In wireless communications, the performance of non-orthogonal sequence sets significantly affects the level of multi-user interference when the number of users surpasses the sequence length. The design of non-orthogonal sequences plays a crucial role in both the non-orthogonality of the pilots in multi-cell systems and the signature sequences in overloaded code-division multiple-access (CDMA) systems. In multi-cell systems, considering the strength disparity between channels originating from the home cell and the neighboring cells, the extended total squared correlation (ETSC) is proposed as a new sequence design criterion, which is defined as the sum of squares of the weighted correlations among sequences. In this paper, we derive a closed-form expression for the lower bound of ETSC for multi-cell systems with a given sequence length $\tau$, where $\tau \leq K$ and $K$ is the number of users per cell. This can be regarded as a generalization of the well-known Welch bound (Welch, 1974, IEEE TIT) and the extended Welch bound (Wang et al., 2021, IEEE TWC). Additionally, from the necessary conditions of the bound, the optimal sequence set can be easily obtained when the interference power factor matrix is positive definite. On the other hand, to address the lack of sequence generation methods under certain parameter conditions, we propose the ETSC-MM algorithm, which generates sequence sets with low ETSC based on a Majorization-Minimization (MM) optimization framework.

翻译：在无线通信中，当用户数量超过序列长度时，非正交序列集的性能显著影响多用户干扰水平。非正交序列的设计对于多小区系统中导频的非正交性以及过载码分多址（CDMA）系统中的签名序列都起着至关重要的作用。在多小区系统中，考虑到来自本小区与相邻小区信道之间的强度差异，本文提出了扩展总平方相关（ETSC）作为一种新的序列设计准则，其定义为序列间加权相关平方之和。本文推导了给定序列长度 $\tau$（其中 $\tau \leq K$，$K$ 为每小区用户数）的多小区系统 ETSC 下界的闭式表达式。该结果可视为著名的 Welch 界（Welch, 1974, IEEE TIT）与扩展 Welch 界（Wang et al., 2021, IEEE TWC）的推广。此外，根据该下界的必要条件，当干扰功率因子矩阵正定时，可轻松获得最优序列集。另一方面，针对特定参数条件下序列生成方法的缺失，我们提出了 ETSC-MM 算法，该算法基于 Majorization-Minimization（MM）优化框架生成具有低 ETSC 的序列集。