Satellite Navigation and Positioning

A satellite-navigation receiver generally needs signals from four satellites because it must determine both its three-dimensional position and the error in its own clock. These are four unknowns: three position coordinates and one receiver-clock offset, so four independent measurements are normally required.

How satellite signals provide distance measurements

Each navigation satellite broadcasts timing information and orbital data from which a receiver can determine the satellite's position when the signal was transmitted. The receiver compares the transmission time with the reception time measured by its own clock.

• Multiplying the signal's travel time by the speed of light gives a distance estimate.
• This estimate is called a pseudorange because it includes clock errors and other measurement errors, rather than representing only the true geometric distance.
• If the distance were known exactly, it would place the receiver somewhere on a sphere centred on the satellite; measurements from other satellites narrow down the possible position.

Why the receiver clock creates a fourth unknown

Navigation satellites carry highly accurate atomic clocks, but an ordinary receiver's clock is not perfectly synchronised with satellite-system time. The receiver therefore cannot initially distinguish extra signal travel time from an error in its own clock.

• The receiver must solve for its three position coordinates, commonly represented as x, y and z.
• It must also solve for its receiver-clock offset, which affects all satellite measurements made at the same measurement epoch.
• Even a timing error of one microsecond corresponds to approximately 300 metres of range error, making clock-offset estimation essential.

How four satellites resolve the problem

Each satellite supplies one pseudorange equation relating its known position to the receiver's unknown position and clock offset. With suitable satellite geometry, four independent equations allow the receiver to estimate all four unknowns together.

• The fourth satellite is not used only to correct time; all four measurements jointly determine position and clock offset.
• Fewer satellites may suffice if an additional independent constraint is available, such as accurately known receiver time or altitude.
• Four satellites are the normal minimum, not a guarantee of accuracy: satellite geometry, signal obstruction and measurement errors also affect the solution.
• Receiving more than four satellites provides additional measurements that can improve the positioning estimate and help check its consistency.