Sparse Array Design Methods via Redundancy Analysis of Coprime Array
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摘要: 互质阵列因具有较低的互耦效应而备受关注，但交替部署的子阵却在一定程度上限制了连续自由度的提升。针对上述问题，该文在分析子阵互差集中冗余虚拟阵元产生条件的基础上，提出了两种子阵移位互质阵列(Coprime Array with Translated Subarray, CATrS)，以改善自由度性能。首先，将子阵平移至适当位置以优化布阵结构，并分析了子阵的平移距离。随后，推导了CATrS结构的自由度、连续自由度、孔洞位置和虚拟阵元权重的闭合表达式。理论分析表明，CATrS结构能够在保持物理阵元数量不变的条件下，有效增加自由度和连续自由度，并抑制阵元互耦。最后，利用仿真实验验证了CATrS结构在波达方向估计中的优越性。Abstract: The coprime array has garnered noticeable attention due to the reduced mutual coupling, but the alternately deployed subarrays limit the enhancement of consecutive degrees of freedom to some extent. To address this deficiency, two types of Coprime Array with Translated Subarray (CATrS) are presented based on the analysis of the conditions for generating redundant virtual arrays in the cross-difference sets. Firstly, the subarray is shifted to an appropriate location to optimize the array geometry, and then the displacement distance is analyzed. Subsequently, the closed-form expressions for the degrees of freedom, consecutive degrees of freedom, hole positions, and weight functions of CATrS are derived. Theoretical analysis indicates that CATrS can effectively increase both the number of degrees of freedom and consecutive degrees of freedom, while suppress the mutual coupling without altering the number of physical sensors. Finally, the superiorities of CATrS in direction of arrival estimation are verified through simulation experiments.
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表 1 不同互质阵列的最佳布阵方式、最大连续自由度和最大自由度
阵列名称 物理阵元数量 最优$M$和$N$ 最大连续自由度 最大自由度 CA $T$为偶数 $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ $2T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{6T} \mathord{\left/ {\vphantom {{6T} 4}} \right. } 4} - 1$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为偶数 $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ $2T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{6T} \mathord{\left/ {\vphantom {{6T} 4}} \right. } 4} - {7 \mathord{\left/ {\vphantom {7 4}} \right. } 4}$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为奇数 $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ $2T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{6T} \mathord{\left/ {\vphantom {{6T} 4}} \right. } 4} - {{19} \mathord{\left/ {\vphantom {{19} 4}} \right. } 4}$ RSRCA-Ⅰ $T$为偶数 $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ $3T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - 3$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为偶数 $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ $3T$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - {{19} \mathord{\left/ {\vphantom {{19} 4}} \right. } 4}$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为奇数 $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ $3T - 2$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - {{39} \mathord{\left/ {\vphantom {{39} 4}} \right. } 4}$ RSRCA-Ⅱ $T$为偶数 $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ $3T + 3$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - 1$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为偶数 $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ $3T + 4$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - {3 \mathord{\left/ {\vphantom {3 4}} \right. } 4}$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为奇数 $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ $3T + 6$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - {7 \mathord{\left/ {\vphantom {7 4}} \right. } 4}$ CATrS-Ⅰ $T$为偶数 $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - 1$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为偶数 $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T - {5 \mathord{\left/ {\vphantom {5 4}} \right. } 4}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - {5 \mathord{\left/ {\vphantom {5 2}} \right. } 2}$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为奇数 $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T - {{21} \mathord{\left/ {\vphantom {{21} 4}} \right. } 4}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - {{17} \mathord{\left/ {\vphantom {{17} 2}} \right. } 2}$ CATrS-Ⅱ $T$为偶数 $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - 1$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为偶数 $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T + {3 \mathord{\left/ {\vphantom {3 4}} \right. } 4}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - {5 \mathord{\left/ {\vphantom {5 2}} \right. } 2}$ $T$为奇数且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$为奇数 $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T - {5 \mathord{\left/ {\vphantom {5 4}} \right. } 4}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - {{17} \mathord{\left/ {\vphantom {{17} 2}} \right. } 2}$ 
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表 2 不同互质阵列的前3个权重的表达式
权重 CA ACA RSRCA-Ⅰ RSRCA-Ⅱ CATrS-Ⅰ CATrS-Ⅱ $ \omega \left( 1 \right) $ 2 2 2 2 1 1 $ \omega \left( 2 \right) $ $ \left\{ {\begin{array}{lllllllllll} {N - 1,}&{M = 2} \\ {2,}&{M \ge 3} \end{array}} \right. $ $ \left\{ {\begin{array}{*{20}{l}} {N,}&{M = 2} \\ {5,}&{M = 3,N = 2} \\ {2,}&{M \ge 3} \end{array}} \right. $ $ \left\{ {\begin{array}{*{20}{l}} {N - 2,}&{M = 2} \\ {2,}&{M \ge 3} \end{array}} \right. $ $ \left\{ {\begin{array}{*{20}{l}} {N - 2,}&{M = 2} \\ {2,}&{M \ge 3} \end{array}} \right. $ $ \left\{ {\begin{array}{*{20}{l}} {N - 1,}&{M = 2} \\ {1,}&{M \ge 3} \end{array}} \right. $ $ \left\{ {\begin{array}{*{20}{l}} {N - 1,}&{M = 2} \\ {1,}&{M \ge 3} \end{array}} \right. $ $ \omega \left( 3 \right) $ $\left\{ \begin{array}{ll}N-1, & M=3 \\ 2, & 其它 \end{array} \right.$ $ \{\begin{array}{ll}N, & M=2 \\ 2M-1, & N=3 \\ 2, & 其它 \end{array} $ $\left\{ \begin{array}{ll}N-2, & M=3 \\ 0, & M=2,N=3 \\ 2, & 其它 \end{array} \right.$ $ \left\{ \begin{array}{ll}N-2, & M=3 \\ 0, & M=2,N=3 \\ 2, & 其它 \end{array}\right. $ $ \left\{ \begin{array}{ll}N-1, & M=3 \\ 1, & 其它 \end{array} \right.$ $ \left\{ \begin{array}{ll}N-1, & M=3 \\ 1, & 其它 \end{array} \right.$
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表 3 不同互质阵列的阵元位置、连续自由度、自由度、前3个权重和耦合泄漏量
阵列名称 阵元位置 连续自由度 自由度 $ \omega \left( 1 \right) $ $ \omega \left( 2 \right) $ $ \omega \left( 3 \right) $ 耦合泄漏量 CA {0,5,6,10,12,15,18,20,24,25} 21 39 2 2 2 0.2392 ACA {0,3,5,6,9,10,12,15,20,25} 35 43 2 2 5 0.2496 RSRCA-Ⅰ {-5,5,6,10,12,15,18,20,24,25} 31 47 2 2 2 0.2371 RSRCA-Ⅱ {-6,5,6,10,12,15,18,20,24,25} 33 49 2 2 2 0.2369 CATrS-Ⅰ {0,6,12,17,18,22,24,27,32,37} 45 59 1 1 1 0.1824 CATrS-Ⅱ {0,5,10,15,16,20,22,25,28,34} 41 55 1 1 2 0.1878
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