Noncircular-PARAFAC for 2D-DOA estimation of noncircular signals in arbitrarily spaced acoustic vector-sensor array subjected to unknown locations

In this paper, we propose a noncircular-parallel factor (NC-PARAFAC) algorithm for two-dimensional direction of arrival (DOA) estimation of noncircular signals for acoustic vector-sensor array. The proposed algorithm enhances the angle estimation performance via utilizing the noncircularity of the signals, and it can be suitable for arbitrary array subjected to unknown locations and achieve automatically paired two-dimensional angle estimation. The proposed algorithm has better angle estimation performance than estimation of signal parameters via rotational invariance technique, PARAFAC algorithm, and propagator method. Furthermore, the proposed algorithm has a lower computational complexity than the PARAFAC algorithm. We also derive the Crámer-Rao bound of DOA estimation of noncircular signal in acoustic vector-sensor array. The simulation results verify the effectiveness of the algorithm.

The binary phase shift keying (BPSK) and amplitude modulation modulated signals, which are called noncircular signal because their statistics are rotationally variants, are widely used in communication systems [22,23]. As far as we know, few works on DOA estimation of noncircular signal in acoustic vector-sensor array has been reported. In this paper, we propose a noncircular-PARAFAC (NC-PARAFAC) algorithm for two-dimensional (2D) DOA estimation of noncircular signals using arbitrarily spaced acoustic vector-sensor array subjected to unknown locations. Compared with PARAFAC algorithm, the proposed algorithm enhances the parameter estimation performance via utilizing the noncircularity of the signals. The proposed algorithm can be suitable for arbitrary array subjected to unknown locations and achieve automatically paired two-dimensional angle estimation. Furthermore, our algorithm has better angle estimation performance than PM, ESPRIT algorithm, and PARAFAC algorithm.
The work in [15] links the acoustic vector-sensor array parameter estimation problem to trilinear model and derives a PARAFAC algorithm whose angle estimation performance is better than the ESPRIT algorithm. There are some differences between our algorithm and the PARAFAC algorithm. Firstly, our algorithm exploits the noncircular property to improve DOA estimation. Secondly, our NC-PARAFAC algorithm is suitable for the DOA estimation of noncircular signal, and our algorithm can be regarded as an extension of PARAFAC. Thirdly, the real trilinear model is used for NC-PARAFAC algorithm, while the complex trilinear model is employed in PARAFAC algorithm. Our NC-PARAFAC algorithm is better for the practical system. A contribution of this paper is to extend PARAFAC method to noncircular signal acoustic vector-sensor array.
The remainder of this paper is structured as follows. Section 2 develops the data model, and Section 3 presents the algorithm and the complexity analysis. In Section 4, we derive the Crámer-Rao bound (CRB) of the DOA estimation of noncircular signal in acoustic vectorsensor array. In Section 5, simulation results are presented to verify improvement for the proposed algorithm, while conclusions are shown in Section 6.
The following notations are used: (.) T , (.) H , (.) −1 , and (.) + denote transpose, conjugate-transpose, inverse, and pseudoinverse operations, respectively. diag(v) stands for diagonal matrix whose diagonal element is a vector v. D n (.) is to take the nth row of the matrix to construct a diagonal matrix. I K is a K × K identity matrix. ⊗, ∘, and ⊙ are the Kronecker product, Khatri-Rao product, and Hadamard product, respectively. Re{.} and Im{.} are to get the real part and imaginary part of the complex, respectively. E[.] is expectation operator.

Data model
We assume that a total of K narrowband plane waves impinge on an array equipped with M acoustic vector sensors, which are all located at arbitrarily unknown three-dimensional positions as shown in Figure 1. We consider the noncircular signals in the far field, in which case the sources are far away enough that the arriving waves are essentially planes over the array. Also assume that the noise is an additive independent identically distributed Gaussian with zero mean and variance σ 2 , which is independent of the sources. The kth signal is arriving from direction (∅ k , φ k ), where ∅ k and φ k stand for the azimuth angle and the elevation angle, respectively. Let θ k = [ϕ k , φ k ] T , which is the 2D-DOA of the kth source.
According to [7], the output of the irregular array containing M acoustic vector sensors is τ mk is the differential time delay of the kth wave between the origin and the mth sensor; s(t) contains K signals; n p (t), n x (t), n y (t), and n z (t) are the received signals from the acoustic vector-sensor array. Ф x , Ф y , and Ф z are shown as Φ y ¼ diag sinϕ 1 cosφ 1 ; sinϕ 2 cosφ 2 ; ⋯; sinϕ K cosφ K ð Þ : Consider that channel state information is constant during L transmitting symbols; we have We construct the following matrix X E ∈£ 4MÂL ∈£ K ÂL is the source matrix for L samples; and N p = [n p (1), n p (2),L, n p (L)], N x = [n x (1), n x (2), L, n x (L)], N y = [n y (1), n y (2), L, n y (L)], and N z = [n z (1), n z (2), L, n z (L)] are the noise matrices. According to noncircular property, the source matrix can be denoted by Figure 1 The array geometry.
We define the matrix H as The matrix V n (n = 1, 2, …, 4) in Equation 10 can be denoted by where D n (.) is to extract the nth row of its matrix argument and construct a diagonal matrix out of it. D n H ð Þ ¼ diag h n;1 ; h n;2 ; ⋯; h n;K Â Ã À Á ∈£ K ÂK , and h m,k stands for the (m,k) element of the matrix H. The noiseless signal in Equation 13 can be denoted as the trilinear model [24], where s l,k is the (l, k) element of matrix S 0 and similarly for the others. The trilinear model displays the reflection for three different kinds of diversity [24]. Equation 13 can be interpreted as slicing the 3D data in a series of slices along the spatial direction. In similar ways, the model symmetry in Equation 14 allows two more matrix system rearrangements, where we have where N ′ l and N ′ m are the received noises. V n , U l , and W l are slices along the different directions when slicing the trilinear model. Remark 1. The trilinear model with M × L × 4 is used in PARAFAC algorithm [19], while that with 2M × L × 4 is employed in our algorithm. Because of noncircularity, the array aperture in our algorithm can be regarded as two times that of the PARAFAC algorithm.

Trilinear decomposition
Trilinear alternating least square (TALS) algorithm is a common data detection method for trilinear model [24]. The detailed steps are shown as follows: Referring to Equation 10, least squares (LS) fitting can be given by LS update for S 0 is shown aŝ Similarly, from the second way of slices: Finally, from the third way of slices: The sum of squared residuals (SSR) in the trilinear fitting is defined as wherex m;l;n is the noisy data,â m;k ,ŝ l;k , andĥ n;k are the estimates of the a m,k , s l,k , and h n,k , respectively. With respect to Equations 18, 20, and 21, the matrices S 0 , A, and H are conditionally updated with least squares, respectively. TALS is quite easy to implement and guaranteed to converge. In this paper, we use the complex parallel factor analysis (COMFAC) algorithm [25] for trilinear decomposition. COMFAC algorithm is essentially a fast implementation of TALS.

Identifiability of trilinear decomposition
In this subsection, we discuss the identifiability of trilinear decomposition.

then
A , S 0 , and H are unique up to the permutation and scaling of columns, which indicates any other triple A;Ŝ 0 ;Ĥ that construct V n (n = 1,…, 4) is related to A, S 0 , and H viâ where ∏ is a permutation matrix, and Δ 1 , Δ 2 , and Δ 3 are diagonal scaling matrices which satisfy Δ 1 , Δ 2 , Δ 3 = I K . N 1 , N 2 , and N 3 are estimation errors. Generically, the matrices are full k-rank, and then the identifiable condition becomes

Two-dimensional DOA estimation algorithm
For the received noisy signal, we attainĤ ¼ HΠΔ 3 þ N 3 through PARAFAC decomposition. The matrixĤ is processed through normalization, which also resolves the scale ambiguity, to get the matrix Finally, the elevation and azimuth angles are estimated bŷ A NC-PARAFAC-based 2D-DOA estimation for acoustic vector-sensor array is proposed in this paper. The detailed steps are shown as follows: Step 1. Construct the data matrix V via Equations 8 and 9.
Step 2. Initial the matrices A, S 0 , and H. Remark 3. Random initialization is used for the proposed algorithm, and we also employ two slices and ES-PRIT algorithm to obtain the initial estimation of parameter matrices.
Remark 4. The number of sources is pre-known, and it can be estimated by some methods in [26][27][28][29].

Complexity analysis and advantages of the proposed algorithm
The algorithm that we propose can have a lower computational complexity compared to PARAFAC algorithm. PARAFAC algorithm in [15] requires O(nK 3 + 4nMLK), where n is the number of TALS iterations, while our NC-PARAFAC algorithm requires O(nK 3 /4 + 2nMLK). Our algorithm has a heavier computational load than ESPRIT algorithm, and the complexity of ESPRIT is O(16M 2 L + 64M 3 + K 3 ). Figure 2 shows the complexity comparison among three algorithms with n = 40, K = 3, and M = 8. From Figure 2 we find that our algorithm has lower complexity than PARAFAC and higher complexity than ESPRIT.
The proposed algorithm in this paper has the following advantages: 1. The angle estimation performance of the proposed algorithm is better than that of ESPRIT algorithm, PM, and PARAFAC algorithm. 2. The proposed algorithm can be suitable for arbitrary array subjected to unknown locations. 3. The proposed algorithm can achieve automatically paired two-dimensional angle estimation. 4. The proposed algorithm has lower computational complexity than PARAFAC algorithm.

Crámer-Rao bound
There are some differences between the CRB of noncircular signal DOA estimation and that of circular signal DOA estimation. The parameters which are needed to estimate can be expressed as ð28Þ where s R (t),t = 1, …, L and s I (t),t = 1, …, L denote the real and imaginary parts of s(t), respectively.
According to Equation 6, the output with L snapshots is rewritten as where x E (l) is the lth column of the matrix X E .
The mean μ and the covariance matrix Γ of y are ; ⋯Γ ¼ From [30], the (i, j) element of the CRB matrix (P cr ) is expressed as where Γ ′ i and μ ′ i are the derivative of Γ and μ on the ith element of ζ, respectively. The covariance matrix is just related to σ 2 , so the first part of Equation 31 can be ignored. Then the (i, j) element of the CRB matrix (P cr ) can be rewritten as According to Equation 32,    where s k (t) is the kth element of s(t), and and According to Equations 35, 36, and 37, Equation 32 can be denoted by Thus, we just consider the elements related to the angles in J −1 . Define We demonstrate that ; so J −1 is written as where κ denotes the part we do not concern. Until now, we give the CRB matrix as follows After further simplification, we rewrite the CRB matrix where ⨀ stands for Hadamard product.
is the power of the noise.

Simulation results
We define root mean square error (RMSE) as whereâ k;n is the estimation of elevation angle/azimuth angle a k of the nth Monte Carlo trial. We assume that there are K = 3 sources with angles (φ 1 , φ 1 ) = (10°, 15°), (φ 2 , φ 2 ) = (30°, 35°), and (φ 3 , φ 3 ) = (50°, 55°).  estimation performance than PARAFAC algorithm since it uses noncircularity to expand the array aperture. Our algorithm requires the noncircularity of signal, while ESPRIT algorithm and PARAFAC algorithm need no knowledge on the noncircularity. Figure 7 presents angle estimation performance of the proposed algorithm with M = 10, K = 3 and different values of L. It is indicated in Figure 7 that the angle estimation performance of the proposed algorithm is improved with L increasing. Figure 8 shows angle estimation performance of the proposed algorithm with L = 100, K = 3 and different values of M. From Figure 8, the angle estimation performance of the proposed algorithm is improved with the number of antennas increasing. Figure 9 displays the angle estimation of the proposed algorithm with two closely spaced sources. We assume two close sources located at angles (φ 1 , φ 1 ) = (10°, 15°), (φ 2 , φ 2 ) = (12°, 17°). M = 12, J = 100, K = 2, and SNR = 12 dB are used in Figure 9. It implies that our algorithm works well for two closely spaced sources.

Conclusions
In this paper, we have presented an NC-PARAFAC algorithm for 2D-DOA estimation of noncircular signals for acoustic vector-sensor array. The proposed algorithm has better angle estimation performance than the ESPRIT algorithm, PARAFAC algorithm, and PM. Furthermore, the proposed algorithm has a lower computational complexity than PARAFAC algorithm. The proposed algorithm enhances the angle estimation performance via utilizing the noncircularity of the signals, and it can be suitable for arbitrary array subjected to unknown locations and achieve automatically paired two-dimensional angle estimation. We also analyze the complexity and derive the CRB of noncircular signal DOA estimation.