Two-dimensional DOA estimation based on thin array towed by a small autonomous platform

JIANG Jiajia，YANG Guoliang，LI Chunyue，LI Yao，WANG Xianquan， SUN Zhongbo，DUAN Fajie，FU Xiao

(1. State Key Lab of Precision Measuring Technology and Instruments，Tianjin University，Tianjin 300072，China； 2. System Engineering Research Institute，China State Shipbuilding Corporation,Beijing 100036，China)

Abstract：The small autonomous platform with a thin line array is an important tool for underwater acoustic mobile surveillance.Generally,only one-dimensional (1-D)direction-of-arrival (DOA)of the source signal can be estimated using a thin towed line array.In this work,the two-dimensional (2-D)DOA estimation is achieved by the thin line array towed by a small autonomous platform due to its flexible maneuver.Two perpendicular tow paths are formed through the fast turning of this array.An L-shaped array is formed by the same towed array on these two tow paths at different times.Using the array on these two straight paths,two 1-D DOAs of the source signal are obtained respectively,and then the 2-D DOA based on the formed L-shaped array can be estimated.The effectiveness of proposed approach is verified by numerical simulations and its theoretical error is analyzed.

Key words：towed line array；autonomous platform；direction-of-arrival (DOA)；underwater signal；passive sonar

0 Introduction

The passive sonars such as the fixed sonar system,sonobuoy,flank sonar and towed array are widely used in underwater acoustic surveillance[1].For the surveillance in a fixed sea area,the fixed sonar system has high direction-of-arrival (DOA)estimation accuracy because the element position of its sonar array can be accurately calibrated.Sonobuoys are consumables that can be deployed by helicopters or airplanes,which are more flexible and cost-effective than fixed sonar systems.However,it is a more common requirement to realize low-cost and flexible surveillance in mobile areas,for instance,vast ocean patrol,the submarine detection around a voyaging ship and marine animal survey[2-4].

For mobile surveillance,the commonly used tools are towed arrays and flank sonars.The flank sonar is installed on the surface of a ship or submarine.Its acoustic aperture is limited by the size of its carriers.The traditional towed array is a single neutrally buoyant line array towed behind a surface ship or submarine.Unlike the flank sonar,the hydrophone array of towed array which is towed by a long tow cable is far away from its towing platform,noise and vibration of the platform hardly have any effect on its acoustic performance.Additionally,the acoustic aperture of hydrophone array is usually made quite large for a good DOA performance,so it is widely used in military applications[5-6].However,the long tow cable and large acoustic aperture of hydrophone array make the traditional array long and heavy.Therefore,the traditional towed array requires immense resources for its deployment and recovery,which results in high operational costs and limits the speed and turn rate of towing platform[7].In order to realize mobile surveillance flexibly and at low cost,thin line arrays have been developed for small autonomous platforms such as the autonomous underwater vehicle (AUV)and the unmanned surface vehicle (USV)in recent years[8-9].Based on this,the excellent coherent beamforming results of the thin array towed by AUV are verified,and the AUV with thin towed array are used as sensing platforms in antisubmarine warfare (ASW)[10-11].Similarly,the USV with thin array is used to detect and track submarine targets and monitor marine mammals[12].

The autonomous platform with a thin line array plays an important role in underwater acoustic surveillance applications due to its low cost,easy deployment and high flexibility.However,only one-dimensional (1-D)DOA estimations of the source signal can be obtained based on a single towed line array,while the two-dimensional (2-D)DOA information is needed to find the position of a source signal in the three-dimensional (3-D)physical space.Generally speaking,planar arrays are used to estimate the 2-D DOA,such as the uniform circular array(UCA),the parallel array and the L-shaped array.By rotating two elements and setting a fixed time delay,a virtual UCA is formed in Ref.[13] to estimate the 2-D DOA.And a circular synthetic array is also formed by two rotating sensors in Ref.[14].In addition,a parallel array is formed by a 1-D array moving vertically along its axis and the 2-D DOA estimation is obtained in Ref.[15].Obviously,the above approaches are not suitable for the thin array towed by a small autonomous platform because the array can only move along its axis.

In order to improve the performance of underwater target positioning using the autonomous platform with a thin array,an approach is proposed to estimate the 2-D DOA of the source signal in this work based on the characteristic that the towed array can only move along it axis.That is,an L-shaped array is formed through a 90-degree turn of the towed array to realize 2-D DOA estimation.

This approach does not require a planar array,nor does it need to rotate and translate the 1-D array.It only uses the maneuver of the thin array towed by a small autonomous platform,which is simpler and more implementable.

1 Basic assumptions and data model

Assuming that a thinM-element uniform linear array (ULA)is towed by an AUV or USV with a constant velocity in a calm water environment,this ULA is in a horizontal and straight state when working.A far-field narrowband source signal impinges on this ULA which is assumed to be on a straight tow path.After the signal is sampled,the ULA need to have a turn to the heading which is perpendicular to the original one.The signal is sampled again after the array is restored to the linear shape.The time interval between these two samples is short due to the small size of the autonomous platform and the thin array,and the source signal is assumed to be static during this period of time.When we establish a 3-D coordinate system based on this pair of vertical straight arrays,an L-shaped array is formed by a subarrayXonx-axis and a subarrayYony-axis,as illustrated in Fig.1.

Fig.1 L-shaped array formed by a single towed line array

The source signal impinges on the L-shaped array from the direction at 2-D angles (θ,φ),whereθandφdenote the elevation and azimuth,respectively.Due to the short time and small space required to form an L-shaped array using the thin array towed by an autonomous platform,the elevation and azimuth are assumed to keep constant during the process of forming the L-shaped array.And in a short time and small space,the changes in marine environment parameters are assumed to be minor,which is not enough to affect the work of the subarrayXand subarrayY.

The DOA of this signal can also be represented by the direction cosine alongx-axis andy-axis in this established coordinate system.Obviously,

cosα=sinθcosφ,

(1)

cosβ=sinθsinφ.

(2)

Due to the different headings of the towed array on these two tow paths,the same signal observed by subarrayXand subarrayYhave different Doppler shifts.Taking the element close to the origin as the reference,the observed vectors at subarrayXandYare given by

x(tx)=As(tx)+nx(tx),

(3)

y(ty)=Bs′(ty)+ny(ty),

(4)

2 2-D DOA estimation

As shown in Fig.1,αandβare the 1-D DOAs when the source signal impinges on subarrayXandY.Therefore,they can be estimated respectively using subarrayXandYby the 1-D DOA estimation algorithms such as estimation of signal parameters via rotational invariance techniques (ESPRIT)[16]and multiple signal classification (MUSIC)[17].Then,according to Eqs.(1)and (2),the 2-D DOA estimation of the source signal can be calculated.

Considering the computational efficiency,the ESPRIT algorithm is used to estimate the 1-D DOA.First,we estimatethe 1-D DOA based on subarrayX.Dividing the subarrayXinto two subarrays,according to Eq.(3),their output can be written as

x1(tx)=A′s(tx)+n1,

(5)

(6)

(7)

(8)

whereRSis the signal covariance matrix,I2(M-1)is a 2(M-1)×1 order identical matrix.H denotes the complex conjugate transpose.According to the eigenvalue ofRx,the signal subspace can be obtained and be divided into

(9)

whereE1andE2are (M-1)×1 vectors,andTis a unique full rank matrix.Therefore,

E2=E1T-1ΦxT=E1Ψx,

(10)

whereΨx=T-1ΦxT.Since a perfect measurement ofRxcannot be obtained,the sample covariance is defined by

(11)

whereLis the number of snapshots.According to the total least squares (TLS)criterion[16],we can get

(12)

(13)

Then,the direction cosine alongx-axis in the established coordinate system can be calculated by

(14)

Finally,according to Eqs.(1)and (2),the elevation and azimuth angles of source signal can be calculated by

(15)

(16)

3 Numerical simulation

For the thin array towed by a small autonomous platform,there is currently no reported effective ways of estimating 2-D DOA of underwater signals.Therefore,there is no comparison between proposed approach and other methods in the numerical simulation.

Firstly,the performance of the proposed approach in terms of the SNR is examined when the number of snapshots is fixed at 500.Fig.2 shows the root mean squared error (RMSE)of the 2-D DOA estimation of the source signal according to the SNR.The RMSE is defined as

Fig.2 RMSEs of elevation and azimuth angle estimates versus SNR

(17)

As shown in Fig.2,when the SNR increases from 0 dB to 50 dB,the RMSEs of 2-D DOA estimation of the source signal gradually decreases,and the performance curves become stabilized as the SNR increases.

Next,the performance of the proposed approach in terms of number of snapshots are examined when the SNR is fixed at 15 dB.

As shown in Fig.3,when the number of snapshots increases from 100 to 2 500,the RMSEs of 2-D DOA estimation of the source signal also gradually decreases.Thus in Fig.2,the curves also become stabilized as the number of snapshots increases.

In Figs.2 and 3,as SNR and number of snapshots increase,the RMSEs of the 2-D DOA estimation of the source signal decrease and stabilize gradually.So the effectiveness of the proposed method is verified.

4 Performance analysis

4.1 Heading estimation error

For the proposed approach,the 3-D coordinate system to define the 2-D DOA of source signal is established based on the headingof towed array which is estimated by the compass or other heading sensors.When these two 1-D DOAs are estimated based on the towed array which is not on thex-axis ory-axis,the 2-D DOAs are calculated with error even the 1-D DOAs are estimated correctly.The heading estimation error makes the towed array deviate from the coordinate axis,causing 2-D DOA estimation error.

Fig.4 Heading estimation error

According to Eqs.(3)and (4),the observed vectors of these two subarrays of nominally L-shaped array are given by

(18)

(19)

When only the heading estimation error is considered,the errors of the direction cosine of the source signal in the established 3-D coordinate system are given by

(20)

(21)

In order to illustrate the relationship between direction cosine estimations and heading estimation errors,the simulations are performed.The RMSE is defined according to Eq.(17).Other simulation parameters are the same as those in the first example except that there are Δhxand Δhy.In order to estimate the 1-D DOA more accurate,no noise is added.The other simulations in this section are also carried out under the same conditions as above.

As shown in Fig.5,when the towed array deviates from thex-axis by 150° to the right and 30° to the left,the heading of towed array is parallel to the projection of the signal incident direction.Therefore,the direction cosine estimation error reaches the extremum.When the array deviates from thex-axis by 60° to the left,the error is the same as that when it moves along thex-axis.

Fig.5 RMSEs of the estimates of direction cosine along x-axis versus heading estimation error

As shown in Fig.6,when the towed array deviates from they-axis by 60° to the right and 120° to the left,the heading of towed array is also parallel to the projection.Therefore,the direction cosine estimation error also reaches extremums.And when the array moves in the direction of 0° and 60°,the error is the same and smallest.

Fig.6 RMSEs of the estimates of direction cosine along y-axis versus heading estimation error

There is no doubt that the direction cosine estimation error is related to the incident angle of the source.Generally speaking,the heading error is varied in a small range.In this range,as the absolute value of the heading error increases,the direction cosine error also increases.

(22)

(23)

{Δhy=Δhx+Kπ} or

(24)

Fig.7 RMSEs of elevation estimates versus heading estimation errors

(25)

Fig.8 RMSEs of azimuth estimates versus heading estimation errors

As shown in Figs.7 and 8,even the heading error is varied in a small rang,from -2 to 2 degrees,its impact on the 2-D DOA estimation is already significant.Therefore,the proposed approach needs to use a higher accuracy heading sensor to ensure better 2-D DOA estimation performance.

4.2 Array shape perturbation

Unlike the fixed arrays,the nominally linear geometry of towed array may be distorted by the varying speed and transverse motion of towing vessel,by the hydrodynamic effects plus oceanic swells and currents[18].For the ESPRIT algorithm,the perturbation of the array shape destroys the rotation invariance between the subarrays and causes the performance of 1-D DOA estimation to decrease.Therefore,the 2-D DOA estimation performance of the approach proposed based on two 1-D DOA estimates also decrease.

Deviations of towed array from the linear shape are also dependent on its construction parameters like length,thickness and number of mechanical section.Generally speaking,for planned tow-ship maneuvers,the shape of array is modeled using a bow[19].And for unplanned variations in the tow-ship trajectory,the array shape is modeled using undamped and damped sinusoidal[20].Due to the short length of the thin towed array used in this work,the array shape is modeled using undamped sinusoidal to investigate the effect of array shape perturbation on 2-D DOA estimation.

The linear thin array is simulated to contain a forward vibration isolation module (VIM)with length of 12d.The first hydrophone is at an length ofdfrom the point where the VIM connects to the hydrophone array.So the total array length is 20d.When there is array shape perturbation,the equation for the array shape can be written as

y=asin(qx),

(26)

whereaandqare the parameters to determine the sinusoidal.Assuming that the total length of the towed array keeps constant,the position of each hydrophone can be determined by the arc-length integral.In proposed approach,two direction cosines are estimated respectively.Assuming that the array deformation is the same in these two processes,seven patterns of array are considered and the array shape is shown in Fig.9.

Fig.9 Array shapes

As Fig.9 shows,the array shapes are named “shape 1”-“shape 7”,whose hydrophone positions are calculated and marked.“shape 1”is a linear array without array perturbation whose hydrophone position is treat as balance position.“shape 2”-“shape 4”are sinusoidal curves whose parameterqare smaller.And their hydrophones are distributed on both sides of balance position.“shape 5”-“shape 7”are also sinusoidal curves whose parameterqare larger.Their hydrophones are distributed on one side of balance position.

Then,the simulations are performed to exam the effect of array shape perturbation on direction cosine estimation and 2-D DOA estimation performance.The other simulation parameters are the same as those in the first example.

As shown in Figs.10 and 11,by comparing the RMSEs of “shape 2”and “shape 4”,“shape 5”and “shape 7”,it is clear that the error of the direction cosine estimation grows with the increase of the value of parametera.And by comparing “shape 2”and “shape 3”,“shape 5”and “shape 6”,we get that the error of the direction cosine estimation decreases as the parameterqincreases.

Fig.10 RMSEs of the estimates of direction cosine along x-axis versus the SNR

Fig.11 RMSEs of the estimates of direction cosine along y-axis versus the SNR

As shown in Fig.12,by comparing the RMSEs of “shape 2”and “shape 4”,“shape 5”and “shape 7”,the RMSE of azimuth angle estimation also grows with the increase of the value of parametera.And by comparing “shape 2”and “shape 3”,“shape 5”and “shape 6”,the RMSE of azimuth angle estimation also decreases as the parameterqincreases

Fig.12 RMSEs of azimuth estimates versus the SNR

However,for the elevation estimation RMSE in Fig.13,the performance of sinusoidal array is similar to that of the linear array whenqis larger.This is because the array elements are distributed on one side of balance position.The array shape is approximately straight.And the towed array with the same perturbation forms the L-shaped array.The headings of the array with largerqare approximately perpendicular.According to Eq.(24),the elevation estimation error is small.

Fig.13 RMSEs of elevation estimates versus the SNR

As Figs.12 and 13 depict,the effect of the array shape perturbation on the 2-D DOA estimation is significant when the hydrophones of towed array are distributed on both sides of balance position.Therefore,the towed array shape should be calibrated before the proposed approach is applied.

5 Conclusions

In this work,we present a 2-D DOA estimation approach to improve the performance of underwater target positioning using the autonomous platform with a thin array.This approach only uses the maneuver of the thin array,which is simpler and more implementable.The effectiveness of this approach is verified through numerical simulations.Through our work,the 2-D DOA estimation performance of the proposed approach is analyzed from heading estimation errors and array shape perturbation.It is clear that the influence of heading error and array shape perturbation on 2-D DOA estimation performance is significant.High-precision heading sensor and array shape calibration are the basis of high 2-D DOA estimation performance of proposed approach.