Another strategy for HBF design is to maximize the minimal SINR among all users subject to a constraint on the sum of all users’ transmitted power. The design problem is formulated as
where P0 > 0 is the upper bound on the total sum power.
Unfortunately, (5.22) can not be formulated as a convex optimization problem. Nev-ertheless, it can be solved with an iterative algorithm in [21] that makes use of the connection between power minimization and SINR maximization. The iterative proce-dure is based on the following theorem quoted from [21].
Theorem 1. [21] The power minimization problem of (4.8) and the SINR maximization
problem of (5.22) are inverse problems, namely
γ0 = S(P(γ0)) (4.15)
P0 = P(S(P0)). (4.16)
Furthermore, the optimal objective value of each optimum problem is continuous, and is strictly and monotonically increasing with its input argument, i.e.
γ0 > ˜γ0 =⇒ P(γ0) > P( ˜γ0) (4.17) P0 > ˜P0 =⇒ S(P0) > S( ˜P0). (4.18)
The proof of this theorem can be found in [21]. Based on Theorem 1, S(P0) can be solved iteratively with the algorithm summarized below.
Algorithm 1
1: Initialize γmin = MinSINR and γmax = MaxSINR 2: repeat
3: γ0 ←− (γmin+ γmax)/2 4: Pb0 ←− P(γ0)
5: if bP0 ≤ P0
6: then γmin ←− γ0 7: else γmax ←− γ0
8: until bP0 = P0 9: return γ0 and wi
The MinSINR and MaxSINR must be adjusted such that bP0 = P0 exists with a feasible γ0 ∈ [MinSINR, MaxSINR].
Chapter 5
The practical implementation for hybrid beamforming
In practice, the phase shifter can only be adjusted to certain predefined values, and different phase shifter also couples an individual random offset to the predefined values.
Thus, the phase differences of the received signals in the desired direction may not be perfectly compensated with phase shifter. To guarantee the SINR of each user under this phase uncertainty, we consider two types of robust beamforming methods to combat the phase uncertainty.
5.1 The finite resolution of the phase shifters for HBF
This section discusses the finite resolution of the phase shifter of each element antenna for HBF. Since considering the practical circuit implementation of the phase shifter, it has the finite resolution of the phase shifters, means that the adjustable direction of the phases for each phase shifter are restricted into the limited kinds.
direction of φ = ±π/2, we define the finite resolution set of the elevation angle θ and azimuth angle φ for each phase shifter as follows
Θ : {θ ∈ Θ|θ = (n + 1/2)(π/2)/K, n = 0, . . . , K − 1}
Φ : {φ ∈ Φ|φ = m(2π)/L, m = 0, . . . , L − 1}
where K denotes the number of the finite resolution for the elevation angle, and L denotes the number of the finite resolution for the azimuth angle. According to the above definition of the phase, there are K × L types of the fixed maneuvered phase applied for each phase shifter. For the generality of discussion, we apply the N × M antenna arrays of the effective antenna for HBF, and the equivalent antennas of all users share the same phase shifter. Hence, only K × L types of the directions for the effective antenna of each user can be adjusted. Therefore, the RF array factor of the effective antenna for the i-th user can be presented as
Ai(φ, θ) =
N −1X
n=0 M −1X
m=0
ej[2n(Ψx+βxi)]ej[2m(Ψy+βyi)] (5.1)
where βxi , −kdxcos ϕisin ϑi and βyi , −kdysin ϕisin ϑi are the maneuvered phases of the i-th user, and (ϕi, ϑi) ∈ (Φ, Θ) are the direction of those maneuvered phases. The above array factor shows that the direction of the maneuvered phase for each element antenna are absolutely included in the (Φ, Θ) set.
In order to find the approximate pattern compared with the idea case, we consider a simple and straightforward concept to decide the maneuvered phase of each user. Since there are K × L possible types of the adjustable directions for each user, we can jointly design the phase of the RF beamforming and the weights of baseband beamforming based on the finite phase condition. Two methods are used to approach the maneuvered phase of each user, and the performance of each method will be shown in the section of the computer simulation. First, the straightforward method for this problem is the
exhausting search described as below.
• Exhausting search: It considers the total possible permutation of (ϕ, ϑ) within the (Φ, Θ) set to decide the maneuvered phase for each user with the best perfor-mance of HBF.
For this method, each user has the K × L possible types of the maneuvered phase, the possible permutation of the maneuvered phases for all users denote the K × L to the power of i. We will decide the maneuvered phase of each user with the minimum transmit power of the power minimization problem in the whole possible permutation.
It means that each discrete phase is considered to the power minimization problem to obtain the weights guaranteed the SINR constraints, thus we can choose the feasible phase with the lowest transmit power in comparison with that of others.
Therefore, the complexity of the computation can be expressed as (K × L)i. Ac-cording to the expression of the complexity with the exponential terms, the exhausting search may have the myriad simulation times. However, the exhausting search takes too numerous computation to apply for the practical implementation. In view of this huge computation, we propose another simplest method to simplify the process of the computation.
• Simplest method : It directly maps the maneuvered phase of each phase shifter into the nearest resolution of the maneuvered phase in comparison with the ideal case.
This method considers a simple and straightforward concept to simplify the exhaust-ing search. The maneuvered phase of each pase shifter is determined as the one with the most adjacent distance compared to the idea case, and the computation rule for distance
can be presented as the function below. idea maneuvered phase of the i-th user at the desired direction (φi, θi). The definition of the distance between the fixed adjustable phases and the idea phase can be defined as the sum of the distances between the phase at the direction (φ, θ) and that of the desired direction (φi, θi) for total phase shifters of the user i. Since only K × L types of the fixed maneuvered directions for each user can be chosen to steer the phase of the phase shifters, we decide the phase of each user with the minimum value of function fi(ϕ, ϑ) where (ϕ, ϑ) is in the (Φ, Θ) set, namely
(ϕi, ϑi) = arg min
(ϕ,ϑ)∈(Φ,Θ)fi(ϕ, ϑ). (5.3)
where (ϕi, ϑi) denotes the determinative direction of all phase shifters for the i-th user.