Constructions of Two-Dimensional Golay Complementary Array Pairs Based on Generalized Boolean Functions

Abstract

The one-dimensional (1-D) Golay complementary set (GCS) has many well-known properties and has been widely employed in engineering. The concept of 1-D GCS can be extended to the twodimensional (2-D) Golay complementary array set (GCAS) where the 2-D aperiodic autocorrelations of constituent arrays sum to zero except for the 2-D zero shift. The 2-D GCAS includes the 2-D Golay complementary array pair (GCAP) as a special case when the set size is 2. In this paper, 2-D generalized Boolean functions are introduced and novel constructions of 2-D GCAPs, 2-D GCASs, and 2-D Golay complementary array mates based on generalized Boolean functions are proposed. Explicit expressions of 2-D Boolean functions for 2-D GCAPs and 2-D GCASs are given. Therefore, they are all direct constructions without the aid of other existing 1-D or 2-D sequences. Moreover, for the column sequences and row sequences of the constructed 2-D GCAPs, their peak-to-average power ratio (PAPR) properties are also investigated. Index Terms Golay complementary pair (GCP), Golay complementary array pair (GCAP), Golay complementary array mate, Golay complementary array set (GCAS), peak-to-average power ratio (PAPR). * In our previous conference paper [40], we provided constructions of 2-D GCAPs and 2-D Golay complementary array mates from Boolean functions which can be found in [40, Th.6] and [40, Th.7], respectively. The result from [40, Th.6] will be described in Theorem 9 in this paper. Then, we provide more general constructions of 2-D GCAPs and Golay complementary array mates in Theorem 12 and Theorem 16, respectively.

Designing Two-Dimensional Complete Complementary Codes for Omnidirectional Transmission in Massive MIMO Systems

“…Against the above background, we propose to use 2D generalized Boolean functions (GBFs) for efficient synthesis of 2D CCCs whose array parameters (e.g., array size, set size, number of constituent arrays) take any power-of-two integer values. This positively settles our open problem raised in [23] on direct constructions of 2D CCCs. Moreover, the proposed 2D CCCs include those from [23,Th.…”

“…This positively settles our open problem raised in [23] on direct constructions of 2D CCCs. Moreover, the proposed 2D CCCs include those from [23,Th. 6 and Th.…”

“…In this work, driven by their modern application for omnidirectional transmission in massive MIMO systems, we have introduced a novel direct design of 2D CCCs based on 2D GBFs in Theorem 1. The proposed 2D (2 k , 2 k , 2 n , 2 m )-CCCs have flexible set sizes and array sizes, which includes our earlier result in [23,Th. 7] as a special case.…”

“…Remark 1: If we take k = 1 in Theorem 1, then the proposed 2-D CCCs will be reduced to (2, 2, 2 n , 2 m )-CCCs given in [23,Th. 7].…”

“…In most of the current literature, nevertheless, 2D CCCs are obtained by recursively applying various sequence operations to certain seed 1D sequences or 2D arrays [13]- [18], [22]. In [23], we presented a construction of 2D mutually orthogonal Golay complementary array pairs (GCAP) which are essentially 2D CCCs with the set size of two only. Recently, [24] constructed 2D Z-complementary array sets (ZCASs) through recursive product of certain matrices of generating polynomials, which may be reduced to 2D CCCs when the set size equals the number of constituent arrays and the zero-correlation zone covers the entire 2D shift domain.…”