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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]).

Source: Reproduced with permission of IEEE, ION.

where and . The clock bias is estimated by solving a weighted least‐squares (WLS) problem, resulting in the estimate

and the associated estimation error variance , where W is the weighting matrix. The true clock bias of the i‐th BTS can now be expressed as , where w_i is a zero‐mean Gaussian random variable with variance .

Since the navigating receiver is using the estimate of the BTS clock bias, which is produced by the mapping receiver, the pseudorange measurement made by the navigating receiver on the i‐th BTS becomes

where and η_i ≜ v_i − w_i models the overall uncertainty in the pseudorange measurement. Hence, the vector is a zero‐mean Gaussian random vector with a covariance matrix ∑ = C + R, where is the covariance matrix of and is the covariance of the measurement noise vector . The Jacobian matrix H of the nonlinear measurements with respect to is given by , where

The navigating receiver’s state can now be estimated by solving a WNLS problem. The WNLS equations are given by

where l is the iteration number, and denotes the nonlinear measurements evaluated at the current estimate .