Research on optimization of the number of acoustic arrays

The reasonable number of base stations in multiple acoustic array azimuth cross location is studied. In this paper, the maximum likelihood estimation (MLE) algorithm is used to discuss the problem of increasing the number of arrays to improve the positioning accuracy, and the positioning error model is given. The experimental results show that with the increase of the number of base stations, the positioning accuracy will gradually improve, but if there are too many base stations, the improvement of positioning accuracy will gradually slow down.


Introduction
In order to improve the positioning accuracy of signal source, it is usually necessary to improve the direction finding accuracy of acoustic array and optimize the positioning algorithm. This paper is based on the maximum likelihood algorithm, and gives the positioning accuracy with the number of acoustic arrays. The structure of the paper is organized as follows. In the second section, we first review the principle of maximum likelihood estimation and the bearing-crossing location. Then introduce how to choose the initial coordinate and depict the accuracy error. The third section gives the simulation results and conclusions. The fourth section summarizes the whole paper.

The principle of DF cross location
The method of DF cross location is also known as triangulation method, it measure the emitter by highprecision DF equipment at two or more stations. The equipment can calculate the source coordinate with the measured angle data and the station coordinate.
The station position is represented by

MLE algorithm based on acoustic location
According to the position relationship, we can get the equation (1) and ( , , , , , , , The estimation value of the signal source can be expressed as The MLE is an nonlinear estimation, so it does not have a closed-form solution and requires a numerical search algorithm. The Gauss-Newton algorithm, which is a batch iterative minimization technique, can be employed to calculate the MLE. The GN algorithm consists of is denoted as the distance between the i th signal source and the k th station.
is a vector which contain the first two entries of i X .The initial coordinate of the signal source can be calculated by PLE.

Position error
The covariance matrix of measurement error can be expressed as Where the position of DF station and measurement error of azimuth ( k  ) and elevation( k  )are independent of each other, so the correlation in equation (9) can be expressed as the model can be written as the following.
Where: 2 The distance error of the j th signal of the i th simulation can be written as The average and variance of M target of N simulation can be written as the following. 1 1

Simulation conditions
In order to find the best station numbers, we use simulation experiment technology to analysis. The experimental parameters are as follows: possible station number is 2 ~ 20, expanding width is 6km and depth is ± 0.2km, high and low is ± 0.2km,showed a random distribution. We assume azimuth ( k  ) and elevation( k  ) has the standard errors, which are separated into two groups, one is 0.1 0 , 0.3 0 , 0.5 0 , the other is 0.5 0 , 1.0 0 , 1.5 0 . The area of target is assumed as: axis x is ±3km, axis y is 1km～7km,axis is ±0.4km.      We can conclude the conclusion from figure 3 and table 2: Comparing the curve of the average on figure 2, we can find that with the station number increasing, the station's DF is more precise and the convergence speed is faster, on the contrary, the station's DF is slightly lower and the convergence speed is slower. The result is the opposite of the figure 1.

Analysis of the simulation results
Comparing curves of the variance on figure 2, we can find that with the station number increasing, the curve is convergence if the station's DF is more precise, however, if the station's DF is lower than a reasonable limit, the curve is divergence. Table 2 and Figure 2 for the general direction finding more conditions, using MLE for Monte-Carlo 200 experiments, the 500 random simulation of target distance error mean and variance of data and the corresponding curve.