Direct position determination of multiple coherent sources using an iterative adaptive approach
Introduction
Localization of sources is a crucial fundamental task in many application fields such as radar, sonar, navigation system and wireless sensor network, and recently attracts much interest of many researchers [1], [2], [3], [4], [5], [6], [7], [8], [9]. Generally, localization methods are mainly categorized as two-step localization and one-step localization. In the two-step localization method, the estimation of a source position is implemented by two processing steps, where sensors firstly measure angles or ranges of the source from received signals, such as angle of arrival (AOA), time of arrival (TOA) and time difference of arrival (TDOA), and then estimate the source position by utilizing the previous measurements according to the geometric relationship between the source and sensors [6], [10], [11], [12], [13], [14], [15]. In the one-step localization method, the position is determined by directly handling the raw received signals without estimating the intermediate measurements like TDOA, etc. On the basis of knowing a received signal model, the one-step localization is generally accomplished by adopting the maximum likelihood (ML) estimation [8], [9], [16], the least square (LS) approach [17], [18], [19] or the sparsity recovery method [20], [21].
The one-step localization method can directly estimate the source position from the raw received signals, and obtain processing gain, so it has an advantage of high-accurate and robust localization over the two-step approach, especially in strong noise environment. In [17], the direct position determination (DPD) approach, belonging to the one-step localization, has shown better accuracy than the two-step localization based on AOA or TOA at low signal to noise ratio (SNR) for a single emitter. Additionally, the DPD approach has also natural superiority in application of multi-target localization, since it needn’t match these AOAs, TOAs or TDOAs to the corresponding targets as the two-step approach does.
Having the advantages over the two-step localization, the DPD approach has been applied in the scene of multiple sources [18], [19], [22]. The DPD approach based on the multiple signal classification (MUSIC) [23], called as DPD-MUSIC, is proposed to locate multiple sources on the condition of knowing the number of sources [18]. DPD-MUSIC only requires two-dimensional spectrum peak search in a planar geometry to determine the positions of all sources without high computational cost. In [19], the high-accurate DPD combined with the minimum variance distortionless response (MVDR) [24], called as DPD-MVDR, is developed to deal with the case where the number of sources are unknown. Similar with DPD-MUSIC, DPD-MVDR can also locate the source positions only by two-dimensional search. But, only when the transmitted signals are uncorrelated or partially correlated, can both DPD-MUSIC and DPD-MVDR obtain high localization accuracy for each source.
In practice, many perfectly correlated (coherent) signals appear in some cases, such as coherent interference, multipath propagation and coherent radiation sources. In the coherent case, many techniques based on eigenstructure and beamforming, like MUSIC and MVDR approaches, fail to work [25]. Due to the property of MUSIC and MVDR, DPD-MUSIC and DPD-MVDR will encounter a serious performance deterioration for coherent signals. Although many algorithms based on the spatial smoothing preprocessing scheme and the iterative approach have been proposed to settle the coherent signal problems [26], [27], [28], [29], [30], [31], they can only provide the intermediate measurements AOAs to determine the source positions by the two-step localization [32], [33], [34], [35]. As mentioned above, the two-step localization methods based AOA confront unstable performance at low SNR, and also have to match measurement parameters in multiple sources scene.
The focus of this paper is to solve the direct localization problem of multiple coherent sources. Our proposed method combines the DPD technique with the iterative adaptive approach (IAA) [30] to estimate the positions of coherent sources. In this paper, we quantitatively analyze the reasons of invalidity of DPD-MUSIC and DPD-MVDR for coherent signals. To avoid the covariance matrix becoming rank-deficient, we propose to employ the IAA to compute it, instead of the traditional array processing. Meanwhile, the received signals are also estimated iteratively. Based on these signals, a spatial power function is derived to compute the intercepted power from which the source positions can be determined by grid search. In addition, an extraction method based on connected domain separation is proposed to obtain the position of each source. Similar with DPD-MUSIC and DPD-MVDR method, the proposed algorithm also requires only two-dimensional search for planar geometry to locate all sources. Unlike the two-step methods, the proposed algorithm is direct without estimating the intermediate measurements. The simulation results are shown to demonstrate the efficiency of the proposed algorithm for coherent sources.
The contributions of our paper are summarized as follow. (1) We propose to employ the IAA to address the direct localization problem of multiple coherent sources. The IAA is developed to apply in the one-step localization without estimating intermediate measurements AOAs. (2) For the position extraction problem that the multi-target localization always encounters, we propose a method based on connected domain separation to obtain the position of each source.
The paper is organized as follows: Sections 2 and 3 state the signal model and the DPD problem formulated for coherent sources at distributed receivers, respectively. Section Sections 4 presents the proposed algorithm in detail, followed by numerical examples in Sections 5. Finally, a conclusion is drawn in Sections 6.
Section snippets
Signal model
Consider a localization system consisting of L widely distributed receivers, each of which equips a M-element antenna array. It is required that K emitters (or sources) transmitting unknown coherent signals are located by the L receivers, shown in Fig. 1. The receivers and emitters are located in a two-dimensional plane. The coordinates of L receivers and K emitters are denoted by and respectively. The transmitted signals from K emitters, represented by
Problem formulation about direct position determination of coherent sources
As denoted in (6), the position parameters of the emitters only appear at the equivalent array response so many spectral methods can be employed to estimate the positions [18], [19]. For brevity, (6) is rewritten as the vector formwhereand “⊤” is the transpose notation. Considering the coherency defined in (1), the correlation matrix of the transmitted signals in frequency
Direct position determination algorithm of multiple coherent sources
In this section, the DPD based on the IAA is derived to solve the localization problem for the coherent sources, called as DPD-IAA below for brevity. DPD-IAA utilizes signal power to estimate the source positions essentially. We first employ the weighted least square (WLS) to estimate the intercepted signals, and then further compute the covariance matrix and the spatial power iteratively by the estimated signals. After determining the convergent spatial power function, the positions of each
Numerical examples
In this section, we investigate the proposed DPD-IAA, DPD-MUSIC and DPD-MVDR algorithms for multiple sources in the two situations of transmitting coherent and noncoherent signals. The Monte Carlo simulation experiments are conducted to examine the performance of three algorithms. In each of the experiments, three sources are distributed randomly at three different rectangular regions respectively with their centers located at coordinates km, (0,18) km and (8,12) km while four receivers
Conclusion
In this paper, we propose the localization algorithm combining the IAA method with the DPD concept to locate multiple coherent sources with distributed receivers. The paper derives the spatial power function to estimate the source positions. The proposed algorithm can still work effectively even though the signals are perfectly correlated. The method based on connected domain separation is also proposed to extract the position of each source. Simulation experiments compare the performance of
Conflict of interest
None.
Acknowledgments
This work was supported in part by the Chang Jiang Scholars Program, in part by the 111 Project No. B17008, in part by the National Natural Science Foundation of China under Grant 61771110, in part by the Fundamental Research Funds of Central Universities under Grant ZYGX2016J031, and in part by the Chinese Postdoctoral Science Foundation under Grant 2014M550465 and Special Grant 2016T90845.
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