Research Article  Open Access
Jianzhong Li, Xiaobo Gu, Ruidian Zhan, Xiaoming Xiong, Yuan Liu, "DOA Estimation without Source Number for CyberPhysical Interactions", Complexity, vol. 2020, Article ID 9768361, 5 pages, 2020. https://doi.org/10.1155/2020/9768361
DOA Estimation without Source Number for CyberPhysical Interactions
Abstract
In this paper, a direction of arrival (DOA) estimator is proposed to improve the cyberphysical interactions, which is based on the secondorder statistics without a priori knowledge of the source number. The impact of noise will firstly be eliminated. Then the relationship between the processed covariance matrix and the steering matrix is studied. By applying the elementary column transformation, an oblique projector will be designed without the source number. At last, a rooting method will be adopted to estimate the DOAs with the constructed projector. Simulation results show that the proposed method performs as well as other methods, which requires that the source number must be known.
1. Introduction
Nowadays, the requirement for different objects to communicate with each other is rapidly rising in many fields of the practical life. However, the network combining all the objects is very complicated, and the communications among different nodes faces many problems [1]. On one hand, energy diffusion of the transmitted signal would lower down the quality of communication. The energy collected by the target receiver is very poor in this situation. On the other hand, the currently adopted omnidirectional antennas also increase the risk of being attacked during the communication. Therefore, it is very important to omit the signal in the desired direction, and the technique of finding direction of arrival (DOA) can help to improve the performance of cyberphysical interactions.
DOA estimation has been an important research topic for array signal processing [2–4]. This research topic is widely applied in many fields such as sonar and electronic surveillance [5], where the signals are often nonstationary [6]. The wavefront of a farfield source signal can be assumed to be plane when it impinges on the receiver array. Each source can be localized with its corresponding DOA.
Plenty of researchers over the world have been making efforts to contribute to the research of DOA estimation, and there are already many achievements, such as the multiple signal classification (MUSIC) in [7], estimating signal parameters via rotational invariance techniques (ESPRIT) in [8], rootMUSIC in [9], and oblique projection operator method (LOFNS) in [10]. However, all these existing methods require a priori knowledge of the source number to guarantee their successful application.
In this paper, we propose a method to localize farfield sources without any priori knowledge of the source number. Firstly, the impact of noise is eliminated by taking advantage of the property of nonstationary signal. Then the relationship between the steering matrix and covariance matrix is studied. The elementary column transformation is applied to get rid of the dependency of the source number and an orthogonal matrix is designed based on this relationship. At last, a rooting method is applied for the estimation of DOA, reducing the computational complexity.
The rest of this paper is organized as follows. Section 2 presents the signal model and some common assumptions. In Section 3, the proposed method is described in detail. The complexity analysis is also given to illustrate the improvement of the proposed method. In Section 4, several simulations are provided. At last, the conclusion of the whole paper is made in Section 5.
In this paper, represents the transpose operation, the conjugate transpose, and the complex conjugate. A bold capital letter symbolizes a matrix, and a bold letter in lowercase stands for a vector, such as and , respectively.
2. Signal Model
As shown in Figure 1, narrowband farfield source signals are received with a uniform linear array (ULA). sensors are distributed in the array with intersensor spacing being . The output of the th sensor can be expressed aswhere is the DOA of the th source, is the wavelength of the signal satisfying , and is the corresponding Gaussian white noise with the variance . The noises are assumed to be independent from each other and from all the source signals. Written in matrix form, the received signal can be expressed aswhere
Without loss of generality, we make the following assumptions, which are the same as those in [10–17]:(1)The kurtosis of the source signal is nonzero(2)All the DOAs are different from each other(3)The Gaussian noise is independent of the source signals, and the source signals are independent of each other
3. Proposed Algorithm
Generally in order to estimate DOAs with an oblique projection, the source number should be known to divide the covariance matrix, like LOFNS in [10]. Indeed, after a deep analysis into the basic principle of the oblique projection based methods, we observed that the division with the source number is mainly to ensure that the desired matrix is full column rank, such that the inverse operation can work properly while constructing the oblique projection. Therefore, we propose a rooting method to localize sources with nonstationary signal that can get rid of the dependence of the source number.
When the signal is nonstationary and the noise is stationary, two covariance matrices can be obtained with two different group of snapshots as follows:where is estimated with the first snapshots and with the last ones. The impact of the noise can be eliminated by constructing another matrix [18]:where .
From (5), it can be learned that all the columns of are the linear combination of the whole steering matrix. The relationship between and can still be maintained if we apply the elementary column transformation to :where , , is the th column of , and is the th element. For every iteration, we fix the parameter , and make traverse all the legal values. The procedure will be repeated before we get a matrix which is upper triangularlike. All the nonzero columns we obtain after the transformation operation can form a maximal linearly independent subsystem of , and can form a full columnrank matrix . Construct an oblique projection aswhere is an identity matrix of . It can be easily calculated that
Due to the fact that and span the same column space, (8) equals
Based on this orthogonality, a spectrum can be used for DOA estimation. In order to avoid the spectrum search which is computationally expensive, we propose the application of a rooting method. A polynomial is designed as follows:where . The roots closest to the unit circle are the desired ones for the estimation of :where is to take the angle of . The DOAs of the sources are estimated through
4. Simulation
In this section, the performance of the proposed method will be studied, which is examined with the root mean square error (RMSE). The definition of RMSE is given bywhere represents the DOA estimate of the th simulation, is the real DOA and means the total number of Monte Carlo simulations. The performance of the proposed method will be compared with other methods such as rootMUSIC in [9] and LOFNS in [10]. Two sources are considered in the simulations, whose DOAs are and , respectively. The inner space between sensors of the array . For the proposed method, the source number can be unknown. For the other methods, must be an exact priori knowledge. Specifically, the CramerRao lower bound (CRLB) is also illustrated to make a better comparison.
The first simulation is designed to examine the relationship between the RMSEs and signalnoise radio (SNR), which is defined aswith being the signal power. Assume that there are 8 sensors in the array (i.e. ) and 400 snapshots are received with the array. The SNR varies form 0 dB to 30 dB. As shown in Figures 2 and 3, by eliminating the effect of noise, both the proposed method and LOFNS outperform rootMUSIC, even though LOFNS does not perform well when the SNR is low. The proposed method shows a robust output for all the SNR. As for the estimates of different sources, we can see that the RMSEs with the same method are almost the same. The different directions of sources do not affect the estimation accuracy.
The second simulation studies the RMSEs in terms of the number of snapshots. Set the SNR at 15 dB, and the number of snapshots changes from 10 to 10000. The array is the same as that in the first simulation. The corresponding results are displayed in Figures 4 and 5. Similar to those RMSEs in Figures 2 and 3, the proposed method provides the most robust performance while rootMUSIC performs the poorest.
The computational efficiency will be studied in the third simulation. 1000 simulations are run in a PC, whose CPU is Intel (R) Core (TM) I7 8700 3.2 GHz and RAM is 8 GB, to get the total proceeding time for different methods. The results with 400 snapshots are shown in Table 1. We can see that the results of proceeding time of the three methods are almost the same. As for the average proceeding time for one simulation, the difference is negligible.

5. Conclusion
In order to get rid of the dependency on the source number, a rooting method based on the oblique projection is proposed in this paper. By taking advantage of the property of nonstationary signal, the effect of noise can be eliminated. The elementary column transformation is applied and a matrix orthogonal with the steering matrix is constructed without the number of sources. At last, a rooting method is applied with the constructed orthogonal matrix. Simulation results verify the effectiveness of the proposed method, which can perform better than other existing methods even with the known exact source number.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the KeyArea Research and Development Program of Guangdong Province (2019B010145001, 2019B010142001, and 2019B010141002) and Science and Technology Program of Guangdong (2019A141401005).
References
 V. Behzadan, “Cyberphysical attacks on uas networkschallenges and open research problems,” 2017, http://arxiv.org/abs/1702.01251. View at: Google Scholar
 F.G. Yan, S. Liu, J. Wang, M. Jin, and Y. Shen, “Fast doa estimation using coprime array,” Electronics Letters, vol. 54, no. 7, pp. 409410, 2018. View at: Publisher Site  Google Scholar
 H. Krim and M. Viberg, “Two decades of array signal processing research: the parametric approach,” IEEE Signal Processing Magazine, vol. 13, no. 4, pp. 67–94, 1996. View at: Publisher Site  Google Scholar
 Z. Zheng, J. Sun, W.Q. Wang, and Y. Haifen, “Classification and localization of mixed nearfield and farfield sources using mixedorder statistics,” Signal Processing, vol. 143, pp. 134–139, 2018. View at: Publisher Site  Google Scholar
 A. Liu, Q. Yang, X. Zhang, and W. Deng, “Modified root music for coprime linear arrays,” Electronics Letters, vol. 54, no. 15, pp. 949–951, 2018. View at: Publisher Site  Google Scholar
 S. Chen, X. Dong, G. Xing, Z. Peng, W. Zhang, and G. Meng, “Separation of overlapped nonstationary signals by ridge path regrouping and intrinsic chirp component decomposition,” IEEE Sensors Journal, vol. 17, no. 18, pp. 5994–6005, 2017. View at: Publisher Site  Google Scholar
 R. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp. 276–280, 1986. View at: Publisher Site  Google Scholar
 H. Chen, W. Zhu, and M. Swamy, “Realvalued esprit for twodimensional doa estimation of noncircular signals for acoustic vector sensor array,” in Proceedings of the 2015 IEEE International Symposium on Circuits and Systems (ISCAS), IEEE, Lisbon, Portugal, May 2015. View at: Publisher Site  Google Scholar
 B. Friedlander, “The rootMUSIC algorithm for direction finding with interpolated arrays,” Signal Processing, vol. 30, no. 1, pp. 15–29, 1993. View at: Publisher Site  Google Scholar
 W. Zuo, J. Xin, N. Zheng, and A. Sano, “Subspacebased localization of farfield and nearfield signals without eigendecomposition,” IEEE Transactions on Signal Processing, vol. 66, no. 17, pp. 4461–4476, 2018. View at: Publisher Site  Google Scholar
 A. Ahmed, Y. D. Zhang, and B. Himed, “Effective nested array design for fourthorder cumulantbased DOA estimation,” in Proceedings of the Radar Conference, pp. 0998–1002, Seattle, WA, USA, May 2017. View at: Publisher Site  Google Scholar
 Y. Hu, Y. Liu, and X. Wang, “DOA estimation of coherent signals on coprime arrays exploiting fourthorder cumulants,” Sensors, vol. 17, no. 4, 2017. View at: Publisher Site  Google Scholar
 Z. Zheng, M. Fu, W.Q. Wang, and H. C. So, “Mixed farfield and nearfield source localization based on subarray crosscumulant,” Signal Processing, vol. 150, pp. 51–56, 2018. View at: Publisher Site  Google Scholar
 R. Challa and S. Shamsunder, “Highorder subspacebased algorithms for passive localization of nearfield sources,” in Proceedings of the Conference Record of the TwentyNinth Asilomar Conference on Signals, Systems and Computers, IEEE, Pacific Grove, CA, USA, November 1995. View at: Publisher Site  Google Scholar
 J. M. Mendel, “Tutorial on higherorder statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proceedings of the IEEE, vol. 79, no. 3, pp. 278–305, 1991. View at: Publisher Site  Google Scholar
 K. Wang, L. Wang, J. R. Shang, and X. X. Qu, “Mixed nearfield and farfield source localization based on uniform linear array partition,” IEEE Sensors Journal, vol. 16, no. 22, pp. 8083–8090, 2016. View at: Publisher Site  Google Scholar
 Z. Zheng, M. Fu, D. Jiang, W.Q. Wang, and S. Zhang, “Localization of mixed farfield and nearfield sources via cumulant matrix reconstruction,” IEEE Sensors Journal, vol. 18, no. 18, pp. 7671–7680, 2018. View at: Publisher Site  Google Scholar
 A. Paulraj and T. Kailath, “Eigenstructure methods for direction of arrival estimation in the presence of unknown noise fields,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 34, no. 1, pp. 13–20, 1986. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2020 Jianzhong Li 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.