Speeding-Up Symbol-Level Precoding Using Separable and Dual Optimizations

Symbol-level precoding (SLP) can fully exploit the multi-user interference in the downlink. This paper investigates fast SLP algorithms for phase-shift keying (PSK) and quadrature amplitude modulation (QAM). In particular, we prove that the weighted max-min signal-to-interference-plus-noise ratio (SINR) balancing (SB) SLP problem with PSK signaling is not separable, which is contrary to the power minimization (PM) SLP problem, and accordingly, existing decomposition methods are not applicable. To tackle this issue, we establish an explicit duality between the SB-SLP and PM-SLP problems with PSK modulation. The proposed duality enables simultaneously obtaining the solutions to the SB-SLP and PM-SLP problems. We concurrently propose a closed-form power scaling algorithm to address the SB-SLP problem by the solution to the PM-SLP problem, via which the separability can be leveraged to decompose the problem. In terms of QAM signaling, a succinct model is used to formulate the PM-SLP problem and convert it into a separable equivalent. The new problem is decomposed into several simple parallel subproblems with closed-form solutions, employing the proximal Jacobian alternating direction method of multipliers (PJ-ADMM). We further prove that the proposed duality can be generalized to the multi-level modulation case, based on which a power scaling parallel inverse-free algorithm is proposed to solve the SB-SLP problem with QAM signaling. Numerical results show that the proposed algorithms offer optimal performance with lower complexity than the state-of-the-art.