Dirty Paper Coding (DPC) is considered as the optimal precoding which achieves capacity for the Gaussian Multiple-Input Multiple-Output (MIMO) broadcast channel (BC). However, to find the optimal precoding order, it needs to repeat N! times for N users as there are N! possible precoding orders. This extremely high complexity limits its practical use in modern wireless networks. In this paper, we show the equivalence of DPC and the recently proposed Higher Order Mercer's Theorem (HOGMT) precoding[1][2] in 2-D (spatial) case, which provides an alternate implementation for DPC. Furthermore, we show that the proposed implementation method is linear over the permutation operator when permuting over multi-user channels. Therefore, we present a low complexity algorithm that optimizes the precoding order for DPC with beamforming, eliminating repeated computation of DPC for each precoding order. Simulations show that our method can achieve the same result as conventional DPC with about 20 dB lower complexity for N = 5 users.

翻译：脏纸编码（DPC）被视为高斯多输入多输出（MIMO）广播信道（BC）中达到容量的最优预编码方案。然而，为确定最优预编码顺序，需要对N个用户重复计算N!次（因存在N!种可能的预编码顺序），其极高复杂度限制了该方案在现代无线网络中的实际应用。本文证明了DPC与近期提出的二维（空间）高阶默瑟定理（HOGMT）预编码[1][2]的等价性，为DPC提供了替代实现方案。进一步研究表明，在多用户信道置换中，所提实现方法对置换算子呈线性特征。据此，我们提出一种低复杂度算法，通过波束成形优化DPC的预编码顺序，消除对每个预编码顺序重复计算DPC的步骤。仿真表明，当用户数N=5时，该方法在保持与传统DPC相同性能的同时，复杂度降低约20 dB。