Multiple-snapshot Compressive Beamforming with Non-convex Sparse Constraints
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摘要: 基于极小极大凹惩罚函数约束的压缩感知波束形成，相对于传统l1范数的压缩波束形成来说，可以增强信号的稀疏性，获得更精确的波达方向估计。然而，该算法在强噪声背景下，方位估计结果不稳定。针对这个问题，该文提出一种基于极小极大凹惩罚函数约束的多快拍压缩感知波束形成(MCP-MCSBF)算法。通过多个快拍的联合处理，提供更好的抗噪性能和更精准的波达方向估计结果。仿真结果表明与其他多快拍波达方向估计算法相比，该文算法提供了更优的精确性和更高的角度分辨率，湖试结果则进一步验证了所提算法的有效性。Abstract: Compressive beamforming based on the minimax-concave penalty function constraint, compared with the traditional
$ {l_1} $ norm compressive beamforming, can enhance the sparsity of the signal and obtain a more accurate Direction Of Arrival (DOA) estimation. However, under the background of strong noise, the azimuth estimation result of this algorithm is unstable. In response to this problem, a Multiple-Snapshot Compressed sensing BeamForming based on the constraint of the Minimax Concave Penalty (MCP-MCSBF) function is proposed. Through the joint processing of multiple snapshots, it provides better anti-noise performance and more accurate direction of arrival estimation results. The simulation results show that compared with other multi-snapshot direction of arrival estimation algorithms, the proposed algorithm provides better accuracy and higher angular resolution. The lake test results verify further the effectiveness of the proposed algorithm. -
表 1 基于MCP函数约束的压缩波束形成算法步骤
输入：观测数据${\boldsymbol{y}}$，参数$ \varepsilon $，超参数$ \gamma $，最大迭代次数$ Q $，终止条
件${\rm{tol}} = {10^{ - 3} }$。参数向量$ {t_0} = 1 $，${{\boldsymbol{u}}^0}$和${{\boldsymbol{s}}^0}$的初始值均设为
${ {\textit{0}}}$向量，且${{\boldsymbol{u}}^0} = {{\boldsymbol{u}}^1}$, ${{\boldsymbol{s}}^0} = {{\boldsymbol{s}}^1}$。(1) 当$ q \le Q $时： (2) 根据式(14)和式(13)更新$ w_u^q $， (3) 获得变量${\tilde {\boldsymbol{u}}^q}$，并按照式(9)更新变量为${{\boldsymbol{u}}^{q + 1} }$， (4) 根据式(14)和式(13)更新$ w_s^q $， (5) 同理获得变量${\tilde {\boldsymbol{s}}^q}$，并按照式(12)更新变量为${{\boldsymbol{s}}^{q + 1} }$。 (6) 如果${{\boldsymbol{s}}^{q + 1} }$和${{\boldsymbol{s}}^q}$的变化$ \le {\rm{tol}} $： (7) 那么停止迭代，且最优估计${{\boldsymbol{s}}^ * } = {{\boldsymbol{s}}^{q + 1} }$， (8) 找到${{\boldsymbol{s}}^ * }$的峰值，并将其对应到DOA估计的角度${{\boldsymbol{\theta}} ^ * }$。 (9) 结束 (10) 结束 输出：DOA估计的角度${{\boldsymbol{\theta}} ^ * }$。 
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表 2 MCP-MCSBF算法步骤
输入：多快拍观测数据矩阵${\boldsymbol{Y}}$。 (1) 根据式(16)和式(17)计算降噪后的观测矩阵${{\boldsymbol{Y}}_{\hat K} }$， (2) 构造加权向量$\hat {\boldsymbol{w}}$，满足$ {\hat w_1} + {\hat w_2} + \cdots + {\hat w_{\hat K}} = 1 $， (3) 根据式(25)计算${{\boldsymbol{y}}_{\hat K} }$， (4) 将${{\boldsymbol{y}}_{\hat K} }$代入式(26)中得到全新的目标函数， (5) 根据表1的算法得到DOA估计角度${{\boldsymbol{\theta}} ^ * }$。 输出：DOA估计的角度${{\boldsymbol{\theta}} ^ * }$。
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