Direction of arrival estimation for more correlated sources than active sensors

• The number of correlated sources can be larger than the number of active antennas.
• Only computation up to second-order statistics is involved.
• A periodic scanning is performed on an underlying uniform array.
• Least squares (LS) reconstruction of the correlation matrix of the underlying array.
• Angular power spectrum reconstruction using LS, sparsity-regularized LS, or MUSIC.

In this paper, a new direction of arrival (DOA) estimation method for more correlated sources than active receiving antennas is proposed. The trick to solve this problem using only second-order statistics is to consider a periodic scanning of an underlying uniform array, where a single scanning period contains several time slots and in different time slots different sets of antennas are activated leading to a dynamic non-uniform array with possibly less active antennas than sources in each time slot. We collect the spatial correlation matrices of the active antenna arrays for all time slots and are able to present them as a linear function of the spatial correlation matrix of the underlying array. We provide a necessary and sufficient condition for this system of equations to be full column-rank, which allows for a least squares (LS) reconstruction of the spatial correlation matrix of the underlying array. Some practical greedy algorithms are presented to design dynamic arrays satisfying this condition. In a second step, we use the resulting spatial correlation matrix of the underlying array to estimate the DOAs of the possibly correlated sources by spatial smoothing and MUSIC. Alternatively, we can express this matrix as a linear function of the correlation matrix of the sources (incoming signals) at a grid of investigated angles, and solve this system of equations using either LS or sparsity-regularized LS (possibly assisted by additional constraints), depending on the grid resolution compared to the number of antennas of the underlying array.