1. Introduction
Direction-of-arrival (DOA) estimation is a vital technology in the field of array signal processing that has been widely applied in radar, sonar, acoustic, navigation, and wireless communication [
1,
2,
3]. Traditionally, uniform linear array (ULA) has been the most commonly used array configuration because of its simplicity for application and well-developed techniques. The main disadvantage of using ULA is that the achievable number of degrees-of-freedom (DOFs) of a ULA consisting of
M sensors is limited to
. To break through this limitation, coprime arrays were proposed in Ref. [
4], which provide a larger array aperture than the ULA with the same number of sensors and can resolve up to
sources with
sensors.
In order to take advantage of the DOF superiority offered by the coprime array, several techniques have been developed for DOA estimation, including spatial smooth [
5], array interpolation [
6] and compressive sensing [
7]. Among them, compressive sensing (CS) is one of the most promising signal-processing techniques [
8], which can accurately recover signal at a sub-Nyquist sampling rate. The CS-based DOA estimation algorithms can be divided into three categories: greedy algorithms, such as orthogonal matching pursuit (OMP) [
9]; convex relaxation algorithms, such as least absolute shrinkage and selection operator (LASSO) [
10]; and Bayesian CS, such as sparse Bayesian learning (SBL). In Ref. [
7], a sparse reconstruction method based on LASSO was proposed for coprime arrays, which fully utilizes the entire virtual array aperture. Combining the compressive measurement method with sparse array, Guo et al. proposed two categories of methods in Ref. [
11] to significantly reduce the system complexity.
However, excellent performance of the CS-based DOA estimation methods is critically dependent on the precondition that all the sensors in the array are properly calibrated. Otherwise, sensor miscalibration will deteriorate the performance of these methods, and even lead to operation failure. Therefore, it is clear that array calibration is important to ensure robust DOA estimation in practice. In recent decades, two branches of the research have been conducted to address this miscalibration problem [
12]. The first branch is called active calibration that requires auxiliary sensors or co-operation signal sources with known DOAs. In Ref. [
13], the good calibrating performance is acquired by placing the known sources in space and storing the array manifold of the uncalibrated array excited by these sources. When the number of calibrated sensors is more than the number of signal sources and the number of signal sources is more than one, the partly calibrated method proposed in Ref. [
14] also has excellent performance. Nevertheless, the requirements for using active calibration methods are often difficult to meet, making them difficult to implement in practice. Unlike active calibration methods, the other branch, referred to as self-calibration methods, does not have the need for assistant sources or sensors. Under the assumption that all the sensor miscalibrations can be modeled as unknown but deterministic gain and phase error parameters, several self-calibration methods have been proposed on the basis of CS methods, which jointly estimate these error parameters with DOAs. Just like the classification of CS methods, these self-calibration methods can be also divided into the same three categories. In Ref. [
15], a method was proposed under the framework of OMP, which treats the gain/phase uncertainties as an additive error matrix by transferring the array signal model with gain/phase uncertainties into an error-in-variables (EIV) model. A total least squares (TLS) problem was then formulated and the simultaneous orthogonal matching pursuit (SOMP) algorithm was introduced to solve this problem. Although the authors claim that their method can be applied to the sparse array, the enhanced DOF offered by the sparse array is not exploited throughout this paper. Besides, since this method discretizes the signal space into grids and assumes that all targets fall on the predefined grid, it may have a grid mismatch problem. To address the mismatch problem, Camlica et al. proposed another OMP-based method in Ref. [
16] for off-grid signals. In this method, a non-linear cost function is approximated by its first order Taylor series for the sake of simplicity. Thus, this method works well only when the perturbations are not severe. These two OMP-based methods have some common shortcomings: first, the number of sources must be known a priori; second, they can only obtain suboptimal solutions through non-convex models. Meanwhile, some other self-calibration methods are developed under the framework of LASSO. In Ref. [
17], the calibration problem is transformed into a sparse matrix completion problem. This problem is non-convex; it is then decomposed into two convex subproblems so that it can be efficiently solved. However, this method adopts the same linearization strategy as the method in Ref. [
16], so it cannot be applied to the cases of serious disturbance either. In Ref. [
18], a convex optimization approach was proposed to blindly calibrate the sensor array. Unfortunately, the applied range of this approach is very limited because it can only work under the noise-free condition. The methods proposed in Refs. [
19] and [
20] exploit the orthogonality property between the signal subspace and noise subspace, so they cannot be implied for the coprime array with the expectation to find more sources than sensors. Under the framework of SBL, Lu et al. proposed a method for nested array calibration [
21], which employs the EM algorithm to solve a non-convex optimization problem and thus jointly estimates the DOAs with the error parameters. This method is able to exploit the enhanced DOF offered by the nested array; however, it is computationally intensive and its convergence is not guaranteed. In Ref. [
22], a robust CS-based DOA estimation method was proposed under the assumption that few sensors in the array are miscalibrated. Unlike the methods that explicitly estimate the error parameters, this method treats the miscalibrated sensor observations as outliers and a weighting factor is adaptively optimized and applied to mitigate the effect of the outliers.
Inspired by Ref. [
22], in this paper, a robust CS-based DOA estimation method is proposed for coprime array in the presence of miscalibrated sensors, which can be used to find more sources than sensors. Similar to Ref. [
22], it is assumed that the sensor miscalibration occurs randomly in the array, where both the number and the positions of the miscalibrated sensors are not known a priori. Signals received by these miscalibrated sensors are viewed as outliers, and the correntropy is employed as the similarity measurement to distinguish them. The sparse signal recovery is formulated as an optimization problem, in which the maximum correntropy criterion (MCC) is employed as the constraint condition to suppress the influence of the sensor miscalibration. Exploiting the property of conjugate function, an iterative algorithm is developed under the convex relaxation framework to effectively solve the optimization problem. Furthermore, a grid refinement strategy is employed to alleviate the grid mismatch problem, with which the grids are adaptively refined at each iteration. The DOA estimation is thus acquired after the convergence of this iterative algorithm. The proposed method fully uses the virtual aperture constructed by the coprime array structure, and achieves accurate DOA estimation with enhanced DOF even when array sensors are severely perturbed. Numerical simulations are conducted to verify the effectiveness and robustness of the proposed method.
The rest of this paper is organized as follows. In
Section 2, the signal model used through this paper is set up, and then the effect of miscalibrated sensors is discussed. The theories of CS-based DOA estimation method and MCC are reviewed in
Section 3. Based on the theories reviewed in
Section 3, a robust DOA estimation algorithm for coprime array is proposed in
Section 4 with a multiresolution grid refinement strategy, where the Cramér-Rao Bound (CRB) of DOA estimation is also given. The simulation results are shown in
Section 5. Finally, the conclusions are made in
Section 6.
Notations: The lower-case boldface characters, upper-case boldface characters, and upper-case characters in blackboard boldface are used to denote vectors, matrices, and sets respectively throughout this paper.
denotes a complex matrix or vector (when
N = 1). The superscripts
,
and
denote the transpose, conjugate transpose, and complex conjugation, respectively.
and
respectively denote the inverse and the trace of a matrix.
indicates the (
i,
j)-th entry of
. The square bracket notation of a vector
represents the
i-th component of
. For
, the triangular bracket notation
denotes the signal value at the support location
n, where the detailed definition is given in Ref. [
23].
denotes the cardinality of a set. The notation
denotes the statistical expectation.
stands for the vectorization operator that sequentially stacks each column of a matrix, and
represents a diagonal matrix with the corresponding elements on its diagonal. The symbols
and
represent the Khatri-Rao product and Kronecker product respectively.
denotes the identity matrix with an appropriate dimension. Finally, the symbol
j represents imaginary unit
.
5. Simulation
In this section, a series of numerical simulations are conducted to examine the performance of the proposed method. In these simulations, the pair of coprime integers is chosen as
to deploy the extended coprime array. There are
sensors in the array, which are located at
. The proposed robust DOA estimation algorithm is compared to several recently-reported DOA estimation algorithms utilizing the coprime array, namely the Spatial Smooth MUSIC algorithm (SS-MUSIC) [
4], the Nuclear Norm Minimization (NNM) algorithm [
31], and the Sparse Signal Reconstruction (SSR) algorithm [
7]. Two CS-based self-calibration methods that do not require prior knowledge of the number of sources are also compared, namely the Sparse-Based Array Calibration algorithm (SBAC algorithm in Ref. [
17]) and the Sparse Bayesian learning Array Calibration algorithm (SBAC algorithm in Ref. [
21]). The sampling interval of the predefined grid is set as
for the SSR algorithm, the SBAC algorithm in Ref. [
17] and the SBAC algorithm in Ref. [
21]. The regularization parameter
and precision parameter
for the SBAC algorithm in Ref. [
17] are set as 0.1 and 0.01 respectively. For the proposed algorithm, the sampling interval of the initial coarse grid is selected to be
, and the minimum spacing length
is selected to be
.The regularization parameter
for the SSR algorithm and the proposed algorithm is empirically set to be 0.25. The tolerance parameter
and the maximum number of iterations
Q for the proposed algorithm are set as
and 50 respectively. The convex optimization problems are solved by using the CVX [
32].
For the first three examples, it is assumed that the
K = 11 equal-power sources uniformly distributed in
impinge on the array. In the first example, the spatial spectra estimated by these algorithms in the presence of miscalibrated sensors are compared. The sensor miscalibrations are assumed to occur on the fourth, sixth and eighth sensors in the array, i.e.,
. The specific values of the distortion parameters of the miscalibrated sensors are
,
and
respectively. It is worth mentioning that both the locations and the distortion parameters of the miscalibrated sensors are unknown a priori. The SNR of all the sources is set to be 30 dB, and
T = 2000 snapshots of the received signals are collected for DOA estimation. The normalized spatial spectra are depicted in
Figure 3, where the vertical dashed lines denote the actual directions of the incident sources.
It is observed from
Figure 3a,b that the number of obvious sharp peaks in the spatial spectra of SS-MUSIC and NNM algorithm is less than the source number
K, which makes it hard to correctly identify the number of sources. Moreover, most of the estimation results of these two algorithms deviate from the actual source directions. As shown in
Figure 3c, the SSR algorithm cannot correctly identify the number of sources either, and only a few of the estimation results of it are close to the actual source directions. However, unlike the SS-MUSIC and NNM algorithm, the number of peaks in the spatial spectrum of SSR algorithm is more than the sources number.
Figure 3d,e show that neither the SBAC algorithm in Ref. [
17] nor the SBAC algorithm in Ref. [
21] can properly calibrate the gain and phase distortions under such serious disturbance. Most of the estimation results of them still deviate from the actual source directions. In contrast, the proposed algorithm is able to correctly resolve all the peaks in the actual source direction.
In the second example, the root mean square error (RMSE) of each algorithm is compared in
Figure 4. Here, the RMSE is defined as
where
is the estimated DOA of the
k-th source in the
Monte Carlo trial, and
is the number of Monte Carlo trials. It can be seen from example 1 that, under some conditions, some of the algorithms cannot correctly distinguish the number of sources. Therefore, for statistical convenience, the number of sources,
K, is considered as known, and the angels corresponding to
K highest peaks in spatial spectra are picked as estimated DOAs. The number of snapshots is fixed at
T = 2000 when the SNR varies, whereas the SNR is fixed at 30 dB when the number of snapshots varies. The locations and the distortion parameters of the miscalibrated sensors are set to the same as in example 1. For each data point,
Monte Carlo trials are conducted to calculate the RMSE. The Cramér-Rao bound (CRB) Equation (39) is also plotted.
It can be seen from
Figure 4a that, when there are some miscalibrated sensors in the array, the estimation accuracy of SS-MUSIC algorithm, NNM algorithm and SSR algorithm cannot be significantly improved by increasing SNR. Compared with these three uncalibrated methods, the RMSE performances of two self-calibration methods show no significant improvement. As for the proposed algorithm, it does not show superiority in estimation accuracy when
. However, when SNR is lager 5 dB, the proposed algorithm obviously outperforms other algorithms in terms of RMSE. The reason lies in that when SNR is low (lower than 0 dB), the outliers cannot be effectively distinguished by using correntropy. As shown in
Figure 4b, increasing the number of snapshots is not helpful to improve the estimation accuracy of the SS-MUSIC algorithm, NNM algorithm and SSR algorithm, either. Meanwhile, neither the SBAC algorithm in Ref. [
17] nor the SBAC algorithm in Ref. [
21] can improve estimation accuracy. The proposed algorithm outperforms other algorithms even when the number of snapshots is small. Comparing
Figure 4a with
Figure 4b, it can be also found that the proposed MCC-based algorithm is more sensitive to SNR than to the number of snapshots. The RMSE of the proposed algorithm still decreases significantly when the SNR increases from 25 dB to 30 dB.
In the third example, both the DOA estimation accuracy and the robustness of the tested algorithms are compared in the presence of miscalibrated sensors. First, the deviation distance of estimation (DDOE) of the
Monte Carlo trial is defined as
For the same reason as stated in example 2, the number of sources,
K, is considered as known. The SNR and the number of snapshots in this example are set as 30 dB and 2000 respectively. It is assumed that there are three miscalibrated sensors in the array, whose locations are randomly selected from
in each Monte Carlo trial. The gain distortion parameter
of each miscalibrated sensor is randomly selected from the interval
, and the phase distortion parameter
of each miscalibrated sensor is randomly selected from the interval
in each Monte Carlo trial.
Figure 5 gives the box plots of DDOE of the tested algorithms, where the statistical data is collected from 500 Monte Carlo trials for each algorithm.
Figure 5 shows that variation range of DDOE of the CS-based DOA estimation algorithms are smaller than the other two algorithms. However, neither the SBAC algorithm in Ref. [
17] nor the SBAC algorithm in Ref. [
21] show higher robustness or better estimation accuracy than the SSR algorithm. Moreover, the proposed algorithm shows the best robustness with the minimum box height and the best estimation accuracy with smallest mean value.
In the fourth example, it is assumed that
K = 18 equal-power sources uniformly distributed in
impinge on the array. The sensor miscalibration is assumed to occur on the fourth and eighth sensors in the array, i.e.,
. The specific values of the distortion parameters of the miscalibrated sensors are
and
respectively. In this case, the number of sources is equal to the number of virtual sensors in the contiguous part of the difference coarray; therefore, the SS-MUSIC algorithm cannot be used. The normalized spatial spectra of the remaining three algorithms are compared in
Figure 6 with the SNR = 30 dB and the number of snapshots
T = 2000, where the vertical dashed lines also denote the actual directions of the incident sources. The comparison results shown in
Figure 6 are similar to those in Example 1, which also verifies the superiority of the proposed algorithm.