## 1 Introduction
The ESPRIT method supposes that the sources are narrowband signals, and also that the number of received signals is less than the number of sensors [20]: the method exploits the rotational invariance of the signal subspace, which is produced by two arrays with a translational invariance structure. It is essential to separate these subarrays translationally and not rotationally. The performances of the algorithms mentioned above deteriorate significantly when the data recorded are few or when the received signals are highly correlated. Resorting to a larger number of snapshots or removing the correlation between the incoming signals increases both complexity and execution time. Additionally, the large numbers of snapshots required are not always available in wireless environments that change rapidly. Recent research on the DOA estimation problem has focused on Bayesian Compressed Sensing (BCS) and Sparse Signal Reconstruction (SSR) theories [21; 22; 23], exploring whether, if specific conditions are satisfied, it is possible to recover signals from fewer snapshots than with classical techniques [24; 25]. Since the incident signals on a sensor array are intrinsically sparse in the spatial domain, the exploitation of Compressive Sensing (CS) has been extended to include the DOA estimation problem in array signal processing, where application of these methods improved estimation accuracy with fewer measurements and robustness to noise as presented in CS-MUSIC [26] and subspace-augmented (SA-) MUSIC [27]. Although these approaches offer good robustness to noise and correlated signals, their estimation performance is degraded unless there is a prior knowledge of the number of arrival signals [28]. The original work [24] exploited spatial sparsity while providing a method different in principle to the \(\ell_{1}\)-SVD approach for DOA estimation. Merging the SVD approach with the \(\ell_{1}\) norm minimization based method can provide high accuracy for both narrowband and wideband scenarios. With single snapshot or single measurement vectors (SMV), the \(\ell_{1}\) optimisation approach is preferred to sparse recovery because of guaranteed recovery resolution [25]. In the case of multiple snapshots, sparse signals at all measurements provide the same support. Thus, such joint sparsity is a favourable approach to improving the average recovery success under a similar uncorrelated matrix condition [29]. A compressive beamforming algorithm using CS techniques was proposed in [30] to solve the problem of DOA estimation, with a Nyquist sampling rate at the sensors. However, this is sensitive to signal correlation. Signal reconstruction from compressive measurements using an efficient technique based on CS theory called Sparse Bayesian Learning (SBL) has been proposed recently for sparse signal recovery [22]. With this SBL method, the signal recovery is formulated from a Bayesian perspective whereas the sparsity information is utilized by considering previously known sparse distributions of the incident signal at all snapshots. Note that the \(\ell_{1}\)-norm regularised optimisation is considered a special case of the SBL algorithm when prior Laplace signal of the impinging signals is adopted with a maximum a posteriori (MAP) optimal estimate approach. Analytical and experimental results both verify that the SBL algorithm can provide better performance than \(\ell_{1}\)-norm regularised optimisation [31]. In general, the measurement matrix in SBL is assumed to be accurately known; unfortunately, this assumption is invalid when perturbations on the measurement matrix are considered [32; 33] or when the array manifold suffers from imperfections [34]. Although the CS-based techniques mentioned above give improved estimation resolution when there is a limited number of snapshots, there are still difficulties in applying them in practical situations where the correct AOAs are not located on the sampling grid.