- Open Access
Computationally efficient near-field source localization using third-order moments
© Chen et al.; licensee Springer. 2014
Received: 8 March 2014
Accepted: 14 June 2014
Published: 25 June 2014
In this paper, a third-order moment-based estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm is proposed for passive localization of near-field sources. By properly choosing sensor outputs of the symmetric uniform linear array, two special third-order 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 direction-of-arrivals (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 high-order ESPRIT method, the proposed algorithm requires a lower computational burden, and it avoids the parameter-match procedure. Computer simulations are carried out to demonstrate the performance of the proposed algorithm.
In the last decades, low-complexity 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 far-field source localization, such as the multiple signal classification (MUSIC) method, estimation of signal parameters via rotational invariance techniques (ESPRIT) method, and their derivatives[5, 6]. In order to locate the near-field sources, 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 two-dimensional (2-D) MUSIC was the original solution to the near-field source localization problem. To cope with the computationally inefficient 2-D searching procedure, a symmetric uniform linear array-based method was proposed. Based on, resorting to the polynomial rooting, an improved method was developed, which has further avoided the one-dimensional (1-D) 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 closed-form solutions. In this version, the high-order 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 direction-of-arrival (DOA) and range estimations, which avoided the computationally inefficient 2-D spectral search. However, two slight disadvantages of the high-order ESPRIT method are that (1) the construction of four cumulant matrices leads to an additional computational burden and (2) a parameter-pairing procedure is required for avoiding the estimation failure problem.
In this paper, we present a computationally more efficient ESPRIT algorithm to locate near-field 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 near-field source. For the sake of reducing the computational load, we compute the third-order moments of sensor outputs and construct only two special (N × N)-dimensional matrices M1 and M2. 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 near-field sources. Instead of estimating two rotational factors from different eigenvalue-decompositions (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 auto-pairing for DOA and range can be realized.
The remainder of this paper is organized as follows: Section 2 describes the near-field source localization model. In Section 3, an efficient third-order moment-based ESPRIT algorithm is proposed to deal with the near-field 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 Near-field signal model
The source signals are statistically independent, zero-mean narrowband stationary processes with nonzero kurtosis;
The sensor noise is zero-mean Gaussian process, and independent of the source signals;
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 third-order moment introduced in. Then, by properly choosing the array outputs, two special third-order moment matrices are constructed, and a joint MUSIC- and ESPRIT-based solution is derived for the near-field 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 third-order moment matrix
where τ = τ1 + τ2, and represents the third-order moment of the m th source signal.
3.2 DOAs estimation for near-field sources
where U ∈ C2N × M is composed of the left singular vectors related to non-zero singular values, which spans the signal subspace of M, U1 ∈ CN×M, and U2 ∈ CN×M span the signal subspaces of M1 and M2, respectively, and the matrix T ∈ CM×M is the unique non-singular one.
where I denotes the identity matrix.
In order to reduce the computational burden and improve the estimation accuracy, we obtain search-free estimator of DOAs based on polynomial rooting.
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 near-field sources
where E12 and E22 are the corresponding partitions of the matrix E, which is generated from the right singular vectors of.
3.4 Theoretical analysis of the proposed algorithm
- 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 high-order 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(24)where T is the snapshot number. The proposed algorithm constructs two (N × N)-dimensional third-order moment matrices M1 and M2, implements the SVD of one (2N × N)-dimensional matrix M, as well as executes once root-polynomial construction and solution for DOA estimation, the resulting multiplications are in order of(25)
It is obvious that the proposed method is computationally more efficient than the high-order ESPRIT method.
Parameter match: The high-order ESPRIT method implements the EVDs of two matrices to separately estimate DOA and range of near-field sources. Although the eigenvectors related to non-zero 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, section 2.4 for details). The proposed method firstly estimates DOAs using (16) and substitutes them into A and Ω to obtain the estimations and, 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 are in one-to-one correspondence with those of. That is, the auto-pairing 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, the Fresnel region of the above array is r ∈ (0.16λ,25λ). The source signals are set as the zero-mean exponentially distributed ones. The sensor noise is assumed to be spatial white complex Gaussian, and the signal-to-noise ratio(SNR) is defined relative to each signal. For comparison, we simultaneously execute the high-order ESPRIT method[11, 12] and the near-field Cramer-Rao bound (CRB). Note that the high-order ESPRIT method is based on a ULA with even-numbered 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.
It can be seen from this figure that the proposed algorithm is computationally less complex than the high-order ESPRIT method.
This paper has presented a third-order moment-based ESPRIT method to cope with the near-field source localization problem. Our investigation has shown that the proposed method is capable of yielding reasonably good estimation of azimuth DOAs of near-field sources. Compared with the high-order ESPRIT method, the proposed method is efficient in the sense that it requires a lower computational burden, as well as realizes the auto-pairing for DOA and range.
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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