How the GPS works?

The Global Positioning System or GPS is much used in daily life to go to an unknown place, to find out where you are. It is used in ships, drones and airplanes to avoid collisions. How it works is this post’s subject.

The satellites

The GPS is currently formed by a constellation of 31 satellites, It was 24 before. Stay in 20200 km of altitude and are distributed in 6 planes.

Each one of the satellites have an atomic clock and send radio waves to the receivers and the time the signal was transmitted. How the satellite know where you are? It is used a technique called trilateration, the receiver receive signals from three satellites and calculate the distance of three, knowing the exact location. A forth satellite can be used to provide the altitude and increase the precision.

The receiver

All smartphones have a GPS antenna and band-pass filter to L1 (1.575 GHz) and L2 (1.227 GHz) frequencies. The received signal by cellphones is too weak, so it is used the technique of Direct Sequence Spread Spectrum. This technique consist in transmit the information in a wider frequency band and with reduced power. In the figure, the red is the information before be mixed with a signal which looks like random, and the green is how the signal is transmitted.

The receiver has a code used in transmission and can recover the signal. Terrestrial stations with parabolic antennas verify if the satellites are in right positions. This figure shows where are the GPS control stations.

Application of Relativity

The GPS wouldn’t be possible without the knowledge of Special and General Relativity. When faster a body, slower the passage of time to this body. The satellite is in space and with speed of 14,000 km/h, much faster than the receiver on Earth.

Using the formula:

$t=\frac{to}{\sqrt{1-\frac{v^2}{c^2}}}$

Where $to$ are seconds per day (86,400 s), $c$ is the light speed 299,792,458 m/s and $v$ is the satellite speed. The atomic clock in the satellite stays 7 microseconds per day slower than a clock in Earth due to speed. The General Relativity says that a body further from a gravitacional field has it time advanced in relation to a body closer to this field. Has this formula:

$t=\frac{to}{\sqrt{1-\frac{2MG}{rc^2}}}$

Where $M$ is the Earth`s mass, $r$ it is Earth’s radius and $G$ is the gravitacional constant which values $6,67408\cdot 10^{-11} \frac{m^2}{(kg\cdot s^2)}$ . The atomic clock stays 45 microseconds per day more advanced than in Earth. In total, stays 38 microseconds advanced than in Earth. May looks like few, but the accumulation of advance can cause an unacceptable error margin. For that, clocks in satellites need to make this correction and send the right time to receivers on Earth.