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.