Non-Orthogonal Training Sequence Design in Two-Cell Interference Networks Based on an Extended Welch Bound

Abstract

Interferences due to non-orthogonality of training sequences usually exist in cellular networks when the number of all users is relatively large compared to the coherence time, such as the case in massive MIMO systems. In this paper, we address this effect from the perspective of non-orthogonal training sequence design in two-cell interference networks with $K$ users per cell. We relax the general assumption in which the cross-correlations of sequences are restricted to be 0 or 1, and target at designing the training sequences to minimize training phase interference with a given pilot length $\tau$ , which is no larger than the total number of users, i.e., $\tau\in[K, 2K]$ . We note that when large scale fading between different cells $\beta\neq 1$ , the strengths of interferences arising from non-orthogonal training sequences within a cell or from the adjacent cell become asymmetric, and optimal design needs to treat the intra-cell sequence correlation and inter-cell correlation differently. To this end, by incorporating $\beta$ into the design, we extend the Welch bound (Welch 1974 [1]) to the two-cell scenario with asymmetric intra-cell and inter-cell interference, and characterize the lower bound of the interference precisely. Specifically, we obtain the result that the sum of the squares of $\beta$ -weighted cross-correlations of the training sequences is lower-bounded by $\frac{2K^{2}(1+\beta^{2})}{K+(\tau-K)\beta^{2}}$ , which can be achieved by the proposed training sequence design in closed-form. Particularly, when $\beta=1$ , this bound reduces to $\frac{(2K)^{2}}{\tau}$ which is exactly the Welch bound. This result is applicable for the uplink design of general interference networks such as the pilot design in massive MIMO and the signature sequence design in multicell CDMA systems.