Efficient Gradient‐based Dictionary Optimisation Method for Time‐varying Array Incorporating with Structure Characteristic

To guarantee good sparsity reconstruction quality, a suitable dictionary should be as orthogonal as possible. Also, to reduce the coherence of a dictionary, a sensing matrix optimisation model should be formulated. In this model, a high-dimension O(N) problem has to be considered because the sparse representation base is overcompleted, thus leading to a heavy computational load. This study proposes an efficient gradient-based method to address the dictionary optimisation problem of the time-varying arrays, whose elements relatively move in an arbitrary but known way. The dictionary optimisation models associated with different array geometries are formulated with distinct structure characteristic, and the Toeplitz and circulant properties are incorporated into the corresponding models to reduce the complexity by dimension reduction from O(N) to O(2) and O(1), respectively. An alternating minimisation approach for sensing matrix design (SMD) is derived by the gradient descent method, and an adaptive stepsize selection method is derived to further reduce complexity. Numerous simulations are conducted, and simulation results demonstrate that the presented methods have lower computational complexity but similar sparse recovery performance as the normalisation-constrained SMD method. Also, the effectiveness of the proposed structured optimisation framework to design a sensing matrix of a time-varying array is verified.