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In this paper, we combine the acoustic vector-sensor array parameter estimation problem with the parallel profiles with linear dependencies (PARALIND) model, which was originally applied to biology and chemistry. Exploiting the PARALIND decomposition approach, we propose a blind coherent two-dimensional direction of arrival (2D-DOA) estimation algorithm for arbitrarily spaced acoustic vector-sensor arrays subject to unknown locations. The proposed algorithm works well to achieve automatically paired azimuth and elevation angles for coherent and incoherent angle estimation of acoustic vector-sensor arrays, as well as the paired correlated matrix of the sources. Our algorithm, in contrast with conventional coherent angle estimation algorithms such as the forward backward spatial smoothing (FBSS) estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm, not only has much better angle estimation performance, even for closely-spaced sources, but is also available for arbitrary arrays. Simulation results verify the effectiveness of our algorithm.

Compared with traditional acoustic pressure sensor arrays, acoustic vector sensors can measure the acoustic pressure as well as all three orthogonal components of the acoustic particle velocity at a single point in space, which offers certain significant advantages in collecting more acoustic information and enhancing the system performance [

The reflected signal of the same sources through different propagation paths will produce multipath signals. Therefore, it is of significant importance to study the coherent angle estimation problem. The angle estimation algorithms listed above are all proposed for incoherent sources. When it comes to coherent sources, the coherency of sources will result in serious degradation or invalidity of the above algorithms. Some conventional coherent angle estimation algorithms, including forward backward spatial smoothing (FBSS) [

This paper combines the acoustic vector-sensor array parameter estimation problem with the so-called PARAllel profiles with LINear Dependencies (PARALIND) model, and proposes a blind coherent 2D-DOA estimation algorithm for arbitrarily spaced acoustic vector-sensor arrays subject to unknown locations by exploiting the PARALIND decomposition approach. Our algorithm can provide coherent and incoherent two-dimensional angle estimation for arbitrary arrays, and it automatically archives paired azimuth and elevation angles, and the paired correlated matrix of the sources can also be acquired. Compared with conventional coherent angle estimation algorithms such as the FBSS-ESPRIT algorithm which only works for uniform arrays, our algorithm has much better angle estimation performance. Furthermore, our algorithm performs considerably well for angle estimation of closely spaced sources. We also derive the Cramér-Rao bound (CRB) of angle estimation for arbitrarily spaced acoustic vector-sensor arrays. Simulation results verify the effectiveness of the proposed algorithm.

The trilinear decomposition, also known as PARAFAC analysis [

Although the ESPRIT algorithm [

The present paper is structured as follows: Section 2 develops the data model for arbitrarily spaced acoustic vector-sensor arrays at unknown locations; Section 3 establishes our PARALIND decomposition-based coherent 2D-DOA angle estimation algorithm in addition with the identifiability issues and complexity analysis; In Section 4, simulation results are presented to verify effectiveness of the proposed algorithm, while the final conclusions are made in Section 5.

We assume that a total of _{m}_{m}_{m}_{m}

We also assume the signals in the far-field. _{1} incoherent sources and _{1} coherent sources, are considered. Assume that the noise is additive white Gaussian, which is independent of the sources. The _{k}_{k}_{k}_{k}_{k} = [_{k}_{k}^{T}_{1}), _{2}),…, _{K}_{1}(_{2}(_{K}^{T}_{1} incoherent sources. _{1}, _{2},…, _{K}

Therefore, the output of

_{1}), _{2}),…, _{J})], _{1}), _{2}),…, _{J})] is an 4

Define ^{T}

According to [

In a no-noise case, according to _{n}^{T}D_{n}^{T}^{J×M},

According to [

According to

For attaining ^{T}

According to ^{T}

According to

Similarly, we have:

Extracting the diagonal elements of the matrices in two sides of the equation, we get:

Then we get:

The matrix ^{−1}(_{n}(

According to

According to ^{T}^{T}^{T}^{T}_{n}_{n}^{−1}(_{n}

Define ^{T}^{T}_{ji}^{−8}, finally we obtain

According to [

The two slices _{i}_{j}_{E} = _{i}^{T}^{T}_{j}((^{T}_{i}^{−}^{1}((^{T}

^{H}^{H}_{1} and _{2} are full rank. According to

_{2}_{1}^{−1}. ^{−1}_{1}, _{1}^{−1}, _{E}^{+}_{i}^{T}_{i}^{−}^{1}((^{T}_{E}

Notably, scale ambiguity and permutation ambiguity are inherent to the separation problem. However, the scale ambiguity can be resolved easily by normalization, while the existence of permutation ambiguity is not considered for angle estimation.

By imposing PARALIND decomposition for the received data matrix, we get the estimate of matrix _{1},_{2}, ⋯,_{K}_{K}_{K}

Finally, the azimuth angle estimate of

Obviously, the azimuth angle and elevation angle are automatically paired. We also obtain the estimate of the correlated matrix

Till now, we have achieved the proposal for the proposed coherent 2D-DOA estimation algorithm for the acoustic vector-sensor array. We show major steps of the proposed algorithm as follows:

Obtain

According to ^{−8}, finally obtain the corresponding estimated

Estimate the two-dimensional DOA through

In our algorithm, the complexity of each iteration is ^{2}_{1}^{2} + 2_{1} + 2_{1}) + 4_{1}^{2} + ^{2} + _{1} + 2^{2}(^{3}_{1}^{3} + 2^{3}+ _{1}^{3} + 5_{1}

_{i}

According to [^{2} stands for the covariance of the noise and _{k}_{j}_{j}

The advantages of the proposed algorithm can be presented as follows, which can be verified by the simulation results in Section 4:

The proposed algorithm is effective for coherent and incoherent two-dimensional angle estimation.

The proposed algorithm does not require precise knowledge of the characteristics of the receiver array.

The proposed algorithm can archive automatically paired angles and the corresponding correlated matrix.

The proposed algorithm has much better angle estimation performance than conventional FBSS-ESPRIT algorithm which only works for uniform array.

The proposed algorithm has considerably performance for angle estimation of closely spaced sources.

In most of the following simulations, we assume that there are 3 sources in which only source 1 and source 3 are coherent, namely the correlated matrix is

The sources are located at angles (_{1},_{1}) = (−15°, 10°), (_{2},_{2}) = (25°,20°), and (_{3},_{3}) = (35°,30°), respectively. _{k}_{k}_{k,l}_{k,l}_{k}_{k}

In order to present the angle estimation performance comparison of the proposed algorithm and FBSS-ESPRIT algorithm, we assume that the acoustic vector-sensor array is a uniform linear array (ULA) in

The following

We assume two coherent closely spaced sources located at angles (_{1},_{1}) = (0°,30°), (_{2},_{2}) = (2°,28°).

For non-coherent sources, the proposed algorithm becomes the PARAFAC algorithm [

In the present paper we have expanded the PARALIND model, originally applied to biology and chemistry, to acoustic vector-sensor array signal processing. The PARALIND model is used for coherent 2D- DOA estimation for arbitrarily spaced acoustic vector-sensor arrays subject to unknown locations. We have derived the PARALIND decomposition and its uniqueness in acoustic vector-sensor arrays, which is convenient for simulation, performance analysis and further study. Our algorithm obtains automatically paired 2D-DOA estimation and the correlated matrix of the sources. Compared with the FBSS-ESPRIT algorithm which only works for uniform arrays, our algorithm has much better angle estimation performance. Furthermore, our algorithm has considerably good performance for angle estimation of closely spaced sources.

This work is supported by China NSF Grants (60801052), Jiangsu Planned Projects for Postdoctoral Research Funds (1201039C), China Postdoctoral Science Foundation (2012M521099), Open Project of Key Laboratory of Underwater Acoustic Communication and Marine Information Technology (Xiamen University), Hubei Key Laboratory of Intelligcnt Wireless Communications (IWC2012002), the Aeronautical Science Foundation of China(20120152001), Qing Lan Project and the Fundamental Research Funds for the Central Universities (NS2013024, NZ2012010, kfjj120115, kfjj20110215).

(·)^{T}^{H}^{−1} and (·)^{+} denote transpose, conjugate transpose, conjugate, matrix inversion and pseudo-inverse operations, respectively.

_{P}

○, ⊗ and ⊙ denote Khatri-Rao, Kronecker, and Hadamard product, respectively.

_{m}(.) is to take the

^{−}^{1}(

_{k}_{k}

The structure of acoustic-vector sensor array [

Angle estimation of our algorithm for ULA in SNR = 15 dB (

Angle estimation performance comparison for ULA (

Angle estimation of our algorithm in SNR = 5 dB (

Angle estimation of our algorithm in SNR = 15 dB (

Angle estimation performance comparison (

Angle estimation performance comparison (

Angle estimation performance of our algorithm with different

Angle estimation performance of our algorithm with different

Angle estimation of our algorithm for closely spaced sources.

Angle estimation performance comparison for non-coherent sources.