Analysis: The coherent baseband discriminator function is defined as
The signal component of the normalized discriminator function
It can be seen from Figure 38.42 that the discriminator function can be approximated by a linear function for small values of Δτk, given by
where kSSS is the slope of the discriminator function at Δτk = 0, which is obtained by
The mean and variance of Dk can be obtained from Eq. (38.26) as
(38.27)
(38.28)
Closed‐Loop Analysis: In a rate‐aided DLL, the pseudorange rate estimated by the FLL‐assisted PLL is added to the output of the DLL discriminator. In general, it is enough to use a first‐order loop for the DLL loop filter since the FLL‐assisted PLL’s pseudorange rate estimate is accurate. The closed‐loop‐error time update for a first‐order loop is shown to be [57]
where Bn, DLL is the DLL noise‐equivalent bandwidth, and KL is the loop gain. To achieve the desired loop noise‐equivalent bandwidth, KL must be normalized according to
Using Eq. (38.13), the loop noise gain for a coherent baseband discriminator becomes
.
Assuming zero‐mean tracking error, that is,
At steady state, var{Δτ} = var {Δτk + 1} = var {Δτk}; hence,
From Eq. (38.30), it can be seen that the standard deviation of the ranging error is related to the correlator spacing through g(teml). Figure 38.43 shows g(teml) for 0 ≤ teml ≤ 2. It can be seen that g(teml) is not a linear function, and it increases significantly faster when teml > 1. Therefore, to achieve a relatively high ranging precision, teml must be set to be less than 1. It is worth mentioning that for the GPS C/A code with an infinite bandwidth, g(teml) = teml.
Figure 38.44 shows the pseudorange error of a coherent DLL as a function of C/N0, with Bn, DLL = {0.005, 0.05} Hz and teml = {0.25, 0.5, 1, 1.5, 2}. It is worth mentioning that in Figure 38.44, the bandwidth is chosen so as to enable the reader to compare the results with the standard GPS results provided in [55].
38.6.3.2 Non‐Coherent DLL Tracking
In a typical DLL, the correlation of the received signal with the early, prompt, and late locally generated signals at time t = kTsub are calculated according to
where x can be either e, p, or l representing early, prompt, or late correlations, respectively. Figure 38.45 represents the general structure of the DLL. This subsection studies the code phase error with two non‐coherent discriminators: dot‐product and early‐power‐minus‐late‐power.
Figure 38.43 The standard deviation of the ranging error Δτ is related to the correlator spacing through g(teml), which is shown as a function of teml (Shamaei et al. [73]).
Source: Reproduced with permission of IEEE, European Signal Processing Conference.
Figure 38.44 Coherent baseband discriminator noise performance as a function of C/N0 for different teml values. Solid and dashed lines represent the results for Bn, DLL = 0.05 Hz and Bn, DLL = 0.005 Hz, respectively (Shamaei et al. [73]).
Source: Reproduced with permission of IEEE, European Signal Processing Conference.
