Background: Uniform Linear Array (ULA) processing is a standard approach for identifying the Direction-of-Arrival (DoA) of a number of signal-emitting sources. A ULA receiver gathers a collection of received-signal snapshots (vector-measurements) which it processes to estimate the autocorrelation matrix of the received-signal model. Typically, the MUltiple SIgnal Classification (MUSIC) algorithm is applied on the estimated autocorrelation matrix to identify the DoAs of the signal-emitting sources. In this setting, the number of identifiable sources is upper-bounded by the number of sensors the array is equipped with. To remedy this limitation of ULAs, researchers proposed new and sparse array geometries which enable the identification of more sources than sensors. Nested arrays are early examples of sparse arrays. Coprime arrays are another class of sparse arrays in which the sensor-element locations are determined by principles of Coprime number theory. In Coprime array processing, the receiver gathers a collection of received-signal snapshots and estimates the autocorrelation matrix of the physical array. Then, with intelligent processing, the receiver processes the estimated autocorrelation matrix of the physical array and estimates a new autocorrelation matrix estimate which corresponds to a larger (virtual) ULA which is known as coarray and, in turn, enables the identification of more sources than sensors. 

Motivation: By the received-signal model and the specific coprime array geometry, the receiver obtains multiple estimates for each element of the coarray’s autocorrelation matrix. Therefore, the receiver has to sample these multiple estimates and return a single estimate per entry. Existing sampling approaches are the selection-sampling and averaging-sampling approaches.  According to the former, the receiver selects arbitrarily a single estimate per entry, while, according to the latter, the receiver returns one estimate per entry by averaging all available estimates corresponding to that entry. Intuitively, one would expect that averaging-sampling will return superior estimates with respect to the Mean-Squared-Error (MSE) estimation criterion. However, there is no formal proof whether or not averaging-sampling returns superior autocorrelation estimates compared to selection-sampling. Moreover, both the selection- and averaging-sampling approaches have been proposed by researchers arbitrarily and are not optimized with respect to any criterion. Finally, by the received-signal model, the autocorrelation matrix of the coarray has to be (1) Positive-Definite, (2), Hermitian, (3) Toeplitz, and (4) its noise-subspace eigenvalues have to be equal. To date, there is no coarray autocorrelation matrix estimate satisfying (1)-(4). 

Contributions: In this line of research, our contributions are as follows. (i) We formally prove that averaging-sampling attains superior autocorrelation estimates with respect to the MSE metric and derive closed-form MSE expressions. (ii) We derive a new autocorrelation sampling approach which, in lieu of any prior knowledge at the receiver, assumes that the DoAs are uniformly distributed in space and, in view of this minor assumption, designs a stochastic sampling approach that minimizes the MSE estimation error. (iii) We propose an algorithm framework for computing an improved and structured estimate that satisfies the structure properties (1)-(4) above of the nominal autocorrelation matrix of the coarray. We conduct extensive numerical studies which, in a one-to-one analogy, (i) validate the derived closed-form MSE expressions, (ii) illustrate that the proposed stochastic sampling approach attains superior MSE performance with respect to the MSE estimation error and DoA estimation performance compared to selection- and averaging-sampling, and (iii) the proposed structured estimate outperforms existing estimates with respect to the MSE estimation error and DoA estimation. 

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