DOA and Noncircular Phase Estimation of Noncircular Signal via an Improved Noncircular Rotational Invariance Propagator Method

We consider the computationally efficient direction-of-arrival (DOA) and noncircular (NC) phase estimation problem of noncircular signal for uniform linear array. The key idea is to apply the noncircular propagator method (NC-PM) which does not require eigenvalue decomposition (EVD) of the covariance matrix or singular value decomposition (SVD) of the received data. Noncircular rotational invariance propagator method (NC-RI-PM) avoids spectral peak searching in PM and can obtain the closed-form solution of DOA, so it has lower computational complexity. An improved NC-RI-PM algorithm of noncircular signal for uniform linear array is proposed to estimate the elevation angles and noncircular phases with automatic pairing. We reconstruct the extended array output by combining the array output and its conjugated counterpart. Our algorithm fully uses the extended array elements in the improved propagator matrix to estimate the elevation angles and noncircular phases by utilizing the rotational invariance property between subarrays. Compared with NC-RI-PM, the proposed algorithm has better angle estimation performance and much lower computational load. The computational complexity of the proposed algorithm is analyzed. We also derive the variance of estimation error and Cramer-Rao bound (CRB) of noncircular signal for uniform linear array. Finally, simulation results are presented to demonstrate the effectiveness of our algorithm.


Introduction
Over the last several decades, the problem of estimating the direction-of-arrival (DOA) of multiple sources in the field of array signal processing has received considerable attention [1][2][3].A variety of DOA estimation algorithms have been developed and applied in many fields, such as mobile communication system, radio astronomy, sonar, and radar [4][5][6][7][8][9][10][11][12][13].Although the maximum likelihood estimator provides the optimum parameter estimation performance, its computational complexity is extremely demanding [8][9][10].Simpler but suboptimal solutions can be achieved by the subspace-based approaches, which rely on the decomposition of observation space into signal subspace and noise subspace.For example, multiple signals classification (MUSIC) method [11] and estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm [12,13] are subspacebased DOA estimation algorithms both well known for their good angle estimation performance.However, conventional subspace techniques necessitate eigenvalue decomposition (EVD) of covariance matrix or singular value decomposition (SVD) of data matrix to estimate the signal and noise subspaces; thus, huge computation complexity will be involved, particularly when the number of sensors is large, such as the large towed arrays in sonar [14,15].It is well known that propagator method (PM) does not require EVD of covariance matrix or SVD of received data; thus, the computational load of PM algorithm can be significantly smaller [16].But the spectral peak searching process is used in conventional PM algorithms [17][18][19][20]; in order to save the complexity, rotational invariance PM (RI-PM) algorithms are proposed [21][22][23], which avoid the spectral peak searching process and can obtain the closed-form solution of DOA.
The binary phase shift keying (BPSK), amplitude modulation (AM), and unbalanced quadrature phase shift keying (UQPSK) modulated signals frequently used in communication systems are noncircular (NC) signals [24].The statistical parameters of noncircular signal, such as first and second moments, are rotational variant.The noncircularity of signal is investigated to enhance the performance of angle estimation algorithm [25].Many DOA estimation methods of noncircular signals have been reported, which contain NC-MUSIC algorithms [26][27][28], NC-ESPRIT algorithms [29][30][31], and noncircular parallel factor (NC-PARAFAC) algorithm [32].These algorithms we mention above can estimate more sources and have better angle estimation performance by introducing the noncircularity of signal into the conventional DOA estimation algorithms.
Many DOA estimation algorithms of noncircular signal based on PM have also been studied [33][34][35][36][37][38].These noncircular propagator method (NC-PM) algorithms can be divided into two kinds, that is, the noncircular spectral peak searching PM algorithms and noncircular rotational invariance PM (NC-RI-PM) algorithms.In fact, Liu et al. have proposed NC-PM algorithm [34] and real-valued NC-PM algorithm [35]; unfortunately, the spectral peak searching process is involved, which generally requires highly computational complexity.Subsequently, Liu et al. proposed NC-root-PM algorithm [36,37] by exploiting the uniform distributed characteristic of the array elements, but its complexity is still high.The NC-RI-PM algorithm has been proposed by Sun and Zhou in [38], which has much lower computational complexity than Liu's algorithm and can obtain the closed-form solution of DOA.However, we find that the reconstructed array output can be optimized, and some array elements in the extended propagator matrix have not been used; thus, the angle estimation performance of NC-RI-PM algorithm can be enhanced further.It is worth noting that the rotational invariance PM (RI-PM) algorithms have also been studied in [39][40][41][42], but the noncircularity of signal has not been considered.
The aim of this paper is to develop a computationally efficient parameter estimation algorithm of noncircular signal.We propose an improved noncircular rotational invariance PM (improved NC-RI-PM) algorithm of noncircular signal for uniform linear array, which combine the noncircularity of signal and rotational invariance property between subarrays.Compared with NC-RI-PM algorithm, the proposed algorithm has the following advantages: (1) It has better angle estimation performance than that of NC-RI-PM algorithm.
(2) The proposed algorithm has much lower computational complexity.(3) It can estimate elevation angles and noncircular phases with automatic pairing.
The reminder of this paper is organized as follows.In Section 2, we describe the data model for uniform linear array.The improved NC-RI-PM algorithm is presented in Section 3, as well as the computational complexity analysis and comparison.Section 4 derives the variance of estimation error and Cramer-Rao bound (CRB) of the proposed algorithm.Simulation results are presented in Section 5, while the conclusions are shown in Section 6.

Data Model
We assume that there are  uncorrelated narrowband signals impinging on a uniform linear array equipped with  antennas as shown in Figure 1, and  > .We also assume that sources are far away from the array; thus, the incoming waves over the antennas are essentially planes.The noise is additive independent identically distributed Gaussian with zero mean and variance  2 , independent of signals.We denote the elevation angles of sources as  = [ 1 ,  2 , . . .,   ], where   is the elevation angle of th source,  = 1, . . ., .
From [38], the array output at time  can be modeled as where Α = [a( where s 0 () ∈ R ×1 .Ψ is a diagonal matrix; where   is the noncircular phase of th signal.

Angle Estimation Algorithm
In [38], Sun and Zhou have proposed the NC-RI-PM algorithm for uniform linear arrays which has much lower computational complexity than NC-PM algorithm based on spectral peak searching and can obtain the closedform solution of DOA.However, we find that NC-RI-PM algorithm just uses part of the array elements in the extended propagator matrix, and the reconstructed array output data can be optimized.Thus, we propose an improved NC-RI-PM algorithm of noncircular signal for uniform linear array, which is presented as follows.

Improved NC-RI-PM Algorithm.
According to (1), we construct the extended array output as where n() = [ n() J  n * () ].J  is the permutation matrix, where Φ = Ω −(−1) is a diagonal matrix, and Consider that channel state information is constant during sampling process.The sample data of extended array output can be expressed as where S = [S( 1 ), . . ., S(  )] and N = [N( 1 ), . . ., N(  )] are the sample signal matrix and sample noise matrix, respectively,  = 1, . . ., , and  is the number of snapshots.We partition the extended steering matrix A as where From [43], A 2 is a linear transformation of A 1 , where P  ∈ C ×(2−) is the improved propagator matrix.
According to (4), the covariance matrix of the extended array output is [38] where R ∈ C 2×2 is the covariance matrix.
Partition the covariance matrix R as where In the noiseless case, we can have Actually, there is always noise, and the propagator matrix can be estimated by the following minimization problem [43]: where ‖ ⋅ ‖  denotes the Frobenius norm.The estimate of Then define where P ∈ C 2× .In the noiseless case, according to ( 9)-( 10), we can have In the following, we estimate the elevation angles and noncircular phases via utilizing the rotational invariance property between subarrays.Partition the matrix P into two parts as where where J  and J  are the selective matrix.

Mathematical Problems in Engineering
Let According to ( 6)-( 7) and ( 21), Define According to ( 22)-( 23), we can have Perform the EVD of P  , which can be written as where The eigenvalues of P  in Σ are corresponding to the diagonal elements of Ω, and the eigenvectors of P  in U are estimates of the column vectors in matrix A 1 .Thus, the elevation angle of th source can be estimated by where θ is the estimate of   .Next, we can also estimate the noncircular phases.According to ( 6) and ( 17)-( 18), we can have where Δ = ΦΨ −2 is a diagonal matrix.Define According to ( 27)-( 28), we can get Then P  can be written via EVD as where V = [ 1 , . . .,   ] and Λ = [ 1 , . . .,   ].The eigenvalues of P  in Λ corresponding to the diagonal elements of Δ and the eigenvectors of P  in V are estimates of the column vectors in matrix A 1 .Note that the EVD of P  and P  are performed, respectively; we should consider the column and scale ambiguity between Λ and Σ before estimating the noncircular phases.According to ( 23) and ( 28), we can find that P  and P  are all computed from the matrix P, so replace P  and P  in (28) with Thus, the estimated noncircular phases can be expressed as where ψ is the estimate of th noncircular phase.

Algorithm
Steps.The implementation of the proposed algorithm with finite array output data is summarized in this section.According to (4), the covariance matrix of sample extended array output data can be expressed as We show the major steps of the improved NC-RI-PM algorithm of noncircular signals for uniform linear arrays as follows.
(1) Construct the sample array output matrix X and corresponding covariance matrix R.
(2) Partition R into two parts as (12), and compute the propagator matrix P  .
(3) Compute P  and P  , and perform the EVD of P  and P  as ( 25) and (30).
(1) According to (34)- (35), y() ∈ C (4−4)×1 , there are 2( − 2) duplicated data in Y.These duplicated data have no benefit to improve the angle estimation performance but increase the computational complexity.The dimension of extended covariance matrix F in NC-RI-PM algorithm is (4 − 4) × (4 − 4), whereas that of R in our algorithm is just 2 × 2.So our algorithm has much lower computational complexity than NC-RI-PM algorithm.
(2) According to (36), NC-RI-PM algorithm just uses part of the extended array elements in the propagator matrix P, that is, the array elements in P 1 and P 3 , but the array elements in P 2 have not been used, whereas the improved NC-RI-PM algorithm fully uses all the elements in the extended propagator matrix.Thus, the angle estimation performance of our algorithm is better than that of NC-RI-PM algorithm.
(3) The proposed algorithm can estimate elevation angles and noncircular phases with automatic pairing, while just the elevation angles are estimated in NC-RI-PM algorithm.
Remark 1.We assume that the number of sources is preknown, and it can be estimated by some methods shown in [44][45][46].
Remark 2. According to ( 23) and ( 28), the elevation angles and noncircular phases are all estimated from the matrix P, and the column and scale ambiguity can be solved by (31).Thus, the elevation angles and noncircular phases can be pairing automatically.

Complexity Analysis.
Regarding the major computational complexity, we just consider matrix complex multiplication operations.It is known that the complexity of computing the covariance matrix R with  snapshots is in the order of ( 2 ) and that of the EVD of  ×  dimension matrix is ( 3 ).In this paper, computational complexity of computing the matrices R, P  , and P  and EVD of P  , P  are analyzed for the improved NC-RI-PM algorithm.Table 1 presents the total computational complexity of the improved NC-RI-PM algorithm, NC-ESPRIT algorithm in [29], and NC-RI-PM algorithm in [38].Figure 2 shows the simulation results of computational complexity comparison versus  and , respectively.From Figure 2, we can find that our algorithm has much lower computational complexity than that of NC-RI-PM algorithm and has approximate but still lower complexity than NC-ESPRIT algorithm.

Error Analysis and CRB
4.1.Error Analysis.This section analyses the variance of estimation error for the improved NC-RI-PM algorithm.We assume that where R is the estimation error of covariance matrix.According to (12), we can have where R 1 and R 2 are the estimation error matrix of R 1 and R 2 , respectively.Combine ( 15) and (38); the estimate of the propagator is Define where is the estimation error of the propagator matrix.
Thus, the estimation error of matrix P is According to ( 16)-( 18) and ( 41), where P  and P  are the estimation error of matrices P  and P  , respectively.Â is the estimate of the steering matrix A.
According to ( 21) and ( 42), P = [ J  (P  + P  ) P = [ J  (P  + P  ) Define P = P  + P  , P = P  + P  .According to [43], we can obtain the estimate of matrix P  as Let ε denote the th eigenvalues of P , which can be written as where   is the estimation error of   ,   =   e   P +  (P  − P  )e  , and e  is a unit column vector, e  = [0, . . ., 1, . . ., 0].
Based on the first-order Taylor series expansion, the variance of estimation error of th elevation angle can be expressed as 4.2.Cramer-Rao Bound.We derive the CRB of noncircular signal for uniform linear array of the proposed algorithm in this section.The parameters needed to be estimated can be denoted as [32]  = [ 1 , . . .,   , S   ( 1 ) , . . ., S   (  ) , S   ( 1 ) , . . ., where S  (  ) and S  (  ) denote the real and imaginary parts of S(  ), respectively.According to (8), the extended array output with  snapshots can be rewritten as The mean  and covariance Γ of Z are . . .
From [47], the (, ) element of the CRB matrix P can be expressed as where    and Γ   denote the first-order derivative of  and Γ with respect to the th element of , respectively.
For the covariance matrix Γ is just related to  2 and the first term of (51) can be ignored.The (, ) element of CRB matrix P can be simplified as According to (48) and ( 52), . . .
where S  (  ) is the th element of S(  ), d  = a  /  , and a  is the th column vector of extended direction matrix A.

S (𝑡
Then we can have According to (49) and ( 54)-( 55), the first-order derivative of  with respect to  is Combine ( 50) with (56); (52) can be rewritten as where where B  and B  are the real and imaginary parts of B, respectively.
According to (56) and (59), we can demonstrate that where We just consider the elements related to the angles.According to (61), we can have J −1 as where  denotes the parts unrelated to the elevation angles.
Until now, we obtain the CRB matrix as follows: After further simplification, the CRB matrix can be rewritten as where , and  2 is the power of noise.⊙ denotes the Hadamard product.

Simulation Results
The Monte Carlo simulations are adopted to evaluate the angle estimation performance of the proposed algorithm.We define the root-mean square error (RMSE) as where θ, is the estimate of   of the th Monte Carlo trial and  = 300 is the number of simulation loops.In the following simulations except Figures 6 and 8, we assume that there are  = 3 sources, which are located at angles of ( 1 ,  2 ,  3 ) = (10 ∘ , 30 ∘ , 50 ∘ ), respectively.The corresponding  noncircular phases are ( 1 ,  2 ,  3 ) = (20 ∘ , 40 ∘ , 60 ∘ ), respectively.We also assume that all sources have the same symbol duration and the same input signal-to-noise ratio (SNR), and the distance between adjacent antennas is equivalent to half of the wavelength,  = /2.Figure 3 presents angle estimation results of elevation angles and noncircular phases of the proposed algorithm. = 8 and  = 300 are used in the simulations, while SNR = 5 dB is used for elevation angles and SNR = 20 dB is used for noncircular phases.From Figure 3, we can see that the elevation angles and noncircular phases can be clearly observed.
Figure 4 shows the angle estimation performance comparison among the proposed algorithm, NC-RI-PM algorithm [38], NC-ESPRIT algorithm [29], and CRB of noncircular signals for uniform linear array. = 6 and  = 200 are used.It is indicated in Figure 4 that the angle performance of the improved NC-RI-PM algorithm is better than NC-RI-PM algorithm, since the proposed algorithm fully uses all the elements in the extended propagator matrix.The angle can be used to compare the angle performance of different algorithms.From Figure 4, we can see that the RMSE of the improved NC-RI-PM algorithm is much closer to CRB compared with that of NC-RI-PM algorithm.It is clearly indicated that the angle performance of the improved NC-RI-PM algorithm is better than that of NC-RI-PM algorithm, since the proposed algorithm fully uses all the elements in the extended propagator matrix.Finally, the angle estimation performance and computational complexity of the proposed algorithm are evaluated by computer simulations.Simulation results illustrate the effectiveness of the proposed algorithm in a variety of scenarios, particularly when the sources are closely spaced.Therefore, the proposed algorithm can be regarded not only as an improvement of the work in [38] but also as an improved DOA and noncircular phase estimation algorithm.

Figure 3 :
Figure 3: Angle estimation result over Monte Carlo simulations.

Figure 5 :
Figure 5: Angle estimation performances with different values of .

Figure 6 :
Figure 6: Angle estimation performances with different values of .

Table 1 :
Computational complexity of the proposed algorithm, NC-RI-PM algorithm, and NC-ESPRIT algorithm.