RRH is constrained or not. For subcarriers powered by a constrained antenna, we get:
𝜕 𝐿
The subcarriers that are not powered by a constrained antenna do not feature an 𝛼𝑖 term as in (3.10). Instead, their partial derivative yields:
𝑚𝑘
𝜎2
ℎ𝑘 , 𝑛,𝑟(𝑛) +𝑃𝑘 ,𝑛,𝑟(𝑛)
=1. (3.11)
3.3. Optimal Power Allocation for OMA HDAS 57
Equations (3.10) and (3.11) can be rearranged in the following form:
𝑃𝑘 ,𝑛,𝑟(𝑛) = 𝑚𝑘 antenna. If OP1 was considered without the power constraints (3.8), the solution of the PA problem would revert to the classical case of power minimization with user-specific rate constraints, resulting in a user-specific waterfilling where the waterline 𝑚𝑘 of user 𝑘 would be the same for all of its subcarriers. However, in HDAS, the waterline becomes specific to the separate classes of subcarriers, grouped according to the transmitting an-tenna. More specifically, equations (3.12) and (3.13) show that, for every user 𝑘, a specific waterlevel 𝑚𝑘 𝑖 ,𝑚𝑘/(1+𝛼𝑖) is assigned for every subset of subcarriers allocated to 𝑘 on the constrained antenna𝑅 𝐿𝑖, whereas the remaining subcarriers of 𝑘 that are not powered by a constrained RRH share a common waterlevel 𝑚𝑘. Furthermore, all the waterlevels of user 𝑘, corresponding to its powering RRHs, are related to 𝑚𝑘 by the factors (1+𝛼𝑖). Replacing the power variables by their expressions from (3.12) and (3.13) in the rate con-straints (3.7), and applying some manipulations yield the following forms, for each user 𝑘: Therefore, 𝑚𝑘 can be written as:
𝑚𝑘 =©
Recall from (2.10) that 𝑊𝑘 is the common waterline that user 𝑘 would have had in a
“classical” waterfilling scheme, i.e. if the power constraints on RL were not taken into account (𝑚𝑘 = 𝑊𝑘 when 𝛼𝑖 = 0,∀𝑖). Therefore, 𝛼𝑖 can be seen as the power correction factors that are applied to the unconstrained waterfilling solution to obtain the new power
Chapter 3. NOMA Mutual SIC for Power Minimization in Hybrid Distributed Antenna
Systems 58
pouring solution in (3.12) and (3.13). Note that if user 𝑘 has all its subcarriers powered by non-constrained antennas (T𝑘 𝑖 = ∅, 1 ≤ 𝑖 ≤ 𝐹), then the allocated power to the user subcarriers is according to (3.13). The user will have a unique waterline 𝑚𝑘 for all of its subcarriers, and 𝑚𝑘 = 𝑊𝑘 since |T𝑘 𝑖| = 0,1 ≤ 𝑖 ≤ 𝐹. Such a user is not affected by the power correction. Also, if a unique constrained RRH 𝑅 𝐿𝑖 is exclusively serving a user (T𝑘 𝑖 = S𝑘, and ∀𝑗 ≠ 𝑖,T𝑘 𝑖 = ∅), the power of all of its subcarriers will be accord-ing to (3.12), meanaccord-ing that a unique waterline is assigned to the user and it is given by 𝑚𝑘/(1+𝛼𝑖). But we have from (3.14)𝑚𝑘 =𝑊𝑘(1+𝛼𝑖), therefore the waterline of the user is equal to𝑊𝑘 and it is not affected by the power correction. At last, another possibility for having unique waterlines (per user) resides in a system where the classical waterfilling solution abides by (3.8). Indeed, if all the 𝛼𝑖 variables were null, the resulting Lagrangian would not account for the power constraints (3.8), hence the solution would be a simple user-based waterfilling: if 𝛼𝑖 = 0, 𝑚𝑘 = (1+𝑚𝛼𝑘
𝑖) =𝑊𝑘 which results in a uniform waterlevel over all the subcarriers of the user 𝑘.
As a consequence, if an initial RA technique verifies the constraints on the antennas as-suming a user-based waterfilling, this is indeed the best PA solution for the given SA. On the other hand, the proposed optimal PA technique can be applied in association with any SA scheme that constitutes a solution to problem OP1. Solving the power optimiza-tion problem reduces to determining the 𝐹 Lagrangian variables 𝛼𝑖 relative to the power constraints. By replacing (3.14) into (3.12), the power constraint (3.8), corresponding to 𝛼𝑖 ≠0 in the KKT conditions, for the 𝑖𝑡 ℎ antenna 𝑅 𝐿𝑖 in RL, can be written as:
Õ
𝑛∈𝑆(𝑅 𝐿𝑖)
𝑊𝑘(𝑛)Î𝐹
𝑗=1(1+𝛼𝑗)
|T𝑘(𝑛)𝑖|
|S𝑘(𝑛)|
(1+𝛼𝑖) − 𝜎2 ℎ𝑘(𝑛),𝑛,𝑟(𝑛)
!
= 𝑃𝑚
𝑖. (3.15)
Equation (3.15) consists of 𝐹 non-linear equations with unknowns 𝛼𝑖. In the sequel, the case of a single power-limited antenna (𝐹 =1) is considered first in order to provide a clear analysis of the hybrid system behavior. Then, the generalized study for higher values of 𝐹 is developed.
3.3.1 Single Power-Limited Antenna
For the special case of a single power-limited antenna, we simply denote by 𝑅 𝐿 the consid-ered RRH and 𝛼 the Lagrangian variable relative to the corresponding power constraint.
For each user, we can identify at most two sets of subcarriers and thus two waterlevels which are related by the factor (1+𝛼). The waterlevel of the subcarrier set that is not powered by the constrained antenna 𝑅 𝐿 is obtained from (3.14) as:
𝑚𝑘 = (1+𝛼)
|T𝑘|
|S𝑘|𝑊𝑘. (3.16)
This equation shows how the introduction of the constraint on one of the antennas affects the PA scheme, compared to the non-constrained case: since |T𝑘|/|S𝑘| ≤ 1, and𝛼 >0, the waterline of the subcarriers in T𝑘 decreases with respect to 𝑊𝑘 (since 𝑚𝑘/(1+𝛼) < 𝑊𝑘), while that of the subcarriers in S𝑘 ∩ T𝑘 increases (since 𝑚𝑘 > 𝑊𝑘). This behavior is depicted in Fig. 3.2.
3.3. Optimal Power Allocation for OMA HDAS 59
Figure 3.2 – Power pouring diagram for a user 𝑘 after power correction.
When the PA solution of the unconstrained problem does not respect (3.8), then a power correction using (3.12) and (3.13) is necessary, and the rate transfer from the con-strained antenna to the unconcon-strained ones translates into an unbalanced power transfer from antenna 𝑅 𝐿 to the other antennas. This gives further justification to why 𝛼 can be seen as the deviation factor from the unconstrained problem. A greater value of 𝛼 translates into a greater deterioration of the performance of the solution towards that of the unconstrained problem, meaning a more important increase of the total power in HDAS compared to DAS.
For 𝐹 = 1, the system of equations in (3.15) reduces to a single equation with a unique unknown 𝛼:
Õ
𝑛∈S (𝑅 𝐿)
𝑊𝑘(𝑛)(1+𝛼)
|T𝑘(𝑛)|
|S𝑘(𝑛)|−1− 𝜎2 ℎ𝑘(𝑛),𝑛,𝑟(𝑛)
!
=𝑃𝑚. (3.17)
There is no a priori guarantee for the existence of a solution to (3.17). An example of a situation with no solution is when every user served by 𝑅 𝐿 is exclusively linked to 𝑅 𝐿. Keeping in mind that such users are not affected by the power correction, if their total power consumption is greater than 𝑃𝑚, then no PA could, at the same time, verify the antenna power constraint and provide the users the rates they are requesting. Therefore, it is of interest to assess the feasibility of a proposed SA before proceeding to the resolution of (3.17) through a numerical solver. By isolating the terms 𝛼 from the others, (3.17) takes the following form:
Õ
𝑛∈S (RL), T𝑘(𝑛)≠S𝑘(𝑛)
𝑊𝑘(𝑛)(1+𝛼)
|T𝑘(𝑛)|
|S𝑘(𝑛)|−1=𝑃𝑚 −Õ
𝑛∈S (𝑅 𝐿), T𝑘(𝑛)=S𝑘(𝑛)
𝑊𝑘(𝑛) + Õ
𝑛∈S (𝑅 𝐿)
𝜎2 ℎ𝑘(𝑛),𝑛,𝑟(𝑛)
,
𝐴(𝛼)=𝐶 .
(3.18)
The first sum is a function of 𝛼, hence the notation 𝐴(𝛼). It includes all the subcarriers in S (𝑅 𝐿) belonging to users that are served by at least one non-constrained antenna.
Chapter 3. NOMA Mutual SIC for Power Minimization in Hybrid Distributed Antenna
Systems 60
𝐶 is constant (for a fixed subcarrier allocation) and accounts for: the power limit, the waterlines relative to the subcarriers of the users exclusively served by 𝑅 𝐿, and finally, the inverse channel gains of all subcarriers powered by 𝑅 𝐿. In order to have a solution, 𝐶 needs to belong to the image of 𝐴 when𝛼 spans the positive real axis. However, 𝐴(𝛼) is a polynomial with negative, fractional exponents and positive coefficients. Thus, it is a strictly decreasing function of 𝛼, its co-domain is ]lim𝛼→inf 𝐴(𝛼)=0;𝐴(0)]. Therefore, the condition that guarantees the existence of a solution is: 0 < 𝐶 ≤ 𝐴(0).
Proposition 3.1. If the system requires a power correction, 𝐶 will be necessarily smaller than 𝐴(0).
Proof. The left hand side of (3.17) is the power on 𝑅 𝐿 for a given value of 𝛼. When 𝛼 = 0, and since the system requires correction, this power is the actual power of 𝑅 𝐿 before any power correction takes place. This value is greater than 𝑃𝑚, that is Í
𝑛∈S (𝑅 𝐿) 𝑊𝑘(𝑛) −𝜎2/ℎ𝑘(𝑛),𝑛,𝑟(𝑛)
≥ 𝑃𝑚. By setting 𝛼 to 0 in (3.18), we directly obtain 𝐴(0) −𝐶 ≥ 0, i.e. 𝐴(0) ≥𝐶 which concludes our proof.
As a result, the existence of a solution is only conditioned by𝐶 being strictly positive.
Finally, the uniqueness of the solution is an immediate result of the monotonic nature of function 𝐴.