基于互質陣列冗余分析的稀疏陣列設計方法
doi: 10.11999/JEIT240348
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空軍工程大學研究生院 西安 710051
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空軍工程大學防空反導學院 西安 710051
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國防科技大學電子對抗學院 合肥 230037
基金項目: 國家自然科學基金(62071476)，中國博士后科學基金(2022M723879)
Sparse Array Design Methods via Redundancy Analysis of Coprime Array
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Graduate College, Air Force Engineering University, Xi’an 710051, China
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Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China
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College of Electronic Engineering, National University of Defense Technology, Hefei 230037, China
Funds: The National Natural Science Foundation of China (62071476), China Postdoctoral Science Foundation (2022M723879)
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摘要: 互質陣列因具有較低的互耦效應而備受關注，但交替部署的子陣卻在一定程度上限制了連續(xù)自由度的提升。針對上述問題，該文在分析子陣互差集中冗余虛擬陣元產(chǎn)生條件的基礎上，提出了兩種子陣移位互質陣列(Coprime Array with Translated Subarray, CATrS)，以改善自由度性能。首先，將子陣平移至適當位置以優(yōu)化布陣結構，并分析了子陣的平移距離。隨后，推導了CATrS結構的自由度、連續(xù)自由度、孔洞位置和虛擬陣元權重的閉合表達式。理論分析表明，CATrS結構能夠在保持物理陣元數(shù)量不變的條件下，有效增加自由度和連續(xù)自由度，并抑制陣元互耦。最后，利用仿真實驗驗證了CATrS結構在波達方向估計中的優(yōu)越性。Abstract:
Objective Sensor arrays are widely used to capture the spatio-temporal information of incident signal sources, with their configurations significantly affecting the accuracy of Direction Of Arrival (DOA) estimation. The Degrees Of Freedom (DOF) of conventional Uniform Linear Array (ULA) are limited by the number of physical sensors, and dense array deployments lead to severe mutual coupling effects. Emerging sparse arrays offer clear advantages by reducing hardware requirements, increasing DOF, mitigating mutual coupling, and minimizing system redundancy through flexible sensor deployment, making them a viable solution for high-precision DOA estimation. Among various sparse array designs, the Coprime Array (CA)—consisting of two sparse ULAs with coprime inter-element spacing and sensor counts—has attracted considerable attention due to its reduced mutual coupling effects. However, the alternately deployed subarrays result in a much lower number of Continuous Degrees Of Freedom (cDOF) than anticipated, which degrades the performance of subspace-based DOA estimation algorithms that rely on spatial smoothing techniques. Although many studies have explored array configuration optimization and algorithm design, real-time application demands indicate that optimizing array configurations is the most efficient approach to improve DOA estimation performance. Methods This study examines the weight functions of CA and identifies a significant number of redundant virtual array elements in the difference coarray. Specifically, all virtual array elements in the difference coarray exhibit weight functions of two or more, a key factor reducing the available cDOF and DOF. To address this deficiency, the conditions for generating redundant virtual array elements in the cross-difference sets of subarrays are analyzed, and two types of coprime arrays with translated subarrays, namely, CATrS-I and CATrS-II are devised. These designs aim to increase available cDOF and DOF and enhance DOA estimation performance. Firstly, without altering the number of physical sensors, the conditions for generating redundant virtual array elements in the cross-difference sets are modified by translating any subarray of CA to an appropriate position. Then, the precise range of translation distances is determined, and the closed-form expressions for cDOF and DOF, the hole positions in the difference coarray, and weight functions of CATrS-I and CATrS-II are derived. Finally, the optimal configurations of CATrS-I and CATrS-II are obtained by solving an optimization problem that maximizes cDOF and DOF while maintaining a fixed number of physical sensors. Results and Discussions Theoretical analysis shows that the proposed CATrS-I and CATrS-II can reduce the weight functions of most virtual array elements in the difference coarray to 1, thus increasing the available cDOF and DOF while maintaining the same number of physical sensors. Comparisons with several previously developed sparse arrays highlight the advantages of CATrS-I and CATrS-II. Specifically, the Augmented Coprime Array (ACA), which doubles the number of sensors in one subarray, and the Reference Sensor Relocated Coprime Array (RSRCA), which repositions the reference sensor, achieve only a limited reduction in redundant virtual array elements, particularly those associated with small virtual array elements. As a result, their mutual coupling effects are similar to those of the original CA. In contrast, the proposed CATrS-I and CATrS-II significantly reduce both the number of redundant virtual array elements and the weight functions corresponding to small virtual array elements by translating one subarray to an optimal position. This adjustment effectively mitigates mutual coupling effects among physical sensors. Numerical simulations further validate the superior DOA estimation performance of CATrS-I and CATrS-II in the presence of mutual coupling, demonstrating their superiority in spatial spectrum and DOA estimation accuracy compared to existing designs. Conclusions Two types of CATrS are proposed for DOA estimation by translating the subarrays of CA to appropriate distances. This design effectively reduces the number of redundant virtual array elements in the cross-difference sets, leading to a significant increase in cDOF and DOF, while mitigating mutual coupling effects among physical sensors. The translation distance of the subarray is analyzed, and the closed-form expressions for cDOF and DOF, the hole positions in the difference coarray, and the weight functions of virtual array elements are derived. Theoretical analysis and simulation results demonstrate that the proposed CATrS-I and CATrS-II offer superior performance in terms of cDOF, DOF, mutual coupling suppression, and DOA estimation accuracy. Future research will focus on further reducing redundant virtual array elements in the self-difference sets by disrupting the uniform deployment of subarrays and extending these ideas to more generalized and complex sparse array designs to further enhance array performance. -
表 1 不同互質陣列的最佳布陣方式、最大連續(xù)自由度和最大自由度
陣列名稱 物理陣元數(shù)量 最優(yōu)$M$和$N$ 最大連續(xù)自由度 最大自由度 CA $T$為偶數(shù) $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ $2T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{6T} \mathord{\left/ {\vphantom {{6T} 4}} \right. } 4} - 1$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為偶數(shù) $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ $2T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{6T} \mathord{\left/ {\vphantom {{6T} 4}} \right. } 4} - {7 \mathord{\left/ {\vphantom {7 4}} \right. } 4}$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為奇數(shù) $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ $2T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{6T} \mathord{\left/ {\vphantom {{6T} 4}} \right. } 4} - {{19} \mathord{\left/ {\vphantom {{19} 4}} \right. } 4}$ RSRCA-Ⅰ $T$為偶數(shù) $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ $3T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - 3$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為偶數(shù) $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ $3T$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - {{19} \mathord{\left/ {\vphantom {{19} 4}} \right. } 4}$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為奇數(shù) $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ $3T - 2$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - {{39} \mathord{\left/ {\vphantom {{39} 4}} \right. } 4}$ RSRCA-Ⅱ $T$為偶數(shù) $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ $3T + 3$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - 1$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為偶數(shù) $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ $3T + 4$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - {3 \mathord{\left/ {\vphantom {3 4}} \right. } 4}$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為奇數(shù) $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ $3T + 6$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + {{5T} \mathord{\left/ {\vphantom {{5T} 2}} \right. } 2} - {7 \mathord{\left/ {\vphantom {7 4}} \right. } 4}$ CATrS-Ⅰ $T$為偶數(shù) $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - 1$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為偶數(shù) $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T - {5 \mathord{\left/ {\vphantom {5 4}} \right. } 4}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - {5 \mathord{\left/ {\vphantom {5 2}} \right. } 2}$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為奇數(shù) $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T - {{21} \mathord{\left/ {\vphantom {{21} 4}} \right. } 4}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - {{17} \mathord{\left/ {\vphantom {{17} 2}} \right. } 2}$ CATrS-Ⅱ $T$為偶數(shù) $M = {T \mathord{\left/ {\vphantom {T 2}} \right. } 2},N = {{\left( {T + 2} \right)} \mathord{\left/ {\vphantom {{\left( {T + 2} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T + 1$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - 1$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為偶數(shù) $M = {{\left( {T - 1} \right)} \mathord{\left/ {\vphantom {{\left( {T - 1} \right)} 2}} \right. } 2},N = {{\left( {T + 3} \right)} \mathord{\left/ {\vphantom {{\left( {T + 3} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T + {3 \mathord{\left/ {\vphantom {3 4}} \right. } 4}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - {5 \mathord{\left/ {\vphantom {5 2}} \right. } 2}$ $T$為奇數(shù)且${{\left( {T + 1} \right)} \mathord{\left/ {\vphantom {{\left( {T + 1} \right)} 2}} \right. } 2}$為奇數(shù) $M = {{\left( {T - 3} \right)} \mathord{\left/ {\vphantom {{\left( {T - 3} \right)} 2}} \right. } 2},N = {{\left( {T + 5} \right)} \mathord{\left/ {\vphantom {{\left( {T + 5} \right)} 2}} \right. } 2}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 4}} \right. } 4} + 2T - {5 \mathord{\left/ {\vphantom {5 4}} \right. } 4}$ ${{{T^2}} \mathord{\left/ {\vphantom {{{T^2}} 2}} \right. } 2} + T - {{17} \mathord{\left/ {\vphantom {{17} 2}} \right. } 2}$ 
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表 2 不同互質陣列的前3個權重的表達式
權重 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 不同互質陣列的陣元位置、連續(xù)自由度、自由度、前3個權重和耦合泄漏量
陣列名稱 陣元位置 連續(xù)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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