Recently, deep unfolding networks have been widely used in direction of arrival (DOA) estimation because of their improved estimation accuracy and reduced computational cost. However, few have considered the existence of a nested array (NA) with off-grid DOA estimation. In this study, we present a deep sparse Bayesian learning (DSBL) network to solve this problem. We first establish the signal model for off-grid DOA with NA. Then, we transform the array output into a real domain for neural networks. Finally, we construct and train the DSBL network to determine the on-grid spatial spectrum and off-grid value, where the loss function is calculated using reconstruction error and the sparsity of network output, and the layers correspond to the steps of the sparse Bayesian learning algorithm. We demonstrate that the DSBL network can achieve better generalization ability without training labels and large-scale training data. The simulation results validate the effectiveness of the DSBL network when compared with those of existing methods.

Target misalignment can cause beam pointing deviations and degradation of sidelobe performance. In order to eliminate the effect of target misalignment, we formulate the jamming sub-space recovery problem as a linearly modified atomic norm-based optimization. Then, we develop a deep-unfolding network based on the alternating direction method of multipliers (ADMM), which effectively improves the applicability and efficiency of the algorithm. By using the back-propagation process of deep-unfolding networks, the proposed method could optimize the hyper-parameters in the original atomic norm. This feature enables the adaptive beamformer to adjust its weight according to the observed data. Specifically, the proposed method could determine the optimal hyper-parameters under different interference noise matrix conditions. Simulation results demonstrate that the proposed network could reduce computational cost and achieve near-optimal performance with low complexity.