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Computationally efficient nearfield source localization using thirdorder moments
EURASIP Journal on Advances in Signal Processing volume 2014, Article number: 98 (2014)
Abstract
In this paper, a thirdorder momentbased estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm is proposed for passive localization of nearfield sources. By properly choosing sensor outputs of the symmetric uniform linear array, two special thirdorder moment matrices are constructed, in which the steering matrix is the function of electric angle γ, while the rotational factor is the function of electric angles γ and ϕ. With the singular value decomposition (SVD) operation, all directionofarrivals (DOAs) are estimated from a polynomial rooting version. After substituting the DOA information into the steering matrix, the rotational factor is determined via the total least squares (TLS) version, and the related range estimations are performed. Compared with the highorder ESPRIT method, the proposed algorithm requires a lower computational burden, and it avoids the parametermatch procedure. Computer simulations are carried out to demonstrate the performance of the proposed algorithm.
1 Introduction
In the last decades, lowcomplexity parameter estimation has become an important topic required for radar, sonar, as well as communication[1, 2]. Various efficient solutions have been developed to cope with this issue. However, most of these algorithms mainly focused on the farfield source localization, such as the multiple signal classification (MUSIC) method[3], estimation of signal parameters via rotational invariance techniques (ESPRIT) method[4], and their derivatives[5, 6]. In order to locate the nearfield sources[7], several effective methods have also been derived, which can be sorted as the following two versions. The first version is named as spectral searching methods, in which the twodimensional (2D) MUSIC[8] was the original solution to the nearfield source localization problem. To cope with the computationally inefficient 2D searching procedure, a symmetric uniform linear arraybased method[9] was proposed. Based on[9], resorting to the polynomial rooting, an improved method[10] was developed, which has further avoided the onedimensional (1D) searching procedure.
The other version, based on the rotational invariance of the underlying signal subspace induced by the translational invariance of the sensor array, is named as the closedform solutions. In this version, the highorder ESPRIT method[11, 12] is especially noteworthy. This method was based on the cumulant domain data and adopted the total least squares (TLS) solution to obtain the directionofarrival (DOA) and range estimations, which avoided the computationally inefficient 2D spectral search. However, two slight disadvantages of the highorder ESPRIT method are that (1) the construction of four cumulant matrices leads to an additional computational burden and (2) a parameterpairing procedure is required for avoiding the estimation failure problem.
In this paper, we present a computationally more efficient ESPRIT algorithm to locate nearfield sources. The main task in the first step is to estimate the electric angle γ, whereas in the second step, ϕ related to γ is estimated for each nearfield source. For the sake of reducing the computational load, we compute the thirdorder moments of sensor outputs and construct only two special (N × N)dimensional matrices M_{1} and M_{2}. Since the steering matrix is the function of only electric angle γ, we can adopt the polynomial rooting solution to obtain the DOA estimations for all nearfield sources. Instead of estimating two rotational factors from different eigenvaluedecompositions (EVDs) encountered in[11, 12], the proposed algorithm estimates the only one rotational factor from the TLS version, which can avoid the unknown permutation ambiguity accompanied by EVD, that is, the autopairing for DOA and range can be realized.
The remainder of this paper is organized as follows: Section 2 describes the nearfield source localization model. In Section 3, an efficient thirdorder momentbased ESPRIT algorithm is proposed to deal with the nearfield source localization problem, and the performance analysis of the proposed algorithm is also addressed. Section 4 shows the simulation results. Section 5 presents the conclusion of the whole paper.
2 Nearfield signal model
Suppose that M nearfield signals impinge on a symmetric uniform linear array (ULA) introduced by Liang and Yang[13]. This sensor array consists of L = 2N + 1 sensors with element spacing d, and its array center is the phase reference point. With a proper rate that satisfies the Nyquist rate, the sampled signal received by the l th sensor can be written as
where s_{ m }(t) is the source signal, n_{ l }(t) is the additive sensor noise, and τ_{ lm } is the delay associated with the m th source signal propagation time from 0th to l th sensor. After being approximated by the Fresnel approximation, τ_{ lm } has the following form[11, 12]:
where γ_{ m } and ϕ_{ m } are called electric angles and given by
where λ represents the wavelength of the source signal and θ_{ m } and r_{ m } denote azimuth DOA and range of the m th nearfield source signal, respectively. For the rest of this paper, the following assumptions are required to hold:

1)
The source signals are statistically independent, zeromean narrowband stationary processes with nonzero kurtosis;

2)
The sensor noise is zeromean Gaussian process, and independent of the source signals;

3)
The sensor array is a symmetric ULA with element spacing d ≤ λ/4, and the source number M is not more than the half of the sensor number L, which means M < N.
3 The proposed algorithm
In this section, we firstly review the definition of the thirdorder moment introduced in[14]. Then, by properly choosing the array outputs, two special thirdorder moment matrices are constructed, and a joint MUSIC and ESPRITbased solution is derived for the nearfield source localization. Finally, the theoretical analysis including computational complexity and parameter match is carried out, which further evaluates the performance of the proposed algorithm.
3.1 Construction of thirdorder moment matrix
The proposed algorithm exploits the richness of the thirdorder moment of array outputs, which is defined as[14]
where n ∈ [N,N], the superscript ∗ denotes conjugate operation, and T represents the number of snapshots. Converging in the meansquare sense, the estimation of (5) is expressed as[14]
With the given signal model (1), we further obtain
where τ = τ_{1} + τ_{2}, and${m}_{3,{s}_{m}}(\tau )$ represents the thirdorder moment of the m th source signal.
Based on (7), we can construct two spatial thirdorder moment matrices M_{1} and M_{2}, in which the (k,q)th elements are respectively given by
In matrix form, we have
where the superscript H is conjugate transpose operation, M_{3,s}(τ) is the thirdorder moment matrix of source signals, A = [a_{1},a_{2},…,a_{ M }] is the (N × M)dimensional steering matrix,${\mathbf{a}}_{m}={\left[1,{e}^{\phantom{\rule{0.3em}{0ex}}j2{\gamma}_{m}},\dots ,{e}^{\phantom{\rule{0.3em}{0ex}}j2\left(N1\right){\gamma}_{m}}\right]}^{T}$ is the steering vector, and Ω Φ is the (M × M)dimensional rotational factor and satisfies
Considering the thirdorder moment matrices in (10) and (11), one can form the (2N × N)dimensional matrix
3.2 DOAs estimation for nearfield sources
In this paper, assume that the value of M is known or correctly estimated by the Akaike information criterion (AIC) of the minimum description length (MDL) detection criterion[15]. Implementing a singular value decomposition (SVD) operation to M yields
where U ∈ C^{2N × M} is composed of the left singular vectors related to nonzero singular values, which spans the signal subspace of M, U_{1} ∈ C^{N×M}, and U_{2} ∈ C^{N×M} span the signal subspaces of M_{1} and M_{2}, respectively, and the matrix T ∈ C^{M×M} is the unique nonsingular one.
Note that the azimuth DOAs for all the incoming sources can be estimated by finding the M peaks from the following 1D MUSIC spectral function
where I denotes the identity matrix.
In order to reduce the computational burden and improve the estimation accuracy, we obtain searchfree estimator of DOAs based on polynomial rooting.
Denote z = e^{j 2γ}, we have
The spectral function (16) can be rewritten in the following polynomial[16, 17]
and the DOAs of all the incoming sources can be estimated from the M closest to the unit circle roots of (18).
3.3 Range estimation for nearfield sources
With the obtained azimuth DOA information, we can easily obtain the estimations (denoted by$\stackrel{~}{\mathbf{A}}$ and$\stackrel{\u0304}{\mathbf{\Omega}}$) of A and Ω. Consider the total least squares (TLS) solution of the relation U_{1}T = A, let V be the (2M × 2M)dimensional matrix of the right singular vectors of$\left[\phantom{\rule{2.77626pt}{0ex}}{\mathbf{U}}_{1}\phantom{\rule{2.77626pt}{0ex}}\stackrel{~}{\mathbf{A}}\phantom{\rule{2.77626pt}{0ex}}\right]$, if the matrix is divided into four (M × M)dimensional partitions as[11, 12]
then the solution for the TLS problem is given by
A similar approach will lead to the solution of Ω Φ
where E_{12} and E_{22} are the corresponding partitions of the matrix E, which is generated from the right singular vectors of$\left[\phantom{\rule{2.77626pt}{0ex}}\stackrel{~}{\mathbf{A}}\phantom{\rule{2.77626pt}{0ex}}{\mathbf{U}}_{2}{\stackrel{~}{\mathbf{T}}}_{\text{TLS}}\phantom{\rule{2.77626pt}{0ex}}\right]$.
Therefore, the estimation${\stackrel{~}{\varphi}}_{m}$ related to${\stackrel{~}{\gamma}}_{m}$ can be determined as follows:
Finally, the range estimation of nearfield sources is given by
3.4 Theoretical analysis of the proposed algorithm
The proposed algorithm is based on a symmetric ULA with oddnumbered sensors, but the highorder ESPRIT method[11, 12] requires even ones. For the sake of simplicity, we assume that the two algorithms adopt an array of 2N + 2 sensors and that of 2N+1 ones in the rest of this paper. In this section, we assess the performance of the proposed method from two ways, i.e, computational complexity and parameter match.

1)
Computational complexity: Regarding the computational complexity, we compare the major multiplications involved in statistical matrix construction, EVD or SVD implementation, and MUSIC spectrum search. The highorder ESPRIT method constructs four (N + 1) × (N + 1)dimensional cumulant matrices, and performs the EVD of one (3N + 3) × (3N + 3)dimensional matrix, so the resulting multiplications are in order of
$$O\left(36{(N+1)}^{2}T+4/3{\left(3N+3\right)}^{3}\right)$$(24)where T is the snapshot number. The proposed algorithm constructs two (N × N)dimensional thirdorder moment matrices M_{1} and M_{2}, implements the SVD of one (2N × N)dimensional matrix M, as well as executes once rootpolynomial construction and solution for DOA estimation, the resulting multiplications are in order of
$$O\left(2{N}^{2}T+8/3{N}^{3}+2\mathit{\text{MN}}\right).$$(25)It is obvious that the proposed method is computationally more efficient than the highorder ESPRIT method.

2)
Parameter match: The highorder ESPRIT method implements the EVDs of two matrices to separately estimate DOA and range of nearfield sources. Although the eigenvectors related to nonzero eigenvalues from one EVD are equivalent to those of another EVD, the order may be different from each other. Therefore, an additional procedure is required to pairing them in a sense (see[7], section 2.4 for details). The proposed method firstly estimates DOAs using (16) and substitutes them into A and Ω to obtain the estimations$\stackrel{~}{\mathbf{A}}$ and$\stackrel{\u0304}{\mathbf{\Omega}}$, respectively. Based on this, the estimation of Ω Φ can be obtained from the TLS version shown in (19). Since there are no EVD operations for estimating the only one rotational factor Ω Φ, the unknown permutation ambiguity accompanied by EVD can be avoided, and the diagonal elements of$\stackrel{~}{\mathbf{\Omega}}\stackrel{~}{\mathbf{\Phi}}$ are in onetoone correspondence with those of$\stackrel{\u0304}{\mathbf{\Omega}}$. That is, the autopairing for DOA and range is realized.
4 Computer simulation results
In this section, we explicit some simulation results to evaluate the performance of the proposed algorithm. For all examples, a symmetric ULA with 15 sensors and element spacing 0.25λ is displayed. According to the definition in[13], the Fresnel region of the above array is r ∈ (0.16λ,25λ). The source signals are set as the zeromean exponentially distributed ones. The sensor noise is assumed to be spatial white complex Gaussian, and the signaltonoise ratio(SNR) is defined relative to each signal. For comparison, we simultaneously execute the highorder ESPRIT method[11, 12] and the nearfield CramerRao bound (CRB)[18]. Note that the highorder ESPRIT method is based on a ULA with evennumbered sensors, so we adopt 16 sensors for it in the following experiments. The presented results are evaluated by the estimated root mean square error (RMSE) from the average results of 500 independent Monte Carlo simulations.
In the first example, we examine the estimation accuracy of the proposed algorithm versus the SNRs. The localization parameters for two nearfield sources are (35°, 0.3 λ) and (20°, 0.5 λ), respectively. The thirdorder moment matrices are constructed using estimates (T = 1,024,τ = 0). When the SNR varies from 5 to 20 dB, Figures1 and2 show the RMSEs of DOA and range estimations using the proposed algorithm, respectively.For comparison, both the highorder ESPRIT method and the nearfield CRB are also presented. It can be seen from Figure1 that the proposed algorithm outperforms the highorder ESPRIT solution in estimating DOAs of two nearfield sources, and the RMSEs are reasonably close to the nearfield CRB. However, for the range estimation, the proposed algorithm slightly underperforms the highorder ESPRIT method. Note that the highorder ESPRIT method requires more computations than the proposed method, and it requires an additional parameterpairing procedure.
In the second example, we assess the performance of the proposed algorithm versus the number of snapshots. The other simulation conditions are similar to the first example except that the SNR is set at 2 dB, and the number of snapshots is varied from T = 400 to T = 2,000. The RMSEs of DOA and range estimations for the proposed method are displayed in Figure3, and compared with the highorder ESPRIT method and the nearfield CRB. We can see from this figure that the results are similar to those of the first example. The RMSEs of the proposed algorithm decrease monotonically with the number of snapshots. For the DOA estimations of both nearfield signals, the proposed method shows a more satisfactory performance than the highorder ESPRIT method, and the related RMSEs are reasonably close to the nearfield CRB.
In the last example, the computational burden of the proposed algorithm is compared with the highorder ESPRIT method. The number of sensors is 2N + 2 = 10. When the number of snapshots is varied from T = 0 to T = 2,000, Figure4 shows the computational complexity of the proposed algorithm and that of the highorder ESPRIT method.
It can be seen from this figure that the proposed algorithm is computationally less complex than the highorder ESPRIT method.
5 Conclusion
This paper has presented a thirdorder momentbased ESPRIT method to cope with the nearfield source localization problem. Our investigation has shown that the proposed method is capable of yielding reasonably good estimation of azimuth DOAs of nearfield sources. Compared with the highorder ESPRIT method, the proposed method is efficient in the sense that it requires a lower computational burden, as well as realizes the autopairing for DOA and range.
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Acknowledgements
This work is supported by the National Nature Science Foundation of China (Grant 61171137) and 2009 New Century Excellent Talents in University (NCET) in China.
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Chen, J., Liu, G. & Sun, X. Computationally efficient nearfield source localization using thirdorder moments. EURASIP J. Adv. Signal Process. 2014, 98 (2014). https://doi.org/10.1186/16876180201498
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Keywords
 Thirdorder moment
 ESPRIT
 Nearfield source localization