The Constructions of Optimal Balanced Quadriphase Almost Mismatched Complementary Pairs with Prime Length
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摘要: 当一对失配序列的所有异相自相关函数和均为同一非0整数时，称该对失配序列为几乎失配互补对。该文提出平衡4元几乎失配互补对的新类型序列，通过Gray映射证明得到了平衡的素数长4元几乎失配互补对的理论界，基于4阶分圆类，提出满足理论界的周期为素数长的理想平衡4元几乎失配互补对的构造方法。通过该文研究扩大了4元互补对的存在范围，弥补了目前已有4元互补对大多只存在偶数长度的缺陷。Abstract: A pair of mismatched sequences is called an almost mismatched complementary pair if their periodic autocorrelation functions sum up to a same nonzero integer for all out-of-phase time shifts. In this paper, a new balanced quadriphase almost mismatched complementary pair is proposed, the theoretical bound of balanced quadriphase almost complementary pair with prime length is proved by Gray mapping, based on the cyclotomic classes of order 4, the optimal balanced quadriphase almost mismatched complementary pair with prime length satisfied theoretical bound is constructed. The existence of quadriphase complementary pairs are expanded and compensated the deficiency of most of the existing ones only have even length at present by investigate in this paper.
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表 1
$p$ 为偶数时，4阶分圆数及计算式${(c,d)_4}$ $s \equiv 1(\boldsymbolod 4)$ $s \equiv 3(\boldsymbolod 4)$ ${(0,0)_4}$ $(Q - 11 - 6s)/16$ $(Q - 11 + 6s)/16$ ${(0,1)_4},{(1,0)_4},{(3,3)_4}$ $(Q - 3 + 2s + 8t)/16$ $(Q - 3 - 2s + 8t)/16$ ${(0,2)_4},{(2,0)_4},{(2,2)_4}$ $(Q - 3 + 2s)/16$ $(Q - 3 - 2s)/16$ ${(0,3)_4},{(1,1)_4},{(3,0)_4}$ $(Q - 3 + 2s - 8t)/16$ $(Q - 3 - 2s - 8t)/16$ $\begin{array}{l} {(1,2)_4},{(1,3)_4},{(2,1)_4}, \\ {(2,3)_4},{(3,1)_4},{(3,2)_4} \\ \end{array} $ $(Q + 1 - 2s)/16$ $(Q + 1 + 2s)/16$ 
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表 2
$p$ 为奇数时，4阶分圆数及计算式$(c,d){ _4}$ $s \equiv 1(\boldsymbolod 4)$ $s \equiv 3(\boldsymbolod 4)$ ${(0,2)_4}$ $(Q + 1 - 6s)/16$ $(Q + 1 + 6s)/16$ ${(0,0)_4},{(2,0)_4},{(2,2)_4}$ $(Q - 7 + 2s)/16$ $(Q - 7 - 2s)/16$ ${(0,1)_4},{(1,3)_4},{(3,2)_4}$ $(Q + 1 + 2s - 8t)/16$ $(Q + 1 - 2s - 8t)/16$ ${(0,3)_4},{(1,2)_4},{(3,1)_4}$ $(Q + 1 + 2s + 8t)/16$ $(Q + 1 - 2s + 8t)/16$ $\begin{array}{l} {(1,0)_4},{(1,1)_4},{(2,1)_4}, \\ {(2,3)_4},{(3,0)_4},{(3,3)_4} \\ \end{array} $ $ (Q - 3 - 2s)/16$ $(Q - 3 + 2s)/16$
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表 3
$p$ 为奇数时，平衡4元失配序列$({x_1},{y_1})$ 与$({x_2},{y_2})$ 的定义参数$[({j_1},{k_1},{l_1},{r_1}),(j_1',k_1',l_1',r_1'),({j_2},{k_2},{l_2},{r_2}),(j_2',k_2',l_2',r_2')]$ $[({x_1}(0),{y_1}(0)),({x_2}(0),{y_2}(0))] \in $ $[((0,2,1,3),(0,2,1,3)),((0,2,1,3),(0,2,1,3))]$ $\begin{array}{*{20}{l} }{\{ [(1, - 1),( - 1,1)];[({\rm{i}}, - {\rm{i}}),( - {\rm{i}},{\rm{i}})];}\\{[({\rm{i}},1),(1, - {\rm{i}})];[( - {\rm{i}}, - 1),( - 1,{\rm{i}})];}\\{[(1,{\rm{i}}),( - 1, - {\rm{i}})];[({\rm{i}}, - 1),( - {\rm{i}},1)]\} }\end{array}$ $[((0,2,1,3),(0,2,1,3)),((0,2,3,1),(0,2,3,1))]$ $\begin{array}{*{20}{l} }{\{ [(1, - {\rm{i}}),({\rm{i}},1)];[( - 1,{\rm{i}}),( - {\rm{i}}, - 1)];}\\{[({\rm{i}},1),(1, - {\rm{i}})];[( - {\rm{i}}, - 1),( - 1,{\rm{i}})];}\\{[(1,{\rm{i}}),( - {\rm{i}},1)];[( - 1, - {\rm{i}}),({\rm{i}}, - 1)];}\\{[({\rm{i}}, - 1),( - 1, - {\rm{i}})];[( - {\rm{i}},1),(1,{\rm{i}})]\} }\end{array}$
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表 4
$p$ 为偶数时，平衡4元失配序列$({x_1},{y_1})$ 与$({x_2},{y_2})$ 的定义参数$[({j_1},{k_1},{l_1},{r_1}),(j_1',k_1',l_1',r_1'),({j_2},{k_2},{l_2},{r_2}),(j_2',k_2',l_2',r_2')]$ $[({x_1}(0),{y_1}(0)),({x_2}(0),{y_2}(0))] \in $ $[((0,2,1,3),(0,2,1,3)),((0,2,1,3),(0,2,1,3))]$ $\begin{array}{*{20}{l} }{\{ [(1, - 1),( - 1,1)];[({\rm{i}}, - {\rm{i}}),( - {\rm{i}},{\rm{i}})];}\\{[(1, - {\rm{i}}),( - 1,{\rm{i}})];[({\rm{i}},1),( - {\rm{i}}, - 1)];}\\{[(1,{\rm{i}}),( - 1, - {\rm{i}})];[({\rm{i}}, - 1),( - {\rm{i}},1)]\} }\end{array}$ $[((0,2,1,3),(0,2,1,3)),((0,2,3,1),(0,2,3,1))]$ $\begin{array}{*{20}{l} }{\{ [(1, - 1),(1, - 1)];[( - 1,1),( - 1,1)];}\\{[({\rm{i} }, - {\rm{i} }),( - {\rm{i} },{\rm{i} })];[( - {\rm{i} },{\rm{i} }),({\rm{i} }, - {\rm{i} })];}\\{[(1, - {\rm{i} }),( - {\rm{i} }, - 1)];[( - 1,{\rm{i} }),({\rm{i} },1)];}\\{[({\rm{i} },1),( - 1,{\rm{i} })];[( - {\rm{i} }, - 1),(1, - {\rm{i} })];}\\{[(1,{\rm{i} }),({\rm{i} }, - 1)];[( - 1, - {\rm{i}}),( - {\rm{i} },1)];}\\{[({\rm{i} }, - 1),(1,{\rm{i} })];[( - {\rm{i} },1),( - 1, - {\rm{i} })]\} }\end{array}$ $[((0,1,2,3),(0,1,2,3)),((0,3,1,2),(0,3,1,2))]$ $\begin{array}{*{20}{l} }{\{ [({\rm{i}},1),( - 1,{\rm{i}})];[( - {\rm{i}}, - 1),(1, - {\rm{i}})];}\\{[(1,{\rm{i}}),({\rm{i}}, - 1)];[( - 1, - {\rm{i}}),( - {\rm{i}},1)]\} }\end{array}$ $[((0,1,2,3),(0,1,2,3)),((0,3,2,1),(0,3,2,1))]$ $\begin{array}{*{20}{l} }{\{ [({\rm{i}},1),( - {\rm{i}}, - 1)];[( - {\rm{i}}, - 1),({\rm{i}},1)];}\\{[(1,{\rm{i}}),( - 1, - {\rm{i}})];[( - 1, - {\rm{i}}),(1,{\rm{i}})]\} }\end{array}$
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表 5 4元周期互补对已有结果总结
文献 周期长度 方法 平衡性 文献[15,20] 2, 4, 8, 10, 16, 20, 26, 32, 34, 40, ··· Gray映射，交织操作 不平衡 文献[21,22] 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 16, 18,
20, 22, 24, 26, 30, 32, 36, 40, 44, 48, ···Gray阵列扩展
Baker序列不平衡 文献[23] 3, 5, 7, 9, 11, 13, ···, 41, 43, 45, 49
4, 6, 8, 10, ···, 30, 34, 36, 38, ···Gray映射
生成序列扩展不平衡 定理2 素数长$ Q=4p+1, p$为奇数 4阶分圆 平衡 定理3 素数长$ Q=4p+1, p$为偶数 4阶分圆 平衡
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