Research Article  Open Access
Joint 2D DirectionofArrival and Range Estimation for Nonstationary Sources
Abstract
Passive localization of nonstationary sources in the spherical coordinates (azimuth, elevation, and range) is considered, and a parallel factor analysis based method is addressed for the nearfield parameter estimation problem. In this scheme, a parallel factor analysis model is firstly constructed by computing five timefrequency distribution matrices of the properly chosen observation data. In addition, the uniqueness of the constructed model is proved, and both the twodimensional (2D) directionofarrival (DOA) and range can be jointly obtained via trilinear alternating least squares regression (TALS). The investigated algorithm is well suitable for nearfield nonstationary source localization and does not require parameterpairing or multidimensional search. Several simulation examples confirm the effectiveness of the proposed algorithm.
1. Introduction
Bearing estimation has been a strong interest in radar and sonar as well as communication. In the last three decades, various highresolution algorithms for direction finding of multiple narrowband sources assume that the propagating waves are considered to be plane waves at the sensor array. However, when the sources are located in the Fresnel region [1] of the array aperture, the wavefronts emitted from these sources are spherical rather than planar at each sensor position and characterized by both the DOA and the range parameters; thus, the existing DOA estimation schemes, such as MUSIC and ESPRIT [2], would fail in estimating nearfield localization parameters.
By applying the Fresnel approximation to the nearfield sources localization, the twodimensional (2D) MUSIC method, the highorder ESPRIT method, and the pathfollowing method were, respectively, proposed in [3–7] in order to cope with the problem of estimating azimuth and range. In recent years, several methods that can obtain azimuth, elevation, and range have been developed. For instance, in [8], an original 3D higher order statistics based localization algorithm has been presented. By translating the 1D uniform linear array of nearfield into a virtual rectangular array of virtual farfield, Challa and Shamsunder [9] proposed a unitaryESPRIT method. Aberd Meraim and Hua [10] used only the secondorder statistics and proposed a higher resolution 3D nearfield source localization; however, a parameterpairing process leading to the poor performance in lower signaltonoise ratio had to be taken into account. References [11, 12] also proposed fourthorder cumulant based algorithms to estimate 2D DOA and range, but they all coincided with [10]. All abovementioned algorithms rely on the assumption that impinging source signals are stationary; when nonstationary FM signals exist, they will show an unsatisfactory performance.
While quadratic timefrequency distribution [13, 14] has been sought out and properly investigated into sensor and spatial signal processing, and its evaluation of the observation data across the sensor array yields spatial timefrequency distribution (STFDs), the main advantage of STFDs is that it can well handle signals of nonstationary waveforms that are highly localized in the timefrequency domain and effectively improve the robustness of localization methods by spreading the noise power into the whole timefrequency domain. The STFDs based algorithm to locate nearfield nonstationary sources has been presented in [15] and showed a satisfactory parameters estimation accuracy; however, it required 2D search and only estimated 1D DOA and range.
In this paper, by exploiting favorable characteristics of a uniform cross array, we present a joint 2D DOA and range estimation algorithm. We first compute five timefrequency matrices to construct a parallel factor (PARAFAC) analysis model. Then, we obtain threedimensional (3D) nearfield parameters via trilinear alternating least squares regression (TALS). Compared with the other methods, the main contribution for the proposed method can be summarized as follows: we obtain 3D nearfield sources parameters (elevations, azimuths, and ranges) of nonstationary signals rather than stationary waves; we creatively incorporate STFDs with parallel factor analysis to well avoid both parameter pairing and multidimensional search.
The rest of this paper is organized as follows. Section 2 introduces the signal model of nearfield localization based on cross array. Section 3 develops a joint estimation algorithm of three parameters in nearfield. Section 4 shows simulation results. Section 5 presents the conclusion of the whole paper.
2. NearField Signal Model Based Cross Array
2.1. NearField Signal Model
We consider a nearfield scenario of uncorrelated narrowband signals impinging on a cross array signed with the  and axes (Figure 1), which consists of elements with interelement spacing . Let the array center be the phase point; the signals received by the th and the th can be, respectively, expressed as with being additive noise, and where , , and indicate elevation, azimuth, and range of th signal, respectively, and is wavelength of source signal.
The th source signal with phase can be modeled as where and is the amplitude of the th source signal.
2.2. Assumption of Signal Model
The main problem addressed in this paper is to jointly estimate the sets of parameters ; then the following assumptions are assumed to hold.(1)The source signal is narrowband, independent, and nonstationary process.(2)The additive noise is spatially white Gaussian with zeromean and independent from the source signals.(3)For unique estimation, we require , , and as well as .
2.3. Parallel Factor Analysis
We need to introduce the following notation that will be used in the sequel.
Definition 1 (see [16]). Let stand for the element of a threedimensional tensor , if where denotes the element of matrix , and similarly for the others. Equation (5) indicates as a sum of triple products, which is variably known as the trilinear model, trilinear decomposition, triple product decomposition, canonical decomposition, or parallel factor (PARAFAC) analysis.
Definition 2 (see [16]). For a matrix , if all columns of are linearly independent but there exists a collection of linearly dependent columns of , then it has Kruskalrank (rank) .
Theorem 3 (see [16]). Consider a threedimensional tensor as defined in (5), and represents the common dimension; if then , , and are unique up to permutation and (complex) scaling of columns.
3. PARAFAC Based 3D NearField Sources Localization
3.1. Computation of the Spatial TimeFrequency Distribution Matrices
The discrete form of Cohen’s class of timefrequency distribution of a signal can be expressed as where is the timefrequency kernel and the superscript denotes complex conjugate. Replacing by the data snapshot and , we obtain
Substituting (1) into (8), can be extended to the following form:
Under the assumptions and , it is obvious that where indicates the STFDs of source .
Using a rectangular window of old length , the pseudo WignerVille distribution (PWVD) of is given by
Assume that the thirdorder derivative of the phase can be negligible over the rectangular window length , and , ; then we obtain the approximated expression as
We construct matrix with the th element being given by
On the other hand, following the same process described above, we can easily obtain And their matrices form becomes where , , , , , and .
3.2. Construction of the Parallel Factor Analysis Model
Considering the situation of limited samples, we build a parallel factor analysis model that uses the spatial timefrequency distribution as
Letting , the KhutrrRao product [16] for (16) shows where denoting a row vector consisting of diagonal matrix .
Similarly, (17) also yields
3.3. Estimation of 2D DirectionofArrival and Range
As it stands, and are both Vandermonde matrices, and then they have Kruskalrank (rank) . On the other hand, the rank of will be . When the condition that the number of signals being holds, then , and are unique up to permutation and scaling of columns. With trilinear alternating least squares regression, we obtain that
Then using these estimates, we can get each pair as follows:
Finally, the sources parameters can be estimated as
4. Computer Simulation Results
In this section, we explicit several simulation results to evaluate the performance of proposed method. For all examples, a symmetrical cross array with a number of 17 elements and interelements spacing of , where is the wavelength of the narrowband source signals. The noise used in this section is zeromean, Gaussian distributed, and temporally white, and the root mean square error (RMSE) is defined as (23). All the following presented results are obtained by averaging the results of 200 independent Monte Carlo simulations. Consider where denotes the number of independent Monte Carlo simulations.
In the first example, we examine the performance of the elevation, azimuth, and range estimations accuracy versus the SNR. The snapshot number is set at 512. Two linear frequencymodulated signals arrival at the sensor array with start and end frequencies and , their 3D nearfield parameters locate at and . For the comparison, the fourthorder cumulant based method [9] is also displayed, in which the source signals are nonGaussian and stationary process. When the SNR varies from 0 dB to 25 dB, the RMSEs of the 2D directionofarrival and range estimations using the proposed method and the fourthorder cumulant based method are shown in Figure 2. From Figure 2, we can see that the proposed method outperforms the fourthorder cumulant based method in elevation and azimuth as well as range estimation for all available SNRs. In addition, the RMSE of range estimations for the first source that is closer to the array is less than the second one. This phenomenon is in well agreement with the theoretical analysis that the sources closer to sensor array would hold a smaller standard deviation than the ones far away from the array.
(a) Elevation
(b) Azimuth
(c) Range
In the second example, the proposed method is used to deal with the situation that two nearfield FM signals are impinging on the sensor array shown in Figure 1. The elevation, azimuth, and range of the nearfield sources are located at and . Moreover, the snapshot number and SNR are set at 512 and 10 dB. Table 1 establishes the mean and variance of the elevation, azimuth, and range estimations using the proposed method. From Table 1, we can see that the proposed method shows a satisfactory performance in localizing the 3D nearfield nonstationary sources.

In the last example, we consider the situation when farfield and nearfield nonstationary sources are incoming on the sensor array mentioned above, and they are located at and , respectively. The snapshot number is fixed at 512. When the SNR varies from 0 dB to 25 dB, the RMSEs of the 2D directionofarrival and range estimations using the proposed method are shown in Figure 3. In addition, the results of the fourthorder cumulant based method are also displayed in the same figure for comparison. From Figure 3, it can be seen that the proposed method still performs a satisfactory estimation accuracy for the case that farfield and nearfield sources exist simultaneously. However, the fourthorder cumulant based method shows a poor performance in the same situation.
(a) Elevation
(b) Azimuth
(c) Range
5. Conclusion
We have developed a spatial timefrequency distribution based algorithm for 3D nearfield nonstationary source localization problems. Additionally, with parallel factor analysis technique, there is no parameter pairing or multidimensional searching. Finally, the computer simulation results indicate that using spatial timefrequency distribution and parallel factor together significantly solves the problem of the joint estimation of elevation, azimuth, and range of nonstationary signals. However, the spatial timefrequency averaging methods may lead to the additional computation load.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (61171137) and Specialized Research Fund for the Doctoral Program of Higher Education (20090061120042).
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Copyright © 2014 Jian Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.