Constructions of Cross Z-Complementary Pairs With New Lengths
Avik Ranjan Adhikary, Zhengchun Zhou, Yang Yang, Pingzhi Fan
Abstract
Spatial modulation (SM) is a new paradigm of multiple-input multiple-output (MIMO) systems. Recently, it is observed that SM training sequences derived from cross Z-complementary pairs (CZCPs) lead to optimal channel estimation performance over frequency-selective channels. Recent paper by Liu et al. discussed only perfect CZCPs. In this paper, we focus on non-perfect CZCPs. We introduce the term cross Z-complementary ratio and re-categorise the CZCPs, both perfect and non-perfect, based on that. We propose a systematic construction of CZCPs based on generalized Boolean functions (GBFs). We further extend the lengths of the CZCPs by using the insertion method. The proposed CZCPs are all of new lengths of the form 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sup> 26 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</sup> + 2(α ≥ 1), 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sup> + 2, 26 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</sup> + 2 and 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sup> 26 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</sup> + 2. Finally we propose optimal binary CZCPs having parameters (12,5)- and (24,11) from binary Barker sequences. These CZCPs are also extended to (12N, 5N)-CZCPs and (24N, 11N)CZCPs, where N is the length of a binary Golay complementary pair (GCP). We also found anew structural property of binary CZCPs and concluded all binary GCPs are also CZCPs. Finally, we give some numerical simulations to confirm that depending on the number of multi-paths, the proposed CZCPs can be used to design SM training matrix which attains the minimum mean square error.