2-D DOA tracking using variational sparse Bayesian learning embedded with Kalman filter
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Abstract
In this paper, we consider the 2-D direction-of-arrival (DOA) tracking problem. The signals are captured by a uniform spherical array and therefore can be analyzed in the spherical harmonics domain. Exploiting the sparsity of source DOAs in the whole angular region, we propose a novel DOA tracking method to estimate the source locations and trace their trajectories by using the variational sparse Bayesian learning (VSBL) embedded with Kalman filter (KF). First, a transition probabilities (TP) model is used to build the state transition process, which assumes that each source moves to its adjacent grids with equal probability. Second, the states are estimated by KF in the variational E-step of the VSBL and the variances of the state noise and measurement noise are learned in the variational M-step of the VSBL. Finally, the proposed method is extended to deal with the off-grid tracking problem. Simulations show that the proposed method has higher accuracy than VSBL and KF methods.
Keywords
2-D direction-of-arrival (DOA) tracking Spherical array Transition probabilities (TP) model Variational sparse Bayesian learning (VSBL) Kalman filter (KF)1 Introduction
Direction-of-arrival (DOA) estimation is an active research field of array signal processing and has been used in various applications, such as radar, channel modeling, tracking, and surveillance [1, 2, 3]. Among the estimation algorithms, multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRIT) are the most representative methods, which employ the signal and noise subspaces. Compared with the conventional beamforming algorithms, these methods enhance the estimation precision. However, the number of impinging signals must be the prior knowledge and the computational complexity of decomposing the covariance matrix increases when the number of array elements rises. Recently, sparse reconstruction methods have attracted substantial attention because the signals impinging on an array are intrinsically sparse in the spatial domain [4, 5]. In these methods, the whole angular domain is divided into some predefined grids and a measurement matrix is constructed by sampling these grids. Based on the singular value decomposition (SVD), an approach named as l_{1}-SVD was proposed to reduce the computational complexity and enforce sparsity using l_{1}-norm [6]. Compared with the l_{1}-SVD method, sparse Bayesian learning (SBL) can model the sparse signals more flexibly and give more accurate recovery results [7, 8, 9]. When SBL was used to estimate DOA, the sparse prior for the interested signals is Gaussian [10] or Laplacian distribution [11]. The SBL method can achieve good estimation results for static DOA estimation.
In the case of tracking moving targets, most methods assume that each source angle is constant during a time interval. However, it may differ from one interval to another because of the moving sources [12]. There are some different approaches to track the DOAs of moving sources, such as classical subspace optimization approach, sparsity recovery theory, and adaptive filter method. The subspace optimization approaches only optimize the signal or noise subspace without using the eigenvalue decomposition (EVD) so that they can reduce the computational complexity and storage requirements. Yang presented a new approach to track the signal subspace using an unconstrained minimization method [13]. The subspace method, which can be used to dynamic DOAs, was extended in L-shaped array [14] and two parallel linear arrays [15], respectively. The performance of the subspace method relies on the number of snapshots. This kind of method is inapplicable for the moving targets when the number of snapshots in each time interval is relatively small.
Vaswani et al. reviewed many algorithms on the analysis of dynamic sparse signal recovery. If the support change is highly correlated and the correlation model is known, we can get an accurate support estimation by using the previous estimation information [16, 17, 18]. In [19, 20], these methods combined the slow signal value change and slow support change to enhance the precision. Most of these methods are the deformation of basis pursuit denoising. A sequential Bayesian algorithm was introduced to estimate the moving DOAs in the time-varying circumstance [21]. It assumed that the sources move at a constant velocity and used this hypothesis to construct the signal-moving model. The locally competitive algorithm (LCA) was proposed to construct a dynamic system and track DOAs [22]. This method mainly introduced a thresholding function to enforce sparsity. A model was proposed to describe a time-varying array response in the frequency domain for each source [23]. The key of this paper is that it utilizes a hidden Markov model to describe the moving signal and uses the posterior inference to estimate the signal positions. It traps into the local optimum when the signals alias at high frequency.
The Kalman filter (KF) method is the most representative method for tracking sources in the adaptive filter theory, which means the value of the predicted state should equal to the value of the actual state by minimizing the Bayesian mean square error of the state vector. However, the variance of the noise and state can be seen as prior in the KF method. In [24], the time difference of arrival (TDOA) information was calculated by a pair of microphones and a distributed unscented KF method was used to track speakers in a nonlinear measurement model. The method combining KF with compression sensing was also used for dynamic DOA estimation [25, 26]. The state transition function was built under the assumption that the bearing change rate had been known [25]. This assumption is hard to be satisfied in real applications. A Bayesian compressive sensing Kalman filter (BCSKF) method [26] was proposed to track dynamic moving sources. This method used the constant DOA changes in the KF prediction, which meant that the source moved to the designated direction with a fixed step. The particle filter was utilized to track the trajectory of target. It adopted a series of particles to represent the posterior distribution of signal stochastic process. For example, an algorithm combining compressive sense and particle filter was introduced in [27]. However, it only used the compressive sense to estimate the original position and utilized particle filter to track the trajectory.
In this paper, a spherical array is used to track the moving sources in 3-D space because the array is 3D symmetric structure and can capture high-order sound field information [28, 29, 30]. Besides, comparing with linear array, spherical array can capture the two-dimension information of signal rather than single-angle information. In addition, due to the special construction of spherical array, the receiving signal can be unfolded in special harmonic domain, which can separate the signal position coordinates and sensor position coordinates conveniently.
A new method is proposed based on the spherical array to track the 2-D dynamic DOAs. It combines the variational Bayesian inference and KF to improve the tracking performance. First, a transition probabilities (TP) model is built to describe the state transition process of a signal. In this model, the source moves to an uncertain situation with an equal probability rather than a determining expected DOA change in [26]. In addition, TP model is convenient to build with a tracking framework and adopts sparse methods to estimate signal position. Based on this model, an alternating iterative method is developed to track DOAs. In the first step, we use the KF method to estimate the signal values. In the second step, we use variational sparse Bayesian learning (VSBL) to learn the variances of the measurement noise and state noise, which are useful to update the signal state in the KF method. This is an interdependent process.
There are three differences between the proposed approach and the method appeared in paper [26]. On the one hand, the proposed method constructs a real-valued steering matrix in signal model rather than splitting the complex value into several real-valued ones, which can reduce a half of the computational quantity approximately. On the other hand, the KF estimation values is used instead of the Bayesian estimation values. The main contribution of paper [26] is putting the estimation parameters of KF to optimize the low bound of relate vector machine. However, the KF estimation will be embedded in VSBL to improve the precision further in this paper. Finally, we introduce the off-grid model to overcome the mismatch problems.
The rest of the paper is organized as follows. The real-valued array signal model is given in Section 2. The variational Bayesian inference is briefly reviewed, and the proposed tracking method is introduced in Sections 3 and 4. Numerical examples and simulation results are given in Section 5. Section 6 concludes the paper.
(a, b, A, B, ⋯), scalar variable | arg(⋅), phase operator |
(A, B, ⋯), the matrix variable | |⋅|, the absolute value |
(a, b, ⋯), column vector | (⋅)^{+}, the Moore-Penrose pseudo inverse |
(⋅)^{ T }, transpose operator | (⋅)^{'}, the derivation operator |
(⋅)^{∗}, complex conjugation operator | ‖⋅‖, l_{2} norm |
(⋅)^{ H }, conjugate transpose operator | ‖⋅‖_{ F }, Frobenius norm |
diag(⋅), diagonal matrix | 〈⋅〉, expectation operator |
blkdiag(⋅), block diagonal matrix | A(n, :), the nth row of matrix A |
Re(⋅), the real parts of a complex value | A(:, n), the nth column of matrix A |
exp(⋅), the exponent signal | Im(⋅), the imaginary parts of a complex value |
k, the wavenumber | D, the number of signals |
(ϑ_{ l }, φ_{ l }), the elevation and azimuth of the sensor | \( \left({\overset{\smile }{\theta}}_{d,t},{\overset{\smile }{\phi}}_{d,t}\right) \), the elevation and azimuth of the dth signal at the tth time interval |
L, the number of sensors | t, time interval |
B, snapshots | R, the radius of sphere array |
G_{1}, G_{2}, the azimuth and elevation range | i, the imaginary unit \( \sqrt{-1} \) |
\( \left(\overset{\smile }{\mathbf{X}},\overline{\mathbf{X}},\mathbf{X}\right) \), the space domain, spherical domain and real-valued receiving signal | \( \left(\overset{\smile }{\mathbf{A}},\widehat{\mathbf{A}},\overline{\mathbf{A}},\mathbf{A}\right) \), the true steering, dictionary, the spherical and real-valued matrix |
\( \left(\overset{\smile }{\mathbf{S}},\overline{\mathbf{S}},\mathbf{S}\right) \), the space amplify, the sparse amplify signal and the real-valued amplify signal | \( \left(\overset{\smile }{\mathbf{V}},\overline{\mathbf{V}},\mathbf{V}\right) \), the space domain, the spherical and the real-valued noise |
h_{ n }, spherical Hankel function of order n | \( {Y}_n^m\left(\cdot \right) \), the spherical harmonic of order n and degree m |
I_{ n }, an n × n identity matrix | j_{ n }, the n-order spherical Bessel function |
J_{ n }, the exchange matrix with ones on its antidiagonal and zeros elsewhere | (⋅)^{'}, the derivation operators |
Γ(⋅), a Gamma function | 0_{ n }, a column vector containing n zeros |
(Ab), take the variables A except variable b |
2 Real-valued array signal model
where 0 ≤ n ≤ N, − n ≤ m ≤ n, and \( {P}_n^m\left(\cos \theta \right) \) are the associated Legendre polynomials [32].
where \( {\overline{\mathbf{X}}}_t={\mathbf{B}}^{-1}(k){\mathbf{Y}}^{+}\left(\boldsymbol{\Omega} \right){\overset{\smile }{\mathbf{X}}}_t \), \( \overline{\mathbf{A}}={\mathbf{Y}}^H\left(\boldsymbol{\Phi} \right) \), and \( {\overline{\mathbf{V}}}_t={\mathbf{B}}^{-1}(k){\mathbf{Y}}^{+}\left(\boldsymbol{\Omega} \right){\overset{\smile }{\mathbf{V}}}_t \). Owing to the special property of spherical harmonic function, the complex-valued model can be transformed into a real-valued one to reduce the computational complexity.
Now, the real-valued signal model (20) is obtained in the spherical harmonics domain.
3 Variational sparse Bayesian learning embedded with Kalman filter (VSBLKF) for 2-D DOA tracking
where E_{ t } is the state noise.
where μ indicates one of the parameter and 〈lnp(X, θ)〉_{ θ μ } is the expectation of the joint probability of the data and latent variables which take over all variables except μ.
where Γ(⋅) is a Gamma function [40]. The expectation of α_{g, t} in (34) is 〈α_{g, t}〉 = a_{g, t}/b_{g, t}. Note that the marginal distribution of S_{ t } can be obtained by integrating over α_{ t }. Note that if the prior distribution and likelihood function conjugate each other, it will make the posterior distribution to have the same form with prior distribution [36].
The variational framework introduces a factorial representation (27) approximating to the posterior distribution p(α_{ t }, S_{ t }, Δ_{ t }| X_{ t }) by Q(α_{ t }, S_{ t }, Δ_{ t }) = Q(S_{ t })Q(α_{ t })Q(Δ_{ t }). We propose the VSBLKF algorithm using the expectation maximization (EM) updates. In order to compute the E-step, it needs to know the posterior distribution of the unknown sparse state signal, which is Gaussian with mean U_{t ∣ t} and covariance Σ_{t ∣ t}. These parameters can be computed by using KF prediction and updating equations as follows:
Comparison of computational load in one iteration
Real multiplication | Real addition | ||
---|---|---|---|
Complex-valued model | \( {\mathbf{A}}_{t,c}^H{\boldsymbol{\Sigma}}_{t\mid t-1} \) | 4JG^{2} | (4J − 2)G^{2} |
\( {\mathbf{A}}_{t,c}^H{\mathbf{S}}_{t,c} \) | 4JG | (4G − 2)J | |
Real-valued model | \( {\mathbf{A}}_{t,r}^H{\boldsymbol{\Sigma}}_{t\mid t-1} \) | 2JG^{2} | 2(J − 1)G^{2} |
\( {\mathbf{A}}_{t,r}^H{\mathbf{S}}_{t,r} \) | 2JG | 2(G − 1)J |
4 Extension to off-grid problem
The initial value of β and γ are set as 0_{ G } (a column vector containing G zeros). The number of source signals is known a priori. Each signal current coarse location can be approximated by VSBLKF and the biases between the reference and estimation are gauged using (59) and (60), respectively. The concrete steps are summarized in Algorithm 2.
5 Results and discussion
In this section, we want to verify the robustness and performance of the proposed algorithm compared with the standard KF, SBL, and VSBL methods. We use a rigid spherical array, which has 32 sensors distributed in a uniform way and radius R = 0.1 m. The maximum order of the spherical harmonics is N = 4. In our experiments, the ranges of elevation and azimuth are defined from 0° to 180° and from 0° to 360°. We divide them into 31 and 62 grids with stationary angular interval respectively. Therefore, there are 1922 grids which are the possible angles for source signals. Note that we only choose the range of azimuth from 0° to 180° in our simulations to reduce the calculate complexity. In the moving process, assuming each source is likely to move to its adjacent grids or stay at current grid, which means a source can move to the different directions with 6° or be static with equal probability. The proposed method can track other trajectories as long as the trajectory can be described by the grids and obey the TP model. The trajectory used in this paper is randomly generated under the frame of TP model. In order to show a series of performance quantitatively, such as on-grid, off-grid, and RMSE vs SNR, we used one trajectory to explain these results. One random realization of this movement model is considered for T = 50 time interval. The number of Monte Carlo trials is 500. The hyperpriors \( {a}_{g,t},{b}_{g,t},{b}_{j,t}^{\left(\Delta \right)},{c}_{j,t} \) are set as 10^{‐3}.
5.1 Example 1: the performance of the proposed method with an on-grid model
5.2 Example 2: the RMSE versus SNR with an on-grid model
5.3 Example 3: the performance of the proposed method with an off-grid model
5.4 Example 4: the RMSE versus SNR with an off-grid model
5.5 Example 5: the RMSE versus different grids
5.6 Example 6: the performance of the proposed method for multiple signals
5.7 Example 7: the cost time versus different methods
Comparison of time cost for different methods
VSBL | VSBLKF (real) | OGVSBLKF (real) | VSBLKF (complex) | OGVSBLKF (complex) | |
---|---|---|---|---|---|
Time(s) | 4.51 | 21.23 | 28.51 | 44.31 | 55.36 |
6 Conclusions
In order to track the 2-D DOAs, we construct the state transition function according to the TP model based on a spherical array. The angular space is divided into grids to model a sparse signal. Through combining VSBL and KF methods, we propose an effective method called VSBLKF to track dynamic DOAs, where the KF estimation is embedded into the VSBL to estimate the signals. Besides, we extend our algorithm to the off-grid model. Simulations show that the proposed method achieves better tracking and anti-noise performance than VSBL and KF. In the future, we will extend the algorithm to wideband signals.
Notes
Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their valuable comments.
Funding
The work was supported by the National Natural Science Foundation (61571279, 61501288) and the Shanghai Science and Technology Commission Scientific Research Project (16010500100).
Authors’ contributions
QH and JH designed and implemented the proposed algorithm and wrote the paper. KL and YF scientifically supervised the work and contributed in implementing the proposed algorithm. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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