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.
4
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