Abstract
This paper focuses on a lowcomplexity onedimensional (1D) directionofarrival (DOA) algorithm with an arbitrary crosslinear array. This algorithm is highly accurate without the performance error usually caused by the uncertainty factor of the wave velocity in the underwater environment. The geometric relationship between two crossed linear arrays is employed to derive the expression of DOA estimation with the finding that this algorithm is capable of excluding the wave velocity variable in the DOA estimation expression. A method without parameter pairing is also proposed to reduce the complexity of this algorithm. Additionally, the influence of wave velocity is analyzed in terms of RMSE_{c} and the CramerRao bound (CRB) for 1D DOA with the arbitrary crosslinear array is established. The simulation results demonstrate that the proposed algorithm can achieve better performance than the traditional algorithm under the condition of an inaccurate estimate of wave velocity. Compared with the velocityindependent DOA algorithm, it exhibits the feature of low complexity.
Introduction
The estimation of the underwater DOA is widely used in many fields, such as underwater target positioning, offshore operations, and military reconnaissance [1]. Compared with an electromagnetic wave, a sound wave undergoes less loss in underwater propagation. Therefore, underwater DOA estimation usually employs sound waves as communication signals and takes wave velocity as a constant. However, in a real underwater environment, the numerical value of wave velocity constantly changes for it is affected by temperature, salinity, and other factors [2]. The actual amount of wave velocity in seawater approximately ranges from 1450 to 1550 m/s. The assumption that the wave velocity is a constant will generate serious errors. Therefore, it is critical to handle the influence factors of the wave velocity in underwater DOA estimation.
The ESPRIT algorithm [3] has been extensively used since it was proposed because, among other characteristics, it features no spectral peak search and low computational requirements. Several enhanced algorithms are based on the ESPRIT algorithm. Qian has proposed the MRESPRIT algorithm [4], which improved the array division of the classical ESPRIT algorithm. This classical algorithm sets down two subarrays with an offset of 1 on a single linear array. It then uses the rotation operator between the two subarrays to obtain the result. By contrast, the MRESPRIT algorithm divides subarrays with an offset greater than 1. It utilizes the received signals of the two subarrays to restore the direction matrix of the array. Finally, the estimation result is obtained by solving the direction matrix. The MRESPRIT algorithm is superior at suppressing the impact of noise. An automatic pairing joint directionofarrival and frequency estimation, abbreviated as AFESPRIT, is presented in [5]. By using the multipledelay output of a uniform linear antenna array (ULA), this algorithm can estimate joint angles and frequencies. In [6], a frequencyangle joint estimation algorithm is proposed. It is based on singular value decomposition (SVD) and trilinear decomposition. This algorithm eliminates the influence of the number of snapshots on the computational complexity through SVD and reduces the estimation complexity without losing estimation performance. Pinto [7] and others proposed an estimation algorithm abbreviated as MSKAIESPRIT that performs better at a low signaltonoise ratio (SNR) and small amounts of snapshots. First, this algorithm sets a convergence factor and roughly estimates the DOA of the source signal through the classical ESPRIT algorithm. Next, it uses this result to correct the data covariance matrix and utilizes the modified covariance matrix to estimate the new DOA angle; cyclic iteration is then applied until the convergence condition is satisfied. Onedimensional DOA estimation with different structured arrays, such as Lshaped arrays [8–10] and a uniform rectangular array, has captured a remarkable amount of attention. The MUSIC algorithm has also aroused notable research interest [11–14] given that it has a high resolution, estimation accuracy, and stability under certain conditions. In [15], a velocityindependent MUSIC algorithm (VIMUSIC) is proposed. It is based on Lshaped arrays. The VIMUSIC algorithm uses an Lshaped array and the socalled Same Peak method to eliminate the effect of using imprecise wave velocity. This algorithm can achieve high accuracy even though imprecise wave velocity is used for DOA estimation.
However, the realtime propagation velocity of the sound wave is unknown because of the instability and complexity of the underwater environment. The ESPRIT algorithms mentioned above calculate the wave velocity as a constant. Adopting imprecise velocity leads to a significant error. Despite being a velocityindependent algorithm, the previously mentioned VIMUSIC algorithm needs to search for realtime wave velocity in the range of 1450–1550 m/s, which implies high and often unacceptable computational complexity.
This paper proposes a 1D velocity independent and low complexity ESPRIT algorithm (VILCESPRIT) based on an arbitrary crosslinear array to solve the above problem. The proposed algorithm uses the geometric relationship between the arbitrary crosslinear array to eliminate the effect of variable wave velocity. Besides, it exhibits low computational complexity without parameter matching.
Signal model and wave velocity influence analysis
Arbitrary crosslinear array structure and data model
As illustrated in Fig.1, K farfield narrowband plane wave signals s_{i}(t),i=1,...,K, impinge on the arbitrary crosslinear array. Arbitrary crosslinear array is structured by two uniform arrays in the xy plane with an cross angle of δ, where \(\delta \in (0,\frac {\pi }{2})\). Each array consists of M identical omnidirectional sensors separated by interelement spacing d, namely, d=λ/2, where λ approximates to be the wavelength of the incident waves. The total number of sensors is N−1, where N=2M. r_{n}=[x_{n},y_{n}]^{T},n=1,...,N is the position vector of sensor n. The position of sensor m on the xaxis is denoted as r_{m},m=1,...,M, and the position of sensor m on the yaxis is denoted as r_{m+M}. The center frequency of the ith signal is f_{i}, and considering the underwater homogeneous isotropic fluid medium environment, their wave velocity is defined as c, where c∈[1450,1550] m/s. Let us note θ_{xi} the DOA of the ith signal on the xaxis, where θ_{xi}∈(0,π), and θ_{yi} the DOA of the ith signal on the yaxis, where θ_{yi}∈(0,π).
Using the array element at O as the reference element. As shown in Fig. 1, the incoming area of the source signals is divided into two areas. In area ①, we have θ_{yi}=δ−θ_{xi}, and in area ②, we have θ_{yi}=θ_{xi}−δ. That is to say, in 1DDOA estimation, the angles between the narrowband signal and arbitrary crosslinear array satisfy
where θ_{xi}=θ_{i}, and θ_{i} is the expected wave direction angle of the ith source signal.
The received signals of the array on the xaxis and the yaxis are written as
where S(t)=[s_{1}(t),s_{2}(t),...,s_{k}(t)]^{T} is the K×1 incoming source signals vector, N_{x}(t) and N_{y}(t) are the Gaussian white noise vectors along the xaxis and the yaxis, respectively. The M×K array manifold matrices A_{x} and A_{y} can be represented as
We define τ_{n}(θ_{xi})=d^{T}(θ_{xi})·r_{n}/c as the propagation delay of the ith signal received sensor n, where d(θ_{xi})=[cosθ_{xi}, sinθ_{xi}]^{T} is the unit vector pointing towards the ith signal, and r_{n} is the position vector of sensor n.
are denoted as M×1 array manifold vectors, which have the form of \({a_{m}(\theta _{xi}) = e^{j 2 \pi f_{i} \tau _{m}(\theta _{xi})}}\phantom {\dot {i}\!}\) on the xaxis and \({a_{m}(\theta _{yi}) = e^{j 2 \pi f_{i} \tau _{m+M}(\theta _{yi})}}\phantom {\dot {i}\!}\) on the yaxis, respectively. We suppose that the source signals are nonGaussian and uncorrelated to each other. And the Gaussian noises with zeromean and variance σ^{2} are assumed to be statistically independent to the signals.
Analysis of the wave velocity influence
After performing 1D RootMusic algorithm or TLSESPRIT algorithm, the roots u_{i} on the xaxis and v_{i}, i=1,...,K on the yaxis are obtained. The relationship between u_{i}, v_{i}, θ_{xi} and the realtime underwater wave velocity c are expressed as follows.
According to Eq. (5), the DOA can be rewritten as
Traditional algorithms such as TLSESPRIT, RootMUSIC, and MSKAIESPRIT perform well on the premise conditions: (1) DOA and frequency parameters are paired; (2) the realtime wave velocity is 1500 m/s. The AFESPRIT algorithm can obtain automatic pairing DOA and frequency parameters by using the multipledelay output of a uniform linear antenna array, and it performs well when realtime wave velocity is 1500 m/s. In summary, the above algorithms must set a constant value of c_{0} as the realtime wave velocity, where c_{0}=1500 m/s.
However, the realtime wave velocity c is a variable in the range of 1450–1550 m/s. Let Δc=c−c_{0}. According to Eq. (6), the greater Δc is, the greater deviation is in DOA estimation comparing to the estimation when Δc=0. In order to evaluate the impact of using inaccurate wave velocity c0 instead of realtime wave velocity c, we define RMSE_{c} using different realtime wave velocity values.
Method of velocityindependent and lowcomplexity DOA estimation
The VILCESPRIT algorithm
Firstly, a crosscorrelation matrix R_{xy} is obtained by Eq. (8).
where R_{s}=E{S(t)S^{H}(t)}. It can be noted that the additive noise is removed by the crosscorrelation operation. Let R_{xy1} and R_{xy2} be the first and last M−1 cols of R_{xy}, respectively, so
In the equation above, A_{y2}=A_{y1}Ω^{H}, A_{y1} and A_{y2} are defined as the first and last M−1 rows of A_{y}, and \(\mathbf {\Omega }=\operatorname {diag}\{e^{j2\pi d f_{1} \cos \theta _{y1} / c}, \cdots,e^{j2\pi d f_{K} \cos \theta _{yK} / c} \}\). Since the K source signals are uncorrelated, it is easy to know that A_{y1}, A_{y2} and R_{s} are full rank matrices.
By combining Eq. (9), a new 2M×(M−1) matrix R is defined by using Eq. (10).
where a new direction matrix B can be expressed as
We can obtain the singular value from the decomposition of matrix R using
where Σ_{s} is a K×K matrix, U_{s} is the signal subspace, and U_{n} is the noise subspace. By combining Eq. (8), it can be noted that Σ_{n}=0. In addition, U=[U_{s},U_{n}] is a unitary matrix, so it can be obtained by
According to the properties of SVD, we have
Considering V_{n} is a fullrank matrix and Σ_{n}=0, and combining Eqs. (10), (14) can be rewritten as
It is easy to know that \(\mathbf {A}_{y 1} \mathbf {R}_{s}^{\mathrm {H}}\) is a fullrank matrix. So, B and U_{n} have the following relationship.
By combining Eqs. (13) and (16), a nonsingular matrix T is defined by Eq. (17).
Then, submatrices B_{1} and B_{2} are defined using
There is a matrix Ψ that satisfies
where Ψ= diag{Ψ_{1},⋯,Ψ_{K}}. Ψ is called the rotation matrix, whereas Ψ_{i} is called a phase rotation operator, and \(\Psi _{i} = e^{j 2 \pi d f_{i} \cos \theta _{xi} / c}\). θ_{xi} can be obtained if Ψ_{i} is determined. The matrix U_{s} is divided into matrix U_{1} and U_{2} in the same way as
According to Eq. (17), we obtain
By combining Eqs. (19) and (21), there is
\(\mathbf {U}_{1}^{+}\) is defined as the MoorePenrose generalized inverse of U_{1}, and Eq. (22) can be rewritten adopting
In Eq. (23), by performing eigenvalue decomposition (EVD) of \(\mathbf {U}_{1}^{+} \mathbf {U}_{2}\), eigenvalues λ_{1},λ_{2},⋯,λ_{K} and corresponding eigenvectors T^{−1} are obtained. Besides, λ_{1},λ_{2},⋯,λ_{K} correspond to the diagonal elements of Ψ. Then, according to the expression of Ψ, the azimuth angles can be expressed as
Here, we obtain the estimated value of the direction matrix B adopting
In addition, combining Eq. (11) and the expression of Ω, \(\mathbf {B}(1,i) \mathbf {B}^{*}(M+1,i)=e^{j 2 \pi d f_{i} \cos {\theta _{yi}} /c}\) can be obtained, where B(k,i) is the element of the kth row and the ith col of B. That is to say
where \(\widehat {\mathbf {B}}(k,i)\) is the element of the kth row and the ith col of \(\widehat {\mathbf {B}}\). The effect of inaccurate wave velocity can be reduced by dividing Eqs. (24) and (26).
Finally, combined with Eq. (1), the azimuth angles can be obtained as follows.
when arg(λ_{i})≠0, and θ_{i}=0 when arg(λ_{i})=0.
Remark: According to the definition of Ω and the construction method of the matrix B in Eq. (11), it can be found that the column vectors of B only contain paired θ_{xi} and θ_{yi}, where cosθ_{yi}= cos(θ_{xi}−δ). From Eqs. (23) and (25), \(\mathbf {U}_{1}^{+} \mathbf {U}_{2} = \mathbf {T}^{1} \mathbf {\Psi T}\), and the estimated value of B satisfies \(\widehat { \mathbf {B}} = \mathbf {U}_{s} \mathbf {T}^{1}\), so the columns of \(\widehat {\mathbf {B}}\) and Ψ have related permutations. Therefore, cosθ_{xi} and cosθ_{yi} are matched in Eq. (27) according to Eqs. (23) to (26).
The summary of the proposed VILCESPRIT algorithm is shown as follows: Step1: Compute R_{xy} and construct R from Eqs. (8) and (10); Step2: Construct U_{s}, U_{1}, and U_{2} from Eqs. (12) and (20); Step3: Estimate λ_{i} and \(\widehat {\mathbf {B}}\) from Eqs. (23) and (25); Step4: Obtain velocity independent azimuth angle from Eq. (28).
CramerRao bound (CRB) analysis
When the data vector is assumed to be Gaussian distributed, a particularly convenient CRB formula is derived in reference [16]. In case of Lshaped array configuration, the CRB formula of 2D DOAs is given in reference [17], where the wave velocity is taken as a constant. The CRB formula of the 1D DOAs using an arbitrary crossline array is considered here, taking into account the variable wave velocity. Construct a new received data matrix from the arbitrary crosslinear array as follows.
Equation (29) can be rewritten adopting
The Fisher information matrix F is with respect to θ=[θ_{1},θ_{2},⋯,θ_{K}]. The i,jth element of F is
The concrete form of F_{ii} is
where A_{ni} is the n,ith element of A, [x_{n},y_{n}]is the position of the nth sensor r_{n}, R_{s}=E[SS^{H}] and (R_{s})_{ii} is the i,ith element of R_{s}.
Then, the CRB matrix can be expressed using
Let C_{ij} be the i,jth element of C and we can obtain the CRB of the ith azimuth angle as Eq. (34).
Thus, the total CRB of 1D DOAs can be expressed by Eq. (35).
According to Eq. (32), it can be found that F_{ii} increases as snapshots L increases and decreases as wave velocity c increases or noise power σ^{2} increases. When δ=90^{∘}, (y_{n} cosθ_{i}−x_{n} sinθ_{i})^{2} has a maximum value, which means that F_{ii} has a maximum value. By combining Eqs. (33) and (35), it can be concluded that the CRB of 1D DOAs increases as c increases, decreases as L or SNR increases, and has a minimum value when δ=90^{∘}.
Complexity analysis
In this section, we mainly compare the performance of the following algorithms: TLSESPRIT, RootMUSIC, AFESPRIT in [5], MSKAIESPRIT in [7], VIMUSIC in [15], and the proposed VILCESPRIT algorithm. K is the number of source signals, N is the number of total sensors, and L is the sample snapshots.
As for the complexity, we analyze it based on matrix complex multiplication, which mainly involves in autocorrelation or crosscorrelation matrix construction, EVD or SVD operation, and pseudoinverse operation. Due to L>>N>K, we mainly study the relationship between the complexity of the five algorithms and L,N. The complexity of the proposed VILCESPRIT algorithm is about O(M^{2}L+20M^{3}+4M^{2}K+2MK^{2}+K^{3}), while that of the classical TLSESPRIT algorithm is about O(N^{2}L+4N^{3}), where N=2M. The running time of the six algorithms is employed to check the performance of complexity. The results of the running times are shown in the following figures.
Figures 2 and 3 show that the complexity of the six algorithms increases gradually with the increase of sensors, but increases slowly with the increase of snapshots. Compared with other algorithms, the VIMUSIC algorithm and the MSKAIESPRIT algorithm have higher computational complexity. The complexity of the TLSESPRIT algorithm is low. In general, the proposed VILCESPRIT algorithm has lower complexity than others and has an advantage when the wave velocity is unknown. Because the VIMUSIC algorithm needs to search the realtime wave velocity, its computational complexity is even hundreds of times that of the proposed VILCESPRIT algorithm. Considering that a large amount of calculation limits the practical value of the VIMUSIC algorithm, its estimation performance is not compared in Section 4.
Simulation results and discussion
In all simulation experiments, the sampling frequency is 40 kHz, the number of snapshots L is 200, the element spacing d is 0.05 m, and 1000 Monte Carlo trials are conducted. Note that Δc=c−c_{0}, where c_{0}=1500 m/s. The array structure used in the VILCESPRIT algorithm is a M×2 arbitrary crosslinear array, and the array structure used in other algorithms is a N×1 linear array, where N=2M. We assume that the source signals are nonGaussian and uncorrelated to each other; the Gaussian noises with zeromean and variance σ^{2} are statistically independent of the signals.
An arbitrary crosslinear array is used in the proposed algorithm. To find a suitable cross angle, we compare the performance of all algorithms at different cross angles in following Section 4.1. The influences of different wave directions and different SNR environments on the algorithm are both important factors in the performance evaluation of the DOA estimation algorithm. Thus, the influence of wave direction on the estimation performance is studied in Section 4.2 to detect the estimation accuracy of all algorithms for different wave directions. And simulations under different SNR are implemented.
Effect of crosslinear angle on estimation performance
In the first experiment, all algorithms are compared in terms of RMSE over cross angles. A farfield narrowband signal with an azimuth angle of 30^{∘} incoming with a center frequency of 15 kHz. The total sensors number N is 10, and the SNR is set to 0 dB.
Figure 4a shows that the proposed algorithm perform better as cross angle δ increases; the performance of other algorithms do not change as δ increase because only a linear array is used; CRB decreases as δ increases, which is consistent with the previous analysis of CRB. Besides, the RootMUSIC algorithm performs rather good when Δc=0, the MSKAIESPRIT algorithm, and the TLSESPRIT algorithm perform moderately, while the AFESPRIT algorithm function poorly. And the performance of the proposed VILCESPRIT algorithm is close to that of AFESPRIT when M=5 and is close to that of RootMUSIC when M=10.
The AFESPRIT algorithm needs to estimate the frequency first, and then use the result of frequency estimation to estimate DOA. Compared with other conventional algorithms, this will bring more errors and lead to worse performance when there is no need to estimate the frequency. Considering that sensors on the yaxis in the proposed algorithm is used to eliminate the effect of wave velocity, it does not play a role in reducing the DOA estimation error. Therefore, the proposed algorithm performs worse than other algorithms when M=5 and performs better than other algorithms when M=10. Also considering that the proposed algorithm performs best when δ=90^{∘}, the cross angle δ is set to 90^{∘} in the following experiments.
Figure 4a and b illustrate that the proposed algorithm is not affected by the wave velocity. The other algorithms perform worse when Δc>0, and the RMSE of them is larger than RMSE_{c}. In addition, CRB at c=1500 m/s is larger than CRB at c=1475 m/s, which is consistent with the previous analysis of CRB.
Effect of wave direction on estimation performance
In the second experiment, all algorithms are compared in terms of RMSE over wave directions. The δ is set to 90^{∘} and the other conditions are the same as those of the first experiment.
Figure 5a shows that the performance of the proposed algorithm hardly changes as wave direction increases, while the other algorithms perform better as wave direction θ increases. According to Eq. (32), F_{ii} increases as θ_{i} increases for a single linear array, and F_{ii} does not change significantly for the arbitrary crosslinear array. Because of C=F^{−1}, it means that the performance of the proposed algorithm is not affected significantly as θ increases, and the other algorithms perform better as θ increases, which is consistent with the experimental results.
From Fig. 5a and b, it can be noted that the estimation performance of the proposed algorithm is well in different wave velocity environments than other algorithms. Besides, the other algorithms perform worse when Δc>0 than that when Δc=0. The simulation result shows the proposed algorithm maintains robust performance in an unknown wave velocity environment.
Comparison of algorithms with different sNR
In the third experiment, the proposed VILCESPRIT algorithm in theoretical analysis and experimental studies, MSKAIESPRIT algorithm, TLSESPRIT algorithm, RootMUSIC algorithm, AFESPRIT algorithm, and CRB are compared in term of RMSE with respect to SNRs in an Gaussian noise situation. The number of sensors N is set to 10. Two uncorrelated equal power signals with azimuth angles θ and frequency from (30^{∘},60^{∘}) and (15000, 16000) kHz.
From Fig. 6a, it can be noted that the RMSE of all algorithms, except MSKAIESPRIT, and the CRB decreases as SNR increases. The proposed algorithm performs better than the others at high SNR, while it is worse at low SNR. Considering that sensors on the yaxis in the proposed algorithm is used to eliminate the effect of wave velocity, it does not play a role in reducing the DOA estimation error. The proposed algorithm performs better than other algorithms at high SNR when using 10×2 sensors. Besides, because the TLSESPRIT algorithm and the RootMUSIC algorithm are not suitable for DOA estimation of multifrequency sources, their performance is worse than the AFESPRIT algorithm when Δc=0.
Figure 6a and b note that the estimation performance of the proposed algorithm at Δc=0 is roughly the same as that at Δc=−25 m/s, while other algorithms perform worse as Δc increases. Also, the performance of AFESPRIT is close to RMSE_{c} at high SNR, which quite matches the theoretical analysis in Eq. (7).
Comparison of algorithms with unknown wave velocity
In the fourth experiment, the above algorithms are compared in terms of RMSE with respect to the realtime wave velocity. The parameters configured in this experiment are the same as the third experiment.
From the result of Fig. 7a, the proposed algorithm maintains robust performance as Δc increases, and far superior to the other algorithms when M=10. The larger Δc is, the worse the estimation performs among other algorithms. And the performance of the AFESPRIT is close to RMSE_{c} which is better than the classical algorithms. Besides, the MSKAIESPRIT algorithm performs worse at multifrequency and inaccurate velocity environments.
Figure 7a and b note that the estimate performance of the proposed algorithm is better at SNR=10 dB than that at SNR=0 dB and the proposed algorithm is also better than others when Δc>0.
Conclusions
This paper proposes a velocityindependent and lowcomplexity 1DDOA estimation algorithm with an arbitrary crosslinear array to solve the problem of unknown underwater wave velocity. The arbitrary crosslinear array is employed to reduce the estimation error caused by assuming a fixed wave velocity. Additionally, the proposed algorithm has low complexity, for it does not require parameter pairing. Compared with the traditional 1D ESPRIT algorithm (e.g., AFESPRIT and MSKAIESPRIT), it can deal with the effect of wave velocity deviation and the computational complexity is much lower than that of the former VIMUSIC algorithm. The simulation results well coincide with the analysis of the RMSE_{c} and the CRB. Simulation experiments bear out the proposed VILCESPRIT algorithm, which is effective in handling the impact of an inaccurate wave velocity value.
Abbreviations
 MRESPRIT:

Manifold reconstruction estimating signal parameter via rotational invariance techniques
 AFESPRIT:

Joint angle and frequency estimating signal parameter via rotational invariance techniques
 MSKAIESPRIT:

Multistep knowledgeaided iterative estimating signal parameter via rotational invariance techniques
 TLSESPRIT:

Total least square estimating signal parameter via rotational invariance techniques
 VIMUSIC:

Velocityindependent multiple signal classification
 VILCESPRIT:

Velocityindependent and lowcomplexity estimating signal parameter via rotational invariance techniques algorithm proposed in this paper
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Acknowledgements
The authors would like to thank all anonymous reviewers and editors for their helpful suggestions for the improvement of this paper.
Funding
This work supported by the National Natural Science Foundation of China (61871191), Science and Technology Planning Project of Guangdong Province (2016A020222003,2017A030313368), and Science and Technology Planning Project of Guangzhou (201804010209).
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GN and XL proposed the original idea of the full text; GN and GJ designed and implemented the simulation experiments; GJ and XZ wrote the manuscript under the guidance of GN. All authors read and approved this submission.
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Ning, G., Jing, G., Li, X. et al. Velocityindependent and lowcomplexity method for 1D DOA estimation using an arbitrary crosslinear array. EURASIP J. Adv. Signal Process. 2020, 28 (2020). https://doi.org/10.1186/s13634020006872
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DOI: https://doi.org/10.1186/s13634020006872
Keywords
 1DDOA
 Velocity independent
 Arbitrary crosslinear array
 CramerRao bound