Non-Orthogonal Pilot Sequence Design for Multi-Cells Interference Networks

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.