A New Coprime Array with High Degree of Freedom Based on the Difference and Sum Co-array
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摘要: 针对均匀线列阵自由度(DOF)受限于阵元数的问题，该文提出一种基于差和共阵的新型互质阵，称为放置互质阵(DCA)，其借助由接收信号的时域和空域信息组合成的共轭增广矩阵得到等价的差和共阵来进行波达方向(DOA)估计。DCA将广义互质阵放置在与原点处单阵元相隔一定距离的位置，实现了和共阵与差共阵的阵元位置互补，从而最大限度上利用和共阵带来的自由度增幅。该文给出了DCA阵元位置和放置距离的闭式表达，随后分别对DCA的差共阵及和共阵的连续阵元及孔洞位置进行了理论分析，同时给出了两者间的关系，说明了DCA的高自由度特性。多个仿真实验验证了所提阵型DOA估计的有效性。Abstract: To deal with the problem that the Degree Of Freedom(DOF) of uniform linear array is limited by the number of elements, a new type of coprime array is proposed called Displaced Coprime Array(DCA).It takes use of the conjugate augmented matrix which is formed by the time and space information of the received signal to obtain the equivalent difference and sum co-array and to estimate the Direction Of Arrival(DOA). DCA places the generalized coprime array at a certain distance from the single array element at the coordinate origin so that the elements of the sum co-array and the difference co-array are complemented. As a result, the use of DOF provided by the sum co-array can be maximized. In this paper, the closed-form expressions of the element positions and the placement distance of DCA are given. Then, the performance of the sum co-array and the difference co-array including the continuous elements and the hole positions is theoretically analyzed, the relationship between the two is given and high DOF of DCA is presented. Multiple simulations verify the effectivity of DOA estimation using DCA.
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表 1 两种阵型的最优设计及最大自由度
阵型 表达式 子阵阵元数最优选择 最大自由度 DCA $P = M + N$
${\rm{DOF}} = 2MN - N + 1$$P$为偶，$P/2$为偶：$M = P/2 + 1,N = P/2 - 1$ $({P^2} - P)/2$ $P$为偶，$P/2$为奇：$M = P/2 + 2,N = P/2 - 2$ $({P^2} - P - 10)/2$ $P$为奇：$M = (P + 1)/2,N = (P - 1)/2$ $({P^2} - P + 2)/2$ CA(VCAM) $P = M + N - 1$
${\rm{DOF}} = MN + M + N - 1$$P$为偶：$M = (P + 2)/2,N = P/2$ $({P^2} + 6P)/4$ $P$为奇，$(P + 1)/2$为偶：$M = (P + 1)/2 + 1,N = (P + 1)/2 - 1$ $({P^2} + 6P - 3)/4$ $P$为奇，$(P + 1)/2$为奇：$M = (P + 1)/2 + 2,N = (P + 1)/2 - 2$ $({P^2} + 6P - 15)/4$ 
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表 2 各阵型同孔径下的阵型配置
阵型 阵元总数 子阵阵元数 孔径 自由度 CACIS 27 $(M,N) = (15,13)$ 182 182 CA(VCAM) 27 $(M,N) = (15,13)$ 182 222 CADiS 26 $(M,N) = (14,13)$ 182 182 NA 26 $({N_1},{N_2}) = (13,13)$ 181 181 DCA 23 $(M,N) = (12,11)$ 182 254
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