In this paper, a direction of arrival (DOA) estimator is proposed to improve the cyber-physical interactions, which is based on the second-order 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.
Research and Development Program of Guangdong Province2019B0101450012019B0101420012019B010141002Guangzhou Science and Technology Program key projects2019A1414010051. 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 cyber-physical 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 far-field 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], root-MUSIC 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 far-field 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, T represents the transpose operation, H 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 A and a, respectively.
2. Signal Model
As shown in Figure 1, K narrow-band far-field source signals are received with a uniform linear array (ULA). M+1 sensors are distributed in the array with intersensor spacing being d. The output of the mth m∈0,M sensor can be expressed as(1)ymt=∑k=1Ksktejωmk+nmt,t=1,2,…,T,ωk=−2πdλsinθk,where θk is the DOA of the kth source, λ is the wavelength of the signal satisfying λ≥2d, and nmt is the corresponding Gaussian white noise with the variance σ2. 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 as(2)yt=Aθst+nt,where(3)yt=y0t,y1t,…,yMtT,st=s1t,s2t,…,sKtT,Aθ=aθ1,aθ2,…,aθK,aθk=ej0ωk,ej1ωk,…,ejMωkT,nt=n0t,n1t,…,nMtT.
Far-field source localization with ULA.
Without loss of generality, we make the following assumptions, which are the same as those in [10–17]:
The kurtosis of the source signal is nonzero
All the DOAs are different from each other
The Gaussian noise nmt is independent of the source signals, and the K 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:(4)R1=2T∑t=1T/2AθytyHtAHθ+ntnHt=AθRs1AHθ+σ2I,R2=2T∑t=T/2+1TAθytyHtAHθ+ntnHt=AθRs2AHθ+σ2I,where R1 is estimated with the first T/2 snapshots and R2 with the last T/2 ones. The impact of the noise can be eliminated by constructing another matrix [18]:(5)R=R1−R2=AθRsAHθ,where Rs=Rs1−Rs2≠0.
From (5), it can be learned that all the columns of R are the linear combination of the whole steering matrix. The relationship between R and Aθ can still be maintained if we apply the elementary column transformation to R:(6)rpM+1−prp+qM+1−prp+q−rp⟶rp+q,where p∈1,M, q∈1,M+1−p, rp is the pth column of R, and rpq is the pqth element. For every iteration, we fix the parameter p, and make q traverse all the legal values. The procedure will be repeated before we get a matrix which is upper triangular-like. All the nonzero columns we obtain after the transformation operation can form a maximal linearly independent subsystem of R, and can form a full column-rank matrix Us. Construct an oblique projection as(7)Un=IM+1−UsUsHUs−1UsH,where IM+1 is an identity matrix of M+1×M+1. It can be easily calculated that(8)UsHUn=0M+1.
Due to the fact that Us and Aθ span the same column space, (8) equals(9)AHθUn=0M+1.
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:(10)fz=z2M+1aTz−1UnUnHaz,where z=ejω. The roots closest to the unit circle are the desired ones for the estimation of ω:(11)ω^k=∠zk,where ∠z is to take the angle of z. The DOAs of the sources are estimated through(12)θ^k=−λωk2πd.
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 by(13)RMSE=∑n=1Nθ^n−θtrue2N,where θ^n represents the DOA estimate of the nth simulation, θtrue is the real DOA and N means the total number of Monte Carlo simulations. The performance of the proposed method will be compared with other methods such as root-MUSIC in [9] and LOFNS in [10]. Two sources are considered in the simulations, whose DOAs are 6° and 23°, respectively. The inner space between sensors of the array d=λ/2. For the proposed method, the source number K can be unknown. For the other methods, K must be an exact priori knowledge. Specifically, the Cramer-Rao lower bound (CRLB) is also illustrated to make a better comparison.
The first simulation is designed to examine the relationship between the RMSEs and signal-noise radio (SNR), which is defined as(14)SNR=10log10∑k=1KPskσ2,with Psk being the signal power. Assume that there are 8 sensors in the array (i.e. M=7) 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 root-MUSIC, 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.
RMSEs versus SNR: 1st source.
RMSEs versus SNR: 2nd source.
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 root-MUSIC performs the poorest.
RMSEs versus number of snapshots.
RMSEs versus number of snapshots.
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.
Total proceeding time with Matlab for 1000 times (in second).
Methods
Root-music
LOFNS
Proposed method
M=5
0.3645
0.3545
0.4575
M=7
0.5551
0.6222
0.6636
M=9
0.6923
0.6929
0.7018
M=11
0.7757
0.8646
0.7518
M=13
1.1122
1.1589
1.1240
M=15
1.7455
1.7939
1.7756
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 Key-Area Research and Development Program of Guangdong Province (2019B010145001, 2019B010142001, and 2019B010141002) and Science and Technology Program of Guangdong (2019A141401005).
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