1. Introduction
The direction of arrival (DOA) estimation is an important problem in array signal processing, and has a wide range of applications in the communication, radar, and sonar fields [
1,
2,
3]. The conventional beamforming (CBF) method, also known as the Bartlett beamforming method [
4], is the earliest known arraybased spatial spectrum estimation algorithm. CBF extends the Fourierbased method of the spectrum estimation in the time domain to the spatial domain, and the angular resolution of CBF is limited by the "
Rayleigh limit". To improve the resolution performance, methods based on the timedomain nonlinear estimation method have been proposed, such as the maximum entropy method (MEM) [
5] and the minimum variance method (MVM) [
6]. Furthermore, the subspacebased methods have been proposed in [
7] to improve DOA estimation performance, including the multiple signal classification (MUSIC) method [
8], RootMUSIC method [
9,
10], and the estimating signal parameters via rotational invariance techniques (ESPRIT) method [
11].
Additionally, the least squaresESPRIT method [
12,
13] provides superior resolution and is asymptotically unbiased and efficient. More recently, maximum likelihood (ML) methods [
14,
15], the weighted subspace fitting method [
16], and multidimensional MUSIC method [
17] have also been proposed for DOA estimation. The research of MLbased methods is developing rapidly—for example, Ref. [
18] proposes an alternating projection (AP) method to solve the optimal solution of the likelihood function with lower computational complexity.
The compressed sensing (CS) theory [
19] asserts that, from far fewer samples than traditional methods used, one can recover certain signals, which means that the signal is uniformly sampled at or above the Nyquist rate. The theory of compressed sensing [
20] mainly includes three aspects: the sparse representation of the signal, the design of the observation matrix, and the reconstruction algorithm. The design of the observation matrix is the key to ensuring the quality of signal sampling. The observation matrices are mainly divided into two categories, one of them being the random observation matrix, and the local Fourier observation matrix is a typical random matrix which can reconstruct the original signal very well [
21].
By studying the uniqueness of the
${\ell}_{0}$ norm solution, and the equivalence of
${\ell}_{1}$ and
${\ell}_{0}$ norm solutions, Donoho first gave a sufficient condition that the
${\ell}_{1}$ norm solution can correspond to only the most sparse solution [
22], and transformed a nondeterministic polynomial problems (NPhard problem) into a convex optimization problem. These contributions make it not only possible, but feasible in reality to reconstruct the sparse signal. Later, Donoho proposed the concepts of “Compressed Sensing” [
20] and “Compressive Sampling” [
23], so that the theory of compressed sensing [
19,
24] was formally established.
Sparse representation provides a new perspective of DOA estimation, and has attracted lots of interest [
25]. By exploiting the sparsity of targets in the spatial domain, the corresponding algorithms have been proposed to estimate DOA. The sparse Bayesian learning (SBL) [
26,
27,
28] and the relevance vector machine (RVM) [
29] were proposed. The SBL method achieves a sparser solution without the additional parameters; however, the high computational complexity of SBL methods means there are limitations to its practical application. Combining this with the application of the CS theory in radar signal processing, an iterative adaptive approach (IAA) [
30] was proposed to achieve higher resolution. When the IAA algorithm is applied to SpaceTime Adaptive Processing, a highresolution clutter spectrum can be obtained without training the distance gate [
31,
32,
33]. The
${\ell}_{1}$singular value decomposition method (
${\ell}_{1}$SVD) [
34,
35,
36] is also proposed to solve the problem of excessive computation of the
${\ell}_{1}$ norm method in the case of faster beat, and estimates the spatial spectrum using the sparse signal reconstruction. However, the estimation performance of
${\ell}_{1}$SVD method reduces with decreasing Signal to Noise Ratio (SNR). The orthogonal matching pursuit method (OMP), which is used in [
8], reconstructs the sparse signal with an observation matrix. The multiple orthogonal matching pursuit (MOMP) [
8] and optimized orthogonal matching pursuit (OOMP) [
37] based on the OMP method were also proposed. The OMP methods greatly reduce the computational complexity and improves the estimation performance.
Since the DOA is estimated by measuring the relative phases among antennas, the channel phase inconsistency cannot be ignored in the practical scenarios. Phase errors among antennas have a significant effect on the performance of DOA estimation. To solve this problem, Ref. [
38] proposed a calibration algorithm to estimate the calibration matrix consisting of the unknown phase errors and a set of calibration sources. An eigenstructurebased method for directionfinding in the presence of phase uncertainties is also presented in [
39,
40,
41] proposed a method based on the eigendecomposition of the covariance and conjugate matrices. However, these methods are difficult to use in practical scenarios owing to the high computational complexity.
In this paper, the problem of DOA estimation is investigated in a scenario with unknown phase errors among the antennas. By using a diagonal matrix, the sparsebased system model with additional phase errors is formulated. Then, the DOA estimation problem is converted into a sparse reconstruction problem using the CS theory. To improve the estimation performance, a novel reconstruction method is proposed, where both the DOA and the unknown phase errors are estimated iteratively. We also calculate the phase errors using a gradient descent method with the theoretical expressions. Finally, we compare the DOA estimation performance of the proposed method with stateoftheart methods. In summary, we make the following contributions:
The sparsebased system model with unknown phase errors: A system model using a diagonal matrix with phase errors among antennas is formulated, and converts the DOA estimation problem into a sparse reconstruction problem.
The CSbased DOA estimation method with unknown phase errors: A novel CSbased DOA estimation method is proposed to estimate both the phase errors and DOA, iteratively. The proposed method is costeffective and can achieve a better performance than stateoftheart methods regarding the DOA estimation with unknown phase errors.
The theoretical expressions of the gradient descent method: In the proposed method, the phase errors are estimated by a gradient descent method iteratively with the theoretical expressions.
The remainder of this paper is organized as follows. The uniform linear array (ULA) system model with phase errors is elaborated in
Section 2. The proposed sparse method with unknown phase errors is presented in
Section 3. Simulation results are given in
Section 4. Finally,
Section 5 concludes the paper.
Notations: Capital letters in boldface denotes matrices, and lowercase letters denotes vectors. ${\parallel \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\parallel}_{F}$ denotes the Frobenius norm. ${\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)}_{\mathrm{T}}$ denotes the matrix transpose. ${\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)}_{\mathrm{H}}$ denotes the Hermitian transpose. ${\mathbb{C}}^{N\times M}$ denotes the set of $N\times M$ matrices with the entries being complex numbers. ${\mathit{I}}_{N}$ denotes an $N\times N$ identity matrix. For a vector $\mathit{a}$, $diag\left\{\mathit{a}\right\}$ denotes a diagonal matrix with the diagonal entries from $\mathit{a}$. For a matrix $\mathit{B}$, $vec\left\{\mathit{B}\right\}$ is a linear transformation which converts $\mathit{B}$ into a column vector. For a vector $\mathit{a}$, ${\mathit{a}}_{n}$ denotes the $n$th entry of $\mathit{a}$.
3. SparseBased Algorithm for DOA Estimation
In this section, a novel directionfinding method based on the Simultaneous Orthogonal Matching Pursuit (SOMP) [
42] method is proposed in a scenario with phase errors. SOMP [
42] is a multiple measurement vector (MMV) extension of the OMP algorithm. With less strict requirements on the signal sparsity, this method can realize the recovery of sparse signals by increasing the number of measurement vectors. The details about the proposed DOA estimation method with unknown phase errors is given in Algorithm 1.
At Step 4 of Algorithm 1, the estimated sparse matrix $\widehat{\mathit{X}}$ is updated by using the SOMP method. The SOMP method for the sparse reconstruction is given in Algorithm 2. The SOMP algorithm is an iterative procedure, and in each iteration of the SOMP method an atom is selected, and the residual is updated. The next was selected to maximize the correlation with the current residual. In this method, we use $\widehat{\mathit{G}}\mathit{A}$ as the dictionary matrix instead of $\mathit{A}$.
At Step 7 of Algorithm 1, we use a gradient descent method to update the phase errors. A gradient descent is a firstorder iterative optimization algorithm applied in finding the minimum of a function. Steps proportional to the negative of the gradient (or approximate gradient) of the function at the current point are taken to find a local minimum of a function using gradient descent. We use the following Lemma to derive the gradient descent method.
Algorithm 1 The proposed algorithm to estimate the DOAs in the scenario with phase errors among the antennas. 
 1:
Input: received signal $\mathit{R}$, the dictionary matrix $\mathit{A}$, the number of signals K, the step length p, and the number of iterations t.  2:
Initialization: the iteration indices ${i}_{1}=1$, the phase error matrix $\mathit{G}={\mathit{I}}_{N}$.  3:
while${i}_{1}\le t$do  4:
Obtain the estimated sparse matrix $\widehat{\mathit{X}}$ by using the SOMP method. In this method, we use $\widehat{\mathit{G}}\mathit{A}$ as the dictionary matrix instead of $\mathit{A}$.  5:
$\mathit{Z}=\parallel \mathit{R}\widehat{\mathit{G}}\mathit{A}\widehat{\mathit{X}}{\parallel}_{F}^{2}$.  6:
Obtain ${\frac{\partial \mathit{Z}\left(\mathit{\varphi}\right)}{\partial \mathit{\varphi}}}^{\mathrm{H}}$ from ( 14).  7:
$\mathit{\varphi}=\mathit{\varphi}p{\frac{\partial \mathit{Z}\left(\mathit{\varphi}\right)}{\partial \mathit{\varphi}}}^{\mathrm{H}}$.  8:
Set all the elements of the matrix $\mathit{\varphi}=\mathbf{0}$, except the maximum.  9:
end while  10:
Output: the estimated sparse matrix $\widehat{\mathit{X}}$.

In order to get the steepest descent direction, the derivation is required. The received signal vector
$\mathit{y}$ can be expressed in another form as:
where
$\mathit{B}=diag\left\{\mathit{A}\mathit{S}\right\}$.
To get the exact step of every iteration, we define the following objective function:
and the derivation of
$f\left(\mathit{\varphi}\right)$ can be expressed as:
Additionally, the derivation of
$\frac{\partial {\mathit{\psi}}^{*}}{\partial \mathit{\varphi}}$ is:
where
${\mathit{\psi}}_{\mathit{R}}$ is the real part of
$\mathit{\psi}$, and
${\mathit{\psi}}_{\mathit{I}}$ is the image part. Therefore, (
12) can be expressed as:
where
and
We adopted
$p\frac{\partial f\left(\mathit{\varphi}\right)}{\partial \mathit{\varphi}}$ as the step for every iteration, where p is a fixed step.
Algorithm 2 SOMP algorithm to estimate the DOAs 
 1:
Input: received signal $\mathit{Y}$, the dictionary matrix $\mathit{A}$, the number of signals K.  2:
Initialization: the iteration indices $t=1$, set of index for column vector of $\mathit{A}$ in the tth iteration ${\Lambda}_{t}=\varnothing $, the set of columns of matrix $\mathit{A}$ selected by index ${\mathit{A}}_{t}=\varnothing $, root mean squared error ${\mathit{R}}_{0}=\mathit{Y}$.  3:
while$t\le K$do  4:
${\lambda}_{t}$ = arg max${}_{j}\left\u2329{\mathit{R}}_{t1},{\alpha}_{j}\u232a\right$.  5:
${\Lambda}_{t}={\Lambda}_{t1}\cup {\left\{\lambda \right\}}_{t}$.  6:
${\mathit{A}}_{t}=\left[{\mathit{A}}_{t1},{\mathit{\alpha}}_{{\lambda}_{t}}\right]$.  7:
$\widehat{{\theta}_{t}}=$ arg min${}_{\theta}\parallel \mathit{Y}{\mathit{A}}_{t}{\theta}_{t}\parallel ={\mathit{A}}^{1}\mathit{Y}$, where ${\mathit{A}}^{1}={\left({\mathit{A}}_{t}^{\mathrm{H}}{\mathit{A}}_{t}\right)}^{1}{\mathit{A}}_{t}^{\mathrm{H}}$.  8:
${\mathit{R}}_{t}=\mathit{Y}{\mathit{A}}_{t}\widehat{{\theta}_{t}}=\mathit{Y}{\mathit{A}}^{1}\mathit{Y}$.  9:
$t\leftarrow t+1$.  10:
end while  11:
Output: the estimated sparse vector $\widehat{\mathit{X}}$.

4. Simulation Results
In this section, the simulation results about the proposed method for DOA estimation in the ULA system with unknown phase errors are given, and the simulation parameters are given in
Table 1. All experiments were carried out in MATLAB R2018b on a PC with a 2.4 GHz Intel Core i7 and 8 GB of RAM.
FISTA: fast iterative shrinkagethresholding algorithm method proposed in [
43]. FISTA provided a new resolution to realize the forwardbackward splitting method (also known as the proximal gradient method) splitting to a broad range of problems appearing in sparse recovery, MMV problems, and many other aspects. Whether the problem solved is simple or complex, FISTA simplifies it efficiently by handling issues like stepsize selection, acceleration, and stopping conditions.
OGSBI: the offgrid sparse Bayesian inference method proposed in [
44]. The OGSBI method deals with the offgrid DOA estimation problem and uses an iterative algorithm based on the offgrid model from a Bayesian perspective, while joint sparsity among different snapshots is exploited by assuming a Laplace prior for signals at all snapshots. The OGSBI method applies to both singlesnapshot and multisnapshot cases and has high estimation accuracy, even under a very coarse sampling grid.
SOMP: for compressed sensing, the critical thing is to extend from the singlemeasurement vector (SMV) problem to the MMV problem.
SOMP, which can recover signals by increasing the number of measurement vectors, is an MMV extension of OMP.
These methods are widely used in DOA estimation, so we made a comparison on the performance of these methods with the proposed method.
The number of antennas, samples, and signals and the value of SNRs are typical simulation parameters. Reference [
45] also used these parameters. As a result, we chose these variables as our simulation parameters. After the received signal is generated in the simulation process, phase errors are randomly added to the arrival angels to establish the antenna array model with phase errors. In this paper, we assumed the values of phase errors were from 25° to 35°. These phase errors are described as a uniform distribution between 25° to 35°. The noise given in this paper is the AWGN.
The simulation results of the paper were obtained after multiple Monte Carlo trials. By using the given parameters in the paper, we can reproduce the results.
For different phase errors existed in antennas, where from
Figure 2 we can see the corresponding performance of DOA estimation using the proposed method and the SOMP method. The rootmeansquare error (RMSE) was used to measure the DOA estimation performance.
where with the
ith experiment,
${\theta}_{i}$ is the vector of a truly received signal,
$\widehat{{\theta}_{i}}$ is that with estimation DOA, and
${N}_{i}$ is the number of the experiment. In
Figure 2, we show the DOA estimation performance with different values of SNR. With the SNR being 20 dB, the DOA estimation performance with different numbers of antennas is given in
Figure 2. As shown in this figure, more antennas brings about a better performance of DOA estimation. When the number of antennas reaches 20, the proposed method achieves the best performance. Furthermore, the proposed method achieves a greater improvement compared with the traditional method including SOMP, FISTA, and OGSBI.
We show the DOA estimation performance with different numbers of phase errors in
Figure 3. The number of phase errors refers to the number of antennas with large phase errors in the array. Not each antenna is associated with the phase error. In this paper, we assumed 2–4 antennas in ten had large phase errors, and phase errors of other antennas is small. The situation of each antenna that has a large phase error can be converted into the situation with no existent error. With a larger number of phase errors, the performance of the proposed method is better. When there are five phase errors on the antenna system, the proposed method achieves the best performance. The Figure also indicates that existing methods, including SOMP, OGSBI, and FISTA, cannot estimate correctly with an increasing number of phase errors. Therefore, the proposed method is very efficient in the case of large numbers of phase number.
For different sample numbers,
Figure 4 shows that further optimization of the sample number can achieve a better resolution. Besides, the Figure shows the proposed method has a much better performance when the sample number changes.
Additionally, for different SNRs, we can see the DOA estimation performance in
Figure 5, where the SNR is from 5 dB to 25 dB. When the SNR is smaller than 15 dB, with the increase of SNR, the DOA estimation performance can be improved. When the SNR is larger than 15 dB, the DOA estimation performance changes slowly with different SNRs. Since the sparse signals and the phase errors are estimated iteratively using the proposed methods, it achieves better performance than the existing methods, including SOMP, FISTA, and OGSBI.
We show the DOA estimation performance with different numbers of target signals in
Figure 6. As shown in this Figure, a better DOA estimation can be achieved by decreasing the number of target signals using the proposed method. However, when the number is getting larger, the estimation error of the proposed method improves faster compared with the traditional SOMP method. When the number is 6, it achieves the same estimation performance as SOMP, which does not consider the phase errors existing in antennas.
Figure 6 indicates that the proposed method has an advantage in that the number of target signals is small, and achieves the best performance among existing methods, including SOMP, FISTA, and OGSBI.
Finally, the computation time of the proposed method is compared with the present three methods in
Table 2. We have not further worked in decreasing the computational time for all the methods. The computational complexities of SOMP in Step 4 and Step 7 are
$\mathcal{O}$ $\left(NKM\right)$ and
$\mathcal{O}$ $({N}^{2}K+{K}^{3}+{K}^{2}N+NKM)$. Without the additional simplifications, the computational complexities of the proposed method in Step 5 and Step 6 are
$\mathcal{O}$ $({N}^{2}K+NKM+MN)$ and
$\mathcal{O}$ $({N}^{2}M+{N}^{2})$. With the conditions
$K\ll N$ and
$K\ll M$, the computational complexities of the proposed method can be simplified as
$\mathcal{O}$ $\left({N}^{2}M\right)$ per iteration. As shown in
Table 2, the method SOMP has the least computation time. However, the proposed method has better performance of DOA estimation. The proposed method is comparable with the FISTA method, but the DOA estimation performance is better in the scenario with phase errors among antennas. The computation time of the OGSBI method is much longer than our proposed method. Thus, the DOA estimation performance in the ULA system with phase errors can be significantly improved in the cost of an acceptable computational complexity.