1. Introduction
In recent years, a new type of antenna system, multiple-input multiple-output (MIMO), has been introduced [
1,
2]. A MIMO antenna system is generally defined as a system that has multiple transmitted linearly independent waveforms and is capable of jointly processing multiple received antenna signals. MIMO radar draws on the idea of MIMO communication [
3]. Its mechanism is to adopt transmit waveform diversity technology to improve the angular (spatial) resolution of the radar system while reducing the number of physical channels and antenna aperture, compared with traditional array antennas.
Estimation of the direction-of-arrival (DOA) of multiple targets in noise-polluted received data is the most important task in the practical application of centralized MIMO radar. The most fundamental DOA estimation method is the delay-and-sum (DAS) method, which can be implemented efficiently by fast Fourier transform (FFT) due to the Fourier structure of the orientation matrix. However, the resolution of this method is poor; to improve the positioning accuracy of the target source, many high-precision DOA estimation methods have been developed for traditional single-input multiple-output (SIMO) radar [
4,
5,
6,
7]. More recently, DOA estimation methods have been introduced into the field of centralized MIMO radar DOA estimation [
8,
9,
10]. Among these methods, estimation of signal parameters via rotational invariance techniques (ESPRIT) and multiple signal classification (MUSIC) are relatively classic because of their simplicity and high-resolution performance [
8,
9]. Based on the orthogonality characteristics of the signal subspace and noise subspace, the multiple signal classification (MUSIC) method can lead to better DOA estimation performance than the DAS method with higher resolution. Taking advantage of the rotational invariance of the spatial correlation matrix signal subspace, the (ESPRIT) algorithm was proposed with excellent resolution and search-free advantages. In reference [
10], the parallel factor analysis (PARAFAC) algorithm was proposed for a colocated MIMO radar with imperfect waveforms. ESPRIT is a low-dimensional and high-efficiency version of PARAFAC [
11], and they have similar resolution performance. However, the resolution of these methods is limited, and noticeable performance degradation occurs when there are a few snapshots or a single snapshot. Recently, deep-learning techniques have been applied to DOA estimation for massive MIMO systems and achieved good system performance [
12]. However, this method requires many training samples and the training phase is complex.
To further improve the resolution performance of MIMO radar DOA estimation, refs. [
13,
14] proposed the iterative adaptive method (IAA) for MIMO radar imaging; tests have demonstrated that the IAA method can function stably with a small number of snapshots or a single snapshot and has better angular resolution and target positioning accuracy. However, due to its high computational complexity, this method is difficult to apply in practical engineering [
6]. Taking advantage of the sparsity of the source distribution, sparse learning via iterative minimization (SLIM) was proposed [
15] for MIMO radar imaging. The method follows an
-norm constraint and thus offers a more accurate estimate. In addition, SLIM has been demonstrated to have a higher angular resolution and lower computational complexity than IAA. Furthermore, a sparse spectrum estimation method with a higher resolution, called SParse Iterative Covariance-based Estimation (SPICE), was proposed based on weighted covariance fitting criteria [
16,
17]. This method is a semiparametric method, which converges globally and requires no selection of user parameters. In the case of a small number of snapshots, its frequency estimation performance is better than those of IAA and SLIM. However, the method has an
-norm penalty (sparse constraint) for both signal and noise, which may result in a singular covariance matrix or many conditions. To avoid this problem, an improved algorithm, named qSPICE, was developed by introducing a
-norm constraint on noise changes, where
[
18]. This method offers better estimation performance than SPICE. However, this method requires solving for the covariance matrix and its inverse, as well as performing the associated matrix multiplication operations to estimate each sampling grid point, which may result in a large computational burden, especially in massive MIMO (m-MIMO) signal processing.
With the increasing demand for higher reliability and higher data rates, novel MIMO antenna technologies are booming. In [
19], an optimized algorithm based on semidefinite programming (SDP) and minimum mean squared error (MMSE) is proposed for s cognitive radio (CR) MIMO system. This method is proven to perform better in terms of total transmitted power and signal-to-interference plus noise ratio (SINR). To alleviate the large channel state information (CSI) feedback in m-MIMO system, a robust channel estimation scheme is proposed based on the separation mechanism of the channel matrix [
20]. Moreover, some MIMO antenna design strategies and beamforming methods have also been well developed, such as m-MIMO antenna technology for 5G communication base stations [
21,
22], 10-element MIMO antenna design for new 5G smartphones [
23,
24], ultra-massive MIMO radar technology for terahertz antenna beamforming [
25], etc. However, m-MIMO antenna arrangement requires the integration of a huge number of antennas at the base station and a large number of antennas at the user terminal, which will undoubtedly cause hardware implementation challenges and extremely high signal-processing complexity: whether it is for the spatial diversity and beamforming of MIMO communication systems [
26], or directions of arrival (DOA) estimation of MIMO radar systems [
17].
To reduce the high signal-processing complexity for m-MIMO systems, various advanced matrix inversion acceleration algorithms, such as the Neumann series (NS) algorithm [
27] and the Jacobi algorithm [
28], may be used to reduce the computational burden of qSPICE or IAA methods. In [
29], the NS algorithm was used for matrix inversion approximation (MIA) for m-MIMO signal processing. This method transforms the matrix inverse problem into a matrix multiplication problem, which is suitable for hardware platforms, but its computational complexity is the same or even higher than those of direct inverse methods (e.g., QR-based methods [
30]). Although the Jacobi method reduces the complexity from
to
, where
L represents the number of iterations, it converges slowly [
31].
The alternating-direction method of multipliers (ADMM) is a powerful technique for solving massive optimization problems. This method is widely used in various fields, such as compressed sensing (CS) [
32], regularization estimation [
33], image processing [
34], and machine learning [
35]. In [
36], the least absolute shrinkage and selection operator (LASSO)-ADMM algorithm was proposed for solving
-norm constrained optimization problems for m-MIMO signal detection. In [
37], ADMM was used to reduce the computational complexity of CS-DOA compared with the traditional interior point method (IPM); this method reduces the computational complexity by dividing the problem into multiple subproblems during the iteration process to reduce the dimension of each problem. In [
38], an imprecise augmented Lagrange multiplier (ALM)-ADMM algorithm was proposed for solving a weighted mixed
-norm penalty minimization problem to improve the DOA estimation performance for MIMO radar signals with missing elements.
Although the ADMM-based methods discussed above can efficiently solve the array DOA estimation problem, the application of an ADMM method introduces two additional user parameters, i.e., a Lagrangian parameter and a sparse regularization parameter [
39]; the simultaneous adjustment of these two parameters is very tricky. To avoid this problem, in the present paper, the basic form of qSPICE is expanded, weighted covariance fitting is performed, and a mean square minimized form of the cost function is obtained by an equivalent transformation of the qSPICE cost function. Then, through the optimization properties, the problem is transformed into a sparse optimization problem in the form of a weighted LASSO. Then, this unconstrained optimization problem is decomposed into three subproblems to reduce the dimension of each problem and thus reduce the computational complexity based on ADMM. Due to the absence of user parameters in qSPICE, the proposed method can eliminate one sparse regularization parameter compared with the traditional ADMM-LASSO method, therefore significantly alleviating the difficulty of parameter selection. Meanwhile, the theoretical computational complexity of the proposed method is one order of magnitude lower than that of the traditional qSPICE method. Simulation results and measured data indicate that the estimation performance of the proposed method is not lower than those of qSPICE and ADMM-LASSO, and it incorporates the advantages of both methods, i.e., low complexity and a single user parameter.
The remainder of this paper is organized as follows.
Section 2 reviews the MIMO radar signal model and the generalized SPICE and ADMM-LASSO methods. Then, the proposed ADMM-qSPICE DOA estimation method for MIMO radar is derived in detail. In
Section 3, simulation and test results are presented, which demonstrate the effectiveness of the proposed method. In
Section 4, the limitations of the proposed method are discussed.
Section 5 presents the conclusions of this study.