Spatial modulation (SM) is a new paradigm of multiple-input multiple-output (MIMO) systems, in which only one antenna at the transmitter is activated during each symbol period. Recently, it is observed that SM training sequences derived from corss Z-complementary pairs (CZCPs) lead to optimal channel estimation performance over frequency-selective channels. CZCPs are special form of sequence pairs which have zero aperiodic autocorrelation zones and cross-correlation zone at certain time-shifts. Recent paper by Liu \textit{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 generalised 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 $2^\alpha10^\beta26^\gamma+2~(\alpha\geq1)$, $10^\beta+2$, $26^\gamma+2$ and $10^\beta 26^\gamma+2$. Finally we propose a construction of 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). During the proof, we also found a new structural property of binary CZCPs and concluded all binary GCPs are CZCPs too. 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.
翻译:空间调制( SM) 是多输出量多输出量( CP) 系统的新模式, 其中每个符号期内只激活发报器上的一个天线。 最近, 观察到SM培训序列来自Cors Z- 补充配对( CZCP), 使得频道在选择频率的频道上产生最佳的频道估计性能。 CZCP 是一种特殊形式的序列配方, 它们具有零周期性自动调整区和在某些时间档中交叉点( CP) 。 最近由 Liu\ textitit{ al.} 讨论的扩展文件只有完美的 CZCP 。 在本文件中, 我们侧重于无效果的 CZCP 。 我们在此的基础上引入了 Exc- 补充比率, 并重新配置 CZCP, 以及 CZ 的不完善值 。 我们提议根据普通化的 Boolean 功能来系统构建 CZCP, 我们通过插入法进一步扩展了 CZCP 的长度。