The elevation angles from a receiver at the test site in Foster City to the DTV transmitters are listed in the rightmost column of Table 40.1, which are around 1° (0.8° to 1.1°). The difference between a 3D (slant) range and a 2D (down) range is illustrated in Figure 40.22(a). The range differences to the six stations at 551, 635, 563, 617, 605, and 683 MHz, which vary from 2.6 to 5.6 m, are listed in Table 40.1. Given the transmitter locations, it is possible to perform compensation of 3D slant ranges into down‐ranges (DRs) for 2D positioning in an iterative manner.
In addition to their effect on navigation errors, the geometry and coverage also impact the receiver design (or architecture). DTV signals are strong in transmission power and have a large coverage as compared to other types of SOOP such as cellular phone and Wi‐Fi signals, as shown in Figure 40.22(b). A DTV receiver can simply tune to a station and acquire and maintain track of the station over a large area while roaming. By contrast, cellular signals are weaker and have a smaller coverage. In particular, a receiver may traverse signals from multiple sectors on the same cell tower after just making a simple turn. For a passive listener (i.e. without an advanced notice from the cellular network), the receiver needs to perform a constant search of new signals with quick acquisition and have short spans for signal tracking before switching. This agility requirement increases the receiver complexity. Nevertheless, the presence of a local cell tower or towers can significantly improve the geometry for accurate positioning and availability of the overall navigation solution.
Table 40.1 Geometry from a receiver at (0, 0, 0) to DTV transmitters
| DTV station | Down‐range (2D), m | Slant range (3D), m | Difference, m | Height/DR ratio | Elevation, ° |
|---|---|---|---|---|---|
| 551 | 20648.24 | 20652.12 | 3.87 | 0.0194 | 1.1098 |
| 635 | 20669.40 | 20673.57 | 4.17 | 0.0201 | 1.1502 |
| 563 | 27135.41 | 27137.98 | 2.56 | 0.0137 | 0.7875 |
| 617 | 27128.47 | 27132.79 | 4.32 | 0.0178 | 1.0221 |
| 605 | 35387.07 | 35391.70 | 4.62 | 0.0162 | 0.9261 |
| 683 | 34687.47 | 34693.08 | 5.61 | 0.0180 | 1.0306 |
Figure 40.22 Geometry and coverage on estimation error and receiver complexity.
40.4.2 Radio Dead Reckoning with Mixed SOOP
A radio receiver can relatively easily measure the TOA of a variety of SOOP such as those DTV signals described in Section 40.2 and AF/FM signals [86–89] and cellular signals [46, 90]. But it needs a means of determining the TOT in order to generate range measurements. Once the ranges to signal sources at known locations are available, the receiver location can be determined. In Section 40.3, we describe a calibration method that uses the initial position information, either known a priori as is the case for many navigation systems or from an aiding source (thus cooperative), to determine the TOT and the clock drift as well. As long as the operations of the signal sources and receiver are not interrupted, the one‐time calibration remains valid for subsequent relative positioning. The aiding source for this method can be a digital map, a visual determination at a known road intersection, or a cooperative navigator (either remote or co‐located).
Instead of the absolute position (x, y), a receiver may calculate its position relative to a reference point (x0, y0) as Δx = x – x0 and Δy = y – y0, respectively, which can be understood as a displacement vector (Δx, Δy). Adding successive displacements onto the initial position yields a continuous navigation solution [91], thus making radio dead reckoning. Like a self‐contained inertial navigation solution, the accuracy of a radio dead‐reckoning solution cannot be better than the initial condition. However, unlike the inertial solution, whose errors keep grow due to time integration of the accelerometer bias and gyro drift, the radio dead‐reckoning solution errors may stay bounded due to direct displacement estimation.
Denote the location of the k‐th transmitter by (xk, yk) and the unknown TOT of this transmitter by TOT k. The TOA measurements at the reference point and a subsequent time, denoted by
where c is the speed of light, and
Similar to Eq. 40.1, we have TOT k = nTfield +