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

Figure 38.53 Experimental hardware setup and environment layout in Riverside, California, showing eNodeBs’ locations and the traversed trajectory as estimated by GPS and LTE signals.

      Map data: Google Earth (Shamaei et al. [65]). Source: Reproduced with permission of IEEE.

38.7 BTS Sector Clock Bias Mismatch

      A typical BTS transmits into three different sectors within a particular cell. Ideally, all sectors’ clocks should be driven by the same oscillator, which implies that the same clock bias (after correcting for the PN offset) should be observed in all sectors of the same cell. However, factors such as the unknown separation between the phase centers of the sector antennas and delays due to RF connectors and other components (e.g. cabling, filters, amplifiers) cause the clock biases corresponding to different BTS sectors to be slightly different. This behavior was consistently observed experimentally in different locations, at different times, and for different cellular providers [18, 22]. In the following sections, a stochastic dynamic model for the observed clock bias mismatch for different sectors of the same BTS cell is derived.

      38.7.1 Sector Clock Bias Mismatch Detection

In order to demonstrate the presence of the discrepancy between sectors’ clock biases, a cellular CDMA receiver was placed at the border of two sectors of the i‐th BTS cell corresponding to the US cellular provider Verizon Wireless and was drawing pseudorange measurements from both sector antennas. The receiver had full knowledge of its state and of the BTS’s position. Subsequently, the receiver solved for the BTS clock biases and observed in sectors p_i and q_i, respectively. A realization of and is depicted in Figure 38.54.

      Figure 38.54 suggests that the clock biases and can be related through

      where ɛ_i is a random sequence that models the discrepancy between the sectors’ clock biases and

      is the indicator function.

      Note that the cdma2000 protocol requires all PN offsets to be synchronized to within 10 μs from GPS time; however, synchronization to within 3 μs is recommended [80]. Since each sector of a BTS uses a different PN offset, then the clock biases and will be bounded according to −10μs and −10μs . Therefore, ɛ_i will be within 20 μs from GPS time, namely

The discrepancy between the clock biases observed in two different sectors of some BTS cell over a 24 hour period is shown in Figures 38.55(a)–(b) for two different BTSs. Both BTSs pertained to the US cellular provider Verizon Wireless and are located near the University of California–Riverside campus. The cellular signals were recorded between September 23 and 24, 2016. It can be seen from Figure 38.55 that ∣ɛ_i∣ is bounded by approximately 2.02 μs and 0.65 μs, respectively, which is well below 20 μs. In the following subsection, a stochastic dynamic model for ɛ_i is identified.

      38.7.2 Sector Clock Bias Discrepancy Model Identification

      The discrepancy for p_i ≠ q_i adheres to an autoregressive (AR) model of order n, which can be expressed as [81]

where ζ_i is a white sequence. The objective is to find the order n and the coefficients that will minimize the sum of the squared residuals . To find the order n, several AR models were identified, and for a fixed order, a least‐squares estimator was used to solve for . It was noted that the sum of the squared residuals corresponding to each n ∈ {1, …, 10} were comparable, suggesting that the minimal realization of the AR model is of first order. For n = 1, it was found that a_{i, 1} = − (1 − β_i), where 0 < β_i < < 1 (on the order of 8 × 10⁻⁵ to 3 × 10⁻⁴). This implies that ɛ_i is exponentially correlated with the continuous‐time dynamics given by

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