Design of sparse reconfigurable array using reweighted atomic norm minimization
-
摘要:
为克服网格失配问题并提升阵列性能, 提出了使用重加权原子范数最小化的稀疏可重构直线阵列设计方法, 将稀疏可重构直线阵列设计问题表示为多测量矢量稀疏优化模型, 并通过重加权原子范数最小化算法解算出阵元位置和阵元激励。区别于经典压缩感知方法, 该方法借助原子范数理论建立了阵元数量、阵元位置和阵元激励联合优化的无网格稀疏优化模型, 从而可以克服网格失配问题, 并提升阵列波束图的匹配精度。仿真实验表明, 与压缩感知类方法相比, 重加权原子范数最小化算法可以设计出波束匹配精度高一个数量级的稀疏可重构直线阵列。
Abstract:To overcome the grid mismatch issue and enhance the performance of beam patterns, a sparse reconfigurable linear array design method based on reweighted atomic norm minimization is proposed. This method formulates the sparse reconfigurable linear array design problem as multiple measurement vectors sparse optimization model. It solves for the element positions and element excitations of the sparse reconfigurable linear array by the reweighted atomic norm minimization. Unlike conventional compressed sensing-based methods, this approach leverages atomic norm theory to establish a gridless sparse optimization model that jointly optimizes the number, positions, and excitations of array elements. As a result, it can overcome the grid mismatch problem and enhance the matching accuracy of the array beam pattern. Simulation experiments demonstrate that compared to compressed sensing methods, the reweighted atomic norm minimization method can design sparse reconfigurable linear arrays with an order of magnitude higher beam matching accuracy.
-
图 4 不同主瓣指向的参考波束和RANM算法重构的波束 (a) 主瓣指向
$ \left[ { \pm 10^\circ ,0^\circ } \right] $ 的参考波束; (b)主瓣指向$\left[ { \pm 20^\circ , \pm 10^\circ ,0^\circ } \right]$ 的参考波束; (c) 主瓣指向$\left[ { \pm 30^\circ , \pm 20^\circ , \pm 10^\circ ,0^\circ } \right]$ 的参考波束; (d) 主瓣指向$\left[ { \pm 10^\circ ,0^\circ } \right]$ 的RANM波束; (e) 主瓣指向$\left[ { \pm 20^\circ , \pm 10^\circ ,0^\circ } \right]$ 的RANM波束; (f)主瓣指向$\left[ { \pm 30^\circ , \pm 20^\circ , \pm 10^\circ ,0^\circ } \right]$ 的RANM波束图 5 不同旁瓣级的参考波束和RANM算法重构的波束 (a)
${\text{PSLL}} = - 45{\text{ dB}}$ 的参考波束; (b)${\text{PSLL}} = - 50{\text{ dB}}$ 的参考波束; (c)${\text{PSLL}} = - 55{\text{ dB}}$ 的参考波束; (d)${\text{PSLL}} = - 45{\text{ dB}}$ 时RANM重构的波束; (e)${\text{PSLL}} = - 50{\text{ dB}}$ 时RANM重构的波束; (f)${\text{PSLL}} = - 55{\text{ dB}}$ 时RANM重构的波束表 1 使用RANM的稀疏可重构直线阵列设计流程表
输入: 采样点数$ J $, 约束项参数$ \eta $, 重加权参数$ \epsilon $, 能量比阈值$ \delta $, 最大迭代次数D=6, 均匀采样的参考可重构波束图${{\boldsymbol{F}}_{{\text{REF}}}}$ 1. 初始化权重参数$ {\boldsymbol{W}} = {\boldsymbol{I}} $ 2. for t=1:D 3. 求解对偶优化方程 $ \begin{array}{*{20}{l}} {\mathop {\min }\limits_{{\boldsymbol{V}},{\boldsymbol{Z}}} }&{2\eta {{\left\| {\boldsymbol{V}} \right\|}_{\text{F}}} + 2{\rm {Re}} \left( {{\rm {Tr}}\left( {{{\boldsymbol{F}}_{{\text{REF}}}}{\boldsymbol{V}}} \right)} \right),} \\ {\rm {s.t.}}&{\left[ {\begin{array}{*{20}{c}} {\boldsymbol{I}}&{{{\boldsymbol{V}}^{\rm H}}} \\ {\boldsymbol{V}}&{\boldsymbol{Z}} \end{array}} \right] \geqslant {\boldsymbol{0}},} \\ {}&{\displaystyle\sum\limits_{n = 1}^{2J - 1 - j} {{Z_{n,n + j - 1}}} =\displaystyle \sum\limits_{n = 1}^{2J - 1 - j} {{W_{n,n + j - 1}}} ,{\text{ }}j = 1, \cdots ,2J - 1.} \end{array} $ 4. 通过优化工具包中特定语法获得对应原始问题的最优解$ {\boldsymbol{F}} $和$ {\boldsymbol{x}} $ 5. 更新下一次迭代需要的重加权矩阵$ W={\left(T\left(x\right) + \epsilon I\right)}^{-1} $ 6. end 7. 对最后一次迭代获得的最优$T\left( {\boldsymbol{x}} \right)$使用特征分解, 确定原子数量$N$, 即$N$需要满足式(15)的条件 8. 使用旋转不变传播算子算法通过$T\left( {\boldsymbol{x}} \right)$估计N维原子频率${\boldsymbol{\widehat f}}$, 并通过式(16)计算对应的N个信号矢量$ {{\boldsymbol{\widehat r}}_1}, \cdots ,{{\boldsymbol{\widehat r}}_N} $ 9. 通过式(6)将原子频率与信号矢量转换为稀疏可重构直线阵列的N维阵元位置矢量${\boldsymbol{\widehat d}}$和对应的N个激励矢量$ {{\boldsymbol{\widehat w}}_1}, \cdots ,{{\boldsymbol{\widehat w}}_N} $ 表 2 M = 2时可重构直线阵性能对比表
方法 阵元数量 匹配误差 笔形波束PSLL (dB) 平顶波束PSLL (dB) 最小阵元间距 ($\lambda $) 阵列孔径 ($\lambda $) 参考波束[8] 20 — −24.93 −25.01 0.50 9.50 MT-BCS 15 3.19 × 10−3 −26.36 −22.50 0.02 9.23 RANM 15 2.03 × 10−4 −25.00 −23.67 0.59 9.26 表 3
$M = 3$ 时可重构直线阵性能对比表方法 阵元数量 匹配误差 笔形波束PSLL (dB) 平顶波束PSLL (dB) 余割平方波束
PSLL (dB)最小阵元间距 ($\lambda $) 阵列孔径 ($\lambda $) 参考波束[8] 20 — −20.03 −19.50 −20.01 0.50 9.50 MT-BCS 17 3.14 × 10−2 −17.87 −16.96 −16.16 0.02 9.00 RANM 17 4.92 × 10−3 −18.93 −17.48 −17.04 0.19 9.42 表 4 不同主瓣指向的可重构直线阵列性能表
参考可重构直线阵列 RANM算法获得的稀疏可重构直线阵列 模式数量M PSLL (dB) 阵元数量 稀疏后阵元数量 匹配误差 PSLL (dB) 最小阵元间距 ($\lambda $) 阵列孔径 ($\lambda $) 3 −40 64 41 2.38 × 10−6 −38.6 0.67 31.47 5 −40 64 47 4.64 × 10−7 −39.9 0.55 31.49 7 −40 64 51 1.01 × 10−6 −38.2 0.58 31.48 表 5 不同旁瓣级的可重构直线阵性能表
参考可重构直线阵 RANM算法设计的稀疏可重构直线阵 模式数量 旁瓣级 阵元数量 稀疏后阵元数量 匹配误差 PSLL (dB) 最小阵元间距 ($\lambda $) 阵列孔径 ($\lambda $) M=3 −45 64 41 4.09 × 10−6 −44.53 0.70 31.45 M=3 −50 64 42 2.97 × 10−7 −49.47 0.64 31.46 M=3 −55 64 42 1.89 × 10−7 −54.59 0.64 31.46 表 6 幅度扰动为12%时稀疏可重构直线阵性能表
方法 阵元数量 匹配误差 笔形波束PSLL (dB) 平顶波束PSLL (dB) 最小阵元间距 ($\lambda $) 阵列孔径 ($\lambda $) 参考波束[8] 20 — −24.93 −25.01 0.50 9.50 RANM 15 5.09 × 10−3 −25.84 −21.87 0.59 9.26 -
[1] Qi Z, Bai Y, Wang Q, et al. Optimal synthesis of reconfigurable sparse arrays via multi‐convex programming. IET Radar Sonar Navig., 2020; 14(8): 1125—1134 doi: 10.1049/iet-rsn.2019.0236 [2] Xia W, Jin X, Dou F. Thinned array design with minimum number of transducers for multibeam imaging sonar. IEEE J. Oceanic Eng., 2016; 42(4): 892—900 doi: 10.1109/JOE.2016.2626558 [3] Pan X, Li S, Pan C. Distributed broadband phased‐MIMO sonar for detection of small targets in shallow water environments. IET Radar Sonar Navig., 2018; 12(7): 721—728 doi: 10.1049/iet-rsn.2017.0381 [4] 王朋, 迟骋, 纪永强, 等. 二维解卷积波束形成水下高分辨三维声成像. 声学学报, 2019; 44(4): 613—625 doi: 10.15949/j.cnki.0371-0025.2019.04.022 [5] 秦留洋, 黄海宁, 王朋, 等. 频域宽带下视多波束三维声成像算法. 声学学报, 2019; 44(4): 604—612 doi: 10.15949/j.cnki.0371-0025.2019.04.021 [6] Chakraborty A, Das B, Sanyal G. Beam shaping using non-linear phase distribution in a uniformly spaced array. IEEE Trans. Antennas Propag., 1982; 30(5): 1031—1034 doi: 10.1109/TAP.1982.1142917 [7] Dürr M, Trastoy A, Ares F. Multiple-pattern linear antenna arrays with single prefixed amplitude distributions: modified Woodward-Lawson synthesis. Electron. Lett., 2000; 36(16): 1345—1346 doi: 10.1049/el:20000980 [8] Gies D, Rahmat-Samii Y. Particle swarm optimization for reconfigurable phase differentiated array design. Microw. Opt. Technol. Lett., 2003; 38(3): 168—175 doi: 10.1002/mop.11005 [9] Mahanti G K, Das S, Chakraborty A. Design of phase-differentiated reconfigurable array antennas with minimum dynamic range ratio. IEEE Antennas Wirel. Propag. Lett., 2006; 5: 262—264 doi: 10.1109/LAWP.2006.875899 [10] 沈晓炜. 基于粒子群算法的稀疏阵列超声相控阵全聚焦成像. 应用声学, 2020; 39(3): 354—359 doi: 10.11684/j.issn.1000-310X.2020.03.005 [11] Li Z, Cai J J, Hao C. Synthesis of sparse linear arrays using reweighted gridless compressed sensing. IET Microw. Antennas Propag., 2021; 15(15): 1945—1959 doi: 10.1049/mia2.12209 [12] 吴国清, 王美刚. 无源声呐稀疏阵无栅瓣性能分析. 声学学报, 2006; 31(6): 506—510 doi: 10.15949/j.cnki.0371-0025.2006.06.005 [13] 霍健, 杨平, 施克仁, 等. 基于遗传算法的二维随机型稀疏阵列的优化研究. 声学学报, 2006; 31(2): 187—192 doi: 10.15949/j.cnki.0371-0025.2006.02.015 [14] Chen P, Zheng Y Y, Zhu W. Optimized simulated annealing algorithm for thinning and weighting large planar arrays in both far-field and near-field. IEEE J. Oceanic Eng., 2011; 36(4): 658—664 doi: 10.1109/JOE.2011.2164957 [15] Gu B, Chen Y, Liu X, et al. Distributed convex optimization compressed sensing method for sparse planar array synthesis in 3-D imaging sonar systems. IEEE J. Oceanic Eng., 2019; 45(3): 1022—1033 doi: 10.1109/JOE.2019.2914983 [16] Liu Y, Liu Q H, Nie Z. Reducing the number of elements in multiple-pattern linear arrays by the extended matrix pencil methods. IEEE Trans. Antennas Propag., 2013; 62(2): 652—660 doi: 10.1109/TAP.2013.2292529 [17] 沈海鸥, 王布宏, 刘新波. 基于酉变换−矩阵束的稀布线阵方向图综合. 电子与信息学报, 2016; 38(10): 2667—2673 doi: 10.11999/JEIT151437 [18] Shen H O, Wang B H, Li L J. Effective approach for pattern synthesis of sparse reconfigurable antenna arrays with exact pattern matching. IET Microw. Antennas Propag., 2016; 10(7): 748—755 doi: 10.1049/iet-map.2015.0628 [19] Yan F, Yang P, Yang F, et al. Synthesis of pattern reconfigurable sparse arrays with multiple measurement vectors FOCUSS method. IEEE Trans. Antennas Propag., 2016; 65(2): 602—611 doi: 10.1109/TAP.2016.2640182 [20] Ma X, Liu Y, Xu K D, et al. Synthesising multiple‐pattern sparse linear array with accurate sidelobe control by the extended reweighted L1-norm minimisation. Electron. Lett., 2018; 54(9): 548—550 doi: 10.1049/el.2018.0333 [21] Yan F, Yang P, Yang F, et al. Synthesis of planar sparse arrays by perturbed compressive sampling framework. IET Microw. Antennas Propag., 2016; 10(11): 1146—1153 doi: 10.1049/iet-map.2015.0775 [22] Park Y, Choo Y, Seong W. Multiple snapshot grid free compressive beamforming. J. Acoust. Soc. Am., 2018; 143(6): 3849—3859 doi: 10.1121/1.5042242 [23] Yang Z, Xie L. Enhancing sparsity and resolution via reweighted atomic norm minimization. IEEE Trans. Signal Process., 2015; 64(4): 995—1006 doi: 10.1109/TSP.2015.2493987 [24] Li Y, Chi Y. Off-the-grid line spectrum denoising and estimation with multiple measurement vectors. IEEE Trans. Signal Process., 2015; 64(5): 1257—1269 doi: https://doi.org/10.1109/TSP.2015.2496294 [25] Yan S, Ma Y, Hou C. Optimal array pattern synthesis for broadband arrays. J. Acoust. Soc. Am., 2007; 122(5): 2686—2696 doi: 10.1121/1.2785037 [26] 鄢社锋, 王文侠. 交替迭代多约束波束优化设计. 声学学报, 2021; 46(6): 896—904 doi: 10.15949/j.cnki.0371-0025.2021.06.011 [27] Chen X, Wang C, Zhang X. DOA and noncircular phase estimation of noncircular signal via an improved noncircular rotational invariance propagator method. Math. Probl. Eng., 2015; 2015: 1—12 doi: 10.1155/2015/235173 [28] Pinchera D, Migliore M D. Comparison guidelines and benchmark procedure for sparse array synthesis. Prog. Electromagn. Res. M, 2016; 52: 129—139 doi: 10.2528/PIERM16092503 -