Initial Estimate Selection Method in Passive TDOA-Based Iterative Position Estimation Algorithms

. 2026 Jul 12;26(14):4431. doi: 10.3390/s26144431

Algorithm 1. Newton–Raphson algorithm

Input:

I n i t i a l . G u e s s

s e n s o r . c o o r d = [\begin{matrix} x_{H_{1}} & x_{H_{2}} & \dots & x_{H_{N}} \end{matrix}]

T D O A = [\begin{matrix} T D O A 1 & T D O A 2 & \dots & T D O A m & T D O A 1 \end{matrix}]

e r r o r . l i m i t

I t . l i m i t

$Calculate F$ and $J$ value for $I n i t i a l . G u e s s$ with (11) and (12)
$Calculate e r r o r = m a x (|F|)$
$Set e r r o r . d i f f = e r r o r$
$Set I t = 0$
$while e r r o r \geq e r r o r . l i m i t$ and $I t < I t . l i m i t$ and $e r r o r . d i f f \neq 0$ do
$Calculate V = - J^{- 1} F$
$Calculate I n i t i a l . G u e s s = I n i t i a l . G u e s s + V$
$Calculate F$ and $J$ value for new $I n i t i a l . G u e s s$ with (11) and (12)
$Calculate e r r o r . d i f f = m a x (|F|) - e r r o r$
$Calculate e r r o r = m a x (|F|)$
$Set I t = I t + 1$
end while

Output:

P o s i t i o n = I n i t i a l . G u e s s