Motivated by the problem of reducing the peak-to-average power ratio (PAPR) of transmitted signals, we consider a design of complementary set matrices whose column sequences satisfy a correlation constraint. The design algorithm recursively builds a collection of $2^{t+1}$ mutually orthogonal (MO) complementary set matrices starting from a companion pair of sequences. We relate correlation properties of column sequences to that of the companion pair and illustrate how to select an appropriate companion pair to ensure that a given column correlation constraint is satisfied. For $t=0$, companion pair properties directly determine matrix column correlation properties. The proposed companion pair-based design can construct binary complementary sets with either a minimum out-of-phase autocorrelation magnitude or a minimum sum-of-out-of-phase autocorrelation magnitude for column sequences of length at least up to $28$. For $t geq 1$, reducing correlation merits of the companion pair may lead to improved column correlation properties. Exhaustive search for companion pairs satisfying a column correlation constraint is infeasible for medium length and long sequences. We instead search for two shorter length sequences by minimizing a cost function in terms of their autocorrelation and cross-correlation merits. In addition, by exploiting the well-known Welch bound, sufficient conditions for the existence of companion pairs which satisfy a set of column correlation constraints are given.
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机译：由于降低发射信号的峰均功率比（PAPR）的问题，我们考虑了其列序列满足相关约束的互补集矩阵的设计。该设计算法从同伴序列对递归地构建$ 2 ^ {t + 1} $个相互正交（MO）互补集矩阵的集合。我们将列序列的相关属性与伴随对的相关性相关联，并说明如何选择合适的伴随对以确保满足给定的列相关约束。对于$ t = 0 $，伴随对属性直接确定矩阵列相关属性。提出的基于伴侣对的设计可以构造长度至少达$ 28的具有最小异相自相关幅度或最小异相自相关幅度的二进制互补集。对于$ t geq 1 $，减少配对对的相关优劣可能会导致改善列相关性。对于中等长度和长序列，穷举搜索满足列相关性约束的伴侣对是不可行的。相反，我们通过在自相关和互相关优点方面最小化成本函数来搜索两个较短的序列。另外，通过利用众所周知的韦尔奇（Welch）界，给出了满足满足一组列相关性约束的伴随对的充分条件。
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