Position, Navigation, and Timing Technologies in the 21st Century

Once the beacon pilot/data signal acquisition is complete, the range measurements as well as the trilateration data can be extracted. Ranging is normally done using the pilot section with known modulation, but can also be done using the data section.

Estimating the TOA in terrestrial channels from the beacon is a different challenge when compared to a GPS satellite channel. The channel responses are quite complex due to blockage, diffraction, and reflection from a variety of obstacles creating a mix of LoS and NLoS paths. Figure 39.14shows sample measured correlation functions measured at the receiver when using a direct‐sequence spread MBS transmission waveform of bandwidth 2MHz. The measurements were carried out in outdoor rooftop locations. The purpose of these figures is to illustrate various common channel scenarios in static outdoor terrestrial scenarios. In the figures, the red vertical line represents the TOA of the true LOS path, whereas the green line represents the TOA of the detected earliest path in the receiver. The x ‐axis represents distance in meters, and the y ‐axis represents the magnitude of the correlation function. Figure 39.14(a) shows the measured correlation function in the case where the LOS is clearly detectable so that the green and red vertical lines overlap each other. Figure 39.14(b) shows a correlation function for the NLoS scenario with a strong early NLoS path, and Figure 39.14(c) shows a correlation function for the NLoS scenario with a weak early NLoS path. Note that in both cases the earliest path is not detectable, as shown by the green vertical line (representing the estimated TOA of the earliest detectable path) being to the right of the red vertical marker (true LoS TOA). Observe that in Figure 39.14(b), the earliest detectable path is actually stronger, whereas in Figure 39.14(c), the earliest detectable path is actually weaker.

Figure 39.13 (a) Shows the MBS preamble search space and (b) shows the MBS beacon search space after preamble detection.

Figure 39.14 (a) Shows a correlation function for a scenario with detectable LoS path, (b) shows a correlation function for the NLoS scenario with a strong early NLoS path, (c) shows a correlation function for the NLoS scenario with a weak early NLoS path.

Some additional examples of measured channel correlation functions are shown in Figure 39.15The x ‐axis represents the correlation lag in units of 122 ns (which corresponds to the time duration of a sample when using the sampling rate = 8 × 1.023 MHz chipping rate). From the various plots, a wide variety of channel spreads and types of channels is observed.

Note that there are cases where the earliest path is weaker than the multipath. In order to retain the channel information, a simple two/three tap early‐late‐prompt correlation will not suffice. A multi‐delay correlation function, as shown in Figure 39.15, is required from the receiver to facilitate accurate ranging.

The channel spread statistics help determine the width of the TOA detection correlation window required on the receiver. The choice of window size directly affects the receiver complexity. In order to help this analysis, the percentage of detectable paths within a certain delay (expressed in meters) relative to the signal peak can be analyzed. Note that the simplest way is to center the window using the signal peak as the center of the window. Figure 39.16shows the channel spread statistics obtained using real measurements for different environments in the San Francisco Bay Area including suburban, urban, and dense urban. The results show that in the suburban environment, a correlation window that includes ±900 m includes 98% of the paths, whereas in a dense urban environment the same window includes only 90% of the paths. 100% percent of paths in all environments fit within the ±1800 m correlation window.

In order to get the best performance in a positioning system, the ranges should correspond to the LoS or the earliest arriving detectable path in the channel response to minimize range bias errors. The MBS system link budget and beacon network plan facilitate high‐resolution range determination to determine the earliest detectable path since the signals are designed to have higher SNRs as compared to GPS systems.

The MBS system facilitates accurate 3D position computation. Since the MBS is a network of tightly synchronized beacons, trilateration can be done using pseudoranges determined from time‐stamped TOA measurements from the beacons and the beacon coordinates available from the beacon data.

The range equation in 3D space from the receiver to the transmitter is given by

(39.1)

The location of the transmitters is given by (x i, y i, z i), and the unknown location of the mobile units is given by (X, Y, Z) in some local coordinate frame. The pseudorange measurement has a receiver time bias additive term as well, so that the usual pseudorange measurement equation can be written as

(39.2)

where c is the speed of light, and Δt corresponds to the receiver time bias. Traditionally, a minimum of four pseudorange measurements would be required for 3D trilateration to solve for the four variables: X, Y, Z, and receiver time bias. In a terrestrial network, estimating the Z coordinate through trilateration is error prone due to limited VDOP. When the z ‐axis is available through barometric techniques, a minimum of three pseudorange measurements is sufficient for 3D trilateration.

There is another aspect of trilateration that is quite different for a terrestrial system of beacons when compared to a GPS satellite system. In a GPS system, traditionally, the trilateration problem is linearized to a weighted least squares (WLS) problem. The linearization works well due to the large distance of the satellites relative to the receiver. In the terrestrial system, the case of when a receiver is close to a beacon has to be carefully considered. In such conditions, locally linearized algorithms can suffer from position divergence. In general, the best estimate of the receiver location when altitude aiding is available can be obtained as the set of (X,Y,Z,Δt) that minimizes the objective function

Figure 39.15 Sample channel responses from MBS beacons.

Figure 39.16 Channel spread statistics.

(39.3)