(38.29) yields
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 Discriminator The closed‐loop code phase error in a dot‐product discriminator can be obtained by substituting Eqs. 38.34 and 38.36 into Eq. (38.40), yielding
(38.41)
(38.42)
Figure 38.47(a) shows gα(teml) for 0 ≤ teml ≤ 2. It can be seen that gα(teml) is a nonlinear function and increases significantly faster for teml > 1. Figure 38.48 shows the standard deviation of the pseudorange error for a dot‐product DLL as a function of C/N0 with teml = 1 and Bn, 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 gβ(teml) for 0 ≤ teml ≤ 2. It can be seen that gβ(teml) is significantly larger than gα(teml). To reduce the ranging error due to gβ(teml), teml must be chosen to be less than 1.5.
Figure 38.48 shows the standard deviation of the pseudorange error for an early‐power‐minus‐late‐power discriminator DLL as a function of C/N0 with Bn, DLL = {0.05, 0.005} Hz and teml = 1. It can be seen that decreasing the loop bandwidth decreases the standard deviation of the pseudorange error. However, very small values of Bn, DLL may 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 gα(teml) shown in (a), while for an early‐power‐minus‐late‐power discriminator it is related through gα(teml) and gβ(teml) shown in (b) (Shamaei et al. [74]).
Source: Reproduced with permission of IEEE.
Figure 38.48 DLL performance as a function of C/N0 for non‐coherent discriminators: dot‐product discriminator (solid line) and early‐power‐minus‐late‐power discriminator (dashed line), Bn, DLL = {0.05, 0.005} Hz, and teml = 1 (Shamaei et al. [74]).
Source: Reproduced with permission of IEEE.
38.6.3.3 Code Phase Error Analysis in Multipath Environments
Sections 38.6.3.1 and 38.6.3.2 evaluated 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].
38.6.4 Cellular LTE Navigation Experimental Results
This section presents experimental results for navigation with cellular LTE signals. Section 38.6.4.1 analyzes the pseudorange obtained with the SSS and CRS signals produced by the receiver discussed in Section 38.6.2. Sections 38.6.4.2 and 38.6.4.3 present navigation results with aerial and ground vehicles, respectively.
38.6.4.1 Pseudorange Analysis
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 38.50 and 38.51 show (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.
