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

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.

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 iand q i, respectively. A realization of and is depicted in Figure 38.54.

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

where ɛ iis 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 −10 μ s and −10 μ s . Therefore, ɛ iwill 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 38.55that ∣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 iis identified.

The discrepancy for p i≠ q iadheres to an autoregressive (AR) model of order n , which can be expressed as [81]

where ζ iis 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 −5to 3 × 10 −4). This implies that ɛ iis exponentially correlated with the continuous‐time dynamics given by

Figure 38.54 (a) A cellular CDMA receiver placed at the border of two sectors of a BTS cell, making pseudorange observations on both sector antennas simultaneously. The receiver has knowledge of its own states and has knowledge of the BTS position states. (b) Observed BTS clock bias corresponding to two different sectors from a real BTS (Khalife et al. [12, 18]; Khalife and Kassas [23, 25]).

Source: Reproduced with permission of IEEE.

Figure 38.55 The discrepancies ɛ 1and ɛ 2between the clock biases observed in two different sectors of some BTS cell over a 24 hour period. (a) and (b) correspond to ɛ 1and ɛ 2for BTSs 1 and BTS 2, respectively. 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 that ∣ɛ i∣ is well below 20 μs (Khalife et al. [12]).