How can signals help a receiver find its position?

A receiver on a mountain can work out its position by listening to satellites it never contacts in return. The satellites do not look down, recognize the receiver and send back a location. They broadcast timing signals and information about their orbits. The receiver turns those broadcasts into a solution to a geometry-and-clock problem.

A radio signal takes time to travel. In an ideal vacuum, distance equals the speed of light multiplied by travel time. A timing error of just one millionth of a second corresponds to about 300 meters of range error. Measuring the delay therefore needs much finer timing than a clock whose display merely shows hours and minutes. GPS satellites carry atomic clocks, and their signals let receivers identify when particular portions of the broadcast were sent.

If the satellite's location and the exact distance to it were known, the receiver could be anywhere on a sphere centered on that satellite. A second distance would narrow the possibilities, and additional independent distances would constrain them further. This is a three-dimensional problem: a point needs three coordinates, not just a spot on a flat street map. Satellite orbit information allows the receiver to calculate where each satellite was when its signal was transmitted.

There is a catch. The receiver's clock is not perfectly synchronized with satellite time. Its measured travel times contain a shared clock offset. Multiplying those times by the speed of light produces pseudoranges: distance-like measurements that include the clock effect. After accounting for satellite clock corrections, a simplified measurement is ρᵢ = distance to satellite i + cΔt + other errors. Here c is the speed of light and Δt is the receiver's clock offset. The same unknown Δt appears in every measurement made at that moment.

The receiver ordinarily solves for four unknowns together: three position coordinates and that clock offset. At least four suitable satellite measurements are normally needed. A proposed position gives a different geometric distance to each satellite; a proposed clock correction shifts all the timing-based ranges together. The receiver searches for values that make these predictions agree with the measurements. The fourth measurement is not a special fourth direction in space. It supplies another constraint that helps separate position from clock error.

Real measurements do not fit perfectly. The atmosphere delays signals, and reflections from buildings can make a received signal appear to have traveled farther. Satellite arrangement matters too: sources spread across the sky generally constrain position better than sources clustered together. Additional measurements allow a receiver to fit the data and assess inconsistencies, but they do not automatically erase these errors.

The result includes time as well as location. Navigation receivers can derive a precise common time reference without carrying their own atomic clocks, which is useful to communication networks and power systems as well as travelers. The moving dot on a map is one visible result of an invisible exchange between light-speed travel, orbital knowledge, geometry and synchronized time.