Pseudo-range multilateration
Pseudo-range multilateration, often simply multilateration 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 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 and times of transmission :. Pseudo-ranges are TOFs multiplied by the wave propagation speed:. 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, 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 is the number of physical dimensions being considered and is the number of signals received, it is required that.
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: determines numerical values for the TOT and vehicle coordinates; or ignores the TOT and forms time difference of arrivals, which are used to find the vehicle coordinates. Almost always, or . 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. A multilateration surveillance system provides vehicle position to an entity "not on" the vehicle. By the reciprocity principle, any method that can be used for navigation can also be used for surveillance, and vice versa.
Systems have been developed for both TOT and TDOA 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.
Background
Prior to deployment of GPS and other global navigation satellite systems, pseudo-range multilateration systems were often defined as TDOA systems i.e., systems that measured TDOAs or formed TDOAs as the first step in processing a set of measured TOAs. However, as result of deployment of GNSSs, two issues arose: What system type are GNSSs ? What are the defining characteristic of a pseudo-range multilateration system?- The technical answer to has long been known: GNSSs are a variety of multilateration navigation systems having moving transmitters. However, because the transmitters are synchronized not only with each other but also with a time standard, GNSS receivers are also sources of timing information. This requires different solution algorithms than TDOA systems. Thus, a case can also be made that GNSSs are a separate category of systems.
- There is no authoritative answer to. However, a reasonable two-part answer is a system whose only measurements are TDOAs or TOAs ; and a system whose station clocks must be synchronized. This definition is used here, and includes GNSSs as well as TDOA systems. TDOA systems are explicitly hyperbolic while TOA systems are implicitly hyperbolic.
Principle
Frequencies and waveforms
Pseudo-range multilateration navigation systems have been developed utilizing a variety of radio frequencies and waveforms — low-frequency pulses ; low-frequency continuous sinusoids ; high-frequency continuous wide-band. Pseudo-range multilateration surveillance systems often use existing pulsed transmitters — e.g., Shot-Spotter, ASDE-X and WAM.Coordinate frame
Virtually always, the coordinate frame is selected based on the wave trajectories. Thus, two- or three-dimensional Cartesian frames are selected most often, based on straight-line wave propagation. However, polar frames are sometimes used, to agree with curved earth-surface wave propagation paths. Given the frame type, the origin and axes orientation can be selected, e.g., based on the station locations. Standard coordinate frame transformations may be used to place results in any desired frame. For example, GPS receivers generally compute their position using rectangular coordinates, then transform the result to latitude, longitude and altitude.TDOA formation
Given received signals, TDOA systems form differences of TOA pairs. All received signals must be a member of at least one TDOA pair, but otherwise the differences used are arbitrary. Thus, when forming a TDOA, the order of the two TOAs involved is not important.Some operational TDOA systems designate one station as the "master" and form their TDOAs as the difference of the master's TOA and the "secondary" stations' TOAs. When, there are possible TDOA combinations, each corresponding to a station being the de facto master. When, there are possible TDOA sets, of which do not have a de facto master. When, there are possible TDOA sets, of which do not have a de facto master.
TDOA principle / surveillance
If a pulse is emitted from a vehicle, it will generally arrive at slightly different times at spatially separated receiver sites, the different TOAs being due to the different distances of each receiver from the vehicle. However, for given locations of any two receivers, a set of emitter locations would give the same time difference. Given two receiver locations and a known TDOA, the locus of possible emitter locations is one half of a two-sheeted hyperboloid.In simple terms, with two receivers at known locations, an emitter can be located onto one hyperboloid. Note that the receivers do not need to know the absolute time at which the pulse was transmitted only the time difference is needed. However, to form a useful TDOA from two measured TOAs, the receiver clocks must be synchronized with each other.
Consider now a third receiver at a third location which also has a synchronized clock. This would provide a third independent TOA measurement and a second TDOA. The emitter is located on the curve determined by the two intersecting hyperboloids. A fourth receiver is needed for another independent TOA and TDOA. This will give an additional hyperboloid, the intersection of the curve with this hyperboloid gives one or two solutions, the emitter is then located at one of the two solutions.
With four synchronized receivers there are 3 independent TDOAs, and three independent parameters are needed for a point in three dimensional space..
With additional receivers enhanced accuracy can be obtained..
For an over-determined constellation a least squares method can be used for 'reducing' the errors. Averaging over longer times can also improve accuracy.
The accuracy also improves if the receivers are placed in a configuration that minimizes the error of the estimate of the position.
The emitter may, or may not, cooperate in the multilateration surveillance process. Thus, multilateration surveillance is used with non-cooperating "users" for military and scientific purposes as well as with cooperating users.
TDOA principle / navigation
Multilateration can also be used by a single receiver to locate itself, by measuring signals emitted from synchronized transmitters at known locations. At least three emitters are needed for two-dimensional navigation ; at least four emitters are needed for three-dimensional navigation. Although not true for real systems, for expository purposes, the emitters may be regarded as each broadcasting narrow pulses at exactly the same time on separate frequencies. In this situation, the receiver measures the TOAs of the pulses. In actual TDOA systems, the received signals are cross-correlated with an undelayed replica to extract the pseudo delay, then differenced with the same calculation for another station and multiplied by the speed of propagation to create range differences.Several methods have been implemented to avoid self-interference. A historic example is the British Decca system, developed during World War II. Decca used the phase-difference of three transmitters. Later, Omega elaborated on this principle. For Loran-C, introduced in the late 1950s, all transmitters broadcast pulses on the same frequency with different, small time delays. GNSSs continuously transmitting on the same carrier frequency modulated by different pseudo random codes.
TOT principle
The TOT concept is illustrated in Figure 2 for the surveillance function and a planar scenario. Aircraft A, at coordinates, broadcasts a pulse sequence at time. The broadcast is received at stations, and at times, and respectively. Based on the three measured TOAs, the processing algorithm computes an estimate of the TOT, from which the range between the aircraft and the stations can be calculated. The aircraft coordinates are then found.When the algorithm computes the correct TOT, the three computed ranges have a common point of intersection which is the aircraft location. If the computed TOT is after the actual TOT, the computed ranges do not have a common point of intersection. It is clear that an iterative TOT algorithm can be found. In fact, GPS was developed using iterative TOT algorithms. Closed-form TOT algorithms were developed later.
TOT algorithms became important with the development of GPS. GLONASS and Galileo employ similar concepts. The primary complicating factor for all GNSSs is that the stations move continuously relative to the Earth. Thus, in order to compute its own position, a user's navigation receiver must know the satellites' locations at the time the information is broadcast in the receiver's time scale. To accomplish this: satellite trajectories and TOTs in the satellites' time scales are included in broadcast messages; and user receivers find the difference between their TOT and the satellite broadcast TOT. GPS satellite clocks are synchronized to UTC, as well as with each other. This enables GPS receivers to provide UTC time in addition to their position.