A co-prime array uses two uniform linear subarrays to construct an effective difference coarray with certain desirable characteristics, such as a high number of degrees-of-freedom for DOA estimation. In this case, the co-prime array concept was generalized with two operations. The first operation was through the compression of the interelement spacing of one subarray and the resulting structure treats the existing variations of co-prime array configurations as well as the nested array structure as its special cases. The second operation exploits two displaced subarrays, and the resulting co-prime array structure allows the minimum inter-element spacing to be much larger than the typical half-wavelength requirement, making them useful in applications where a small interelement spacing is infeasible. This allowed derivation of the analytical expressions for the coarray aperture, the achievable number of unique lags, and the maximum number of consecutive lags for quantitative evaluation, comparison, and design of co-prime arrays.

The co-prime array, which utilizes a co-prime pair of uniform linear sub-arrays, provides a systematical means for sparse array construction. On the other hand, utilizing spectrum bandwidth in co-prime array can achieve a number of advantages. (a) Utilizing multiple frequencies equivalently provides multiple virtual arrays to achieve a higher number of degrees of freedom; (b) Fusing multi-frequency signals improves the robustness of DOA estimation; and (c) The use of signal bandwidth and co-prime array provides DOA-range resolution for target localization.

In one of the proposed schemes, a co-prime array is operated at multiple frequencies in order to fill the missing coarray elements, thereby enabling the co-prime array to form consecutive coarray lags and effectively utilize all of the offered degrees of freedom (DOFs) with subspace-based DOA estimation methods. In another proposed scheme, a single sparse uniform linear array is used to exploit two or more continuous-wave signals whose frequencies satisfy a co-prime relationship. This extends the co-prime array and filtering to a joint spatio-spectral domain, thereby achieving high flexibility in array structure design to meet system complexity constraints.

The DOA estimation is obtained using group sparsity-based compressive sensing techniques. The achievable number of DOFs is derived for the two-frequency case, and an upper bound of the available DOFs is provided for multi-frequency scenarios. The third scheme considered the frequency diverse array (FDA) radar, which offers a range-dependent beampattern capability.

The spatial and range resolutions of an FDA radar are fundamentally limited by the array geometry and the frequency offset. This limitation was overcome by introducing a novel sparsity-based multi-target localization approach incorporating both co-prime arrays and co-prime frequency offsets. The covariance matrix of the received signals corresponding to all sensors and employed frequencies was formulated to generate a space-frequency virtual difference coarray. The joint DOA and range estimation was cast as a two-dimensional sparse reconstruction problem and is solved within the Bayesian compressive sensing framework. The superiority of the proposed approach in terms of DOA-range resolution, localization accuracy, and the number of resolvable targets were evidently demonstrated.

DOA estimation of a mixture of coherent and uncorrelated targets was performed using sparse reconstruction and active nonuniform arrays. The data measurements from multiple transmit and receive elements can be considered as observations from the sum coarray corresponding to the physical transmit/receive arrays. The vectorized covariance matrix of the sum coarray observations emulates the received data at a virtual array whose elements are given by the difference coarray of the sum coarray.

Sparse reconstruction is used to fully exploit the significantly enhanced degrees-of-freedom offered by the difference coarray of the sum coarray for DOA estimation. Simulated data from multiple-input multiple-output minimum redundancy arrays and transmit/receive co-prime arrays were used for performance evaluation of the proposed sparsity-based active sensing approach.

This work was done by Moeness G. Amin, Fauzia Ahmad, and Yimin D. Zhang of Villanova University for the Office of Naval Research. For more information, download the Technical Support Package (free white paper) here under the Information Science category. NRL-0077