Positioning Error Probabilities for Some Forms of Center-of-Gravity Algorithm Calculated with the Cumulative Distributions. Part II

TL;DR

This paper derives the probability density functions for errors in center-of-gravity positioning algorithms using cumulative distribution functions, addressing complex combinations of random variables for improved error analysis.

Contribution

It introduces a detailed method to calculate the complete and partial probability density functions for specific center-of-gravity error expressions, expanding on previous work with new cumulative distribution calculations.

Findings

Derived probability density functions for complex error expressions

Calculated cumulative probability distributions for the first time

Applicable to general observation noise distributions

Abstract

To complete a previous work, the probability density functions for the errors in the center-of-gravity as positioning algorithm are derived with the usual methods of the cumulative distribution functions. These methods introduce substantial complications compared to the approaches used in a previous publication on similar problems. The combinations of random variables considered are: $X_{g3}=\theta(x_2-x_1) (x_1-x_3)/(x_1+x_2+x_3) + \theta(x_1-x_2)(x_1+2x_4)/(x_1+x_2+x_4)$ and $X_{g4}=(\theta(x_4-x_5)(2x_4+x_1-x_3)/(x_1+x_2+x_3+x_4)+ \theta(x_5-x_4)(x_1-x_3-2x_5)/(x_1+x_2+x_3+x_5)$ The complete and partial forms of the probability density functions of these expressions of the center-of-gravity algorithms are calculated for general probability density functions of the observation noise. The cumulative probability distributions are the essential steps in this study, never calculated elsewhere.