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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 (x^k, y^k) and the unknown TOT of this transmitter by TOT ^k. The TOA measurements at the reference point and a subsequent time, denoted by (with ) and TOA^k(with TOT ^k), respectively, are given by

(40.8a)

(40.8b)

where c is the speed of light, and and w^k are uncorrelated measurement errors assumed to be zero‐mean Gaussian with variances and (σ^k)², respectively. This assumption becomes invalid in the presence of NLOS signals, as further discussed in Section 40.4.3.

Similar to Eq. 40.1, we have TOT ^k = nT_field + + cΔt, with the last term accounting for the unknown clock offset between the transmitter and receiver to estimate. Taking the difference between Eqs. 40.8a and 40.8b gives

(40.8c)

where is the combined measurement error, a Gaussian with zero mean and variance +.

By single difference, Eq. 40.8c eliminates but requires estimation of Δt instead. The equation can be further linearized around (x₀, y₀) as

(40.9a)

(40.9b)

where

(40.9c)

With Eq. 40.9b from at least three non‐co‐located transmitters, an instantaneous solution for (Δx, Δy, Δt) can be obtained via, say, the iterative least squares method. With a kinematic model for the displacement and clock state, a sequential solution for (Δx, Δy, Δt) can be obtained via the extended Kalman filter as formulated in [23]. In other words, from the relative ranges, we can derive displacement vectors as a form of relative positioning. The concept of time‐differencing measurements to a transmitter in order to remove the unknown time of transmission is similar to time differencing of GNSS carrier phase measurements in order to remove ambiguity. It has been applied to GNSS‐only positioning [92] and to GNSS/INS positioning [28, 93].

In the above formulation, a constant nominal period T_field is used. However, a practical radio transmitter is subject to a certain frequency error. Experimental data in Figure 40.20(f) show such a clock drift. While omitted for short data sets, the frequency error can be accounted for in the time‐differencing formulation (Eq. 40.8c) by introducing a slow‐varying drift term per transmitter to be then estimated jointly.

In the present setting of dead reckoning from a known location, the initial stationary TOA measurements can be accumulated into a “deterministic” reference. The time difference with respect to such a reference at a fixed time point as in Eq. (40.8) is akin to applying a well‐calibrated bias to all subsequent measurements, which can then be treated as approximately “uncorrelated.” This initial measurement error, if not calibrated out to an insignificant level, would make subsequent time‐difference measurements time‐correlated. Because it is constant, albeit random, it can be viewed as an unknown bias and modeled as an extra state to the positioning Kalman filter.

Strictly speaking, no matter how well a parameter is calibrated, the subsequent measurements can be time‐correlated via the residual calibration errors. Furthermore, time differencing can also be applied continuously to consecutive measurements, that is, between t+1 and t, rather than between t and t₀ = 0 as in Eq. (40.8). Consecutive time differences between times (t+1 and t) and (t and t−1) are correlated through the common epoch t. The standard Kalman filter cannot be applied to consecutive time‐difference measurements because the measurements are correlated. In [92], a fixed‐lag smoother was used where the previous position is introduced as an extra state. With both the current and previous positions in the filter, the position difference becomes directly observable by the time differences of carrier phase measured at the current and previous epochs [92]. A similar formulation can be used for SOOP when processing consecutive time differences of phase and/or range measurements.

In the field test environment depicted in Figure 40.17, a radio receiver with seven synchronous channels is installed in a minivan together with a battery‐powered data recording system. One channel is assigned to GPS L1, which serves as the reference; one channel to a CDMA cell tower in the PCS band; and the other five channels to five DTV stations. One DTV station (605 MHz) is on Monument Peak in the south, four DTV stations (563, 617, 623, and 647 MHz) are on Sutro Tower in the north, and one CDMA cell tower (1933.75 MHz) is also in the north. Since we were driving from the south to north, the range to the DTV station on Monument Peak was increasing while the ranges to the other six sources in the north were decreasing.

Figure 40.23(a) shows the relative ranges to the DTV and CDMA sources. The vehicle was stationary for the first 40 s and then moving about 300 m in 50 s (at a speed of 15 mph on average). As shown in Figure 40.23(b), the initial transients settled after 0.5 s for the DTV channels and after 3 s for the CDMA channel. Except for the DTV channel at 617 MHz, which exhibits large variations (−5 to 10 m), the others are small (±2 m). In fact, the standard deviations during the stationary period are 0.78, 0.88, 1.67, 4.13, 1.15, and 1.49 m, respectively.

Figure 40.23(c) shows the relative ranges during motion. Due to fast fading, the range measurements are noisier in motion than during stationary (about four times larger). As discussed above, the range to the DTV channel at 605 MHz on Monument Peak (in the south) is opposite to the other ranges (in the north), with a maximum peak‐to‐peak variation of 70 m but mostly of 20 m. The DTV channel at 647 MHz shows a sharp change of 120 m after moving and toward the end. The other three DTV channels on Sutro Tower show similar behaviors. The largest separation in range is about 50 m, which is consistent with their distances calculated from their coordinates. It is remarkable to note that the CDMA range also fits nicely with the DTV ranges.

In the following processing, we excluded the DTV channel at 647 MHz due to large deviations. Figure 40.23(d) shows the least squares (LS) solution (green) based on the raw relative range measurements shown in Figure 40.23(a), the Kalman filter (KF) estimate (red), and the GPS solution (blue). As explained in Section 40.4.1, large errors are expected in the cross‐range direction due to the poor geometry.

Figure 40.23 Radio dead-reckoning (relative positioning) with mixed signals of opportunity (SOOP).

Figure 40.23(e) shows the LS solution (green) after averaging with an equivalent interval of 1 s to be compatible with the GPS solution at 1 Hz. Large deviations in the middle of trajectory are likely caused by the range measurements of the DTV channel at 605 MHz. After a threshold testing (±15 m) applied to the relative range predictions to remove measurement outliers, the large deviations appearing in Figure 40.23(e) are eliminated, and the resulting trajectory is shown in Figure 40.23(f). Averaged along the trajectory, the mean error of the SOOP‐alone solution versus the GPS‐alone solution is 1.68 m with a standard deviation of 3.78 m and a maximum error of about 12 m.