Interference Exploitation Precoding Made Practical: Optimal Closed-Form Solutions for PSK Modulations

Abstract

In this paper, we propose closed-form precoding schemes with optimal performance for constructive interference (CI) exploitation in the multiuser multiple-input single-output (MU-MISO) downlink. We first consider an optimization where we maximize the distance between the constructive region and the detection thresholds. The cases of both strict and non-strict phase rotation are considered and can further be formulated as convex optimization problems. For optimization with strict phase rotation, we mathematically derive the optimal beamforming structure with Lagrangian and Karush-Kuhn-Tucker (KKT) conditions. By formulating its dual problem, the optimization problem is further shown to be equivalent to a quadratic programming (QP) over a simplex, which can be solved more efficiently. We then extend our analysis to the case of non-strict phase rotation, where it is mathematically shown that a Kdimensional optimization for non-strict phase rotation is equivalent to a 2K-dimensional optimization for strict phase rotation in terms of the problem formulation. The connection with the conventional zero-forcing (ZF) precoding is also discussed. Based on the above analysis, we further propose an iterative closedform scheme to obtain the optimal beamforming matrix, where within each iteration a closed-form solution can be obtained. Numerical results validate our analysis and the optimality of the proposed iterative scheme, and further show that the proposed closed-form scheme is more efficient than the conventional QP algorithms with interior-point methods, which motivates the use of CI beamforming in practical wireless systems.

A Tutorial on Interference Exploitation via Symbol-Level Precoding: Overview, State-of-the-Art and Future Directions

“…In addition, PM optimization and weighted SB optimization based on CI are further discussed in [125]. It is worth mentioning that for CI precoding designs in [124] and [125], the received signals are forced to be strictly aligned to the desired data symbols with an increase in the amplitude for achieving CI, which follows the CI metric in [118] and is later shown to be sub-optimal and termed 'strict phase rotation' in [143] (Fig. 4a).…”

“…As a comparison, CI precoding does not have this issue and guarantees that the constraint is strictly satisfied for each data [123] CI-based SB Non-strict phase rotation PSK + QAM [124], [125] CI-MRT + PM + weighted SB Strict phase rotation PSK [126], [127] CI-based PM + SB Non-strict phase rotation PSK [128], [129] CI-based PM + weighted SB Relaxed detection region PSK [130], [131] CI-based PM Symbol scaling QAM [132] CI-based Per-antenna PM Strict phase rotation PSK [133] CI-MMSE precoding Non-strict phase rotation PSK [134], [139] CI-based PM + SB Noise-robust CI PSK [135] CI-based weighted per-antenna PM Strict phase rotation APSK [137], [138], [141], [142] CI-based PM + SB Distance preserving CI region Any constellation [143], [144], [145] Closed-form CI solutions Non-strict phase rotation + Symbol scaling PSK + QAM symbol combination during transmission, because CI precoding works on a symbol level and accordingly the constraints are enforced on a symbol level. This can be mathematically observed in [127] as well as in Section III for optimization problems formulated based on CI precoding, where the data symbols have been included in the constraints.…”

“…OA = t·s k represents the scaled data symbol, and t = √ Γ k σ 2 for PM optimizations, where Γ k is the corresponding SINR target for user k [127]. For SB optimizations, t is the objective function to be maximized [143]. Based on the observation that OB = OA + AB, we obtain that AB = h T k Ws − t · s k is the interfering signals.…”

“…For the 'strict phase-rotation' CI metric, as considered in [124], [125] and [143], it is obvious that each λ k should be purely real to guarantee that the phase of the noiseless received signal for user k is exactly the same as that of s k . Furthermore, the received signals should enhance the useful signal power to achieve CI, and therefore the mathematical condition for the 'strict phase-rotation' metric can be expressed as…”

“…where θ th = π M for a M-PSK constellation. Following [127] and [143], this CI condition in (6) can be further expressed as a function of the complex scalar λ k , given by…”