Extending the Welch Bound: Non-Orthogonal Pilot Sequence Design for Two-Cell Interference Networks

Ji Wang, Jun Sun, Xiaodong Wang, Kai Yang, Yingzhuang Liu

Abstract: Interferences due to non-orthogonality of signals usually exist in wireless networks when the number of users is larger than the sequence length, such as non-orthogonality of the pilots in multi-cell systems and non-orthogonality of the signature sequences in overloaded code-division-multiple-access (CDMA) systems. We address this effect from the perspective of non-orthogonal sequence design in a two-cell multiple-antenna network. Specifically, we aim at designing pilot sequences to minimize the sum mean-squared-error (MSE) of channel estimation with a given sequence length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau \in [K,2K]$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> is the number of users per cell. Considering the strength disparity between channels originating from the home cell and the neighbor cell, this problem boils down to minimizing the sum of squares of weighted correlations among sequences, whose lower bound is obtained in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">closed form</i> and can be regarded as a generalization of the well-known Welch bound (Welch, 1974). We prove this extended Welch bound is achievable, and design an algorithm based on the Davies-Higham method to generate the interference-minimizing sequences. Three fundamental properties of the proposed sequences are presented. Finally, we derive closed-form expressions of the average signal-to-interference-plus-noise-ratio (SINR) and rate for data transmission, based on which the optimal training duration can be found.