Based on a dual-size shift invariance sparse linear array, this paper presents a novel algorithm for the localization of mixed far-field and near-field sources. First, by constructing a cumulant matrix with only direction-of-arrival (DOA) information, the proposed algorithm decouples the DOA estimation from the range estimation. The cumulant-domain quarter-wavelength invariance yields unambiguous estimates of DOAs, which are then used as coarse references to disambiguate the phase ambiguities in fine estimates induced from the larger spatial invariance. Then, based on the estimated DOAs, another cumulant matrix is derived and decoupled to generate unambiguous and cyclically ambiguous estimates of range parameter. According to the coarse range estimation, the types of sources can be identified and the unambiguous fine range estimates of NF sources are obtained after disambiguation. Compared with some existing algorithms, the proposed algorithm enjoys extended array aperture and higher estimation accuracy. Simulation results are given to validate the performance of the proposed algorithm.

In recent years, passive source localization has become a key topic in array signal processing [

However, both FF and NF sources may coexist in many interested situations such as speaker localization using microphone arrays, seismic exploration, and electronic surveillance. Most of the algorithms, which deal with pure NF or pure FF sources, may fail in the scenarios of mixed sources.

Recently, mixed source localization problem has been an important research topic in array signal processing [

As is well known, the estimation accuracy is directly correlated with the array aperture size that a larger array would produce more precise estimates. However, most of the existing methods limit the array element spacing to be within a quarter wavelength to avoid DOA ambiguity. Recently, a mixed-order MUSIC (MOMUSIC) algorithm [

In this paper, a novel fourth-order cumulant-based dual-size shift invariance (CDSSI) algorithm is presented to solve the mixed source localization problem. Our technique utilizes a SLA of the dual-size spatial invariance method [

The rest of the paper is organized as follows. In Section

The complex conjugate, transpose, Hermitian transpose, and pseudoinverse are denoted by

Suppose that

Sparse linear array geometry.

In Figure

Let the array center be the phase reference point; the output of the

In a matrix form, the array data can be written as

In the above equations,

Given the array data

The incoming signals are mutually independent, narrowband stationary, and non-Gaussian, as well as nonzero kurtosis.

The DOAs of all source signals differ from each other.

The noise is zero-mean, additive (white or color) Gaussian, and statistically independent from all impinging sources.

In order to avoid the phase ambiguity, the intersensor spacing

As is shown in Figure

Flow graph of the CDSSI algorithm.

Similar to the TSMUSIC algorithm in [

According to the definition in [

We can construct a

Note that

From (

Therefore,

The reason to rewrite it in this manner is that it has a similar form of dual-size invariance and therefore, ESPRIT could be applied to yield coarse and fine estimates of DOAs.

We firstly estimate the DOAs of the mixed FF and NF sources. By taking an eigendecomposition of (

To obtain high-accuracy DOA estimates, we may form two matrices that are related by the extended intersubarray spacing

Construction of 2 subspace matrix elements for fine DOA estimation.

From (

For the purpose of disambiguation, unambiguous but coarse DOA estimates must be obtained as references to the ambiguous fine estimates in the previous subsection. From Figure

Construction of 2 subspace matrix elements for coarse DOA estimation.

From (

By taking the first and the last

From (

The coarse estimates

Note that

In order to derive the range estimates, another fourth-order cumulant matrix needs to be constructed to overcome the rank-deficient phenomenon described in [

If the

In this section, a new fourth-order cumulant matrix

Note that

Therefore,

From (

By taking an eigendecomposition of

According to [

Substituting the disambiguated DOA estimates

Actually, (

In order to generate the coarse range estimates unambiguously,

Similarly, through the extended aperture size between each subarray, fine range estimates with ambiguity can be generated. From (

Therefore, we can find a series of ambiguous range estimates

In an analogous manner to that of Section

Note that

In this section, we compare CDSSI with some recently developed mixed source localization algorithms, including TSMUSIC [

Both TSMUSIC and GESPRIT employ an ULA which requires the intersensor spacing to be within a quarter wavelength. Therefore, with the same sensor number, SLA enjoys extended array aperture size, producing better estimation accuracy for the proposed algorithm. MOMUSIC also utilizes a special nested SLA in the development of the algorithm. However, according to the array model described in [

In CDSSI, the coarse and fine DOA estimates are derived from two

Only the major computation load is considered in this comparison, including construction of the cumulant matrices, eigenvalue decomposition (EVD), and spectral search. The searching steps for the angle parameter

Computational complexity of different algorithms.

Algorithm | Cumulant matrix | EVDs | Spectral search |
---|---|---|---|

CDSSSI | |||

TSMUSIC | |||

GESPRIT | |||

MOMUSIC |

Compared with the second-order-based methods, CDSSI requires more computations in constructing the fourth-order cumulant matrix and EVDs. However, it avoids any 1-D or 2-D complicated spectral search. Since the search steps need to be dense enough for the spectral search-based algorithms to approach their theoretical bounds, the computation load of these algorithms will in turn increase dramatically.

Both TSMUSIC and the proposed algorithm are capable of suppressing additive Gaussian colored noise since they apply fourth-order cumulant in the whole estimation procedure, while GESPRIT and MOMUSIC, which rely on the second-order statistics, will degrade in the presence of spatially correlated noise.

Furthermore, knowledge of the “manifold shape” not only is essential for the investigation of ambiguities and assessment of the detection-resolution capabilities of an array but it may also prove useful in developing new and more effective methods for its search process. We studied on the two array manifold properties, namely, arc length and first-order curvature, and analyse the accuracy and resolution capabilities of mixed sources [

It has been shown that the response of sparse array towards a far-field/near-field source emitting narrowband spherical wavefront from azimuth

For the far-field source scenario, the range parameter approaches to

The rate of change of arc length

The Cramer-Rao lower bound under the assumption of spatially and temporally uncorrelated large number of snapshots

The expression of the asymptotic (

The variance asymptotically approaches the CRB for high SNR or large

Figures

Theoretical CRB with respect to the change of

Asymptotic variance with respect to the change of

Theoretical CRB with respect to the change of snapshot.

In addition, the angle and range estimation accuracy is not only related to the intersubarray spacing of sparse but also related to the possibility of disambiguation. As the intersubarray spacing increases, the possibility of wrong disambiguation increases simultaneously. The MIE method [

In this section, numerical simulations are conducted to validate the performance of the proposed algorithm relative to TSMUSIC, GESPRIT, and MOMUSIC. In the following experiments, we consider a SLA composed of

In the first experiment, we consider two equipower sources that are located at

RMSEs of DOA and range estimates versus SNR.

In the second experiment, we investigate the RMSEs of the four algorithms with the variation of the number of snapshots. The parameter settings are the same as those of the first experiment except that SNR is set equal to 10 dB and the number of snapshots varies from 100 to 10,000. From Figure

RMSEs of DOA and range estimates versus snapshot number.

In the third experiment, the scenario of two NF sources is investigated, with the source location parameters being

RMSEs of DOA and range estimates versus SNR.

In the fourth experiment, we study the dependence of DOA and range estimation accuracy upon the angular gap between two NF sources.

RMSEs of DOA and range estimates versus DOA gap of two NF sources.

In the fifth experiment, the range parameter of the first source

RMSEs of DOA and range estimates versus range gap of two NF sources.

In the last experiment, two FF sources are considered with their location parameters being

RMSEs of DOA estimates versus SNR.

In this paper, an efficient and high-performance algorithm is proposed for the mixed far-field and near-field source localization problems. Based on a sparse linear array of dual-size spatial invariance, the proposed algorithm can offer enhanced accuracy due to the extended aperture size. Moreover, the proposed method has lower computational complexity because it does not require any 1-D or 2-D spectral search. According to the simulations, the proposed algorithm outperforms the conventional ones in the performance of both angle and range estimation.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work is supported by Natural Science Foundation of China (no. 60971108 and no. 61601372).