where μi is the mean of ζi, and λi is the parameter of the Laplace distribution, which can be related to the variance by
Figure 38.57 The (a) acf and (b) psd of ei with a sampling frequency of 5 Hz (Khalife et al. [12]; Khalife and Kassas [23]).
Source: Reproduced with permission of IEEE.
Figure 38.58 Distribution of ζi from experimental data and the estimated Laplace pdf via MLE. For comparison, a Gaussian (dashed) and Logistic (dotted) pdf fits are also plotted (Khalife et al. [12]; Khalife and Kassas [23]).
Source: Reproduced with permission of IEEE.
It was noted, from several batches of collected experimental data, that μi ≈ 0; therefore, ζi is appropriately modeled as a zero‐mean white Laplace‐distributed random sequence with variance
The identified model was consistent at different locations, at different times, and for different cellular providers. To demonstrate this, tests were performed twice at three different locations. There was a six‐day period between each test at each of the three locations. A total of three carrier frequencies were considered, two of them pertaining to Verizon Wireless and one to Sprint. The test scenarios are summarized in Table 38.5 and Figure 38.59. The date field in Table 38.5 shows the date in which the test was conducted in MM/DD/YYYY format.
Figure 38.60 shows six realizations, 5 min each, of the discrepancy corresponding to Tests (a)–(f) in Table 38.5. It can be seen from Figure 38.60 that the behavior of the discrepancy is consistent across the tests. The initial discrepancy is subtracted out so that all realizations start at the origin. The inverse of the time constant for each realization was found to be
Table 38.5 Test dates, locations, and carrier frequencies
| Test | Date | Location | Frequency (MHz) | Provider |
|---|---|---|---|---|
| (a) | 01/14/2016 | 1 | 882.75 | Verizon |
| (b) | 01/20/2016 | 1 | 882.75 | Verizon |
| (c) | 08/28/2016 | 2 | 883.98 | Verizon |
| (d) | 09/02/2016 | 2 | 883.98 | Verizon |
| (e) | 08/28/2016 | 3 | 1940.0 | Sprint |
| (f) | 09/02/2016 | 3 | 1940.0 | Sprint |
38.7.3 PNT Estimation Performance in the Presence of Clock Bias Discrepancy
The clock bias discrepancy will degrade the navigation solution in two scenarios: (i) whenever the receiver is receiving signals from both sector antennas within a BTS cell and (ii) whenever the receiver is exchanging pseudorange measurements with another receiver in a different sector (e.g. in a mapper/navigator framework or a collaborative navigation framework). A practical upper bound on the introduced error in the navigation solution due to this discrepancy as well as theoretical lower bounds on the estimation error covariance for static and batch estimators are derived in [25].
Figure 38.59 Locations of the cellular CDMA BTSs: Colton, California; Riverside, California; and the University of California–Riverside (UCR).
Map data: Google Earth (Khalife et al. [12]; Khalife and Kassas [22]). Source: Reproduced with permission of Institute of Navigation, IEEE.
Figure 38.60 Six realizations, five minutes each, of the sector clock bias discrepancy for the tests in Table 38.5 (Khalife et al. [12]; Khalife and Kassas [22]).
Source: Reproduced with permission of Institute