基于深度展开ISTA网络的混合源定位方法

针对嵌套阵列下近场和远场混合源定位问题，本文通过构建和训练深度展开迭代收缩阈值算法（Iterative Shrinkage Thresholding Algorithm，ISTA）网络实现混合源的波达方向（direction of arrival， DOA）和距离参数估计。首先考虑到近场源协方差矩阵具有Hermitian矩阵形式，远场源协方差矩阵具有Hermitian和Toeplitz矩阵形式，通过将混合源协方差矩阵进行差分可以得到近场源差分向量，其中近场源差分向量转换到实数域，可以显著降低深度展开ISTA网络的计算复杂度。接着将不同参数下的近场源差分向量和近场源真实空间谱进行配对作为训练样本，对近场源深度展开ISTA网络进行训练，其中深度展开ISTA网络的隐藏层对应模型驱动ISTA方法的迭代步骤。然后利用估计出的近场源DOA和距离参数，通过子空间差分方法得到远场源协方差向量。最后将不同参数下的远场源协方差向量和远场源真实空间谱进行配对作为训练样本，对远场源深度展开ISTA网络进行训练，其中远场源协方差向量同样转换到实数域。在深度展开ISTA网络的训练过程中，损失函数只与重构误差和网络输出的稀疏性有关，不需要混合源的标签，可以认为是无监督学习。仿真实验表明所提出的深度展开ISTA网络能够实现混合源识别和定位。此外所提出的深度展开ISTA网络具有可解释参数，对近场和远场混合源的离网格参数估计具有泛化能力。相较于模型驱动ISTA方法，经过训练的深度展开ISTA网络的收敛速度更快，对近场和远场混合源定位的精度更高。 

Mixed Source Localization Based on Deep Unfolded ISTA Network

? ?Aiming at the problem of mixed near-field and far-field source localization under the geometry of nested array， this paper constructed and trained the deep unfolded Iterative Shrinkage Thresholding Algorithm （ISTA） networks to estimate the direction of arrival （DOA） and range parameters of mixed sources. Firstly， considering that the covariance matrix of near-field sources had the form of a Hermitian matrix and that of far-field sources had the form of a Hermitian Toeplitz matrix， the difference vector of near-field sources can be obtained by differencing the covariance matrix of mixed sources. Herein， the difference vector of near-field sources was transformed into real domain， which can significantly reduce computational complexity in the deep unfolded ISTA network. Then， the difference vector and the corresponding spatial spectrum of near-field sources with different parameters were paired as the training samples， which were utilized to train the deep unfolded ISTA network of near-field sources. Herein， the hidden layers of deep unfolded ISTA network were corresponding to the iterative steps of model-based ISTA method. Next， based on the estimated DOA and range parameters of near-field sources， the covariance vector of far-field sources can be obtained by exploiting the subspace difference method. Finally， the covariance vector and the corresponding spatial spectrum of far-field source with different parameters were paired as the training samples， which were utilized to train the deep unfolded ISTA network of far-field sources. Herein， the covariance vectors of far-field sources were also transformed into real domain. In the training process of deep unfolded ISTA networks， the loss function was only related to the reconstruction error and the sparsity of network output， which did not require the labels of mixed sources and can be considered as unsupervised learning. Simulation experiments show that the proposed deep unfolded ISTA networks can realize the classification and localization of mixed sources. Moreover， the proposed deep unfolded ISTA networks with interpretable parameters have generalization ability for the off-grid parameter estimation of mixed near-field and far-field sources. Compared with the model-driven ISTA method， the trained deep unfolded ISTA networks have faster convergence speed and higher localization accuracy for mixed near-field and far-field source localization.