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Localization and identification of unknown target signal using oblique projection
 Liping Huo^{1, 3},
 Huijun Hou^{1, 2} and
 Xingpeng Mao^{1, 4, 5}Email authorView ORCID ID profile
https://doi.org/10.1186/s1363801811479
© The Author(s) 2018
 Received: 16 September 2017
 Accepted: 10 May 2018
 Published: 4 June 2018
Abstract
The problem of source localization and waveform identification is the key of array signal processing. In this paper, an oblique projectionbased localization and identification (OPLI) algorithm is proposed without known prior DOA or waveform information of the sources. The proposed OPLI is implemented iteratively. In each iteration, oblique projection is employed to separate the multiple incident signals into a series of single signal groups. After that, the procedure of waveform and DOA estimation for each single signal is implemented. Theoretical analysis and simulation result verify the performance and effectiveness of the proposed OPLI.
Keywords
 Beamforming
 Direction of arrival estimation
 Oblique projection
 Interference suppression
1 Introduction
Source localization and waveform identification are central problems in antenna array processing, including in particular radar, sonar, or wireless communication [1, 2]. To this end, various sensor signal processing tools have been developed over the last several decades, ranging from directionofarrival (DOA) estimation algorithms to spatial beamforming algorithms [3–13]. It is noted that, in practical applications, the localization and identification of target signal could be difficult, when any a priori knowledge of signalofinterest (SOI) is unavailable or the SOI is sheltered by adjacent interference signals. One of the general approaches to solve this problem is realized as follows: The DOA estimation is firstly implented for all the array receiving signals, and then, the waveform of SOI is obtained via beamforming techniques [14–19].
No matter in the process of DOA estimation or in beamforming, the influence of the interference signals on target localization and identification is generally not trivial, especially when the interference signals are spatially adjacent with the SOI in the presence of large power. If the interference signals are suppressed or attenuated in advance, the performance of localization and waveform identification for SOI would be better. Haimovich and BarNess [20], Haimovich [21], and Honig and Goldstein [22] proposed eigenanalysisbased interference canceler. Similarly, Gu and Leshem [23], Huang et al. [24], Chan and Chen [25], Boyer [26], Xu et al. [27], Mak and Manikas [28], Shi and Lin [29], Behrens and Scharf [30], Mao et al. [31], and Mao et al. [32] employed beamforming techniques to suppress inference signals. Hassanien et al. [33], Vaccaro and Harrison [34], and Han and Zhang [35] designed matrix filters to cancel outofsector interference signals. It should be noted that these mentioned algorithms are based on accurate prior DOA of the SOI, and some of them may also require necessary DOA information of interference signals.
The estimation accuracy of DOA has major influence on the beamforming performances. Generally, the performance of a beamformer decreases severely when the DOA error increases [26, 31, 32]. Different approaches, including the linearly constrained minimum variance (LCMV), diagonal loading, convex optimization, and covariance matrix taper approaches, are developed [16, 36–46] to combat DOA errors. These algorithms can combat DOA uncertainties, but only suitable for small DOA errors. Lam and Singer [47], Bell et al. [48], and Han and Zhang [49] developed Bayesian beamforming, which is able to implement waveform identification when the DOA is uncertain or unknown, while it requires prior statistics that describes the level of DOA uncertainty.
Even if an accurate DOA of the SOI is given, the optimal performance of beamforming is hard to achieve since most of the beamformers rely on the sampled matrix inversion to replace the theoretical one, especially when in the presence of short data samples [3, 4, 44, 46]. With limited number of snapshots, in this paper, we discuss the problem of localization and identification for unknown target signal, where there exist spatially adjacent interference signals and the DOAs of interference signals are also completely unknown. The wellknown RELAX discuss the similar problem [50–52], where a simultaneous realization of DOA estimation and spatial beamforming for all receiving signals is achieved, but the performance of beamforming is not theoretically deduced.
With spatially adjacent interference signals, an oblique projectionbased localization and identification (OPLI) algorithm is proposed for unknown target signals. Firstly, the OPLI employs oblique projection to separate the mixed array receiving signals into individual signal groups, where each signal group contains only one signal. Then, the OPLI sequentially estimates DOA and waveform of the signal in each group. Finally, the OPLI recursively reduces DOA and waveform errors via minimizing the optimal maximum likelihood cost function. Theoretical analysis is provided to show the beamforming performance of the proposed OPLI, and simulation results indicate that the OPLI is computationally effective for source localization and waveform identification; besides, it is superior to the counterpart conventional algorithms at moderate to high input signaltonoise ratio (SNR) region in terms of output signaltointerferencenoise ratio (SINR) and root mean square error (RMSE).
The paper is organized as follows. Section 2 presents the signal model for source localization and waveform identification. The proposed OPLI algorithm as well as its complexity analysis is developed in Section 3. Section 4 discusses the proposed OPLI, and Section 5 presents the simulation results. The conclusion is drawn in Section 7.
2 Signal model and problem description
2.1 Signal model
where s_{ k }(t) refers to the baseband signal waveform of the kth incident signal, and a(θ_{ k }) denotes the corresponding steering vector. n(t) represents the additive noise component with covariance \(\mathbf {R}_{n} = \mathbb {E}\left [\mathbf {n}(t)\mathbf {n}^{\mathrm {H}}(t)\right ]\). \(\mathbb {E}[\cdot ]\), (·)^{T} and (·)^{H} stand for statistical expectation, transposition and Hermitian transposition, respectively.
where \(\mathbf {R}_{S} = \mathbb {E}\left [\mathbf {S}(t)\mathbf {S}^{\mathrm {H}}(t)\right ]\).
2.2 Problem description
This subsection formulates the localization and waveform identification problem. Without loss of generality, the signal s_{0}(t) is assumed to be the SOI and the remaining K−1 received signals are treated as interference signals. For the SOI and the interference signals, neither the source locations (i.e., θ_{1},θ_{2},⋯,θ_{K−1}) nor the signal waveforms (i.e., s_{0}(t),s_{2}(t),⋯,s_{K−1}(t)) are prior known.
The localization and waveform identification problem addressed in this paper is to obtain the DOA θ_{0} and waveform s_{0}(t) from multiple snapshots \(\left \lbrace \mathbf {x}\left (t_{l}\right) \right \rbrace _{l=1}^{L}\), where L represents the number of snapshots.
And, according to (6), the location of the SOI is given by the conventional method of single target localization.
3 Proposed OPLI algorithm
Derivation and implementation of the proposed OPLI are presented in this section. The proposed OPLI attempts to employ oblique projection to separate the multiple incident signals into a series of single signal groups. As a result, the source localization and waveform identification are implemented on each separated single signal. Note that the procedure of oblique projection requires the DOA parameters for source separation. To this end, the OPLI is implemented iteratively and the method of maximum likelihood approximation [53] is employed to evaluate the convergence.
3.1 Basic principle
where the symbol ∖ signifies set difference.
where \(\mathcal {R}\left \lbrace \cdot \right \rbrace \), (·)^{⊥}, and (·)^{ † } denote range space, orthogonal complement, and MoorePenrose pseudoinverse, respectively.
where \(\mathbf {P}_{\mathbf {a}(\theta)}^{\bot } = \mathbf {I}  \mathbf {a}(\theta)\mathbf {a}^{\dagger }(\theta)\), and \(\mathbf {R}_{0} = \mathbb {E}\left [\mathbf {x}_{0}(t)\mathbf {x}^{\mathrm {H}}_{0}(t)\right ]\).
where (·)^{−1} denotes inverse.
Using (5), (7), (10), (17), and (18), both the location θ_{0} and the waveform s_{0}(t) of the SOI can be computed. Besides, the localization and waveform identification processes for the SOI can also be extended, i.e., the localization and waveform identification process for the interference signals are feasible if each of the interference signals is regarded as the SOI.
3.2 Practical considerations
Recall that the localization and waveform identification process in Section 3.2 requires known DOAs of all the incident signals, which is infeasible in practical applications. In this paper, it is considered that the DOA of the SOI is prior unavailable, and the prior DOAs of the interference signals are also unknown. Toward the purpose, the presented localization and waveform identification process is implemented iteratively. The proposed OPLI firstly estimate the parameters of the SOI, so that a rough information of the SOI is available. And then, the parameters of each interference signals are updated one after another. Through iteration, not only the parameters of the SOI are estimated, but also the parameters of the interference signals are gradually achieving high precision.
At the ith iteration, the estimated DOAs \(\hat {\theta }_{{0}}^{(i)}\), \(\hat {\theta }_{1}^{(i)}\), ⋯, \(\hat {\theta }_{k1}^{(i)}\), \(\hat {\theta }_{k}^{(i1)}\), \(\hat {\theta }_{k+1}^{(i1)}\), ⋯, \(\hat {\theta }_{{{K1}}}^{(i1)}\) are utilized as initial values to compute \(\hat {\theta }_{k}^{(i)}\) and \(\hat {s}_{k}^{(i)}(t)\), where i≥1 and k=0,1,2,⋯,K−1. \(\hat {\theta }_{k}^{(i)}\) and \(\hat {s}_{k}^{(i)}(t)\) denote the estimation of θ_{ k } and s_{ k }(t) at the ith iteration (similarly hereinafter), respectively.
Iteration procedure of OPLI

3.3 Computational complexity
In this subsection, the computational complexity of the proposed OPLI is analyzed.
Using L number of snapshots, the computation of matrix \(\hat {\mathbf {R}}_{k}^{(i)}\) in (20) can be given as \(\sum _{l = 1}^{L}\hat {\mathbf {x}}_{k}^{(i)}\left (t_{l}\right) \left (\hat {\mathbf {x}}_{k}^{(i)}\left (t_{l}\right)\right)^{\mathrm {H}}/L\), which takes about O(LN^{2}+N^{2}) flops. Herein, a flop is defined as a complex floatingpoint addition or multiplication operation. The number of flops roughly required to compute (21) is O(N(K−1)L+NL) flops. The calculation of (22) requires approximately O((K−1)^{3}+2N(K−1)^{2}+N^{2}(K−1)+(K−1)NL) flops, where the calculation of \(\mathbf {E}_{\hat {\mathbf {B}}_{k}^{(i)}\mathbf {a}\left (\hat {\theta }_{k}^{(i1)}\right)}\) additionally takes about O(3N^{2}(K−1)+2N(K−1)^{2}+(K−1)^{3}+2N^{2}+N) flops. Thus, the computational complexity of (20) ∼ (22) is roughly O(N^{2}(L+4K−1)+N(4(K−1)^{2}+(2K−1)L+1)+2(K−1)^{3}) flops in total.
The computation of \(\hat {\theta }_{k}^{(i)}\) in (19) requires roughly \(O\left (\left (N^{3} + 2N^{2} + 2N\right)\tilde {N}_{\hat {\theta }_{k}^{(i)}}\right)\) flops, where \(\tilde {N}_{\hat {\theta }_{k}^{(i)}}\) denotes the number of potential source locations in the region scope \(\left [\hat {\theta }_{k}^{(i1)}\epsilon _{k}, \hat {\theta }_{k}^{(i1)}+\epsilon _{k}\right ]\). The computation of \(\hat {s}_{k}^{(i)}(t)\) in (25) requires roughly O(NL) flops, where the calculation of \(\hat {\mathbf {W}}_{k}^{(i)} \) additionally takes about O(2N+1) flops. The computation of \(\mathcal {L}^{(i)}\) in (27) requires roughly O((K+2)NL) flops.
Therefore, according to Table 1, the computational complexity of the proposed OPLI is roughly \(O\left ((K + 2)NL\right) + O\left (\sum _{k=0}^{K1}\left (\left (N^{3} + 2N^{2} + 2N\right)\tilde {N}_{\hat {\theta }_{k}^{(0)}} + N^{2}(L+4K1) +\right.\right. N\left (4(K1)^{2} +(2K1)L+1\right) + 2(K1)^{3} + NL + 2N + 1 +\sum _{i=1}^{N_{r_{k}}}\left (\sum _{j=0}^{k}\left ((N^{3} \,+\, 2N^{2} \,+\, 2N)\tilde {N}_{\hat {\theta }_{j}^{(i)}} \,+\, N^{2}(L\,+\,4K1) +\right.\right. N(4(K1)^{2} +(2K1)L+1) + 2(K1)^{3} + NL + 2N \!\left.\left.\left.\left.\!\!\!\!{\vphantom {\left (\sum _{k=0}^{K1}\left (\left (N^{3} + 2N^{2} + 2N\right)\tilde {N}_{\hat {\theta }_{k}^{(0)}} + N^{2}(L+4K1) + N\left (4(K1)^{2} +(2K1)L+1\right) + 2(K1)^{3} + NL + 2N + 1\right.\right.}} + 1 \right)+ (K + 2)NL\right)\right)\right)\)flops in total, where \(N_{r_{k}}\) denotes the number of iterations employed in the kth outer loop. Particularly, the total complexity is approximately \(O\left (\sum _{k=0}^{K1}\left (N^{3}\tilde {N}_{\hat {\theta }_{k}^{(0)}} + \sum _{i=1}^{N_{r_{k}}} \sum _{j=0}^{k}N^{3}\tilde {N}_{\hat {\theta }_{j}^{(i)}} \right)\right)\) flops, when L≫N>K, \(\tilde {N}_{\hat {\theta }_{k}^{(i)}} \gg L/N\), which occurs often in practical applications.
4 Discussion
The proposed OPLI involves the procedure of DOA estimation, hence it is applicable for localization of unknown target signal. Additionally, the OPLI also can obtain accurate locations of the interference signals, which is valuable in practical applications.
Substituting (34) into (31), it is obtained that \(\boldsymbol {\mathcal {W}}_{\text {opt},0} = \left [\mathbf {a}^{\dagger }\left (\theta _{0}\right) \mathbf {E}_{\mathbf {a}(\theta _{0})\mathbf {B}_{0}}\right ]^{\mathrm {H}}\). So, the optimum beamformer of the proposed OPLI is equivalence to the wellknown oblique projection beamformer which has excellent performance and has been well researched in [26, 31, 32].
where \(\sigma _{0}^{2} = \mathbb {E}\left [\left s_{0}(t)\right ^{2}\right ]\) refers to the power of the SOI.
Comparing (41) with (37), it follows that the output SINR of the proposed OPLI can coincide well with the optimal SINR, when the power of the incident signals are equal and much higher than that of the noise.
5 Simulation results
Extensive simulation results are provided to verify the effectiveness of the proposed OPLI for source localization and waveform identification. The combinational algorithm, which firstly utilizes DOA estimator to obtain source locations and then utilizes beamformer for waveform identification, is compared with OPLI. Besides, the wellknown RELAX [51] is also employed for performance comparison. Specially, for source localization, the stochastic CramérRao bound (CRB) [54] is also used for performance evaluation.
where M_{ c } denotes the number of Monte Carlo trials, \(\hat {\theta }_{k,m}\) refers to the DOA estimation of θ_{ k } in the mth Monte Carlo trial (similarly hereafter). In the following simulations, M_{ c }=200, and a halfwavelength spaced uniform linear array composed of N=6 sensors, is considered.
5.1 Waveform identification
This subsection evaluates the waveform identification performance of the proposed OPLI. The minimum variance distortionless response (MVDR) beamformer [36], the diagonally loaded sample matrix inversion (LSMI) beamformer [38], the general linear combination (GLC)based beamformer [39], and the optimal beamformer which is based on the maximum output SINR principle [46] are employed for performance comparison. The diagonal loading factor of the LSMI beamformer is set to be equal to the noise power.
Note that both OPLI and RELAX implement waveform identification without prior DOA information. Whereas the MVDR, LSMI, and GLC beamformers require prior DOA of the SOI to calculate beamforming weights. Unless otherwise stated, in the subsection, it is assumed that the prior DOA is considered without error.
5.2 Source localization
This subsection evaluates the source localization performance of the proposed OPLI. The wellknown RELAX [51], multiple signal classification (MUSIC) [55], and perturbed SBL (PSBL) [56] are included for performance comparison.
5.3 Comprehensive performance evaluation
Besides, the computational complexity of the proposed OPLI, which is evaluated by the running time of algorithm, is also compared with those of the combinational algorithms.
6 Methods
The oblique projectionbased localization and identification (OPLI) algorithm is proposed without known prior DOA or waveform information of the sources. The proposed OPLI employs oblique projection to separate the multiple incident signals into a series of single signal groups. Then, the source localization and waveform identification are implemented on each separated single signal. To this end, the OPLI is implemented iteratively. The method of maximum likelihood approximation is employed to evaluate the convergence.
7 Conclusions
A new OPLI algorithm, which is based on oblique projection, is proposed for localization and waveform identification of unknown target signal. The oblique projection is employed to separate the SOI from the received data of the array, and can also be used to suppress the interference signals at the same time.
The OPLI requires no prior information of the DOA or the signal waveforms, and it estimates the DOA and waveform of the SOI iteratively. Comparing to the well known RELAX and the combinational algorithms, which estimates DOA via employing MUSIC or PSBL, and realizes beamforming via employing MVDR, LSMI or GLC, the simulation results show that the proposed OPLI exhibits a better performance when the angular separation is small. Especially, when the input SNR within moderate to high region, the proposed OPLI not only shows an attractive output SINR which can coincide well with the optimal one, but also can achieve a high estimation accuracy for source localization since its RMSE could coincide well with CRB.
Both OPLI and RELAX are implemented iteratively, and the running time of the former is lower. Whereas, when compared to MUSICbased counterpart combinational algorithms, the OPLI requires a higher computational complexity to implement source localization and waveform identification. Extensive experiments have been undertaken to verify the effectiveness and superiority of the OPLI with uncorrelated sources. Future work includes extension of the OPLI to correlated and coherent sources, and so on.
Declarations
Funding
This work was supported by the National Natural Science Foundation of China (grant no. 61171180), by the Fundamental Research Funds for the Central Universities (grants nos. HIT. MKSTISP. 2016 13 and HIT. MKSTISP. 2016 26), and by a fund from the Science and Technology on Electronic Information Control Laboratory.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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