^{2}Prior

^{1}

^{2}

^{1}

^{3}

^{1}

^{2}

^{3}

Direction of arrival (DOA) estimation algorithms based on sparse Bayesian inference (SBI) can effectively estimate coherent sources without recurring to extra decorrelation techniques, and their estimation performance is highly dependent on the selection of sparse prior. Specifically, the specified sparse prior is expected to concentrate its mass on the zero and distribute with heavy tails; otherwise, these algorithms may suffer from performance degradation. In this paper, we introduce a new sparse-encouraging prior, referred to as “Gauss-Exp-Chi^{2}” prior, and develop an efficient DOA estimation algorithm for a mixture of uncorrelated and coherent sources under a hierarchical SBI framework. The Gauss-Exp-Chi^{2} prior distribution exhibits a sharp peak at the origin and heavy tails, and this property makes it an appropriate prior to encourage sparse solutions. A three-layer hierarchical sparse Bayesian model is established. Then, by exploiting variational Bayesian approximation, the model parameters are estimated by alternately updating until Kullback-Leibler (KL) divergence between the true posterior and the variational approximation becomes zero. By constructing the source power spectra with the estimated model parameters, the number and locations of the highest peaks are extracted to obtain source number and DOA estimates. In addition, some implementation details for algorithm optimization are discussed and the Cramér-Rao bound (CRB) of DOA estimation is derived. Simulation results demonstrate the effectiveness and favorable performance of the proposed algorithm as compared with the state-of-the-art sparse Bayesian algorithms.

Direction of arrival (DOA) estimation has been a crucial issue in various application areas involving radar, wireless communication, and navigation [

In order to solve the aforementioned problem, several preprocessing techniques are developed for decorrelation. In the related studies, these preprocessing techniques are mainly classified into two categories: spatial smoothing (SS) [

Unlike traditional subspace-based DOA estimation algorithms, the emerging sparse source reconstruction (SSR) algorithms [

In this paper, we develop a new sparse-encouraging prior (called Gauss-Exp-Chi^{2} prior) whose pdf distribution exhibits a sharp peak at the origin and heavy tails. With the proposed prior, the DOA estimation for a mixture of uncorrelated and coherent sources is performed under the hierarchical SBI framework using a uniform linear array (ULA). A three-layer hierarchical Bayesian model is established based on the Gauss-Exp-Chi^{2} prior. Subsequently, according to the variational Bayesian approximation, the model parameters (including the mean and variance of sparse sources and hyperparameters) keep alternately updating until the KL divergence between the true posterior and the variational approximation tends to be zero. By exploiting the estimated model parameters, the source power spectra is constructed, from which the number and locations of the highest peaks are extracted to obtain source number and DOA estimates. Simulation results show that the proposed algorithm has superior estimation performance. Now we briefly summarize the contributions of this work as follows:

To encourage sparse solutions, we develop a new sparse-encouraging prior, called Gauss-Exp-Chi^{2} prior, whose pdf distribution has a sharp peak at the origin and heavy tails.

By constructing the source powers of all the potential directions in the angular space, both source number and DOA estimates are obtained.

Variational approximations are adopted for the estimation of the hierarchical sparse Bayesian model parameters.

Several implementation details for algorithm optimization including Woodbury matrix identity for dimension-reduction, pruning a basis function and the third kind Bessel function approximation are discussed, and the CRB of DOA estimation is derived.

The remainder of this paper is organized as follows. The DOA estimation model for mixed sources is formulated in Section ^{2} prior and DOA estimation algorithm for a mixture of uncorrelated and coherent sources within the hierarchical SBI framework. The algorithm optimization and CRB of DOA estimation are discussed in Section

Vectors and matrices are denoted by lowercase and uppercase boldface letters, respectively.

Consider a total of

Divide the entire angular space into

In this section, a DOA estimation algorithm for a mixture of uncorrelated and coherent sources is proposed within the hierarchical SBI framework. A Gauss-Exp-Chi^{2} prior is developed to encourage sparse solutions, and then the parameters of three-layer hierarchical Bayesian model are estimated via variational Bayesian approximations. By constructing source power spectra, the source number and DOA estimation are obtained.

In the Bayesian model, the pdf of a joint distribution

Combining (

From a Bayesian perspective, the pdf distribution of an assigned prior is appealing to exhibit a sharp peak at the origin and heavy tails, which favors strong shrinkage of noise sources and avoids overshrinkage of the interest sources. This property is generally considered as a desirable property for enforcing sparsity and variable selection [^{2} prior, for

The first two layers (^{2} prior. Thus, the proposed prior has more free parameters to control the degree of sparseness as compared with the Laplace prior [

Based on the above analysis, the directed graph of the sparse Bayesian model is shown in Figure

Directed graph of the sparse Bayesian model.

Since ^{2} prior is a sparsity-encouraging prior, which can help to improve the performance of source reconstruction [

Note that

The reason for choosing the Gauss-Exp-Chi^{2} prior is twofold: its pdf distribution has a sharp spike at the origin and heavy tails; it forms in a conjugate manner since the exponential and Chi2 distributions belonging to the exponential distribution family are chosen as the 2nd layer and the 3rd layer of this prior, which significantly simplifies the form of posterior distributions [^{2} prior, Laplace prior, Student’s-^{2} prior is a sparsity-encouraging prior which has sharper peak and heavier tails than the existing priors.

Four kinds of pdf curves with the standard derivation: (a) compares four distributions at origin and (b) compares four distributions in tail.

It is well known that Bayesian inference operates on the basis of the posterior distribution

Denote a set

(1)

(2)

Denoting

The posterior distribution of

(3)

Thus, it can be known that

The estimation of model parameters is implemented by alternately updating the mean

1.

2.

3. update

4. update

5. update

6. update

7.

8.

The DOAs of the impinging sources are estimated via the following two steps: (1) form spatial spectra using the estimated source powers of all the potential directions and (2) extract the number and the corresponding locations of the peaks beyond a power threshold from the spectra. Let

Therefore, the source powers of all the potential directions in the angular space are estimated by calculating

For alleviating the computational complexity and speeding up the update efficiency, some implementation details are adopted for algorithm optimization in this subsection.

At each iteration, it is inevitable to calculate a

In order to speed up the update efficiency,

It is known that (

To evaluate the DOA estimation performance, the CRB of the DOA estimation for a mixture of uncorrelated and coherent sources is derived in this subsection. Consider the sparse form of the array output vector

Let

Then, the Fisher information matrix (FIM) [

It is noted that the following equality holds

In this section, several simulations are presented to illustrate the performance of the proposed algorithm as compared with the BCS [_{,} and power threshold

In the first simulation, we evaluate the effectiveness of the proposed algorithm. Assume that two uncorrelated sources from

Spectra for the proposed algorithm with the fixed SNR 0 dB and number of snapshots 200.

As can be seen from Figure

The second simulation compares the estimation performance of three algorithms, including the proposed algorithm, BCS, and NP-1 algorithms. Consider one uncorrelated source from ^{2} prior is a three-layer sparsity-encouraging prior, which can help to improve the accuracy of source reconstruction.

Success rates versus number of snapshots with the fixed SNR 0 dB.

Success rates versus SNR with the fixed number of snapshots 200.

RMSE versus number of snapshots with the fixed SNR 0 dB.

RMSE versus SNR with the fixed number of snapshots 200.

The third simulation investigates the RMSE versus grid number. The simulation settings are the same as those of the second simulation, except that grid number in this simulation is ranged from 61 to 361. Figure

RMSE versus grid number with the fixed SNR 10 dB and number of snapshots 500.

It is observed that the RMSE curves of the three algorithms rapidly decrease as grid number increases from 61 to 181, and these curves tend to slowly decrease with the increase of grid number when grid number is beyond 181. In addition, the proposed algorithm has minimum estimation errors among the three algorithms. Note that the three algorithms recover sparse sources from a Bayesian perspective; thus, they are not restricted to the restricted isometry property. This implies that the grid number can continue to increase to further improve the estimation accuracy and the computational complexity would increase accordingly. Therefore, for the purpose of balancing estimation accuracy and computational complexity, the reasonable range of grid number is from 181 to 361.

In the last simulation, we test the angular resolution of the three algorithms. Consider two coherent sources from 10.1° and

RMSE versus angular separation with the fixed SNR 10 dB and number of snapshots 500 for coherent sources.

In this paper, we develop a DOA estimation algorithm for a mixture of uncorrelated and coherent sources using SBI. In the Bayesian framework, a Gauss-Exp-Chi^{2} prior is introduced to promote sparse solutions, and the corresponding three-layer hierarchical Bayesian model is established. Then, the model parameters are estimated via variational Bayesian approximations. Finally, we form the source power spectra by using the estimated model parameters, from which the number and the locations of the highest peaks are extracted to achieve source number and DOA estimates. Simulation results show that the proposed algorithm can effectively estimate the DOAs of mixed sources and has better estimation performance than the state-of-the-art BCS algorithm and NP-1 algorithm in terms of estimation accuracy, success rate, and angular resolution.

In this Appendix, we prove that (

The expression on the right hand side of (

Thus, we first calculate the integral over

By substituting (

Thus, we have

The authors declare that they have no competing interests.

Pinjiao Zhao proposed the main idea and conceived the proposed approach. Pinjiao Zhao and Weijian Si discussed and designed the proposed algorithm. Pinjiao Zhao performed the experiments and wrote the paper. Guobing Hu and Liwei Wang reviewed and revised the manuscript. All authors read and approved the manuscript.

This work was financially supported by the High Level Talent Research Starting Project of Jinling Institute of Technology (Grant no. jit-b-201724), the National Natural Science Foundation of China (Grant no. 61671168), and the Natural Science Foundation of the Jiangsu Province (Project no. BK20161104) and the Six Talent Peaks Project of the Jiangsu Province (Project no. DZXX-022).

_{1}-norm penalty