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

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

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(38.45)

where , τ iis 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 iis given by . Figure 38.56shows an experimental realization of ɛ iand 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 idefined as the difference between the measured data ɛ′ iand predicted value from the identified model ɛ iin Eq. (38.46), that is, e i≜ ɛ i′ − ɛ i, were computed [81]. Figure 38.57shows the acf and psd of e icomputed from a different realization of ɛ i. The psd was computed using Welch’s method [82]. It can be seen from Figure 38.57that the residual error e iis nearly white; hence, the identified model is capable of describing the true system.

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

(38.47)

where μ iis the mean of ζ i, and λ iis 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 iand λ iof p ( ζ i) [83]. Figure 38.58shows 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 iwith a sampling frequency of 5 Hz (Khalife et al. [12]; Khalife and Kassas [23]).

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Figure 38.58 Distribution of ζ ifrom 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]).

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It was noted, from several batches of collected experimental data, that μ i≈ 0; therefore, ζ iis 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 38.5and Figure 38.59. The date field in Table 38.5shows the date in which the test was conducted in MM/DD/YYYY format.

Figure 38.60shows six realizations, 5 min each, of the discrepancy corresponding to Tests (a)–(f) in Table 38.5. It can be seen from Figure 38.60that 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 icorresponding to the six realizations of ɛ ifrom Figure 38.60exhibited very quick decorrelation, validating that ζ iis approximately a white sequence [12]. Also, a histogram of each realization of ζ ialong 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

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].