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We explore the estimation of a two-dimensional (2D) nonsymmetric coherently distributed (CD) source using L-shaped arrays. Compared with a symmetric source, the modeling and estimation of a nonsymmetric source are more practical. A nonsymmetric CD source is established through modeling the deterministic angular signal distribution function as a summation of Gaussian probability density functions. Parameter estimation of the nonsymmetric distributed source is proposed under an expectation maximization (EM) framework. The proposed EM iterative calculation contains three steps in each cycle. Firstly, the nominal azimuth angles and nominal elevation angles of Gaussian components in the nonsymmetric source are obtained from the relationship of rotational invariance matrices. Then, angular spreads can be solved through one-dimensional (1D) searching based on nominal angles. Finally, the powers of Gaussian components are obtained by solving least-squares estimators. Simulations are conducted to verify the effectiveness of the nonsymmetric CD model and estimation technique.

Although point source models are commonly used in applications such as wireless communications, radar and sonar systems, distributed source models which have considered multipath propagation and the surface features of targets tend to perform better. Position localization estimators based on distributed source models have proved to be more precise in multipath scenarios compared with point source models. With information of surface features, the distributed source models have potential application for high-resolution underwater acoustical imaging.

Sources may be classified into coherently and incoherently distributed sources [

Some classical point source estimation techniques have been extended to CD sources. DSPE [

Based on DSPE, several spectral searching methods for 2D CD sources have been proposed. In [

Using an array with sensors oriented along three axes of the Cartesian coordinates, the authors of [

In [

Several low-complexity algorithms have been presented in [

Utilizing two closely spaced parallel ULAs and L-shaped arrays, two estimators for CD noncircular signals are proposed [

Estimation techniques for distributed sources mentioned above are performed based on the assumption that the spatial distributions of sources are symmetric. Nevertheless, scatters are distributed irregularly or nonuniformly around targets; the distributions of scatters in practice are generally nonsymmetric. Considering complexity, people have paid less attention to nonsymmetric distributed sources.

Based on the principle that a nonsymmetric probability distribution can be composed of several symmetric distributions, some methods for 1D nonsymmetric ID sources have been presented. In [

To the best of our knowledge, there are no algorithms for a 2D nonsymmetric distributed source. In this paper, we are concerned on the estimation of a 2D nonsymmetric CD source. Compared with a symmetric source, the modeling and estimation of a nonsymmetric source are more complex. Through modeling the nonsymmetric deterministic angular signal distribution function by Gaussian mixture, we have presented a parameter estimation method under an EM framework based on L-shaped arrays. In general EM frameworks, the maximization step is to maximize the likelihood function to get the best parameters which are hidden in expectation step results. In our method, the DOA parameters of Gaussian components of the 2D CD source are obtained through two approximate rotational invariance relations. By solving least-squares estimators, powers of Gaussian components are obtained.

Figure

The L-shaped array configuration.

The

The noise is assumed to be uncorrelated between sensors and Gaussian white with zero mean

In this study, the deterministic angular signal distribution function of the CD source can be modeled as a summation of Gaussian distributions in order to express nonsymmetry, so the deterministic angular signal distribution function is obtained as

To normalize

The summation of Gaussian distributions in (

The deterministic angular signal distribution function characterized by (

On the basis of the latent variable model described by the authors of [

Define the generalized steering vectors of the complete data for arrays

Combine the incomplete data

The noise is assumed to be distributed in the complete data equally. Combine the complete data

Define two data selection matrices as follows:

The output vectors of the subarrays

The complete data of output vectors of the subarrays

The generalized steering vectors of the complete data for

Assume that the angular spreads of Gaussian angular signal distribution in the complete data are small. The authors of [

The sample covariance matrices of the complete data

The negative logarithm of the likelihood function can be simplified as

In formula (

As

To simplify the estimation, we assume that the azimuth angular spread and elevation angular spread are the same;

The sample covariance matrix of the complete data

Let

From formula (

According to the orthogonality of the subspace,

Under the condition that the parameters nominal azimuth

The least-squares fits of the theoretical and sample covariance can be expressed as

Differentiating (

Let

From

At last, the power of the CD source can be obtained from

Now, we analyze the computational complexity of the proposed method in comparison with DSPE [

The stopping criterion of the EM algorithm is all parameters no longer changing [

Computational complexity of different methods.

Method | Calculation of the sample covariance matrix | Eigendecomposition | Searching | Total |
---|---|---|---|---|

DSPE | ||||

TLS-ESPRIT | ||||

Proposed | o |

Now, our algorithm can be summarized as follows.

Determine the number of components

Repeat the following substeps for

Compute the sample covariance matrices of complete data

Find the eigenvectors

Calculate the nominal azimuth

Estimate the power of each component

Repeat substeps 1 to 4 for

It is noteworthy that the initial positions of the Gaussian components can be obtained by existing algorithms for a symmetric CD source and set uniformly around the guessed values. The distribution of a 2D nonsymmetric CD source is unknown so the true number of Gaussian components

In this section, four simulation experiments are conducted to verify the effectiveness of the proposed technique. We consider the array geometry as depicted in Figure

We use the root-mean-squared error (RMSE) to evaluate the estimation performance. The RMSE of the nominal angles is defined as

The nominal angle in the proposed algorithm is defined as the value corresponding to the maximum point of the deterministic angular signal distribution function, which can be obtained by the partial derivative of the function.

To examine the difference between the estimated and true nonsymmetric distributed source, the estimation of nominal angles is only part of the problem; in addition, the spatial distribution of the source should be emphasized compared with the estimation of a symmetric distributed source. To evaluate the estimation of the spatial distribution, the RMSE of the distributed function is defined as

Figure

Probability density function (PDF) of the constructed nonsymmetric angular signal distribution.

In the first example, we investigate the performance of the proposed method as well as a comparison with DSPE and TLS-ESPRIT. Figures _{a}_{f}_{a}_{f}_{f}

(a) RMSE_{a}_{a}

(a) RMSE_{f}_{f}

The estimation of nonsymmetric angular signal distribution function by the proposed method.

In the second example, we examine the changing processes of Gaussian components during the EM iterations. The initial nominal azimuth and nominal elevation of components

(a) The changing process of central azimuths in EM iterations; (b) the changing process of central elevations in EM iterations; (c) the changing process of angular spreads in EM iterations; (d) the changing process of weights in EM iterations.

In the third example, we examine the relationship between the number of Gaussian components and estimation performance. The number of snapshots is set at 200 and SNR is 15 dB. Figure _{f}_{a}_{f}_{a}

(a) RMSE_{a}_{f}

The initial position parameters may be set around the guessed values; there will be initial Gaussian components outside the distributed source inevitably. To increase the robustness of the algorithm, the number of Gaussian components should be selected from a larger range; meanwhile, computing cost is a matter of balance.

In the fourth example, we examine the relationship between the initial positions of Gaussian components and estimation results. The initial positions of component _{f}_{f}

Squares of taking initial positions for each component.

Figure _{f}

(a) Number of iteration versus initial position for component _{f}

Figure _{f}

(a) Number of iteration versus initial position for component _{f}

From Figure _{f}

(a) Number of iteration versus initial position for component _{f}

Figure _{f}_{f}

(a) Number of iteration versus initial position for component _{f}

In this study, we have considered the problem of estimating a nonsymmetric 2D CD source under L-shaped arrays. The method we proposed is developed by modeling the nonsymmetric deterministic angular signal distribution function as Gaussian mixture. Parameters of a nonsymmetric CD source are more than those of a symmetric CD source. Parameter estimation based on an iterative EM framework has been introduced in detail. The computational cost of estimation for a nonsymmetric CD source is much higher compared with that of a symmetric CD source. For the sake of evaluating the estimation, two indicators are defined, and one reflects nominal angles and another for spatial distribution. Simulation results indicated that the proposed method is effective and robust for the estimation of a nonsymmetric CD source.

Change the variables (

Considering

Thus, we can obtain following relationship

Similarly,

Consider the following relationship:

So we have

The authors declare that they have no conflicts of interest.

This work was supported by the Chinese National Science Foundation under Grant no. 61471299.