Direction of arrival (DOA) estimation from the perspective of sparse signal representation has attracted tremendous attention in past years, where the underlying spatial sparsity reconstruction problem is linked to the compressive sensing (CS) framework. Although this is an area with ongoing intensive research and new methods and results are reported regularly, it is time to have a review about the basic approaches and methods for CS-based DOA estimation, in particular for the underdetermined case. We start from the basic time-domain CS-based formulation for narrowband arrays and then move to the case for recently developed methods for sparse arrays based on the co-array concept. After introducing two specifically designed structures (the two-level nested array and the co-prime array) for optimizing the virtual sensors corresponding to the difference co-array, this CS-based DOA estimation approach is extended to the wideband case by employing the group sparsity concept, where a much larger physical aperture can be achieved by allowing a larger unit inter-element spacing and therefore leading to further improved performance. Finally, a specifically designed uniform linear array structure with associated CS-based underdetermined DOA estimation is presented to exploit the difference co-array concept in the spatio-spectral domain, leading to a significant increase in degrees of freedom. Representative simulation results for typical narrowband and wideband scenarios are provided to demonstrate their performance.

In this work, we present a new coprime array structure termed thinned coprime array (TCA), which exploits the redundancy in the structure of existing coprime array and achieves the same virtual aperture and degrees of freedom (DOFs) as the conventional coprime array with much fewer number of sensors. In comparison to other sparse arrays, thinned coprime arrays possess more unique lags (total number of difference co-arrays) than the nested arrays, while the number of consecutive lags (connected co-arrays) generated is close to 75 percent of the consecutive lags of the nested arrays with hole-free co-arrays. The resulting structure is much sparser and the number of sensor pairs with small separation is significantly reduced. Theoretical properties and proofs are provided and simulations are presented to demonstrate its robustness against heavy levels of mutual coupling using compressive sensing (CS) based direction of arrival (DOA) estimation as well as certain additional desirable characteristics.

A class of low-complexity compressive sensing-based direction-of-arrival (DOA) estimation methods for wideband co-prime arrays is proposed. It is based on a recently proposed narrowband estimation method, where a virtual array model is generated by directly vectorizing the covariance matrix and then using a sparse signal recovery method to obtain the estimation result. As there are a large number of redundant entries in both the auto-correlation and cross-correlation matrices of the two sub-arrays, they can be combined together to form a model with a significantly reduced dimension, thereby leading to a solution with much lower computational complexity without sacrificing performance. A further reduction in complexity is achieved by removing noise power estimation from the formulation. Then, the two proposed low-complexity methods are extended to the wideband realm utilizing a group sparsity based signal reconstruction method. A particular advantage of group sparsity is that it allows a much larger unit inter-element spacing than the standard co-prime array and therefore leads to further improved performance.

An effective sparse array extension method for maximizing the number of consecutive lags in the fourth-order difference co-array is proposed, leading to a novel enhanced sparse array structure based on co-prime arrays (CPAs) with significantly increased number of degrees of freedom (DOFs). One method to exploit the increased DOFs based on nonstationary signals is also proposed, with simulation results provided to demonstrate the effectiveness of the proposed structure.

Based on the high order difference co-array concept, an enhanced four level nested array (E-FL-NA) is first proposed, which optimizes the consecutive lags at the fourth order difference co-array stage. To simplify the formulations for sensor locations for comprehensive illustration and also convenient structure construction, a simplified and enhanced four level nested array (SE-FL-NA) is then proposed, whose performance is compromised but still better than the four level nested array (FL-NA). This simplified structure is further extended to the higher order case with multiple sub-arrays, referred to as simplified and enhanced multiple level nested arrays (SE-ML-NAs), where significantly increased degrees of freedom (DOFs) can be provided and exploited for underdetermined DOA estimation. Simulation results are provided to verify the superior performance of the proposed E-FL-NA, while a higher number of detectable sources is achieved by the SE-ML-NA with a limited number of physical sensors.

Sparse linear arrays (SLAs) can be designed in a systematic way, with the ability for underdetermined DOA estimation where a greater number of sources can be detected than that of sensors. In this paper, as the first stage, a new systematic design named multi-subarray dilated nested array (MDNA), whose difference co-array (DCA) can be proved to be hole-free, is firstly proposed by introducing a sparse ULA and multiple identical dense ULAs with appropriate sub-ULA spacings. The MDNA will degenerate into the nested array under specific conditions, and the uniform degrees of freedom (uDOFs) of MDNA is larger than that of its parent nested array. On the basis of MDNA, to reduce the mutual coupling effect, an augmented multi-subarray dilated nested array (AMDNA) is constructed by migrating some elements of the dense segments of MDNA, without reducing the number of uDOFs. Several theoretical properties of the proposed array structures are proved, and simulation results are provided to demonstrate the effectiveness and superiority of the proposed AMDNA over some existing sparse arrays.