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

The discriminator function can be approximated by a linear function for small values of Δ τ k(cf. Eq. (38.33)) with

(38.37)

The mean and variance of D kare calculated to be

(38.38)

(38.39)

Closed‐Loop Analysis:An FLL‐assisted PLL produces a reasonably accurate pseudorange rate estimate, making first‐order DLLs sufficient. At steady state, var{Δ τ } = var {Δ τ k + 1} = var {Δ τ k} and using Eq. (38.29)yields

(38.40)

In the following, the closed‐loop statistics of the code phase error are derived for a dot‐product and an early‐power‐minus‐late‐power discriminator functions.

Dot‐Product DiscriminatorThe closed‐loop code phase error in a dot‐product discriminator can be obtained by substituting Eqs. 38.34and 38.36into Eq. (38.40), yielding

(38.41)

(38.42)

Figure 38.47(a) shows g α( t eml) for 0 ≤ t eml≤ 2. It can be seen that g α( t eml) is a nonlinear function and increases significantly faster for t eml> 1. Figure 38.48shows the standard deviation of the pseudorange error for a dot‐product DLL as a function of C / N 0with t eml= 1 and B n, DLL= {0.005, 0.05} Hz, chosen so as to enable comparison with the GPS pseudorange error standard deviation provided in [55, 73].

Early‐Power‐Minus‐Late‐Power Discriminator:The variance of the ranging error in an early‐power‐minus‐late‐power discriminator can be obtained by substituting Eqs. (38.37)and (38.39)into Eq. (38.40), yielding

(38.43)

(38.44)

Figure 38.47(b) shows g β( t eml) for 0 ≤ t eml≤ 2. It can be seen that g β( t eml) is significantly larger than g α( t eml). To reduce the ranging error due to g β( t eml), t emlmust be chosen to be less than 1.5.

Figure 38.48shows the standard deviation of the pseudorange error for an early‐power‐minus‐late‐power discriminator DLL as a function of C / N 0with B n, DLL= {0.05, 0.005} Hz and t eml= 1. It can be seen that decreasing the loop bandwidth decreases the standard deviation of the pseudorange error. However, very small values of B n, DLLmay cause the DLL to lose lock in a highly dynamic scenario.

Figure 38.47 Variance of the ranging error in a dot‐product discriminator is related to the correlator spacing through g α( t eml) shown in (a), while for an early‐power‐minus‐late‐power discriminator it is related through g α( t eml) and g β( t eml) shown in (b) (Shamaei et al. [74]).

Source: Reproduced with permission of IEEE.

Figure 38.48 DLL performance as a function of C / N 0for non‐coherent discriminators: dot‐product discriminator (solid line) and early‐power‐minus‐late‐power discriminator (dashed line), B n, DLL= {0.05, 0.005} Hz, and t eml= 1 (Shamaei et al. [74]).

Source: Reproduced with permission of IEEE.

Sections 38.6.3.1and 38.6.3.2evaluated the ranging accuracy with coherent and non‐coherent baseband discriminators in the presence of additive white Gaussian noise. However, multipath is another significant source of error, particularly for ground receivers. Multipath analysis and mitigation for navigation with LTE signals is an ongoing area of research [3, 11, 15, 19, 63, 73, 74, 76–79].

This section presents experimental results for navigation with cellular LTE signals. Section 38.6.4.1analyzes the pseudorange obtained with the SSS and CRS signals produced by the receiver discussed in Section 38.6.2. Sections 38.6.4.2and 38.6.4.3present navigation results with aerial and ground vehicles, respectively.

This section evaluates the pseudorange obtained by the receiver discussed in Section 38.6.2. To this end, the pseudorange variation from GPS is compared with the pseudorange variation due to (i) only tracking the SSS and (ii) aiding the SSS tracking loops with the CRS. The receiver was mounted on a ground vehicle and was tuned to the carrier frequencies of 1955 and 2145 MHz, which are allocated to the US LTE providers AT&T and T‐Mobile, respectively [63]. The transmission bandwidth was measured to be 20 MHz. The vehicle‐mounted receiver traversed a total trajectory of 2 km while listening to the 2 eNodeBs simultaneously as illustrated in Figure 38.49. The position states of the eNodeBs were mapped beforehand. Figures 38.50and 38.51show (a) the change in the pseudoranges between the receiver and the 2 eNodeBs, (b) the error between the GPS pseudorange and the LTE pseudoranges, and (c) the distance error cumulative distribution function (CDF) of the LTE pseudoranges.

Figure 38.49 LTE environment layout and experimental hardware setup. Map data: Google Earth (Shamaei et al. [63]).

Source: Reproduced with permission of Z. Kassas (International Technical Meeting Conference).

Figure 38.50 (a) Estimated change in pseudorange and estimated CIR at t = 13.04 s for eNodeB 1. The change in the pseudorange was calculated using (1) SSS pseudoranges, (2) SSS+CRS pseudoranges, and (3) true ranges obtained using GPS. (b) Pseudorange error between (1) GPS and SSS and (2) GPS and SSS+CRS. (c) CDF of the error in (b) (Shamaei et al. [63]).