Excerpt

Passive Source Localization Algorithm of an

Underwater Sound Source Using Time Difference of

Arrival (TDOA) and Bearing Estimation

Prithvijit Chattopadhyay

Electrical Engineering (3

rd

Year)

Delhi Technological University (Formerly DCE)

New Delhi, India

**Abstract**-- In this paper we present a method to localize a**sound source after TDOA analysis and bearing estimation has**

**been done. The localization is done by completely determining**

**the coordinates of the source as a linear function of the distance**

**of the hydrophone array from the source. The distance of the**

**sound source from the hydrophone array is determined using a**

**quadratic.**

**Index Terms**--TDOA, Bearing, Range, Hydrophones, Passive**SONAR**

I. I

NTRODUCTION

The problem of passive source localization using TDOA

estimation is a widely active research area. The main problems

which are encountered in the algorithms based on this method

can be classified into two categories accuracy of TDOA

estimation , and an efficient localizing algorithm .The basic

approach to the problem is to form equations in terms of the

cartesian coordinates of the source and the hydrophones

(sensors). This comprises of three variables for each

coordinates

. This results in equations that

describe hyperboloids in three dimensional space. With the

number of sensors being , we'd naturally have to solve

hyperboloids for the region of intersection in

which the source is expected to lie [6]. This is usually a very

easy thing to do via numerical mathematical techniques.

However, this method has a disadvantage. Often it usually

simple to give the monitoring AUV (Autonomous Underwater

Vehicle) or the ROV (Remotely Operating Vehicles) the range

and bearing of the source instead of the coordinates. This type

of situation forces us to abandon the hyperbolic equation

approach and look upto more linear approaches. Measurement

errors in TDOA in certain cases can also make the existence of

a hyperbolic equation a problem.

II. TDOA ESTIMATION

The generalized cross correlation technique using phase

transform (GCC-PHAT) is used to calculate the time difference

of arrival corresponding to the correlation peak [5]. Estimated

TDOAs give bearing estimation for far field approximation. A

lot of other methods are available for estimating the TDOA

[3],[4].

III. TETRAHEDRAL ARRAY OF HYDROPHONES FOR

LOCALIZATION OF A PASSIVE SOURCE

As a minimum of 4 hydrophones are required for 3-D

localization of the source in a real-time situation , we will

describe the theory taking into account 4 hydrophones

arranged in tetrahedral array. A tetrahedral array implies that

the hydrophones are arranged in a tetrahedral fashion.

**Fig.1 Dot Modeling of the tetrahedral array**

Consider the above figure. It represents a dot modeling of

the tetrahedral array of hydrophones .

0,1,2,3 are the respective hydrophones and S represents the

sound source .

are the distances of the respective

hydrophones from the source.

The distances

define the specifications of the array.

1

The hydrophones have been so arranged that 0 forms the

origin of 3 perpendicular axes 01, 02 and 03 being the z,x and

y axes respectively. This is merely an arrangement to give us

ease in localizing the coordinates by using symmetry

considerations. The idea is to calculate the position of the

sound source with respect to a coordinate frame formed by the

array of hydrophones. Thus we have the coordinates of

hydrophones 1, 2 and 3 as

.

If

is the TDOA for hydrophones i , j and c is the speed

of sound and we define

as the path difference and thus

as

Then we get the following equations :

10

20

30

12

23

31

...(1)

Now note that each of

are in fact equal

to expressions such as,

Thus when expanded in terms of

each of the

equations are very difficult to handle. They represent

hyperboloids. This has been discussed in detail in [6].

We need to keep track of the fact that

are known to us while designing the array and

are known

to us from TDOA measurements which are usually done using

generalized cross-correlation phase transform (GCC-PHAT)

methods. The localization algorithm is presented in the next

section.

IV. LOCALIZATION ALGORITHM

Consider the figure below.

**Fig.2.Dot Modeling for determining**

Above shown is the source hydrophone triangle AOS. It

consists of only the hydrophones along the z-axis.

Note that

and

Note that here is my bearing. It has been obtained from

TDOA estimation techniques.

is drawn such that

giving us

By the TDOA equations ,

And by the cosine rule in triangle

from (1) and (2) we get ,

(2)

and writing

(3)

We get ,

(4)

Thus we get two quadratics in

. However we have to

determine the coefficients that depend on .

Now let

.

Using cosine rule in triangle

we get

(5)

and using sine rule in triangle OZN. We arrive at,

(6)

Thus the coefficients have been determined.

For the quadratic to admit real solutions the condition

(7)

must be satisfied.

2

Solving for ,

(8)

Note that for feasible solutions one must have the radical

term with a positive sign as the first term on RHS is very small

in comparison to the radical. Thus ,

(9)

Once we have determined the values of the ranges we shall

move on to determining the coordinates. We observe that the z-

coordinate of the sound source is given by ,

(10)

Also note that the cylindrical radius coordinate of the sound

source is given by,

(11)

Now if we look upon the equation ,

20

We can write

We can write,

(12)

We can similarly write down an expression for too.

Considering the sign in the expressions for

,

only one of the values should be feasible. Thus ,

(13)

And

(14)

Thus we have successfully obtained the coordinates of the

sound source with respect to the array of hydrophones. Once

we have determined the Cartesian coordinates

, we

can also determine the cylindrical coordinates

which

are more easier to handle from a control point of view for the

AUV.

Summarizing the algorithm we have,

**Fig.3 Flowchart of the algorithm**

V. RESULTS

The exact coordinates of the sound source with respect to

the hydrophone array are given by (9), (13) and (14) :

and

Where ,

And is given by (6),

dK

' W, d

^

^

3

Using these results in MATLAB to simulate the

localization of a sound source, the results obtained are as

follows:

**Actual**

**Pinger**

**Position**

**(metres)**

**Calculated**

**Pinger**

**Position**

**(metres)**

**Percentage**

**Error**

**(%)**

1

.

10 10.008548

0.08548

11 11.009415

0.0855909

12 12.01036

0.0863333

2

.

20 20.027426

0.13713

5 5.0139030

0.27806

11 11.01511

0.1373636

3

.

7 7.057355

0.8193571

25 25.020858

0.083432

9 9.0075176

0.0835289

4

.

30 30.017234

0.0574467

20 20.011458

0.05729

4 4.0023016

0.0005754

**Table 1 Results of localization using the algorithm**

Corresponding results for are as follows :

S.No.

**Actual**

**(metres)**

**Calculated**

**(metres)**

**Percentage**

**Error (%)**

1.

19.1049732 19.1214688 0.08634192

2.

23.3666429 23.3987402 0.1373638

3.

27.4772633 27.5002147 0.0835287

4.

36.2767143 36.2975883 0.05754104

**Table 2 Results of localization using the algorithm**

Note that here we have taken :

Thus we have the coordinates of hydrophones 1, 2 and 3

as

respectively. The

fourth hydrophone is at

.

Also the input signals are chirp signals in the above case.

They have an initial frequency

and a

final frequency of

, the central

frequency being

. The method of

GCC-PHAT is used to estimate the TDOA values.

We have taken,

VI. C

ONCLUSION

The difficulties in hyperboloid formulation of the problem

in the passive source localization have been discussed. To get

over the discussed difficulties a linear model has been

employed. The presented formulation utilizes the geometry of

the source-hydrophone triangles in conjunction with the

bearing estimation. The solution that has been presented here is

considering a situation of four sensors. The results of the

simulation of the algorithm in MATLAB have been presented

and we can observe that maximum errors obtained are

.

The maximum error obtained in

. These

results imply that the method is accurate to a good degree for

localization purposes. In conclusion, the methods presented

here can be applied in localization technologies and good

accuracy can be achieved in real-time situations.

R

EFERENCES

[1] Xiaoyan Lu,Sound source localization based on microphone

array, Dalian: Dalian University of Technology,Master's

thesis,March 2003.

[2]

H. Wang and M. Kaveh,Coherent signal-subspace processing

for the detection and estimation of angles of arrival of

multiple wide-bandsources, IEEE Transactions on Acoustics,

Speech and Signal Processing,vol. 33,no. 4,August

1985,pp. 823-831.

[3] Darning Wang, Wei Tan,Jianxin Guo and Dongming

Bian,Research on the Time Difference Measurement of Signal

Arrival, Journal of Information Engineering University,vol.

4,no. 2,June 2003,pp.

[4] B. G. Ferguson, "Time-delay estimation techniques applied to

acoustic detection of jet aircraft transits,"

*J. Acoust. Soc. Amer.*,

vol. 106, no. 1, p. 255, 1999.

[5] C.H.Knapp and G.C. Carter , "The generalized correlation

method for estimation of time delay", IEEE Transactions on

Acoustics, Speech and Signal Processing,vol. 24,no. 4,

1976,pp. 320-327.

[6] Leo Singer University of Maryland, SONAR Cookbook an

Underwater Primer for TORTUGA II, 3

rd

edition , 2008.

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- Prithvijit Chattopadhyay (Author), 2014, Passive Source Localization Algorithm of an Underwater Sound Source Using Time Difference of Arrival (TDOA) and Bearing Estimation, Munich, GRIN Verlag, https://www.grin.com/document/281962

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