12. CONSIDERACIONES Y RECOMENDACIONES DE POLITICA
12.5. Consideraciones para las licitaciones de las bandas de 1.7/2.1 GHz y 700 MHz
Based on problem (6.3), we can formulate an optimization problem for computing the robust digital precoders for a given analog precoding matrix ˆA as
minimize b ˆ Ab 2 (6.5a) subject to + Im ¯hTk( ˆA E)b≤Re ¯hTk( ˆA E)b− γk
tan θ, ∀E ∈ E, ∀k ∈ K, (6.5b) − Im ¯hTk( ˆA E)b≤Re ¯hTk( ˆA E)b− γk tan θ, ∀E ∈ E, ∀k ∈ K, (6.5c) which is a semi-infinite convex problem. In the above problem, the constraints in (6.5b) enforce the received signal to lie below the anti-clockwise boundary and the constraints in (6.5c) enforce the received signal to lie above the clockwise boundary of the corre- sponding CI-region at each user and ∀E ∈ E (see Figure 6.2). The optimal solution b?
of the above problem represents the optimal worst-case robust digital precoder.
Re
Im
Anti-clockwise decision boundary Anti-clockwise boundary of CI-region CI-region
Clockwise boundary of CI-region Ei+k
Ei−k
Figure 6.2. Clockwise and anti-clockwise boundaries of the CI-region of a rotated symbol, along with the clockwise and anti-clockwise decision boundaries.
The objective function in problem (6.5) is a quadratic function in optimization vector b, and the constraints in (6.5b) and (6.5c) are linear constraints. Nonetheless, there are infinite constraints in this problem, and hence, it cannot be solved using standard convex optimization algorithms such as the interior-point method. Assuming that problem (6.5) is feasible, in the following, we develop an iterative algorithm to
86 Chapter 6: Interference Exploitation-Based Robust Hybrid Precoding
solve it optimally based on the cutting plane method and alternating procedure [GK73, WFL98]. By exploiting a structure in the problem—namely, the elements of phase error matrix E ∈ E have constant magnitudes—the devised iterative algorithm solves the above semi-infinite program efficiently.
We initialize the algorithm (iteration number i = 1) with sets Ei+
k = {1} and
Ei−
k = {1}, ∀k ∈ K, where 1 is an N×R matrix with all elements equal to 1. The
proposed algorithm comprises two stages in each iteration. In the first stage of the i-th iteration, we solve the following convex quadratic problem, which corresponds to the non-robust precoding problem in the first iteration.
minimize bi ˆ Abi 2 (6.6a) subject to + Im ¯hTk( ˆA E)bi ≤Re ¯hTk( ˆA E)bi − γk tan θ, ∀E ∈ Ei+ k , ∀k ∈ K, (6.6b) − Im ¯hTk( ˆA E)bi≤Re ¯hTk( ˆA E)bi− γk tan θ, ∀E ∈ Ei−k , ∀k ∈ K. (6.6c) In the subsequent iterations, this problem comprises a finite subset of constraints of problem (6.5): the constraint (6.6b) for every error matrix E ∈ Ei+
k ; the constraint
(6.6c) for every error matrix E ∈ Ei−
k , ∀k ∈ K. Problem (6.6) can be readily solved opti-
mally using standard convex optimization algorithms, such as the interior-point method [BV04, BDP12] or commercial solvers such as SDPT3 [TTT99]. In Section 6.2.2, we develop a customized scheme, which exploits special structures in the above problem, and solves it more efficiently. Let bi? denote the optimal solution of problem (6.6) in
the i-th iteration.
In the second stage of the i-th iteration, we compute the worst-case error matrices of constraints (6.5b) and (6.5c) at b = bi?, ∀k ∈ K. The worst-case error matrix Ei+
k of
constraint (6.5b) is defined as an error matrix E ∈ E that violates the constraint (6.5b) with the largest margin, or fulfills it with the smallest margin when the constraint is satisfied ∀E ∈ E, for the k-th user at b = bi?. Equivalently, the error matrix Ei+
k ∈ E
6.2 Robust Hybrid Precoding Against Phase Errors 87
the anti-clockwise direction (see Figure 6.2), when the digital precoder is set to bi?.
Similarly, the worst-case error matrix of constraint (6.5c) for the k-th user, denoted as Ei−k , drives the received signal yk the farthest away from the corresponding CI-region
in the clockwise direction. The closed-form expressions to compute Ei+
k and E i− k are
presented below. Now, if Ei+
k violates the constraint (6.5b), then it is added to the
corresponding set of error matrices, i.e.,
E(i+1)+k = Ei+k ∪ Ei+
k . (6.7)
Similarly, if the error matrix Ei−
k violates the constraint (6.5c), then it is included in
set E(i+1)− k , i.e.,
E(i+1)−k = Ei−k ∪ Ei−
k . (6.8)
When both Ei+
k and E i−
k , ∀k ∈ K, satisfy the constraints (6.5b) and (6.5c) respectively,
we conclude that the solution of problem (6.6) is the global optimal solution of problem (6.5), and accordingly terminate the algorithm.
Optionally, in order to reduce the number of constraints of problem (6.6) in the subsequent iteration, the redundant constraints can be excluded [WFL98]. To this end, we identify the error matrices E ∈ Ei+
k that result in strict inequality of the
corresponding constraint in (6.6b) for the given digital precoder bi? and discard them
from the set Ei+
k . Similarly, the error matrices E ∈ E i−
k that cause strict inequality of
the corresponding constraints in (6.6c) are excluded from the set Ei− k .
Closed-Form Expressions for the Worst-Case Error Matrices: The worst- case error matrices, Ei+
k and E i−
k for k ∈ K, of constraints (6.5b) and (6.5c) for a given
digital precoder bi? can be obtained by solving the following problems, respectively.
Ei+k = argmax |enr|=1,|∠enr|≤δ + Im ¯hTk( ˆA E)bi?−Re ¯hTk( ˆA E)bi?− γk tan θ. (6.9) Ei−k = argmax |enr|=1,|∠enr|≤δ − Im ¯hTk( ˆA E)bi?−Re ¯hTk( ˆA E)bi?− γk tan θ. (6.10)
88 Chapter 6: Interference Exploitation-Based Robust Hybrid Precoding
The above problems are nonconvex problems due to the nonconvex domain of opti- mization variables enr, ∀n ∈ N , ∀r ∈ R. We exploit the constant magnitude property
of the optimization variables and derive closed-form expressions to the worst-case error matrices (see Appendix C.1), which are given by
Ei+k = U++ jW+, (6.11)
Ei−k = U−+ jW−, (6.12)
where elements of the above matrices are computed as u+nr = max
cos δ, Im(znr) cos θ − Re(znr) sin θ |znr|
, (6.13a)
w+nr = Re(znr) + Im(znr) tan θ |Re(znr) + Im(znr) tan θ|
p
1 − (u+
nr)2, (6.13b)
u−nr = max
cos δ, − Im(znr) cos θ − Re(znr) sin θ |znr|
, (6.13c)
w−nr = − Re(znr) + Im(znr) tan θ |− Re(znr) + Im(znr) tan θ|
p 1 − (u− nr)2, (6.13d) with Z , ¯hk(bi?) T ˆA.
Note: The indices k and i are omitted from the matrices U+, W+, U−, W−, and Z for
notational simplicity.
Substituting the optimal solutions Ei+
k and E i−
k in the objective functions of prob-
lems (6.9) and (6.10), we obtain the corresponding optimal values vi+ k and v i− k respec- tively, i.e., vki+ = + Im ¯hTk( ˆA Ei+k )bi?−Re ¯hTk( ˆA Ei+k )bi?− γk tan θ, (6.14) vki− = − Im ¯hTk( ˆA Ei−k )bi?−Re ¯hTk( ˆA Ei−k )bi?− γk
tan θ. (6.15) A non-positive vi+
k implies that bi? satisfies the constraint (6.5b) for the k-th user
∀E ∈ E. On the other hand, a positive value for vi+
k implies that the constraint (6.5b)
is violated at b = bi? for the error matrix Ei+
k . Similarly, a positive v i−
k means the
constraint (6.5c) is violated at b = bi? for the error matrix Ei−
k , for the k-th user.
The above algorithm to design the worst-case robust digital precoding is summa- rized in Alg. 8.
6.2 Robust Hybrid Precoding Against Phase Errors 89
Algorithm 8: Optimal robust digital precoding
1: input: ˆA, θ, δ, ¯hk, γk, ∀k ∈ K
2: initialization: i ← 1, Ei+k ← {1}, Ei−k ← {1}, ∀k ∈ K
3: loop
4: compute bi? by solving problem (6.6) [e.g., using the proposed scheme in Section 6.2.2]
5: compute Ei+k and Ei−k using Eq. (6.11) and Eq. (6.12) respectively, ∀k ∈ K 6: compute vki+ and vki− using Eq. (6.14) and Eq. (6.15) respectively, ∀k ∈ K 7: if vi+k > 0, then execute Eq. (6.7); if vi−k > 0, then execute Eq. (6.8), ∀k ∈ K 8: break, if both vki+ and vki− are non-positive ∀k ∈ K
9: i ← i + 1 10: end loop
11: compute d?k, ∀k ∈ K from bi? using Eq. (6.4)
12: return: d?k, ∀k ∈ K
Theorem 1: When Alg. 8 terminates after an I-th iteration, the optimal solution
bI? of problem (6.6) is equal to the optimal solution b? of problem (6.5).
Proof: see Appendix D.1.
Theorem 2: The sequence, b1?, b2?, . . . , of optimal solutions of problem (6.6)
generated by Alg. 8 converges to the optimal solution b? of problem (6.5).
Proof: see Appendix D.2.