# BeiDou System Positioning Principle

BeiDou Navigaiton Satellite sends out navigation message constantly, whose ephemeris data is received by user terminal. Satellite time is extracted from the data and is compared with the user terminal time, so that the time difference between the satellite and the user terminal is known. The satellite ephemeris data in the navigation message can be also used to calculate the coordinates of the satellite when it sends out the message. According to the distance between two points in the space formula ，three equations can be listed to calculate the coordinates of the user terminal.

Known: satellite coordinates (x1, y1, z1), (x2, y2, z2),(x3, y3, z3)

Time difference between the satellite and the user terminal (t1, t2, t3)

To solve: user terminal coordinates (x, y, z)

Solving the three-element equation：

During the process of solution, when the time difference between satellite and user terminal is measured, the satellite clock and user terminal clock cannot be completely synchronous with the BeiDou Time. The satellite clock is atomic clock with picosecond-level accuracy. The time synchronization among different satellite clocks is also very precise. The user terminal clock is often quartz clock, which could have an error of a hundred thousandth second per second when its accuracy is relatively high; when its accuracy is low, the error could be a ten thousandth second per second. The internationally recognized speed of radio wave is c=299792458meters/second. So every ten picoseconds the radio wave can move 2.99 millimeters. Every hundred thousandth second it moves 3 kilometers. Every ten thousandth second it moves 30 kilometers. To ensure precise positioning, the fourth unknown number is introduced, Δt, which is the error between user terminal and satellite terminal. It’s a correction to the user terminal time when this index is taken into consideration during the process of solving the equation. The distance precision is serviceable enough. Four equations can be solved to find out the coordinates (x, y, z) of the user terminal and the precise time different Δt, which is used for accurate time synchronization.

Known: satellite coordinates (x1, y1, z1), (x2, y2, z2),(x3, y3, z3), (x4, y4, z4)

Time difference between the satellite and the user terminal (t1, t2, t3, t4)

To solve: user terminal coordinates (x, y, z), Δt

Solving the four-element equation：