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

Discretizing the dynamics ( Eq. (38.1)) at a sampling interval T yields the discrete‐time‐equivalent model

where is a discrete‐time zero‐mean white noise sequence with covariance Q clk, and

(38.2)

Table 38.2 Typical h 0and h −2values for different OCXOs [36]

Source: J. Curran, G. Lachapelle, and C. Murphy, “Digital GNSS PLL design conditioned on thermal and oscillator phase noise,” IEEE Transactions on Aerospace and Electronic Systems , vol. 48, no. 1, pp. 180–196, January 2012.

BTS positions can be readily obtained via several methods, for example, (i) from cellular BTS databases (if available) or (ii) by deploying multiple mapping receivers with knowledge of their own states, estimating the position states of the BTSs for a sufficiently long period of time [37–39]. These estimates are physically verifiable via surveying or satellite images. Unlike BTS positions, which are static, the clock error states are stochastic and dynamic, as discussed in Section 38.3, and are difficult to verify.

Estimating the BTSs’ states can be achieved via two frameworks:

Mapper/NavigatorThis framework comprises (i) receiver(s) with knowledge of their own states, referred to as mapper(s), making measurements on ambient BTSs (e.g. pseudorange and carrier phase). The mappers’ role is to estimate the cellular BTSs’ states. (ii) A receiver with no knowledge of its own states, referred to as the navigator, making measurements on the same ambient BTSs to estimate its own states, while receiving estimates of the BTSs’ states from the mappers.

Radio SLAMIn this framework, the receiver maps the BTSs simultaneously with localizing itself in the radio environment.

To make the estimation problems associated with the above frameworks observable, certain a priori knowledge about the BTSs’ or receiver’s states must be satisfied [27, 40–42]. For simplicity, a planar environment will be assumed, with the receiver and BTS three‐dimensional (3D) position states appropriately projected onto such a planar environment. The state of the receiver is defined as , where is the position vector of the receiver, δt ris the receiver’s clock bias, and c is the speed of light. Similarly, the state of the i ‐th BTS is defined as , where is the position vector of the i ‐th BTS, and is its clock bias. The pseudorange measurement to the i ‐th BTS, ρ i, can be expressed as

(38.3)

where and v iis the measurement noise, which is modeled as a zero‐mean Gaussian random variable with variance [27]. The following sections outline the calculations associated with each navigation framework assuming pseudorange measurements from cellular towers. Frameworks with carrier phase measurements are discussed in [43].

Assuming that the receiver is drawing pseudoranges from N ≥ 3 BTSs with known states, the receiver’s state can be estimated from ( Eq. (38.3)) by solving a weighted nonlinear least‐squares (WNLS) problem. However, in practice, the BTSs’ states are unknown , in which case the mapper/navigator framework can be employed [18, 25].

Consider a mapper with knowledge of its own state vector (by having access to GNSS signals, for example) to be present in the navigator’s environment as depicted in Figure 38.3.

The mapper’s objective is to estimate the BTSs’ position and clock bias states and share these estimates with the navigator through a central database. For simplicity, assume the position states of the BTSs to be known and stored in a database. In the sequel, it is assumed that the mapper is producing an estimate and an associated estimation error variance for each of the BTSs.

Consider M mappers and N BTSs. Denote the state vector of the j ‐th mapper by , the pseudorange measurement by the j ‐th mapper on the i‐ th BTS by , and the corresponding measurement noise by . Assume to be independent for all i and j with a corresponding variance . Define the set of measurements made by all mappers on the i ‐th BTS as

Figure 38.3 Mapper and navigator in a cellular environment (Khalife et al. [18]; Khalife and Kassas [25]).