sectors from a real BTS."/>
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 ɛ1 and ɛ2 between the clock biases observed in two different sectors of some BTS cell over a 24 hour period. (a) and (b) correspond to ɛ1 and ɛ2 for 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 ∣ɛi∣ is well below 20 μs (Khalife et al. [12]).
Source: Reproduced with permission of IEEE.
Figure 38.56 (a) A realization of the discrepancy ɛi between the observed clock biases of two BTS sectors and (b) the corresponding residual ζi (Khalife et al. [12]; Khalife and Kassas [23]).
Source: Reproduced with permission of IEEE.
where
where
Residual analysis is used to validate the model (Eq. (38.46)). To this end, the autocorrelation function (acf) and power spectral density (psd) of the residual error ei defined as the difference between the measured data ɛ′i and predicted value from the identified model ɛi in Eq. (38.46), that is, ei ≜ ɛi ′ − ɛi, were computed [81]. Figure 38.57 shows the acf and psd of ei computed from a different realization of ɛi. The psd was computed using Welch’s method [82]. It can be seen from Figure 38.57 that the residual error ei is nearly white; hence, the identified model is capable of describing the true system.
Next, the probability density function (pdf) of ζi will be characterized, assuming that ζi is an ergodic process. It was found that the Laplace distribution best matches the actual distribution of ζi obtained from experimental data; that is, the pdf of ζi is given by
(38.47)
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
