Constructive Interference based Block-Level Precoding for Scene Expansion: Closed-Form Solutions

We study closed-form constructive interference based block-level precoding (CI-BLP) for scene expansion in the downlink of multi-user multiple-input single-output (MU-MISO) systems. We extend the analysis on CI-BLP to the case where the number of symbol slots in a block is smaller than the number of user. To this end, we mathematically prove the feasibility of using the pseudo-inverse to express the closed-form expression of the CI-BLP optimal precoding matrix. Building upon this, a quadratic programming (QP) optimization on simplex is obtained without being limited by the relationship between the number of symbol slots in a block and the number of users. We study the low complexity algorithm of large scale QP problem. We first mathematically obtain the rank of the quadratic coefficient matrix in the QP problem. Although the iterative closed-form algorithm for QP problems in CI-based symbol-level precoding (CI-SLP) can be used in certain scenarios, the complexity of the iterative closed algorithm for large-scale QP problems is impractical. In addition, we design a low complexity algorithm based on alternating direction method of multipliers (ADMM), which can efficiently solve large-scale QP problems. We further analyze the convergence and complexity of the proposed algorithm. Numerical results validate our analysis and the optimality of the proposed algorithm, and further show that the proposed algorithm offers a flexible performance-complexity tradeoff by limiting the maximum number of iterations, which motivates the use of CI-BLP in practical wireless systems.