Pseudo-range multilateration

Pseudo-range multilateration, often simply multilateration (MLAT) when in context, is a technique for determining the position of an unknown point, such as a vehicle, based on measurement of biased times of flight (TOFs) of energy waves traveling between the vehicle and multiple stations at known locations. TOFs are biased by synchronization errors in the difference between times of arrival (TOA) and times of transmission (TOT): TOF=TOA-TOT. Pseudo-ranges (PRs) are TOFs multiplied by the wave propagation speed: PR=TOF ⋅ s. In general, the stations' clocks are assumed synchronized but the vehicle's clock is desynchronized.

In MLAT for surveillance, the waves are transmitted by the vehicle and received by the stations; the TOT is unique and unknown, while the TOAs are multiple and known. When MLAT is used for navigation (as in hyperbolic navigation), the waves are transmitted by the stations and received by the vehicle; in this case, the TOTs are multiple but known, while the TOA is unique and unknown. In navigation applications, the vehicle is often termed the "user"; in surveillance applications, the vehicle may be termed the "target".

The vehicle's clock is considered an additional unknown, to be estimated along with the vehicle's position coordinates. If ${\displaystyle d}$ is the number of physical dimensions being considered (e.g., 2 for a plane) and ${\displaystyle m}$ is the number of signals received (thus, TOFs measured), it is required that ${\displaystyle m\geq d+1}$.

Processing is usually required to extract the TOAs or their differences from the received signals, and an algorithm is usually required to solve this set of equations. An algorithm either: (a) determines numerical values for the TOT (for the receiver(s) clock) and ${\displaystyle d}$ vehicle coordinates; or (b) ignores the TOT and forms ${\displaystyle m-1}$ (at least ${\displaystyle d}$) time difference of arrivals (TDOAs), which are used to find the ${\displaystyle d}$ vehicle coordinates. Almost always, ${\displaystyle d=2}$ (e.g., a plane or the surface of a sphere) or ${\displaystyle d=3}$ (e.g., the real physical world). Systems that form TDOAs are also called hyperbolic systems, for reasons discussed below.

A multilateration navigation system provides vehicle position information to an entity "on" the vehicle (e.g., aircraft pilot or GPS receiver operator). A multilateration surveillance system provides vehicle position to an entity "not on" the vehicle (e.g., air traffic controller or cell phone provider). By the reciprocity principle, any method that can be used for navigation can also be used for surveillance, and vice versa (the same information is involved).

Systems have been developed for both TOT and TDOA (which ignore TOT) algorithms. In this article, TDOA algorithms are addressed first, as they were implemented first. Due to the technology available at the time, TDOA systems often determined a vehicle location in two dimensions. TOT systems are addressed second. They were implemented, roughly, post-1975 and usually involve satellites. Due to technology advances, TOT algorithms generally determine a user/vehicle location in three dimensions. However, conceptually, TDOA or TOT algorithms are not linked to the number of dimensions involved.