Pseudo-range multilateration
For example, GPS receivers generally compute their position using rectangular coordinates, then transform the result to latitude, longitude and altitude.
(Specifically, for GPS and other GNSSs, the atmosphere does influence the traveling time of the signal and more satellites does give a more accurate location).
Although not true for real systems, for expository purposes, the emitters may be regarded as each broadcasting narrow pulses (ideally, impulses) at exactly the same time on separate frequencies (to avoid interference).
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.
GNSSs continuously transmitting on the same carrier frequency modulated by different pseudo random codes (GPS, Galileo, revised GLONASS).
With the advent of GPS and subsequently other satellite navigation systems: (1) TOT as known to the user receiver provides necessary and useful information; and (2) computing power had increased significantly.
The latter is applicable to low-frequency radio waves, which follow the earth's surface; the former applies to higher frequency (say, greater than one megahertz) and to shorter ranges (hundreds of miles).
(A variation: air traffic control multilateration systems use the Mode C SSR transponder message to find an aircraft's altitude.
Steven Bancroft was apparently the first to publish a closed-form solution to the problem of locating a user (e.g., vehicle) in three dimensions and the common TOT using four or more TOA measurements.
A more refined technique involves directly solving a "constrained least squares" problem, while also addressing modified noise statistics.
The constrained least squares solution for TDOA systems was apparently initially proposed by Huang et al.[18] and further explored by subsequent researchers.
Robust version such as the "constrained least absolute deviations" is also discussed and shows superior performance to least squares in scenarios involving non-Gaussian noise and contamination from outlier measurements.
[9][23] Multilateration systems and studies employing spherical-range measurements (e.g., Loran-C, Decca, Omega) utilized a variety of solution algorithms based on either iterative methods or spherical trigonometry.
[11][9] Examples of 2-D Cartesian multilateration systems are those used at major airports in many nations to surveil aircraft on the surface or at very low altitudes.
When necessitated by the combination of vehicle-station distance (e.g., hundreds of miles or more) and required solution accuracy, the ellipsoidal shape of the Earth must be considered.
Chapter 15 in Numerical Recipes[32] describes several methods to solve linear equations and estimate the uncertainty of the resulting values.
Almost always, such systems implement: (a) a transient 'acquisition' (surveillance) or 'cold start' (navigation) mode, whereby the user's location is found from the current measurements only; and (b) a steady-state 'track' (surveillance) or 'warm start' (navigation) mode, whereby the user's previously computed location is updated based current measurements (rendering moot the major disadvantage of iterative methods).
unknown quantities – e.g., 5 or more GPS satellite TOAs – the iterative Gauss–Newton algorithm for solving non-linear least squares (NLLS) problems is often preferred.
Except for pathological station locations, an over-determined situation eliminates possible ambiguous and/or extraneous solutions that can occur when only the minimum number of TOA measurements are available.
Additionally, a configuration of the sensors that minimizes a metric obtained from the Cramér–Rao bound can be chosen so as to optimize the actual position estimation of the target in a region of interest.
To illustrate, consider a hypothetical two-station surveillance system that monitors the location of a railroad locomotive along a straight stretch of track—a one dimensional situation
Simply put, inside the stations' perimeter, consecutive TDOAs will typically amplify but not double vehicle movement
When analyzing a 2D or 3D multilateration system, dilution of precision (DOP) is usually employed to quantify the effect of user-station geometry on position-determination accuracy.
conveys the notion that there are multiple "flavors" of DOP – the choice depends upon the number of spatial dimensions involved and whether the error for the TOT solution is included in the metric.
That is, ?DOP is the rate of change of the standard deviation of a solution quantity from its correct value due to measurement errors – assuming that a linearized least squares algorithm is used.
Figure 5 illustrates the approximate service area of two-dimensional multilateration system having three stations forming an equilateral triangle.
Their roughly equal spacing (outside of the three V-shaped areas between the baseline extensions) is consistent with the rapid growth of the horizontal position error with distance from the stations.
The multilateration technique was apparently first used during World War I to locate the source of artillery fire using audible sound waves (TDOA surveillance).
Multilateration surveillance is related to passive towed array sonar target localization (but not identification), which was also first used during World War I.
Longer distance radio-based navigation systems became viable during World War II, with the advancement of radio technologies.
