## 1. Introduction

Direction of arrival (DOA) estimation is a popular research topic in array signal processing and has been widely used in various fields, such as antenna [

1], radar [

2], and vehicle localization [

3]. For example, driverless vehicle technologies are now quickly emerging with the development of artificial intelligence (AI) and Internet of Things (IoT). DOA can offer a robust and accurate solution for driverless vehicle localization, which has shown many advantages over the conventional global positioning system (GPS) [

4]. DOA can also be applied in patient tracking which is a crucial part of the elderly healthcare systems, playing a significant role in today’s aging era [

5]. Among the conventional DOA estimation techniques, MUSIC (multiple signal classification ) [

6] and ESPRIT (estimation of parameters by rotational invariant techniques ) [

7] have become mainstreams due to their super-resolution. However, prior knowledge of source number and a sufficient number of snapshots are required for these subspace-based algorithms. Moreover, the performance of these methods deteriorates seriously when the sources are correlated.

In recent years, the rapid development of compressed sensing (CS) [

8] and its application in array antenna and array signal processing [

9,

10,

11,

12,

13,

14,

15,

16,

17,

18,

19], has provided a new solution for DOA estimation. A great deal of research has emerged, identifying the DOAs by formulating the problem as a sparse signal recovery problem. Assuming the sparsity of the signal in the spatial domain, we can process the array output directly instead of estimating a sensor covariance matrix. Therefore, sparse signal recovery algorithms can address the limitations of subspace-based algorithms and can be applied in several demanding scenarios with no prior knowledge of the source number, correlated sources, and a limited number of snapshots.

Many sparse signal recovery algorithms have been proposed in CS literature, which can be applied to estimate the DOAs by exploiting the spatial sparsity [

20,

21,

22,

23,

24]. FOCUSS (FOcal Underdetermined System Solver) [

25] uses an iterative method which is based on weighted norm minimization. However, it cannot guarantee global optimal.

${l}_{1}$-SVD [

26] enforces sparsity by imposing penalties based on the

${l}_{1}$-norm, which can achieve high accuracy, but has the difficulty of choosing regularized parameters. SPICE [

27] is obtained by the minimization of a covariance matrix fitting criterion and is useful in both many-snapshot cases and single-snapshot situations. SBL [

28] uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors, which has superior performance because of their use of data adaptive priors and capability of automatic regularization parameter selection.

For these CS methods mentioned above, which are called on-grid methods, true DOAs are assumed as lying on a set of fixed grid points. Therefore, the existing sparse representation techniques can be directly applied. However, in practice, the true DOAs may not be exactly on the fixed grid. The off-grid gap, which is the gap between true DOAs and its nearest grid point, always exists. Besides, the grid interval should be determined empirically. On the one hand, small grid interval brings not only a high computational workload but also a strong correlation with adjacent steering vectors. On the other hand, large grid interval leads to large model error.

To address the off-grid gap, many off-grid methods, in which a sampling grid is still required but true DOAs are not restricted to be on the grid, are proposed. Sparse total least-squares (STLS) [

29] approach can yield a MAP optimal estimate if the matrix perturbation caused by the basis mismatch is Gaussian. OGSBI [

30] takes a Bayesian perspective on off-grid methods. The model of OGSBI is based on a first-order Taylor series expansion, and a Laplace prior is assumed to exploit the spatial sparsity of signals. PSBL [

31] takes an off-grid model based on a perturbed sparse Bayesian learning, in which a linear interpolation between two adjacent grid points is adopted. However, these off-grid methods mentioned above are still faced with the trade-off between accuracy and computational complexity. A dense sampling grid is needed to achieve high accuracy, which, however, will slow its speed. On the contrary, a coarser grid can greatly reduce the computational workload but will introduce more model errors. Root SBL (RSBL) [

32] decreases computational workload by using a root method. It also maintains high accuracy with a coarse grid. However, if more than one DOA exists in the same grid interval, RSBL may fail to discriminate these DOAs in the case of coarse grid. GEDOA [

33] combines off-grid methods and grid refinement to make the grid nonuniformly evolve from coarse to dense, which can discriminate closely spaced DOAs as well as achieve higher efficiency than RSBL. However, its performance will get worse in the low SNR condition.

Both on-grid and off-grid methods are grid-based methods. Another kind of methods to handle the off-grid gap is gridless methods [

34,

35,

36,

37]. They operate in the continuous domain directly so that they can avoid the grid mismatch problem. They are convex and have strong theoretical guarantees as well. However, this kind of methods is only applied to the uniform or sparse linear arrays.

In this paper, an off-grid method named grid reconfiguration direction of arrival (GRDOA) based on sparse Bayesian learning is proposed. Unlike most off-grid methods, in which uniformly sampling and fixed number of grid points are used, the grid number of GRDOA is varied during the reconfiguration process, and the final grid is nonuniform. It has two integral parts: the initial estimation and the fine estimation. Compared with ${l}_{1}$-SVD, OGSBI, and RSBL, GRDOA has the advantages of less computational complexity and remains sufficiently accurate. Besides, it can successfully discriminate DOAs that are in the same grid interval. Furthermore, GRDOA has better robustness than GEDOA, especially in low SNR condition.

The rest of this paper is organized as follows. In

Section 2, we examine the off-grid DOA model. In

Section 3, we introduce the proposed GRDOA algorithm. In

Section 4, we present our simulation results. In

Section 5, we conclude this paper.

## 2. Data Model

Assume that

K narrow-band and far-field source signals with DOAs (

${\theta}_{1},{\theta}_{2},\dots ,{\theta}_{K}$) impinge on a ULA with

M sensors, where

$K<M$. The signal received at the output of the array can be written as

where

$\mathbf{y}\left(t\right)={[{y}_{1}\left(t\right),{y}_{2}\left(t\right),\dots ,{y}_{M}\left(t\right)]}^{T}$,

$\mathbf{s}\left(t\right)={[{s}_{1}\left(t\right),{s}_{2}\left(t\right),\dots ,{s}_{K}\left(t\right)]}^{T}$,

${(\xb7)}^{T}$ is the transpose,

T is the number of snapshots,

$\mathbf{A}=[\mathbf{a}\left({\theta}_{1}\right),\mathbf{a}\left({\theta}_{2}\right),\dots ,\mathbf{a}\left({\theta}_{K}\right)]$ is an

$M\times K$ matrix of steering vectors with

$\mathbf{a}\left({\theta}_{k}\right)=[1,{v}_{{\theta}_{k}},\dots ,{v}_{{\theta}_{k}}^{M-1}{]}^{T}$,

${v}_{{\theta}_{k}}={e}^{-j2\pi d/\lambda sin\left({\theta}_{k}\right)}$,

d is the distance between adjacent sensors,

$\lambda $ is the wavelength of the source, and

$\mathbf{e}\left(t\right)={[{e}_{1}\left(t\right),{e}_{2}\left(t\right),\dots ,{e}_{M}\left(t\right)]}^{T}$ is an unknown noise vector. For simplicity, Equation (

1) can be written as

with the definitions of

$\mathbf{Y}=[\mathbf{y}\left({t}_{1}\right),\mathbf{y}\left({t}_{2}\right),\dots ,\mathbf{y}\left({t}_{T}\right)]$,

$\mathbf{S}=[\mathbf{s}\left({t}_{1}\right),\mathbf{s}\left({t}_{2}\right),\dots ,\mathbf{s}\left({t}_{T}\right)]$, and

$\mathbf{E}=[\mathbf{e}\left({t}_{1}\right),\mathbf{e}\left({t}_{2}\right),\dots ,\mathbf{e}\left({t}_{T}\right)]$.

To cast the DOA estimation as a sparse representation problem, the sparse signal model is constructed. Uniform sampling over DOA range is used in the conventional grid-based methods. Let

$\tilde{\mathit{\theta}}=[{\tilde{\theta}}_{1},{\tilde{\theta}}_{2},\dots ,{\tilde{\theta}}_{N}]$ be a fixed sampling grid in the range

$[-\frac{\pi}{2},\frac{\pi}{2}]$, where

N denotes the grid number

$(N\gg M)$ and

$r={\tilde{\theta}}_{2}-{\tilde{\theta}}_{1}$ denotes the grid interval. If the grid is fine enough, the true DOAs will lie on (or, practically, close to) the grid. Then, the data model can be written as

where

$\tilde{\mathbf{A}}=[\mathbf{a}({\tilde{\theta}}_{1}),\mathbf{a}({\tilde{\theta}}_{2}),\dots ,\mathbf{a}({\tilde{\theta}}_{N})]$,

$\mathbf{X}$ is set as a zero-padded extension of

$\mathbf{S}$ whose non-zero rows correspond to the true DOAs at

${\theta}_{k},k=1,2,\dots ,K$.

Usually, the DOAs cannot be right on the grid, which leads to the off-grid gap problem, as shown in

Figure 1 [

38]. To handle this problem, there are two kinds of off-grid methods: one is based on a fixed grid with a joint estimation of the sparse signal and the grid offset, and the other relies on a dynamic grid. For the former, a linear approximation is usually used to model the off-grid problem [

30]. Dynamic grid methods [

32,

33,

39] were proposed in which the grid is considered as the adjustable parameters. For this kind of methods, the computational complexity is significantly reduced, and the modeling error can be almost eliminated. In this paper, we give a novel dynamic grid method to eliminate the off-grid gap.