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Aiming at the direction-of-arrival (DOA) estimation of two-dimensional (2D) coherently distributed (CD) sources which are coherent with each other, we explore the propagator method based on spatial smoothing of a uniform rectangular array (URA). The rotational invariance relationships with respect to the nominal azimuth and nominal elevation are obtained under the small angular spreads assumption. A propagator operator is constructed through spatial smoothing of sample covariance matrices firstly. Then, combination of propagator and identical matrix is divided according to rotational operators, and the nominal angles can be obtained through eigendecomposition lastly. Realizing angle matching automatically, the proposed method can estimate multiple DOAs of 2D coherent CD sources without spectral peak searching and prior knowledge of deterministic angular signal distribution function. Simulations are conducted to verify the effectiveness of the proposed method.

In the field of array signal processing, traditional DOA estimation is based on point source models. In the real surroundings of radar and sonar systems, because of multipath propagation between receive arrays and targets, especially when the distances of targets and receive arrays are short, the spatial scattering of targets cannot be ignored, and the assumed condition of point source models is no longer valid. In such a condition, DOA estimation based on distributed source models has presented better accuracy [

According to coherence of scatterers, distributed sources can be classified as coherently and incoherently distributed sources [

No matter CD or ID sources, most estimators suppose that signal from different sources is uncorrelated. In the field of underwater detection, the general process of detection is emitting narrowband pulse sound signal firstly, followed by receiving and analyzing the target’s backscatter signal. Thus, different CD sources are supposed to be coherent, which is more reasonable on account of the coherent multipath characteristics of underwater acoustic channel.

As to ID sources, utilizing different array configurations representative estimators have been proposed in [

In [

Figure _{i}, _{i}) (_{i} is nominal azimuth angle and _{i} is nominal elevation angle of _{i} ∈ (0, π), _{i} ∈ (0, π). The noise is assumed to be additive Gaussian white with zero mean and uncorrelated between sensors.

The uniform rectangular arrays configuration.

Denote _{mk}(_{il}, _{il}) is the azimuth and elevation of the _{il} is complex gain of the _{il}) = _{i}. _{mk}(

_{i}) denotes the deterministic angular signal distribution function (ASDF) of the _{i} = (_{i}, _{i}, _{θi}, _{ϕi}) with four elements denoting the nominal azimuth, nominal elevation, azimuth angular spread, and elevation angular spread, respectively. Nominal azimuth and nominal elevation represent the center of the target which also can be expressed as DOA; azimuth spread and elevation spread indicating 2D spatial extension of a target are collectively called angular spreads.

For a Gaussian CD source, which means scatterers of the source obey Gaussian distribution, deterministic ASDF of the source can be expressed as

For a uniform CD source, which means scatterers of the source obey uniform distribution, deterministic ASDF of the source can be expressed as

Then, the received signal of (_{i}(_{i} is the reflected signal of the

Apparently, when _{θi} = _{ϕi} = 0, _{1}(_{i} is the complex correlation coefficient between the _{mk}(_{i}, _{i}) of (

Reflecting the responses of (

Thus, the received signal of (

As shown in Figure _{0} and _{0} sensors along the direction of the _{s} = _{0} + 1 times along the direction of the _{s} = _{0} + 1 times along the direction of the

The subarray grouping of URA on the

Considering the (1, 1)th subarray, the receive vector of _{0} sensors along the

Reflecting the responses of the

According to equations (_{z} is the rotation operator, which can be written as

The received vector of the (1, 1)th subarray along the

From equations (

Considering the (1, 1)th subarray and defining _{0} sensors along the

According to equations (_{x} is the rotation operator, which can be written as

The received vector of the (1, 1)th subarray along the

From equations (

Define _{0}_{0} − _{0} row of _{0} + 1 to _{0}_{0} row of _{0}_{0} − _{0}:) denotes vectors from 1 to _{0}_{0} − _{0} row of _{0}_{0} − _{0} row of _{0} + 1 to _{0}_{0} row of

Considering the (

Similarly, sub vectors

The generalized steering vector of

Considering the (

Define the covariance matrices of the received signal of the (1, 1)th subarray as^{H} denotes the Hermitian transpose and _{1}, _{2}, _{3,} and _{4} are covariance matrices of the noise vector. The spatial smoothing covariance matrices of the received signal can be obtained as follows:

_{0} > _{0} > _{S} ≥ _{S} ≥

Construct matrix _{1} is a _{2} is a (4_{0}_{0} − 4_{0} − _{0}_{0} − 4_{0} −

Let_{q×q} is

Sample covariance matrices with

Thus, spatial smoothing of sample covariance matrices can be obtained as

Combine

Divide (4_{0}_{0} − 4_{0}) × (_{0}_{0} − _{0}) dimensional matrix _{0}_{0} − _{0}) dimensional matrix _{0}_{0} − 4_{0} − _{0}_{0} − _{0}) matrix

The propagator operator ^{+} denotes the pseudoinverse. Divide _{0}_{0} − _{0}) × _{1}, _{2}, _{3,} and _{4}.

From this equation, the following relationship can be obtained:

So, we have

We can obtain eigenvalue _{i} (_{i} of _{1} by means of eigendecomposition. From equations (_{1} and _{2} have the same eigenvector; so the eigenvector of _{2} can be obtained as follows:_{i} and nominal azimuth angle _{i} of

Now, our algorithm can be summarized as follows:

Step 1: compute spatial smoothing of sample covariance matrices using equations (

Step 2: estimate the propagator operator _{1}, _{2}, _{3}, _{4} from equation (

Step 3: find eigenvalue _{i} (_{i} through eigendecomposition of _{1}.

Step 4: obtain eigenvalue _{i} of

Step 5: calculate the nominal azimuth _{i} and nominal elevation _{i} from equation (

The computation cost of the proposed method mainly consists of three parts: the calculation of the sample covariance matrix _{1} and _{2} where computation cost mainly lies in pseudoinverse operation _{0}_{0} − 2_{0})^{3}], and eigendecomposition of _{1} and _{2}^{3}). Suppose 2D CD sources are incoherent, which means decoherence by spatial smoothing is unnecessary. Then, computation cost mainly consists of calculation of the sample covariance matrix ^{3}], and eigendecomposition ^{3}). It can be concluded that using spatial smoothing to realize decoherence of 2D coherent CD sources does not change the magnitude of computational complexity in essence.

In this section, five simulation experiments are conducted to verify the effectiveness of the algorithm we proposed. All simulation experiments are based on array configuration as shown in Figure

In the first example, we investigate the performance of the proposed method versus three 2D CD sources which are uncorrelated with each other. The first source is Gaussian CD source with parameter set (30°, 45°, 2°, 2°). The second and third sources are uniform with parameter sets (50°, 45°, 2°, 2°) and (50°, 60°, 2°, 2°). The number of snapshots is set at 200. The subarray number parameter is _{s} = _{s} = 4, subarray elements parameter is _{0} = _{0} = 4. RMSE is the mean

RMSE estimated by three methods for 2D CD sources uncorrelated with each other.

In the second example, we investigate the estimation of 2D coherent CD sources versus SNR and number of snapshots. We consider three full coherence sources with equal power. The first source is Gaussian with parameter set [30°, 45°, 2°, 2°]. The second and third sources are uniform with parameter sets [50°, 45°, 2°, 2°] and [50°, 60°, 2°, 2°]. _{s} = _{s} = 4 and _{0} = _{0} = 4. RMSE takes mean values of the three sources. Figure

(a) RMSE estimated by three algorithms versus SNR; (b) RMSE estimated by three algorithms versus number of snapshots.

In the third example, we investigate the performance of the proposed method versus angular spreads. We consider two scenarios. One scenario has three full coherence Gaussian sources with equal power, and the nominal angles are (30°, 45°), (50°, 45°), and (50°, 60°). The other has three uniform sources with the same nominal angles as the first scenario. To simplify the analysis, we assume that azimuth angular spread is equal to elevation angular spread within each source, and angular spreads of three sources are the same; _{s} = _{s} = 4 and _{0} = _{0} = 4. RMSE of the two trails is defined as the mean values of three sources. From Figure

RMSE estimated by the proposed versus the angular spread for 2D coherent CD sources.

In the fourth example, we investigate the performance of the proposed method versus subarray parameters. We consider three full coherence sources with equal power. The source parameter sets are the same as the second example. The number of snapshots is set at 200, and SNR is 15 dB. For the convenience of analysis, _{s} is supposed equal to _{s,} and _{0} is supposed equal to _{0}. Figure _{s} with _{0} = 4, while Figure _{0} with _{s} = 4. From Figures _{0} > _{0} > _{S} ≥ _{S} ≥ _{0} or _{s} increases to a certain extent.

RMSE estimated by the proposed versus _{s} with _{0} = 4.

RMSE estimated by the proposed versus _{0} with _{s} = 4.

In the fifth example, we investigate the performance of the proposed method near the boundary region in comparison with the method [_{s} = _{s} = 3 and _{0} = _{0} = 3, and the total sensor number of URA is 25. DPLA is also set with total sensor number as 25. RMSE takes the mean values of the two sources. Figures

(a) RMSE estimated versus azimuth boundary region; (b) RMSE estimated versus elevation boundary region.

In this study, we have considered the problem of estimation of 2D coherent CD sources utilizing URA. The rotation invariance relationships within and between subarrays have been deduced under the small angular spreads assumption. Decoherence can be realized by virtue of 2D spatial smoothing of URA. Propagator method of sample covariance matrices based on spatial smoothing has been introduced in detail. Simulation investigates experiment conditions containing SNR, number of snapshots, angular spreads, and subarray parameters. Simulation outcomes indicated that the proposed method is effective for DOA estimation of 2D coherent CD sources and has a better performance near the boundary regions. The method proposed is also effective for the 2D CD sources which are incoherent as well as point sources.

Considering equation (_{il}, _{il}) which is DOA of

As the scatterers of the

Then, _{mk}(

Change the variables (_{i} and _{i}. Thus, cos

Because of the following relationship

Consider the relation between

Noticing the following relationship

So, we have

Similarly, we have

The data used to support the findings of this study are available from email addresses

The authors declare that they have no conflicts of interest.

Tao Wu and Yiwen Li contributed equally to this work. Tao Wu and Yiwen Li designed the algorithm scheme. Tao Wu and Xiaofeng Zhang designed the software and performed the experiments. Tao Wu and Chaoqi Fu wrote the first draft. Yijie Huang and Qingyue Gu performed proofreading and editing. All authors read and approved the final manuscript.

This research was funded by the National Natural Science Foundation of China (Grant numbers 61471299 and 51776222).