Error control and adjustment method for underwater wireless sensor network localization

Applied Acoustics 130 (2018) 293–299

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Error control and adjustment method for underwater wireless sensor network localization Yunfeng Han, Jucheng Zhang, Dajun Sun ⇑ College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 25 May 2017 Accepted 5 August 2017

Keywords: Underwater wireless sensor network Data post-processing Consistent precision

a b s t r a c t Underwater sensor position information is very important in Underwater Wireless Sensor Networks (UWSNs). The sensor collected information is only meaningful with the correct positions. While current method only processes underwater nodes individually, which causes the localization precision for each nodes varying severely. This problem is very common in both centralized network and distributed network. The varying localization precision causes real trouble especially in data fusion, aided navigation, and signal processing. To solve this problem and achieve a consistent localization precision in the network, we propose an error control and adjustment method to make all the sensor localization precision refined. We use the distance information between nodes to establish a network error adjustment model, which can improve the localization precision and make the network localization error consistent. The simulation and actual experiment proves that this method can make all the node positioning more precise, the localization precision can be lower than decimetres. This method can be widely used in UWSNs and show a favourable potential in ocean engineering applications. Ó 2017 Published by Elsevier Ltd.

1. Introduction Underwater wireless sensor networks (UWSNs) attract increasing interests in ocean exploration and underwater applications, such as environmental monitoring, offshore exploration, disaster prevention, and military surveillance [5,22,8,17,1,21]. Underwater sensors equipped with wireless communication systems provide a cost-effective means to meet stringent requirements in those applications, and to measure infrastructure parameters, collect environmental data, and transmit information in real time between each nodes or to base stations. In many UWSN applications, it is desirable to have the location information of each sensor node, but accurate node localization is one of the most challenging tasks for UWSNs due to several challenges [10,7]. First, underwater nodes are often unstable and they can drift/float around even if they are physically anchored. Second, Global Positioning System (GPS) signals are severely attenuated underwater, therefore it is expensive to equip sensor nodes at deep water with GPS receivers[3,9,20]. Furthermore, It is also difficult to

⇑ Corresponding author at: College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China. E-mail addresses: [email protected] (Y. Han), zhangjucheng2006@163. com (J. Zhang), [email protected] (D. Sun). https://doi.org/10.1016/j.apacoust.2017.08.007 0003-682X/Ó 2017 Published by Elsevier Ltd.

provide accurate timing/clock reference for low-cost sensor nodes because atomic clocks are expensive and high-quality ovenized crystal oscillators are power hungry. Apart from a few high-end reference nodes whose locations may be measured with high accuracy and precision with high-end instrumentation [14,13], ordinary nodes have to rely on communication signals in the network to localize themselves [17,1,8,4]. Thus make the reference node localization precision different with ordinary nodes. The ordinary node localization precision also differs with each other because of using different reference node information. In UWSN, especially large-scale UWSNs, the nodes cannot be localized by one-hop because of the limitation of communication distance. In this situation, the recursive localization methods are proposed. The method divides the nodes into two kinds, reference nodes and ordinary nodes. The reference nodes are moored underwater by a fixed structure, which can be localized by surface to achieve a high accuracy. The ordinary nodes use the reference nodes as known positions, measuring the distances to references and achieving positions. After being localized, the ordinary nodes can upgrade to reference nodes to help other unlocalized nodes positioning. In this process, the underwater nodes will result in different localization precision because of using different reference nodes. The inconsistent localization precision in UWSN will bring huge trouble in data fusion and post-process.

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The information collected by the sensor should be used for fusion with the position data after underwater node data information obtained [11]. No matter centralized systems or distributed systems, the underwater positions with Geodetic coordinates can only be obtained by surface GPS and the measured distance to GPS. This process can deliver GPS information to underwater. While the Geodetic coordinates are always uncertain due to the time-varying or space varying changes in sound speed, flows, tides, etc. Because of the limited underwater characteristic, such as long propagation delay, severe multi-path inference and limited signal bandwidth, the underwater node can be only localized individually to achieve high precision. In this way, the large-scale network with multiple nodes will be result in different precision because of time or spatial varying. To solve the problem that the sensor nodes in UWSN have inconsistent localization error, we propose an error control and adjustment method. This method using the distance information between nodes to establish the math model, which uses the time-delay measurements combining with actual sound speed profile. After processing, the localization precision of each node can be improved with a same level network localization precision. This paper is organized as follows. Section 1 gives the background of UWSN and the purpose of data post-processing. Section 2 gives the related works of UWSN localization method. Section 3 gives the mathematical models of the proposed data postprocessing method and simulation analysis are listed in Section 4. Section 5 gives the experiment results to prove the method and conclusion is listed in Section 6. 2. Related works A recursive position estimation algorithm is designed for large scale RF WSN [2]. This algorithm can estimate their locations from a small set of anchor nodes using only local information. System coverage increases iteratively with newly estimated positions join the anchor nodes. Large-scale localization (LSL) extends it to UWSN and modifies the algorithm to hierarchical approach that allows multi hop links in distance measurement [22]. They integrate the 3-dimensional Euclidean distance estimation method with the recursive location method for distributed scheme. The results achieve high localization coverage with relatively small localization error and communication cost. But this scheme has the problem of the convergence property tightly linked with the message limit. Beyond the value of critical message, the communication costs increase significantly without any corresponding improvement in localization coverage. Underwater positioning system (UPS) uses TDoA techniques by multiple beacon intervals which has the advantage of not requiring time synchronization [6]. Ordinary nodes listen to the broadcasts of anchor nodes to localize itself. They use Saleh-Valenzuela model for underwater acoustic channel, the results perform low localization error and utilize very few anchor nodes while this scheme has limitations in harsh and dynamic underwater acoustic channels. Wide coverage positioning system (WPS) [18], GPS-less localization protocol (GPS-less) [16], motion-aware sensor localization (MASL) [15], underwater sparse positioning (USP) [19] and 3D underwater localization (3DUL) [12] can be used for different application environments with ToA techniques. They analyse the measurement errors due to sound speed variation, NLOS signals, time non-synchronization and node mobility with the assumption that the direct path signal has the strongest signal, which is not always the case in reality. 3. Mathematical models With the positions of sensor nodes obtained previously being the initial values, the UWSN can achieve optimization of the global

error after the accurate range measurement information is obtained. While the direct observation is time-delay measurement (measured by time of arrival between nodes), we use the actual sound speed profile and ray-tracing model to calculate the accurate distance. We establish the error equation based on distance measurement information. We assume that there are N nodes in UWSN, and the number of range measurements within the communication distance range is L. Based on the calculated distance, the distance error equation established is as follows:

v ij ¼ 

Dyij Dyij Dxij Dxij b b b b xi  yi þ xj þ y j  Dij rij r ij r ij r ij

ð1Þ

where ðb xi ; b y i Þ and ðb xj; b y j Þ are the corrected coordinate values of the initial node positions ðxi ; yi Þ and ðxj ; yj Þrespectively. rij is the measured distance between two nodes; lij is the calculated distance between two nodes using initial positions; Dxij ¼ xi  xj and Dyij ¼ yi  yj are the coordinate difference between the two nodes in x direction and y direction; Dij ¼ r ij  lij is the difference between the measured value and the calculated value. Therefore, L error equations can be established for L range measurements.

bD V ¼ Bx

ð2Þ

where V is the coefficient of the error equation (vector dimension is L  1); B is the coefficient matrix constituted by various corrections b is the vector constituted by various (matrix dimension is L  2N); x coordinate corrections (vector dimension is 2N  1); D is the vector constituted by the calculated value and the measurement differences (dimension is L  1). We assume that the range measurement error is a random error of the same precision conforming to the Gaussian distribution. The position optimization is solved under the VT V ¼ min criterion. As L, the number of the range measurements obtained during calculation, is determined by the node density, there are two cases: (1) The number of equations is larger than the number of unknown numbers (L > 2N). The corrected coordinate value is 1

b ¼ ðBT BÞ BT D x

ð3Þ

The coordinate of each node after correction is

b ¼ x0 þ x b X

ð4Þ

(2) The number of equations is smaller than the number of unknown numbers (L < 2N). We obtain a problem of underdetermined equation solving with the number of unknown numbers larger than the number of equations. For this purpose, constraint conditions should be added to the model to constitute the solution conditions. We use a weighted benchmark constraint condition added to the model:

ST Px b x¼0

ð5Þ

It is assumed that the following relationship is true:

(

T

rank½B ST  ¼ rankðBÞ BS ¼ 0

ð6Þ

where ST is a non-singular matrix. Meanwhile, various equations are not correlated. According to the principle of least squares, the function is

u ¼ VT V þ 2KT ðST Px bx Þ ¼ min

ð7Þ

The normal equation is:

(

Nb x þ Px SK ¼ D ST Px b x¼0

ð8Þ

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Thus,

where N ¼ ðBT BÞ . T

We multiply S to Eq. (8). Meanwhile, BS ¼ 0 is true. We can obtain:

ST Px SK ¼ 0

ð9Þ T

As the quadratic form S Px S cannot be zero, only when K ¼ 0 it can meet:

u ¼ VT V ¼ min

ð10Þ

ST Px b x þ SDÞ ¼ 0 x P ¼ ST Px ðb

ð22Þ

Therefore: 1

D ¼ ðST Px SÞ ST Px b x

ð23Þ

Eq. (23) is the conversion formula for the transformation of b x into b xP . Eq. (23) is substituted into Eq. (21) to obtain:

After multiplication and in consideration of the K ¼ 0, the second equation of Eq. (8) and the first equation are added to obtain:

b x x P ¼ ðE  SðST Px SÞ ST Px Þb

ðN þ Px SST Px Þb x¼D

where E is the unit matrix. Given the transformation matrix is:

ð11Þ 1

ð12Þ

the parameter estimation is:

b xP ¼ Q P D

1

Q xp xp ¼ Q P NQ P

ð13Þ

ð14Þ

The estimated value of the variance of unit weight is:

VT V r

ð15Þ

For the ordinary node P, the localization precision is:

r2P ¼ rb 20 ðQ Pxx þ Q Pyy Þ

ð16Þ

where Q Pxx and Q Pyy are the diagonal coefficients of the coordinated factor matrix Q xP xP . It can be seen from the above calculation process that reference conditions should be added to the calculation model. Addition of any criterion meeting Eq. (5) to the above underdetermined equation can meet the least square criteria VT V ¼ min. All of the parameter estimations obtained under different criteria are least square solutions and meet the normal equation Nb x . The corrections V does not change due to different criteria selected. Thus, different criteria can be transformed with the similarity transformation method. It is assumed that the two solutions for the ordinary nodes under different criteria are b x 1 and b x 2 . The following equations are true:

Nb x 1 ¼ D; N b x2 ¼ D

ð17Þ

The two equations are subtracted to obtain:

Nðb x2  b x1 Þ ¼ 0

ð18Þ

Given NS ¼ 0, Eq. (18) has infinitely many sets of solutions. Thus:

b x 1 þ SD x2 ¼ b

ð19Þ

where D is a conversion factor in different solutions. When the constraint condition of the minimum twox P ¼ min, any dimensional criterion serves as the criterion b x TP Px b least square solution b x can be determined by similarity transformation to b x P . It is assumed that the coordinated factor matrix of any solution b x is Q bxbx . According to Eq. (19), we can obtain:

b x þ SD xp ¼ b

ð20Þ

where D is the conversion vector in the process where b x transforms into b x P . D should meet:

@b x TP Px b xP ¼ 2b x TP Px S ¼ 0 @D

ð25Þ

The coordinated factor matrix of b x P is:

Q bx bx ¼ HP Q bxbx HTP

ð26Þ

P P

According to the error propagation law, the coordinated factor of b xP is:

rb 20 ¼

ð24Þ

HP ¼ E  SðST Px SÞ ST Px

On the premise of

Q P ¼ ðN þ Px SST Px Þ

1

ð21Þ

Eqs. (24)–(26) are the criterion transformation formula. To guarantee the consistency between the processing and the distributed positioning result, the initial positioning results of various nodes should be sufficiently considered, that constraining its position correction to be the minimum in the new calculation model. The minimum two-dimensional norm criterion in the above calculation process is the optimal criterion meeting the restraint condition. Under the criterion, the corrected model is a set of solutions making the quadratic sum of the position adjustment amount of the ordinary nodes smallest. Meanwhile, to guarantee the mutual independence between the reference conditions and the coefficient matrix of the above error equation, we have

2

1 6 4 0 y1

 

3

0

1

0

1

0

1

7  5

x1

y2

x2

 

ð27Þ

Eq. (27) is substituted into Eqs. (12)–(14) to obtain the correction results of the ordinary nodes meeting the minimum twodimensional norm criterion. It can be seen from Eq. (27) that the addition of the criteria makes the sum of the x direction corrections of the ordinary nodes and the sum of the y direction corrections zero. Meanwhile, the relative errors of the x direction and the y direction are the same, that dx ¼ dy . X Y It can be seen from the above calculation process that the process applies to the Geodetic Coordinates of the underwater sensor nodes and relative coordinates of the underwater sensor nodes. The major difference between the two cases lies in the setting of the correction of reference nodes. With a focus on the accurate relative position, all nodes are considered as the ordinary nodes, then the error equation is as shown by Eq. (1). With a focus on the accurate typing coordinates, the correction value of the coordinates corresponding to the reference node is 0. 4. Simulation analysis To verify the performance of the above method, we make simulations to compare the post processing results and the raw localization results. The simulation conditions are as follows: (1) 200 nodes are randomly deplyed in the area 100 m  100 m  100 m (Fig. 1 gives the deployed sketch). (2) 20% nodes are reference nodes and the others are ordinary nodes. (3) The node density is 6–16 with the communication distance of 21–31 m (the relationship between node density and communication distance are shown in Fig. 2).

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Fig. 1. The sketch of deployed nodes.

(4) The nodes in the network have the ability to measure distance, the distance measurement error obeys Gaussian Distribution with the true value as mean and 1% distance as standard deviation. The positions of all data collected and ordinary nodes are used as the initial values after the positions of nodes are obtained. The average positioning error of the network is shown in Fig. 3. As shown in Fig. 3, the average positioning error of the network decreases substantially after the above mathematical model is used for post-processing of the data. The error accumulation in

Fig. 2. The relationship between node density and communication distance.

Fig. 3. The compare between initial localization results and post processing results.

the last several steps in the distributed step-by-step positioning method is equally distributed onto the nodes of initial positioning after optimization of the range measurement. Therefore, it can effectively inhibit error accumulation and obtain the global optimization result. 5. Processing of experimental data To test the positioning performance of the underwater node positioning method we propose, an experiment was conducted in the Songhua Lake in October, 2012 with experimental site as shown in Fig. 4. The whole experiment involves arrangement of the underwater nodes, data collection, determination of geological locations, and recovery of nodes. The underwater sensor network nodes used in the experiment are researched and developed by Harbin Engineering University (HEU), China. The underwater sensor nodes carry the pressure sensors, tilt sensors, etc. They have multiple functions such as release, response, distance measurement, and communication. It is capable of meeting the experimental requirements to localize the sensor network nodes. The experiment system mainly includes the water surface processing unit, shipborne transmit-receive transducer, 5 underwater sensor nodes, and differential GPS, etc. The underwater nodes are located at the bottom of the lake by mooring and floating body. They are connected with the release mechanism through a length of flexible cord 5 m away from the bottom of the lake (as shown in Fig. 5). The underwater nodes are provided with high-precision pressure sensors (the measurement precision of the pressure is 0.01% F.S.). The surface transmit-receive transducer is installed onto a small pleasure-boat of the experimental platform. It is installed downward with a mechanical rod. The vertical directionality can reduce the vibration noise of the boat and the water surface reflection sound. It is fixed onto the starboard away from the engine and propeller of the boat. Such isolation materials as rubber, plastic, etc. to reduce vibration during measurement. The transmitreceive transducer is 3 m below the water. The model Leica 1200 GPS moving station antennas are vertically installed above the transducer. It is set to a RTK mode. Its localization precision is 2– 3 cm under a low-speed motion mode. In the experiment, five nodes are arranged underwater as shown in Fig. 6. The solid circles represent the reference nodes and the hollow circles represent the ordinary nodes to be positioned. The relative distance between every two underwater nodes within the communication distance range is measured. The communication distance is set to 120 m (the corresponding time delay

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Fig. 4. The experiment cite of Songhua Lake.

Fig. 5. The moored underwater node.

Fig. 7. The sound speed profile.

is 80 ms). The phase-coded signal is transmitted in a fixed cycle. The distance between every two nodes is estimated by measuring the time delay difference and acoustic velocity value. The estimated precision of time delay of individual nodes can be better than 10 us after the broadband signal processing technique used. In the experiment, Node 1 and Node 2 serve as the reference nodes. 8 ms broadband coding signal is transmitted with the surface transmit-receive transducer. The interrogating signal is transmitted in a fixed cycle. The node positions are calculated with the

GPS information. The measured acoustic velocity profile is shown in Fig. 7. The positions of the underwater sensor nodes can be accurately obtained using the acoustic ray technique. The depths of the reference nodes can be measured with the high-precision pressure sensor. The positions of the reference nodes are listed in Table 1. A total of 50 measurements were performed in the experiment. A positioning result could be obtained in each group. Various nodes uploaded their respective positioning results to the surface following distributed step-by-step positioning. To calculate the average positioning error of the network, the other three ordinary nodes were positioned respectively by obtaining the positions of the reference nodes. The positioning results and depths of the three ordinary nodes were shown in Table 2. They were used as the true values of the underwater sensor nodes in the network. Based on the 50 measurement results, the RMSE error of the three nodes was calculated. The calculation results were listed in Table 2. As seen in Table 2, the RMSE errors of the 3 ordinary nodes excluding ordinary node 3 were within 1 m, indicating that the above distributed step-by-step positioning method could achieve a rather high localization precision. The above 5 sensor nodes were recovered and the data were verified. Fig. 8 presented the measured time delay values between Ordinary node 2 and Ordinary node 3. As the nodes of various sensors are fixed underwater through a mooring structure, the relative distance between two

Table 1 Reference node positions.

Fig. 6. The deployed underwater node.

Reference node 1 Reference node 2

East (m)

North (m)

Depth (m)

315690.36 315697.82

4841955.07 4841835.04

61.73 58.50

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Table 2 Ordinary node localization results.

Ordinary node 3 Ordinary node 4 Ordinary node 5

East (m)

North (m)

Depth (m)

RMSE (m)

315792.54 315786.94 315736.50

4841865.09 4841969.76 4841896.65

60.12 60.12 60.40

10.83 0.64 0.17

Fig. 8. The measured time delays.

nodes must be a constant value. It is roughly believed that the measured value conforms to the average true value and the system measurement errors are the normal distribution of the standard deviation given that the time delay measurement error of the system is 10 microsecond. As seen in the figure, the information of time delay measured between Node 2 and Node 3 includes two values. The values with more sampling points serve as true values. There are 46 true values. The other 4 values serve as outliers. The presence of outliers is one of the major causes responsible for the RMSE error in calculation of ordinary node 3. The collected time delay values were processed again. The outliers in the time delay values measured with Node 2 and Node 3 were rejected. It was believed that Reference Node 1 and Reference Node 2 were true values. The distances measured between other nodes and distances between the reference nodes were used to jointly establish an error equation for determination of accurate values of the geodetic coordinates. The RMSE errors between the coordinate accuracies of Node 3, 4, and 5 based on calculation and true values in Table 2 were listed in Table 3. The east precision referred to the difference between the maximum of the positional scattered points in the eastern direction and the minimum of the positional scattered points in the eastern direction. The north precision referred to the difference between the maximum of the localization scattered points in the northern direction and the minimum of the positional scattered points in the northern direction. As seen in Table 3, the precisions in the eastern direction and northern direction in various stages were the centimetre level, which was attributable to the high-precision time delay measurement and the acoustic rate sensors installed on the sensor nodes. As the distance measuring accuracy of the system was 10 microsecond, the distance measuring accuracy was 1.5 cm in the

case of a acoustic velocity of 1500 m/s. Meanwhile, various sensor nodes were arranged at an equal depth. The depth measured by the high-accuracy pressure sensor was generally 60 m. The acoustic velocity did not change significantly in the horizontal direction. Thus, a positioning precision of centimetre level could be obtained. Meanwhile, based on a comparison of the RMSE errors in Tables 2 and 3, the average localization precision of the network could be effectively improved after our proposed method is used for post processing. The errors of the nodes except Node 5 were improved. The errors of Node 5 were the same due to the fact that Node 5 was determined on the basis of the positions of Reference Node 1 and 2 and the measured distance. Thus, the amount of information used for its positioning did not increase and the positional results were the same. The above method could effectively improve the absolute precision of the positioning of ordinary nodes after post processing. Reference Node 1 and Reference Node 2 were considered as ordinary nodes. All distance information was optimized to obtain a higher relative localization precision within the network. In the above experiment, a total of 6 distances were measured. 5 nodes and 10 coordinates needed to be calculated. It was a calculation for the undetermined equation. The method we proposed was used to obtain the coordinate accuracies of various nodes and the RMSE errors, as shown in Table 4. The east precision referred to the difference between the maximum of the localization scattered points in the east direction and the minimum of the positional scattered points in the east direction. The north precision referred to the difference between the maximum of the localization scattered points in the north direction and the minimum of the positional scattered points in the north direction. As seen in the table, the localization precision obtained by using the above method improved significantly. The localization precision of various nodes were the millimetre magnitude. It indicated that the application of the method could effectively improve the relative positional accuracies of various nodes. Meanwhile, the scattered points obtained with the above method were more convergent and similar to the two-dimensional normal distribution, indicating that the method can eliminate the node positioning deviation arising from inaccurate positions of the reference nodes. It can be seen from the above experiment that the distributed step-by-step positioning method and the post processing method can effectively achieve positioning of large-scale network nodes. However, the current experimental data can only be applied to a limited number of nodes and further research will be conducted to achieve verification of multiple nodes due to the limited experimental conditions.

Table 4 Underwater node localization precisions. Table 3 Ordinary node localization precision.

Ordinary node 3 Ordinary node 4 Ordinary node 5

East precision (m)

North precision (m)

RMSE (m)

0.011 0.013 0.015

0.025 0.025 0.009

0.38 0.42 0.17

Node Node Node Node Node

1 2 3 4 5

East precision (m)

North precision (m)

RMSE (m)

0.0066 0.0068 0.0074 0.0053 0.0087

0.0085 0.0137 0.0106 0.0099 0.0083

0.1603 0.1692 0.1748 0.2447 0.1041

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6. Conclusion In response to the inconsistency of positional accuracies of various nodes in the process of centralized/distributed network positioning, the node positions obtained by the centralized/distributed network are readjusted to obtain accurate relative positions enabling various nodes to have a consistent localization precision. The simulation and experimental results show that the method can effectively improve the average localization precision of the network and well apply to the subsequent fusion of the data of high-precision network node positions. Acknowledgment This work was supported by the Key Program of the National Natural Science Foundation of China (Grant No. 61531012), the Heilongjiang Science Fund for Distinguished Young Scholars of China (Grant No. JC2016013), and the Fundamental Research Funds for the Central Universities of the Ministry of Education of China (Grant No. HEUCFT1303). References [1] Akyildiz IF, Pompili D, Melodia T. Underwater acoustic sensor networks: research challenges. Elsev. Ad Hoc Netw. 2005;3:257–79. [2] J. Albowicz, A. Chen, L. Zhang, Recursive position estimation in sensor networks, in: Proceedings Ninth International Conference on Network Protocols, ICNP 2001, 2001, pp. 35–41 . [3] J. Cao, Y. Han, D. Zhang, D. Sun, Linearized iterative method for determining effects of vessel attitude error on single-beacon localization, Appl. Acoust. 116 (2017) 297–302, doi:https://doi.org/10.1016/j.apacoust.2016.10.001 (acoustic localization; Attitude error; Beacon localization; Least square methods; Linear trajectory; Long baseline array; Underwater operations; Virtual beacons). [4] Carroll P, Zhou S, Zhou H, Xu X, Cui JH, Willett P. Underwater localization and tracking of physical systems. J. Electr. Comp. Eng. 2012;2012:2. [5] V. Chandrasekhar, W.K. Seah, Y.S. Choo, H.V. Ee, Localization in underwater sensor networks, in: Proceedings of the 1st ACM International Workshop on Underwater Networks - WUWNet ’06, 2006, pp. 1–8, . [6] Cheng X, Shu H, Liang Q, Du DH. Silent positioning in underwater acoustic sensor networks. IEEE Trans. Veh. Technol. 2008;57(3):1756–66. [7] Duan W. Bidirectional soft-decision feedback turbo equalization for mimo systems. IEEE Trans. Veh. Technol. 2016;65(7):4925–36.

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