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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

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 ∣ɛ_i∣ is well below 20 μs (Khalife et al. [12]).

Source: Reproduced with permission of IEEE.

Figure 38.56 (a) A realization of the discrepancy ɛ_i between the observed clock biases of two BTS sectors and (b) the corresponding residual ζ_i (Khalife et al. [12]; Khalife and Kassas [23]).

Source: Reproduced with permission of IEEE.

(38.45)

where , τ_i is the time constant of the discrepancy dynamic model, and is a continuous‐time white process with variance . Discretizing Eq. (38.45) at a sampling period T yields the discrete‐time model

(38.46)

where . The variance of ζ_i is given by . Figure 38.56 shows an experimental realization of ɛ_i and the corresponding residual ζ_i.

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 e_i defined as the difference between the measured data ɛ′_i and predicted value from the identified model ɛ_i in Eq. (38.46), that is, e_i ≜ ɛ_i ′ − ɛ_i, were computed [81]. Figure 38.57 shows the acf and psd of e_i computed from a different realization of ɛ_i. The psd was computed using Welch’s method [82]. It can be seen from Figure 38.57 that the residual error e_i is nearly white; hence, the identified model is capable of describing the true system.

Next, the probability density function (pdf) of ζ_i will be characterized, assuming that ζ_i is an ergodic process. It was found that the Laplace distribution best matches the actual distribution of ζ_i obtained from experimental data; that is, the pdf of ζ_i is given by

(38.47)

where μ_i is the mean of ζ_i, and λ_i is the parameter of the Laplace distribution, which can be related to the variance by . A maximum likelihood estimator (MLE) was adopted to calculate the parameters μ_i and λ_i of p(ζ_i) [83]. Figure 38.58 shows the actual distribution of the data along with the estimated pdf. For comparison purposes, a Gaussian and Logistic pdf fits obtained via an MLE are also plotted.

Figure 38.57 The (a) acf and (b) psd of e_i 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 ζ_i 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 μ_i ≈ 0; therefore, ζ_i 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 Hz. The process noise driving the discrepancy was calculated from Eq. (38.45) with and T = 0.2 s. The acf of each of the six realizations of ζ_i corresponding to the six realizations of ɛ_i from Figure 38.60 exhibited very quick decorrelation, validating that ζ_i is approximately a white sequence [12]. Also, a histogram of each realization of ζ_i along with the estimated pdf p(ζ_i) demonstrated that the Laplace pdf consistently matched the experimental data [12].

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