【 Notes 】ML ALGORITHMS of TOA

阿新 • • 發佈：2018-12-21

ML方法是NLS方法的一個推廣版本，具體接著看：

Assuming that the error distribution is known, the ML approach maximizes the PDFs of TOA measurements to obtain the source location. When the disturbances in the measurements are zero - mean Gaussian distributed, it is shown in the following that maximization of Equations 1 will correspond to a weighted version of the NLS scheme.

第 $l$ 個 $r_{TOA,l}$ 的PDF：

向量形式：

（1）

the covariance matrix for $\bold{r_{TOA}}$

To facilitate the maximization of Equation (1) , we consider its logarithmic version:

(2)

As the first term is independent of x , maximizing Equation (2) is in fact equivalent to minimizing the second term, the ML estimate is

(3)

or we can write

(4)

where $\bold{ J_{ML,TOA}(\tilde x)}$ denotes the ML cost function for TOA - based positioning, which has the form of

(5)

Comparing Equations (4) and (5) , it is observed that in the presence of zero - mean Gaussian noise, the ML estimator generalizes the NLS method because the former is a weighted version of the latter.

比較等式（4）和（5），觀察到在存在零均值高斯噪聲的情況下，ML估計器推廣了NLS方法，因為前者是後者的加權版本。

Intuitively speaking, when $\sigma^2_{TOA,l}$ is large, which corresponds to a large noise in $r_{TOA,l}$ , a small weight of $1/\sigma^2_{TOA,l}$ is employed in the squared term of

and vice versa.

When $\bold{C^{-1}_{TOA}}$ is proportional to the identity matrix or $\sigma^2_{TOA,l},l = 1,2,..,L$ are identical, the ML estimator is reduced to the NLS method. To compute Equation (4) , we can follow the numerical methods discussed in the NLS approach. In particular, the Newton – Raphson procedure for Equation (4) is