Construction of Golay Complementary Matrices and Its Applications to MIMO Omnidirectional Transmission

Abstract

Golay complementary sequences (GCS) have found wide applications in radar and communications thanks to their prominent property of complementary autocorrelation. This paper studies the extension of Golay complementary sequences to Golay complementary matrices (GCM), i.e, a pair of matrices whose two-dimensional autocorrelations add to be a two-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> -function. While the construction of (real) binary GCM are available in the literature, how to construct the quaternary GCM (and the more general unimodular GCM) remains unknown. We put together this missing piece, and show the feasible size that allows for binary and quaternary GCM. The feasible sizes can be further expanded if the phases of the matrices’ entries can be arbitrary. In addition, we consider the extension of the GCM to the set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N\geq 2$</tex-math></inline-formula> ) matrices, whose two-dimensional autocorrelations add to be a two-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> -function, which we refer to as autocorrelation complementary matrices (ACM). For <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N=4$</tex-math></inline-formula> , the ACM allows for matrices of arbitrary size. The GCM/ACM can be applied to omnidirectional transmission for a multi-input multi-output (MIMO) communication system with a uniform rectangular array (URA), which is useful for broadcasting cell-wide common messages. We propose two schemes for omnidirectional transmission, one is based on space-time block code (STBC) and the other is based on the dual cross-polarization of the antenna array. Both schemes can be implemented in the radio frequency (RF) analog domain, generating (theoretically) perfect omnidirectional beampattern with low hardware cost.