Recurrent Algorithm for TDOA Localization in Sensor Networks

: Using the mathematical apparatus of the extended Kalman Filter, the recurrent algorithm of the passive location in sensor networks — based on the Time Difference of Arrival method in case of correlated errors of measurements — is developed. The initial estimates of Radio Frequency Sources coordinates and the correlation matrix of the vector estimation are determined based on the method of the least squares in case of 3 difference measurement distances. Efficiency analysis of recurrent adaptive algorithm and its comparison with the quadratic correction one are performed by statistical modeling. A comparison of them with the lower limit of the Cramer-Rao was carried out. The implementation of the recurrent adaptive algorithm requires 2.7 times less computational cost than the quadratic correction one.


INTRODUCTION
The problem of passive position determination of Radio Frequency Sources (RFS) is widely met in the monitoring of the surrounding space, disaster management, in intelligent transport and security systems.Currently, sensor networks are used for its solution (Rullan-Lara et al. 2013;Amar and Leus 2010).
One of the main approaches of passive position determination of RFS is based on the application of the Time Difference of Arrival method (TDOA), which uses the time difference of reception of signals received by the various sensors and the network reference sensor.This method has a significant advantage in the ease of implementation, being widely used in practice ITU (2014).
The accuracy of determining the RFS coordinates based on the TDOA depends on the errors from measuring time signal reception sensors of the sensor network.The errors from the determination of the difference in signal reception times are correlated, because they contain the error in the reference sensor measurement.
In the known methods for determining the RFS coordinates based on the TDOA (Buzuverov 2008), the coordinate calculation is performed after receiving the measurements from all sensors.In this study, based on the mathematical apparatus of the extended Kalman Filter, the algorithm is developed, which -after the formulation of the initial conditions based on the measurement of time for receiving signals from 4 sensors -allows to recurrently specify the location of RFS as the measurement proceeds from the other sensors.The developed algorithm also evaluates the time error from receiving the signal by the reference sensor measurement that allows considering the synthesized algorithm as adaptive.

FORMULATION OF THE PROBLEM
The sensors network transducers have the coordinates (x S i , y S i ), i = 0, n.The position of the RFS is characterized by a point with coordinates (x, y).When determining the coordinates of RFS on the x-y plane, the sensor network should consist of no less than 4 sensors.Figure 1 shows block diagrams of sensor networks on the x-y plane, consisting of 9 (n = 8) and 4 (n = 3) sensors.
When using TDOA, the time difference in the reception of signals between the sensors i = 1, n and the reference probes is measured: distances v i0 , i = 1,n , are correlated because they contain the reference sensor measurement error v 0 .
The presence of correlated errors makes it difficult to use traditional recurrent target coordinates evaluation algorithms.This difficulty can be avoided through the introduction of v 0 into the state vector of the estimated parameters.
It is believed that, during the difference measurement of the distances between the RFS and the network sensors, its coordinates do not change.Synthesizing a recurrent algorithm is required, and, after the formation of the initial conditions based on the measurement of signals receiving time from 4 sensors, it allows to specify recurrently the location RFS in process of receipt of measurements from the other sensors and to estimate the error of the reference sensor measurement.

DEVELOPMENT
The coordinates of the RFS position (x k , y k ) should be assessed; the measurement time of distance difference between the RFS and the network sensors, as well as its coordinates, do not change.The error in measurements of the reference sensor from Eq. 2 does not change as well in 1 measurement cycle, so the equation describing the dynamics of the estimated parameters has the form where: t i is the time of signal reception of the i − m sensor; t 0 is the time of signal reception by the reference sensor; n i is the uncorrelated error of time measurements of signal by the reception of the i − m sensor (Buzuverov 2008) Because of the transformation in Eq. 1, the TDOA equations for network can be represented as: ( ) (1) ( ) (1) (2) S 2 S 2 2 2 0 ) ( ) ( ) ( ) (1) ( ) (1) (1) where: R i is the distance between the i − m sensor and the RFS, i = 1, n; R 0 is the distance between the reference sensor and the RFS; d i0 is the measured difference of distances i = 1, n; c is the spreading speed of electromagnetic waves; t is the moment of RFS signal emission; v i0 is the error of difference measurement between the distances: where: v i are the uncorrelated random variables having the meaning of the distance measurement errors between the RFS and the network sensors with a dispersion In Eq. 4, the coordinates of the reference sensor were relied to be 0. The errors of difference measurement between the where: u k = (x k , y k , v 0k ) T is the state vector including position coordinates of the RFS and the measurement error of the reference sensor v 0k on the current time step k.The index k characterizes the incomings sequence of measured differences in distances.
The measurement equation describing the measured k th distance difference considering Eqs. 2, 4 and 5 can be viewed as where: h(u k ) is a non-linear function described by the nonlinear measurement expression: where: v k is the measurement error of the sensor with coordinates x i , y i .
Using the model (Eqs.6 and 7), a recurrent estimation algorithm of the state vector can be obtained based on the extended Kalman Filter (Welch and Bishop 2006), being described by The correlation matrix Ω of the vector estimation error ω is defined (Amar and Leus 2010) by ( ) S 2 S 2 2 2 0 ( ) ( ) ( ) ( ) S 2 S 2 2 2 0 ( ) ( ) ( ) ( ) ( ) ( ) (3) 2 2 x y ( ) where: u ˆk is the estimation of the state vector u k at the k th step; P ˆk is the correlation matrix of the state vector of the estimation error u k at the k th step; K k is the coefficient of the filter gain: The resulting algorithm (Eqs.8 -10) is non-linear and belongs to the class of the adaptive ones because it along with the estimation of the RFS the coordinates, an estimation of the unknown error v 0 is determined.The initial conditions of u ˆ0 and P ˆ0 must be set for the implementation of adaptive filtering.The initial evaluation of the vector u ˆT 0 is u ˆT 0 = (x ˆ0, y ˆ0, 0) The initial estimates of RFS coordinates x ˆ0, y ˆ0 are determined based on the method of least squares (LS) in case of 3 difference measurement distances (Amar and Leus 2010) using the Eq.12: where: ω T = (x ˆ0, y ˆ0, R ˆ0) is the vector consisting of the RFS assessment; S 2 S 2 0 0 ˆˆ( ) (y ) ; 1,3 The initial correlation matrix 0 P has the block form , where Ω 2×2 is determined based on Ω, S 2 S 2 0 0 ˆˆ( ) (y ) ; 1,3 The initial correlation matrix 0 P has the block form , where Ω 2×2 is determined based o S 2 S 2 0 0 ˆˆ( ) (y ) ; 1,3 The initial correlation matrix 0 P has the block form , where Ω 2×2 is determined based on Ω, where: H is the matrix determined by the formula H = B∑B; B = diag {R ˆ1, R ˆ2, R ˆ3}; R ˆj = √(x ˆ0 -x S 0 ) 2 + (y ˆ0 -y S 0 ) 2 ; j = 1, 3. The initial correlation matrix P ˆ0 has the block form where Ω 2×2 is determi 56'[p-;;;;;;n 56734r5 ned based on Ω, by deleting the third row and the third column.
After the formation of the initial conditions based on the time measurements to receive signals from the 4 sensors, the synthesized algorithm (Eqs.8 -10) allows to recurrently specify, at each step k, the location of the RFS as the measurement proceeds from the other sensors k = 1, n -3.

THE EFFECTIVENESS OF THE ALGORITHM ANALYSIS
The efficiency analysis of recurrent adaptive algorithm (Eqs.8 -10) and its comparison with the quadratic correction one is performed by statistical modeling.The quadratic correction algorithm provides the highest accuracy among those considered in Amar and Leus (2010).It consists of 2 stages: (i) evaluate the RFS coordinates, which depends on the distance R 0, substitute in the initial functionality then re-solve the linear optimization problem; (ii)-adjust the solution for a quadratic relation.
Figure 4 shows the dependence of the actual σ ˆ MC QC (curve 1; where QC is the quadratic correction) and theoretical σ ˆFK (curve 2) circular RMS of the RFS location estimation error for the quadratic correction algorithm.
Figure 5 shows a circular RMS of the RFS location estimation error, which corresponds to the lower limit of the Cramér-Rao bound and characterizes the potential accuracy of the possible RFS coordinates.The values of circular RMS error of the RFS positioning of the recurrent adaptive and quadratic correction algorithms, positioned close to the corresponding values of circular RMS of the Cramér-Rao bound lower limit, indicate their high efficiency.
Figure 6 shows the change of the actual (curve 1) and theoretical (curve 2) circular RMS error estimation of the RFS locations using recurrent adaptive algorithm for fixed RFS  coordinates (x, y).The use of recurrent adaptive algorithm reduces the circular RMS of the RFS location estimation error in 2.5 times relative to the initial conditions.
It is interesting to compare the calculation of the costs required in the implementation of the recurrent adaptive and quadratic correction algorithms.They can be assessed by determining the number of required multiplications (divisions), since they are performed from 100 to 150 times slower than addition and subtraction.For the example with the implementation of the recurrent adaptive algorithm, 461 multiplications are required and, for the quadratic correction one, this value is 1,246.Thus, the application of the developed algorithm reduces the computational cost by 2.7 times.

CONCLUSIONS
Using the mathematical apparatus of Kalman filtering, the algorithm is developed, which, after the formation of the initial conditions, is based on measurements of the receiving time of the signals from 4 sensors.This allows specifying the location of the recurrent RFS as the measurement proceeds from the other sensors.It belongs to the class of adaptive algorithms, because, along with the estimation of the RFS coordinates, it determines the estimation of unknown error of reference sensor measurement.According to the simulation results, the use of recurrent algorithm can reduce the circular RMS of the RFS location estimation error by 1.5 -4.2 times, providing characteristics similar to the potentially achievable.
The implementation of the recurrent adaptive algorithm requires 2.7 times less computational cost than the quadratic correction one.The resulting algorithm can also be easily extended to the case of RFS filtering trajectory at which its motion parameters are estimated (Chiang et al. 2012).Introduction, Tovkach IO and Zhuk SYa; literature review, Tovkach IO and Zhuk SYa; Development, Tovkach IO and Zhuk SYa; Discussion, algorithm analysis, Tovkach IO and Zhuk SYa; Conclusion, Tovkach IO and Zhuk SYa.

Figure 2 .
Figure 2. Circular RMS of the RFS position estimation error for the initial conditions.

Figure 1 .
Figure 1.Configuration of the sensor network with (a) 9 and (b) 4 sensors.

Figure 3 .
Figure 3. Circular RMS of the RFS position estimation error for recurrent adaptive algorithm.

Figure 4 .Figure 6 .
Figure 4. Circular RMS of the RFS position estimation error for the quadratic correction algorithm.

Figure 5 .
Figure 5. Circular RMS of the RFS position estimation error of the Cramér-Rao bound lower limit.