Position, Navigation, and Timing Technologies in the 21st Century. Группа авторов

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(38.1)

Figure 38.2 Clock error states dynamics model.

      Source: Z. Kassas, Analysis and synthesis of collaborative opportunistic navigation systems, Ph.D. Dissertation, The University of Texas at Austin, USA, May 2014. Reproduced with permission of Z. Kassas (University of Texas at Austin).

where the elements of are modeled as zero‐mean, mutually independent white noise processes, and the power spectral density of is given by . The power spectra and can be related to the power‐law coefficients , which have been shown through laboratory experiments to be adequate to characterize the power spectral density of the fractional frequency deviation y(t) of an oscillator from the nominal frequency, which takes the form [32, 33]. It is common to approximate the clock error dynamics by considering only the frequency random walk coefficient h₋₂ and the white frequency coefficient h₀, which lead to and [34, 35]. Typical OCXO values for h₀ and h₋₂ are given in Table 38.2.

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 h₀ and h₋₂ values 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.

h 0	h −2
2.6 × 10^−22	4.0 × 10^−26
8.0 × 10^−20	4.0 × 10^−23
3.4 × 10^−22	1.3 × 10^−24

38.4 Navigation Frameworks in Cellular Environments

      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_r is 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_i is 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].

      38.4.1 Mapper/Navigator Framework

Assuming that the receiver is drawing pseudoranges from N ≥ 3 BTSs with known states, the receiver’s state can be estimated