12. CONSIDERACIONES Y RECOMENDACIONES DE POLITICA
12.6. Recomendaciones para la licitación de la banda de 700 MHz
The major part of the computations involved in the proposed robust digital precoding algorithm, given in Alg. 8, is contributed from the optimization problem (6.6), which needs to be solved in every iteration. In this section, we develop a low-complexity scheme that exploits structures in problem (6.6) and Alg. 8 to compute the robust precoders efficiently. The developed scheme can also solve problem (6.6) in a distributed manner in order to reap the benefits of any available parallel hardware, and accordingly further speed-up the algorithm.
In the following, firstly, we transform the complex-valued problem (6.6) into an equivalent real-valued problem. Next, for the reformulated problem we derive a dual problem. Subsequently, the dual problem is solved iteratively, using a similar procedure
90 Chapter 6: Interference Exploitation-Based Robust Hybrid Precoding
employed in [YP17, HYSP16, YPCO19, YPCO18], to obtain the optimal solution of the primal problem by performing the following steps: first, an approximate problem is constructed for the dual problem that delivers a descent-direction of the dual problem at a given point. Secondly, the approximate problem is decomposed into multiple independent subproblems, which can be solved in parallel. Afterward, a closed-form expression is derived for the optimal solutions of the subproblems. Finally, we derive a closed-form expression to compute the step-size, which is required to update the current point in the descent-direction.
Let Fc2r(X)be a function that transforms a complex matrix X to a real matrix Y
such that Y = Fc2r(X) , Re(X), −Im(X) Im(X), Re(X) . (6.16)
Moreover, let fc2r(x) be a function that transforms a complex vector x into a real
vector y as
y = fc2r(x) ,
Re(x)T,Im(x)TT
. (6.17)
Let M0 , Fc2r( ˆA) stand for the nominal analog precoding matrix in the real domain.
We define the sets of possible corrupted analog precoding matrices ˆA Eof problem (6.6) in the real domain as below (note: the iteration index i is omitted for notational convenience).
M+k , {M = Fc2r( ˆA E) | E ∈ E+k}, ∀k ∈ K, (6.18)
M−k , {M = Fc2r( ˆA E) | E ∈ E−k}, ∀k ∈ K. (6.19)
Furthermore, we define the following:
g , fc2r(b), (6.20) fk , fc2r(¯hk), (6.21) Π1 , I, 0 0, −I , Π2 , 0, I I, 0 , (6.22) pk , Π2fk, qk, Π1fktan θ, (6.23) rk , γktan θ. (6.24)
6.2 Robust Hybrid Precoding Against Phase Errors 91
In Eq. (6.22), I and 0 are N × N identity and zero matrices respectively. Now, we can reformulate problem (6.6) in the real domain as
minimize g ||M0g|| 2 (6.25a) subject to (+pk− qk) T Mg + rk ≤ 0, ∀M ∈ M+k, ∀k ∈ K, (6.25b) (−pk− qk)TMg + rk ≤ 0, ∀M ∈ M−k, ∀k ∈ K. (6.25c)
The Lagrangian function of the above problem can be written as
L (g, λ) = ||M0g||2− (Ψλ)Tg + rTλ, (6.26)
where λ denotes the vector of Lagrange multipliers. The vector r in the above problem is given by r ,r111×L + 1, . . . , r K11×L + K, r 111×L − 1, . . . , r K11×L − KT, (6.27) where L+ k , #{M + k} and L − k , #{M −
k} represent the total number of elements in
sets M+
k and M −
k respectively. The matrix Ψ in the above problem is given by Ψ ,h(M+1,1)T(q1− p1), . . . , (M+ 1,L+1) T(q 1− p1), . . . , (M+K,1) T(q K− pK), . . . , (M+ K,L+K) T(q K− pK), (M−1,1)T(q1+ p1), . . . , (M− 1,L−1) T(q 1+ p1), . . . , (M−K,1) T(q K+ pK), . . . , (M− K,L−K) T(q K+ pK) i , (6.28) where M+
k,m is the m-th element of set M +
k, and M −
k,m is the m-th element of set
M−k. Taking the infimum of the Lagrangian function L (g, λ) w.r.t. g, we obtain the dual function in terms of λ. Subsequently, we formulate a dual problem to the primal problem (6.25) as minimize λ ||Nλ|| 2 − rTλ (6.29a) subject to λ ≥ 0, (6.29b)
where the matrix N is given by
N , (M
† 0)TΨ
2 . (6.30)
Note that problem (6.25) is convex and it comprises only affine inequalities in g. There- fore, according to the Slater’s condition strong duality holds for this problem when it
92 Chapter 6: Interference Exploitation-Based Robust Hybrid Precoding
is feasible [BV04]. Moreover, one of the KKT conditions dictates that the Lagrangian function (6.26) has a vanishing gradient w.r.t. g at an optimal primal point g? and an
optimal dual point λ? [BV04]. By setting ∂L (g?,λ?)
∂g = 0, we obtain the expression for
an optimal primal point g? of problem (6.25) in terms of the corresponding optimal
dual point λ? as g? = (M T 0M0)−1 2 Ψλ ? . (6.31)
In the following, we design an iterative algorithm to solve the dual problem (6.29) optimally.
• Approximate Problem: Let W be the total number of elements in vector λ and W , {1, . . . , W }. In problem (6.29), the objective function is convex in each optimization variable λw for w ∈ W. Based on the Jacobi theorem
[PC06, SFS+14] we construct an approximate problem for the original problem
(6.29) in the p-th iteration around a given point λp as
minimize {λw}w∈W W X w=1 ||N−wλp−w+ nwλw||2− rT−wλ p −w− rwλw (6.32a) subject to λw ≥ 0, ∀w ∈ W, (6.32b)
where N−w denotes the matrix obtained by discarding the w-th column nw from
matrix N, i.e.,
N−w , [n1, . . . , nw−1, nw+1, . . . , nW]. (6.33)
Similarly, λp
−w indicates the vector obtained by discarding the w-th element from
vector λp, and r
−w stands for the vector obtained by eliminating the w-th element
rw from vector r. Let ˆλ , [ˆλ1, . . . , ˆλW]T denote the optimal solution of the
above problem. According to the Jacobi theorem, ˆλ − λp represents a descent-
direction of the objective function (6.29a) in the domain of problem (6.29) [PC06]. Consequently, the current point λp can be updated to a new point λp+1 in the
descent-direction of the objective function (6.29a) as
6.2 Robust Hybrid Precoding Against Phase Errors 93
where ηp is an appropriate step-size, with 0 < ηp ≤ 1. When ˆλ = λp, the iterative
algorithm has converged to the global optimal solution λ? of problem (6.29).
• Decomposition of the Approximate Problem: The objective function (6.32a) comprises W summands, where each summand contains only one op- timization variable λw. Moreover, the constraint set in (6.32b) is a Cartesian
product of W convex sets, with each convex set defined by only one optimiza- tion variable λw. Thus, we can decompose problem (6.32) into W independent
subproblems [YP17], each containing only one optimization variable λw, as
ˆ
λw = argmin λw≥0
||N−wλp−w+ nwλw||2− rwλw, (6.35)
∀w ∈ W. Note: In the objective function of the above problem, the constant term rT
−wλ p
−w has been omitted, owing to that it does not have any influence on
the optimal solution of the problem.
• Closed-Form Solution of the Subproblem: The objective function in sub- problem (6.35) is convex in the optimization variable λw, and it comprises only
an affine inequality, namely, λw ≥ 0. According to the Slater’s condition, the
strong duality holds for the subproblem and its dual, and KKT conditions are satisfied by the primal and dual optimal points of the subproblem [BV04]. The Lagrangian of subproblem (6.35) can be written as
L (λw, µw) = ||N−wλp−w+ nwλw|| 2
− rwλw− µwλw, (6.36)
where µw is the Lagrange multiplier. Using the KKT conditions, we derive a
closed-form expression for ˆλw as
ˆ λw = max 0, 1 ||nw||2 rw 2 − n T wN−wλ p −w . (6.37)
• Optimal Step-Size Computation: Based on the exact line search method [YP17], we can formulate an optimization problem to compute the optimal step- size ηp that minimizes the objective function (6.29a) between the current point
λp and the descent-direction ˆλ as ηp = argmin 0≤η≤1 N λp+ η( ˆλ − λp) 2 − rTλp + η( ˆλ − λp) | {z } ˚ f (η) . (6.38)
94 Chapter 6: Interference Exploitation-Based Robust Hybrid Precoding
The function ˚f (η)in the above problem is convex and differentiable in η. Differ- entiating ˚f (η)w.r.t. η and equating the gradient to zero, we obtain a closed-form expression for the optimal solution ηp of problem (6.38) as
ηp = −2(Nλp)TN( ˆλ − λp) + rT( ˆλ − λp) 2 N( ˆλ − λp) T N( ˆλ − λp) 1 0 . (6.39)
• Termination: When ˆλ = λp, the iterative algorithm has converged to the
global optimal solution of problem (6.29) [YP17]. In practical applications where a finite numerical precision is sufficient, the iterations can be terminated when ||λp+1− λp|| ≤ ε, where ε is a small positive scalar that controls the numerical precision of the algorithm.
The above-proposed scheme to solve problem (6.6) is summarized in Alg. 9. Algorithm 9: Low-complexity parallel implementation scheme
1: input: ˆA, θ, ¯hk, γk, E+k, E −
k, ∀k ∈ K
2: initialization: p ← 1, λ1 ← any non-negative values
3: loop
4: λ ←ˆ using Eq. (6.37) [each element in ˆλ can be computed independently in parallel] 5: ηp ←using Eq. (6.39) 6: λp+1 ←using Eq. (6.34) 7: break, if ||λp+1− λp|| ≤ ε 8: p ← p + 1 9: end loop 10: g? ← using Eq. (6.31)
11: b? ← using the relation in Eq. (6.20) 12: return: b?