Acoustic,Velocity-Independent,2-D,DOA,Estimation,for,Underwater,Application

Gengxin Ning,Zhenfeng Liao,Xiaopeng Li,Cui Yang

Abstract—In this paper,an acoustic velocityindependent two-dimensional direction of arrival(2-D DOA)estimation for underwater application is presented to eliminate the effect of the inaccurate acoustic velocity estimation.According to the geometric relationship between the linear arrays,the proposed method employs the cross correlation matrix(CCM)of the data received by three crossed linear arrays to remove the acoustic velocity factor.The simulation results demonstrate that the proposed method is not susceptible to the acoustic velocity.For a single source,the proposed method outperforms the conventional method in all conditions.For multiple sources,there is a little performance degradation for the proposed method compared with the conventional method.However,the proposed method displays a better performance than the conventional method in situations where the signal to noise ratio(SNR)is extremely low or the acoustic velocity estimation error is non-negligible.Furthermore,the computational complexity of the proposed method is lower than that of the conventional method using the same amount of sensors in total,while the performance is still acceptable.

Keywords—velocity-independent,underwater,2-D DOA

Two-dimensional direction of arrival(2-D DOA)estimation can estimate the azimuth and elevation angle of the target.Lots of array structures have been developed for 2-D DOA estimation,such as uniform plane array[1,2],uniform circular array[3,4],L-shaped array[5,6],two L-shaped arrays[7-9],etc.Based on the uniform plane array,Krekel et al.[10]proposed a two-dimensional estimating signal parameter via rotational invariance techniques(ESPRIT)algorithm without spectral peak search,which converts a two-dimensional estimation problem into two one-dimensional estimation problems.But this method involves the pairwise matching of the azimuth and elevation angles.To address the matching problem,Zhang et al.[11]proposed an automatic pairwise ESPRITlike two-dimensional DOA estimation algorithm.However,the method requires high computational complexity and can only be effective when there are few targets.Ref.[12]proposed a computationally efficient 2-D DOA estimation(CEDE)for an L-shaped sparse array with automatic pairing,based on the propagation method(PM)[13]and the ESPRIT method.The CEDE method exploits the conjugate symmetry property of the direction matrix to enlarge the effective aperture of the array,which contributes to a significant improvement of the estimation performance.Previously,the cross correlation matrix(CCM)has been used to estimate the 2-D DOA of an L-shaped sparse array,which is free from noise and able to automatically match the azimuth and elevation angles.Ref.[14]presented a computationally efficient algorithm exploiting the CCM of the sensor data and shows the effectiveness and validation of advantages in comparison to the PM method.The method proposed by Wu et al.[15]can be used to estimate pairwise DOA angles utilizing CCM on an L-shaped sparse array structure.In the meanwhile,the L-shaped sparse array structure[16]can achieve better performance owing to its higher degree of freedom(DOF)and larger array aperture.In terms of space and power budget,sparse arrays are of high value since they can reduce the cost,or achieve better performance for the same cost.Recently,the concept of compressive sensing(CS)has improved the performance and analysis of DOA estimation.Based on compressive sensing and least square optimization,Ref.[2]proposed an effective and efficient 2-D DOA estimation method by developing a modified 2-D off-grid model and bringing out a solution for the multisnapshot case in the separable observation model.With the popularity of deep learning,many researchers combine the DOA estimation with deep learning.Ref.[17]presented a novel deep ensemble learning method to carry out fast and accurate 2-D DOA estimation.Tian et al.[18]proposed a deep neural network(DNN)based covariance matrix completion method,which is able to expand a virtual uniform circular array(UCA)so that it can approach the performance of a complete UCA.

When more of those methods are applied to the underwater environment,the underwater acoustic velocity is usually assumed to be a known constant in the calculations.In more realistic scenarios,the acoustic velocity is affected by factors such as temperature and salinity.Its value varies constantly between 1450∼1550 m/s[19],which makes it more appropriate to be a variable instead of a known constant.Assuming the acoustic velocity as a known constant is equivalent to causing errors in the estimation of the acoustic velocity,which leads to serious errors in the underwater 2-D DOA estimation.To overcome the influence of the acoustic velocity,a velocityindependent and low complexity approach were proposed in Ref.[20].However,it is only applicable to one-dimensional DOA estimation.

In this paper,we propose an acoustic velocity-independent 2-D DOA estimation for underwater application.According to the geometric relationship between the linear arrays,the proposed method eliminates the effect of the acoustic velocity.The simulation results show that the proposed method is not affected by acoustic velocity.For a single source,the proposed method displays better performance than the conventional method in any case.For multiple sources,the proposed method presents a little performance degradation in comparison to the CEDE.Nevertheless,the estimation of the proposed method outperforms the CEDE at a very low signal to noise ratio(SNR)level.It is also superior to the CEDE when the acoustic velocity estimation deviates significantly from the real-time acoustic velocity.In addition,the computational complexity of the proposed method is lower than that of the CEDE for the same total number of sensors.At the same time,the performance is still acceptable.

The rest of the paper is organized as follows.The data model is described in section II.The proposed method is developed in section III.Section IV shows simulation results and section V makes conclusions.

The notations(·)∗,(·)T,(·)H,(·)−1,and(·)+stand for conjugate,transpose,conjugate transpose,inverse,and pseudoinverse.E{·}is the expectation operator.Diag{·}represents the diagonalization operator,and arg(·)indicates the phase angle of the complex number.The scalar is denoted byx,vector byx,and matrix byX.

Consider the two L-shaped uniform linear arrays(ULAs)in thex−zandy−zplanes using three ULAs placed on thex,yandzaxes,as shown in Fig.1.Thexandyaxes form an arbitrary cross angleδ,whereδ∈(0,π).Each ULA consists ofMelements,with a spacingdbetween the adjacent elements.The element placed at the origin is common for referencing purposes.

Fig.1 Two L-shaped arrays configuration with an arbitrary cross angle

Suppose that there areK(K

The baseband signal representation of thetth snapshot measured by the array along the three axes is expressed as

wheres(t)=[s1(t),s2(t),···,sK(t)]T.The vectorsnx(t),ny(t),nz(t)are the additive white Gaussian noise,with zero mean,independent of each other and the signal samples.Ax,Ay,Azare the manifold matrices represented by

The relationship between angles can be known based on the theory of solid geometry as shown in Fig.1.We have

A.Elevation and Azimuth Angles Estimation

where vectorx1(t)and matrixAx1are the firstM−1 rows ofx(t)andAx,accordinglyx2(t)andAx2are the lastM−1 rows ofx(t)andAx.The vectorsy1(t),y2(t),z2(t)and matricesAy1,Ay2,Az2are similarly defined.

The two sub-arrays of the manifold matrix of each axis have the following relationship:

Based on(6),the following cross-correlation matrices can be derived:

Algorithm 1 Acoustic velocity-independent 2-D DOA estimation Input:Received signals of sub-arrays of three axes{x1(t),x2(t),y1(t),y2(t),z2(t)}Lt=1.Output:The estimation of azimuth and elevation angles{(ˆθi,ˆϕi)}Ki=1.1:Construct the cross-correlation matrices Rzx,1,Rzx,2,Rzy,1,Rzy,2 by(8);2:Formulate matrices Rx and Ry via(9),and then perform SVD on Rx and Ry to get their left singular vector groups Uxs and Uys;3:Divide Uxs into matrices Uxs1 and Uxs2,Uys into matrices Uys1 and Uys2 as(15);4:Perform EVD on U+xs1Uxs2 and U+ys1Uys2 to obtain eigenvalues{λxi}Ki=1 and{λyi}Ki=1,invertible matrices Tx and Ty;5:Compute vectors ϕz1 and ϕz2 according to(20);6:Reconstruct manifold matricesˆBx andˆBy using(21)to get vectors ϕx and ϕy in(24),then rearrange ϕy into ϕ′y;7:Calculate DOA estimation{(ˆθi,ˆϕi)}Ki=1 via(26)and(27).

B.Performance Analysis

The Cramer-Rao bound(CRB)provides an unbeatable performance limit for any unbiased estimator and therefore can be used as a benchmark for evaluating the performance of a particular estimator.According to Ref.[21],the CRB on the two L-shaped arrays for a source from(θi,ϕi)is given by

whererm=[xm,ym,zm]Tis the position vector of sensormthat depends on the geometry of the two L-shaped arrays.

The computation complexity of the proposed method isO((M−1)2L+12(M−1)3+4(2M−3)K2+K3),while that of the CEDE isO(M2L+8MK2+4M(4M−K)K+8(M−1)K2).When employing the same amount of sensors,the proposed method has lower complexity than the CEDE.

In this section,we evaluate the performance of the proposed method and compare the results with its CRB and the CEDE method.All simulation results are obtained for the case of sources withf=15 kHz center frequency by means ofP=1000 Monte-Carlo trials withL=800 snapshots andfs=40 kHz sampling frequency.The spacingdbetween the adjacent elements in an array is 0.05 m.Define the estimation error of real-time acoustic velocity∆c=c−c0,where c is the value of real-time acoustic velocity andc0is the estimated value of acoustic velocity.The performance of the proposed method and the CEDE is evaluated in terms of root mean square error(RMSE).RMSE is defined as followed:

A.Effect of Cross Angle on Estimation Performance

Fig.2 plots the RMSE versus the variation of cross angleδin the range of 30◦to 150◦with 20◦step size for the proposed method.Suppose there are 1 to 2 sources impinging on the two L-shaped arrays composed ofM=19 elements per branch(totally 55)at∆c=0 m/s,SNR=−10 dB,0 dB and 10 dB,separately.The signal comes from(40◦,50◦)for one source.Signals come from(40◦,50◦),(50◦,45◦)for two sources.Asδincreases,the performance first gets better and then gets worse.In the case of one source,the performance is considered to be best whenδis about 90◦.In the case of multiple sources,the best performance comes up whenδis approximately 120◦.The pattern for a single source or multiple sources stays the same even if the SNR changes.

Fig.2 RMSE versus cross angle from DOA=(40◦,50◦)for one source;DOAs=(40◦,50◦),(50◦,45◦)for two sources,at∆c=0 m/s,SNR=−10 dB,0 dB and 10 dB,separately

The CEDE employs one ULA less than the proposed method.The following simulations are performed on the Lshaped array composed ofM=19 elements per branch(totally 37)for the CEDE,the two L-shaped arrays composed ofM=19 elements per branch(totally 55),and the two L-shaped arrays composed ofM=13 elements per branch(totally 37)both for the proposed method.

B.Comparison of Algorithms with Different DOA

Fig.3 exhibits the scatter results for the proposed method and the CEDE at∆c=0 m/s and∆c=10 m/s,respectively.Theδis set at 90◦.The SNR is set at 0 dB.There is one signal impinging on the arrays from DOA=(10◦,10◦),(10◦,50◦),(30◦,30◦),(30◦,70◦),(10◦,50◦),(50◦,50◦),(30◦,70◦),(70◦,70◦),separately.The results for different DOA are presented in the same figure.It can be inferred from Fig.3(a),Fig.3(c)and Fig.3(e)that the scatter of the proposed method is more concentrated than that of the CEDE,which means the proposed method owns better performance as compared to the CEDE,especially for utilizing the same number of sensors in a branch.When the acoustic velocity estimation is inaccurate,the performance of the CEDE drops significantly,while the proposed method still holds the same performance.

Fig.3 Scatter for the CEDE and the proposed method from DOA=(10◦,10◦),(10◦,50◦),(30◦,30◦),(30◦,70◦),(10◦,50◦),(50◦,50◦),(30◦,70◦),(70◦,70◦)at∆c=0 m/s and∆c=10 m/s,respectively:(a)CEDE M=19,∆c=0 m/s;(b)CEDE M=19,∆c=10 m/s;(c)Proposed M=13,∆c=0 m/s;(d)Proposed M=13,∆c=10 m/s;(e)Proposed M=19,∆c=0 m/s;(f)Proposed M=19,∆c=10 m/s

C.Comparison of Algorithms with Different SNR

Fig.4 shows the RMSE versus the variation of SNR in the range of−20 dB to 15 dB with 5 dB step size for the proposed method and the CEDE atδ=90◦,∆c=0 m/s and∆c=20 m/s,respectively.The signal comes from(30◦,50◦).From Fig.4(a),we can learn that without any estimation error of the real-time acoustic velocity,the performance of these two methods is comparable in the case of a single source.Obviously,the proposed one performs even better,whether using the same amount of elements per branch or in total.In Fig.4(b),it is clear that with a non-negligible estimation error of the real-time acoustic velocity,the gap between these two methods is getting larger as the SNR increases.

Fig.5 also depicts the RMSE versus the variation of SNR for the proposed method and the CEDE.Different from the signal source in Fig.4,there are 2 sources impinging on the arrays from DOAs=(40◦,60◦),(70◦,40◦)in Fig.5.The cross angleδis also switched from 90◦to 120◦for multiple sources.Other conditions remain unchanged.We observe from Fig.5(a)that in the case of multiple sources,the performance of the proposed method degrades a little in comparison with the CEDE.However,at a very low SNR level,the CEDE shows no better competence than the proposed method since we strive to avoid the introduction of noise terms in the proposed method.In the presence of a non-negligible estimation error of the real-time acoustic velocity,the performance of the proposed method is still superior to that of the CEDE significantly.

Fig.4 RMSE versus SNR from DOA=(30◦,50◦),at δ=90◦:(a)∆c=0 m/s;(b)∆c=20 m/s

Fig.5 RMSE versus SNR from DOAs=(30◦,50◦),(60◦,30◦),at δ=120◦:(a)∆c=0 m/s;(b)∆c=20 m/s

D.Comparison of Algorithms with Unknown Acoustic Velocity

Fig.6 illustrates the RMSE versus the estimation error of the real-time acoustic velocity,where DOA=(40◦,40◦),δ=90◦,SNR=0 dB and SNR=−20 dB respectively.The estimation error∆cvaries from−100 m/s to 100 m/s.As shown in Fig.6(a),for a single source,the proposed method exhibits better performance than the CEDE,especially when it comes to an inaccurate estimation of the real-time acoustic velocity.Fig.6(b)reveals that the proposed method still maintains the advantage even at the extremely low level of SNR.

Fig.6 RMSE versus estimation error of the real-time acoustic velocity from DOA=(40◦,40◦),at δ=90◦:(a)SNR=0 dB;(b)SNR=−20 dB

As shown in Fig.7,the RMSE is plotted against the estimation error of the real-time acoustic velocity,where DOAs=(30◦,50◦),(60◦,30◦),δ=120◦,SNR=0 dB and SNR=−20 dB respectively.It can be inferred from Fig.7(a)that for multiple sources,at a normal level of SNR,the proposed method outperforms the CEDE except in the situation of an ignorable∆c.A more non-negligible estimation error of the acoustic velocity makes a more clear difference in the performance of these two methods.From Fig.7(b),it is clear that at the extremely low level of SNR,whether the estimated value of the acoustic velocity is accurate or not,the proposed method displays excellent performance compared to the CEDE.In general,the proposed method is not susceptible to the acoustic velocity.E.Comparison of Algorithms on Computational Com

Fig.7 RMSE versus estimation error of the real-time acoustic velocity from DOAs=(30◦,50◦),(60◦,30◦),at δ=120◦:(a)SNR=0 dB;(b)SNR=−20 dB

plexity

Now,we compare the computational complexity of the proposed method with two L-shaped arrays and the CEDE with a single L-shaped array.The computational complexity is mainly related to the number of snapshotsL,the number of sourcesKand the number of sensorsMon a single ULA.

Tab.1 displays the runtime versus the number of sensorsMeach branch of the CEDE for these two methods from DOA=(40◦,40◦),at SNR=0 dB,δ=90◦,∆c=0 m/s.Tab.2 exhibits the runtime versus the number of sourcesKfor the two methods,where SNR=0 dB,δ=120◦,∆c=0 m/s.The signal comes from(50◦,50◦)for one source.Signals come from(40◦,60◦),(70◦,40◦)for two sources;(40◦,60◦),(60◦,45◦),(80◦,35◦)for three sources;(40◦,60◦),(50◦,50◦),(60◦,45◦),(80◦,35◦)for four sources.As we have seen in Tab.1,using the same number of total elements for these two methods,the runtime of the proposed method is less than the other.Using the same number of elements per branch,the proposed method requires more runtime than the CEDE.In general,the runtime of the two methods is on the same order of magnitude.

Tab.1 Runtime(s)versus M

Tab.2 Runtime(s)versus K

Tab.3 presents the runtime versus the number of snapshots for the two methods from DOAs=(40◦,60◦),(70◦,40◦),at SNR=0 dB,δ=120◦,∆c=0 m/s.The number of snapshotsLvaries from 200 to 800.As we can see,the runtime increases slowly with a certain fluctuation as the number of snapshots increases,which means the computational complexity of the two methods is less affected by the number of snapshots.

Tab.3 Runtime(s)versus L

In this paper,an acoustic velocity-independent 2-D DOA estimation for underwater application is proposed.The proposed method utilizes the cross-correlation matrix of the data received by three crossed linear arrays to eliminate the acoustic velocity factor.Simulation results verified that the proposed method is not susceptible to the acoustic velocity.For a single source,the proposed method outperforms the conventional method under any circumstances.For multiple sources,there is a little performance degradation for the proposed method in comparison to the conventional method.However,the proposed method displays better performance than the conventional method at the extremely low level of SNR or in an inaccurate estimation of the acoustic velocity.Besides,using the same amount of sensors in total,the computational complexity of the proposed method is lower than the conventional method,while the performance is still acceptable.