Multi-Input Multi-Output (MIMO) radar [1–3] is a new type of radar in recent years. Compared with traditional antenna arrays, the MIMO radars have potential advantages. The reason is that the MIMO radars make use of multiple antenna elements to transmit diverse waveforms and receive echo signals simultaneously in similar ways [4–6]. The detection of Direction Of Departure (DOD) and Direction Of Arrival (DOA) for MIMO radars has been actively studied and widely applied in many technical fields [7–10]. Subspace-based algorithms are an important kind of methods for DOD and DOA detection, which has the advantages of strong resolution and high detection accuracy [11, 12].

Among the subspace-based methods the most representative ones are the MUltiple SIgnal Classification (MUSIC) [13] and the Estimation Signal Parameter via Rotational Invariance Techniques (ESPRIT) [14]. The MUSIC approach is well-known for its high-resolution capability, and can be used for arrays with any form of geometry to detect DOA and DOA of each target [15, 16]. However, the massive spectral peak search makes it difficult to popularize this algorithm in the engineering field. Without spectral peak search (search-free detection), the ESPRIT algorithm can detect the DOA and DOA in a closed form solution [8, 17–19]. However, this type of methods can only be applied to the Uniform Linear antenna Array (ULA), or the antenna arrays themselves possess a same structure of (shift invariant) subarrays. This limitation makes it difficult to extend such algorithms to practical applications [20–22].

Thus, fast detection of the DOD and DOA in the MIMO radar system with an arbitrary antenna array has always been the pursuit for researchers. In [18], an ambiguity function-based algorithm is designed to detect the DOD and DOA in a MIMO Radar system. By constructing a spatial time-frequency distribution matrix, this method uses ESPRIT and Root-MUSIC to realize the joint DOD and DOA fast detection. This process does not involve multiple-dimensional spectrum peak searching and the parameters can also be paired automatically. In [20], a robust method for joint DOD and DOA detection in a non-Gaussian noise environment is proposed. This method uses a robust M-estimator to form an estimate of the covariance matrix of the array output and then utilizes the random matrix theory and polynomial rooting method to capture the DOD and DOA in a large scale MIMO radar system. By exploiting the banded complex symmetric Toeplitz structure of the mutual coupling matrices [21], a robust sparse Bayesian learning algorithm for DOD and DOA detection is also developed, and this method is demonstrated to work well and have better detection performance in unknown non-uniform noise and mutual coupling.

In this study, a joint DOD and DOA detection scheme through signal subspace optimal reconstruction is developed. In the developed scheme, we first use the beamforming [23–25] to set an objective space in mathematics for the targets to be detected. Then, we randomly select a collection of DODs and DOAs from the objective space to establish a potential signal subspace with the steering matrix of the MIMO array. Considering of the orthogonality between the signal subspace and the noise subspace and a reconstruction error of the signal subspace, we build a multi-dimensional objective function, which contains all information of DODs and DOAs. Subsequently, the Quantum-Behaved Particle Swarm Optimization (QPSO) [26, 27] is employed to optimize the multi-dimensional objective function so as to obtain the optimal reconstruction of the signal subspace. Finally, the DOD and DOA can be fast detected with a high accuracy. A detailed derivation and comprehensive analysis of the proposed method are presented. The simulation results verify that the proposed algorithm outperforms the other methods, especially under low signal to noise ratio (SNR) and small snapshots. To the best of our knowledge, the proposed scheme has not been considered in previous studies.

This paper is structured as follows. The signal model for a MIMO array is formulated in Section 2. A fast DOD and DOA detection method is discussed in detail in Section 3. Section 4 includes experimental setup and analysis of simulation results. Finally, Section 5 concludes the paper.

Consider a bistatic MIMO radar array composed of M _t transmitter sensors and M _r receiver sensors, in the transmit array and receive array, respectively. Both of these arrays are ULAs, and the inter-element spacings of adjacent sensor elements are d _t and d _r (not larger than half wavelength of the signals), respectively. Assume that there are P far-field uncorrelated targets with the DODs and DOAs φ p , θ p ,   p = 1,2 , L , P in the target space. Figure 1 shows the signal model of the bistatic MIMO array. The output of the matched filters at the receiver can be expressed in the following manner: X t = a r φ 1 ⊗ a t θ 1 , a r φ 2 ⊗ a t θ 2 , L , a r φ P ⊗ a t θ P b t + n t = A b t + n t (1) where A is the direction matrix (steering matrix), b t = b 1 t , b 2 t , L , b P t T is a vector containing the reflection coefficients and Doppler phase shifts of the targets, and n t is the complex additive white gaussian noise vector. a r and a t are receive and transmit steering vectors which have the following structures: a r φ p = 1 , e j π d r ⁡ sin ⁡ φ p λ , L   , e j π M r − 1 d r ⁡ sin ⁡ φ P λ T a t θ p = 1 , e j π d t ⁡ sin ⁡ θ p λ , L   , e j π M t − 1 d r ⁡ sin ⁡ θ p λ T (2) where T stands for the transpose operation, and λ is the wavelength of the signals. The covariance matrix of the array output is computed by R X = E X t X H t (3) where H represents the complex conjugate transpose. The signal and noise subspaces are commonly obtained through the eigenvalue decomposition of the covariance matrix of the array output. That is, the eigenvectors corresponding to the P largest eigenvalues span the signal subspace U _s , and the remainder eigenvectors form the noise subspace U _n . The MUSIC method detects the DOD and DOA by constructing and optimizing the following spatial spectrum function f M U S I C = 1 a r φ i ⊗ a t θ i H U n U n H a r φ i ⊗ a t θ i (4)

The spatial spectrum function is expected to show a large positive value if φ and θ are a true DOD and DOA, which is due to the orthogonality between the signal subspace and noise subspace. Obviously, the peak search implies a huge amount of computation. How to reduce the amount of computation generated in the search process has always been the goal of researchers.

In this section, a fast joint DOD and DOA detection scheme through signal subspace optimal reconstruction and matching will be developed in detail. The developed scheme, as we will demonstrate, has excellent detection performance under the scenarios of low SNR and small snapshot.

To reduce the search scope, we firstly use the beamforming to capture the potential regions (including the DODs and DOAs) of the targets. Let Ω _k (k = 1, 2, … , K) be the K spectrum peaks determined by the beamforming. We take the neighbourhoods of the K spectrum peaks in mathematics as the potential regions of the targets, and the following searching will be carried out on these regions (observation space).

Then, we randomly select P (P can obtained by the Minimum Descriptive Length, MDL) criterion directions to form a direction vector Ω ∼ to construct an initial steering matrix A ∼ according to Eq. 2, with which an initial signal subspace can be reconstructed in the following way: U ∼ s = A ∼ Ω ∼ A ∼ H Ω ∼ A ∼ Ω ∼ − 1 2 (5)

Ideally, we consider that this signal subspace and the estimated signal subspace with the eigenvalue decomposition of the covariance matrix of the array output should be equal, i.e., U ∼ s U ∼ s H − U s U s H 2 → 0 (6) where g 2 denotes l ₂ norm. On the other hand, consider the orthogonality between the signal and noise subspaces, the following equation should also hold: U ∼ s H U n 2 = 0 (7)

In order to detect more accurate DODs and DOAs, the above two equations should hold at the same time. Thus, we build a multi-dimensional objective function in the following form: J = U ∼ s U ∼ s H − U s U s H 2 + U ∼ s H U n 2 (8)

The DODs and DOAs can be obtained through optimizing the objective function in a 2P dimensional space. In this study, we use the Quantum-Behaved Particle Swarm Optimization (QPSO) as an optimization method to minizine the multi-dimensional objective function and to solve the DODs and DOAs of the signals. Suppose there are N particles, and the optimization process can be described as:

1) Initialize the particle position vector Ω ∼ n (n = 1, 2, … , N) and the best previous position Ω ∼ b e s t n of each particle.

In the implementation of the QPSO, the approach is run for nmax = 300 iterations with 30 particles by considering the dimension of the target space. However, we allow the method to be terminated if no changes in gbest is 20%nmax consecutive iterations. The particle position vector generated at algorithm termination is the DODs and DOAs we want to detect.

In order to evaluate the performance of the developed scheme, in this section we present two groups of simulations to assess the DOD and DOA detection performance of our algorithm in comparison with the other two commonly methods (MUSIC and ESPRIT). In the comparison, the Root-Mean-Square Error (RMSE) criterion [28, 29] is used as an evaluation indicator which is define as 1 N s ∑ n = 1 N s 1 P ∑ p = 1 P α ^ p n − α 2 (10) where N _s denotes the number of the independent simulations, α p is the pth DOD or DOA of the P impinging sources, and α ^ p is the estimate value of α p captured in the nth simulation.

In the simulation, we normally adopt the bistatic MIMO radar system with 𝑀 _t = 8 and M _r = 8, and assume that there are three non-coherent signals located at φ 1 , θ 1 = 5 ∘ , 15 ∘ , φ 2 , θ 2 = 10 ∘ , 30 ∘ , and φ 3 , θ 3 = 30 ∘ , 10 ∘ , respectively.

In this paper, we put forward a novel signal subspace reconstruction model to match the signal subspace obtained based on the covariance matrix of the array output to enhance the performance of the DOD and DOA detection. In the design progress, the problem of DOD and DOA detection is transformed into a signal subspace reconstruction and matching problem. A multi-dimensional objective function is built and optimized to solve the DOD and DOA. Simulation results prove the performance of the developed algorithm compared with the other subspace-based methods.

At the current stage, we have completed a theoretical analysis and offered a comprehensive suite of experiments. Some practical experiments (including hardware) would be an interesting avenue to explore in future studies.