Design of sparse reconfigurable array using reweighted atomic norm minimization
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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.
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图 1 不同参数对使用RANM的稀疏可重构直线阵设计方法的性能影响 (a) 采样点数
$J$ 对性能的影响; (b) 约束参数$\eta $ 对性能的影响; (c) 重加权参数$\epsilon$ 对性能的影响; (d) 能量比阈值$\delta $ 对性能的影响
图 2
$M = 2$ 时稀疏可重构直线阵波束方向图及阵元激励分布图 (a) 笔形波束方向图对比; (b) 平顶波束方向图对比; (c) RANM设计的稀疏可重构直线阵阵元激励幅度; (d) RANM设计的稀疏可重构直线阵阵元激励相位
图 3
$M = 3$ 时稀疏可重构直线阵波束方向图及阵元激励分布图 (a) 笔形波束方向图对比; (b) 平顶波束方向图对比; (c) 余割平方波束方向图对比; (d) RANM设计的稀疏可重构直线阵阵元激励幅度; (e) RANM设计的稀疏可重构直线阵阵元激励相位
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
图 6 不同采样区间下RANM设计的稀疏可重构直线阵列的波束图 (a) 采样区间[−1, 1]时RANM的设计结果; (b) 采样区间[−1.5, 1.5]时RANM的设计结果; (c) 采样区间[−2, 2]时RANM的设计结果
图 7 幅相扰动对稀疏可重构直线阵的波束模式影响 (a) 阵元幅度扰动对匹配误差的影响; (b) 阵元相位扰动对匹配误差的影响; (c) 幅度扰动为12%时的笔形波束对比图; (d) 幅度扰动为12%时的平顶波束对比图
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} $ 
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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
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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 $) 参考波束[8] 20 — −20.03 −19.50 −20.01 0.50 9.50 MT-BCS 17 3.14 × 10−2 −17.87 −16.96 −16.16 0.02 9.00 RANM 17 4.92 × 10−3 −18.93 −17.48 −17.04 0.19 9.42
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表 4 不同主瓣指向的可重构直线阵列性能表
参考可重构直线阵列 RANM算法获得的稀疏可重构直线阵列 模式数量M PSLL (dB) 阵元数量 稀疏后阵元数量 匹配误差 PSLL (dB) 最小阵元间距 ($\lambda $) 阵列孔径 ($\lambda $) 3 −40 64 41 2.38 × 10−6 −38.6 0.67 31.47 5 −40 64 47 4.64 × 10−7 −39.9 0.55 31.49 7 −40 64 51 1.01 × 10−6 −38.2 0.58 31.48
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表 5 不同旁瓣级的可重构直线阵性能表
参考可重构直线阵 RANM算法设计的稀疏可重构直线阵 模式数量 旁瓣级 阵元数量 稀疏后阵元数量 匹配误差 PSLL (dB) 最小阵元间距 ($\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
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