Deep Learning-Based Single-Snapshot DOA Estimation With a Unit-Nuclear-Norm Structured Covariance Prior

Direction-of-arrival (DOA) estimation from a single snapshot is more challenging than its multi-snapshot counterpart, yet it remains crucial for sensing applications. This paper proposes a deep learning-based method to reconstruct a Toeplitz and positive-definite array covariance matrix with the unit nuclear norm property. Rather than emphasizing the neural network architecture, our focus is placed on the design of the network output layer and the loss function. The output layer is designed to generate a learned spatial spectrum consisting of a fixed-size set of DOAs and their corresponding signal-to-noise ratios, based on which a Toeplitz and positive-definite covariance matrix is reconstructed. The unit nuclear norm constraint is hard-encoded into the covariance matrix without hyperparameters. The loss function is derived from a whitened fitting error, followed by a projected multiplicatively symmetric transformation. The Toeplitz and positive-definite structure fully captures the properties of a true covariance matrix, while the unit nuclear norm constraint promotes sparsity and eliminates the need to add an extra output head for noise variance estimation. The new loss function enhances the accuracy of estimating the smaller eigenvalues of the reconstructed matrix. Based on the reconstructed covariance matrix, the signal number and DOA estimation can be performed using conventional methods. Numerical simulations and outdoor measurements are utilized to demonstrate the effectiveness and superiority of the proposed method with the uniform linear array (ULA) and non-uniform linear array (NULA), as well as its robustness to array nonidealities.