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

Figure 38.41 Structure of a DLL employing a coherent baseband discriminator to track the code phase (Shamaei et al. [73]).

Source: Reproduced with permission of IEEE, European Signal Processing Conference.

Figure 38.42 Output of the coherent baseband discriminator function for the SSS with different correlator spacing (Shamaei et al. [73]).

Source: Reproduced with permission of IEEE, European Signal Processing Conference.

Open‐Loop Analysis:The coherent baseband discriminator function is defined as

The signal component of the normalized discriminator function is shown in Figure 38.42for t eml= {0.25, 0.5, 1, 1.5, 2}.

It can be seen from Figure 38.42that the discriminator function can be approximated by a linear function for small values of Δ τ k, given by

(38.26)

where k SSSis the slope of the discriminator function at Δ τ k= 0, which is obtained by

The mean and variance of D kcan 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 B n, DLLis the DLL noise‐equivalent bandwidth, and K Lis the loop gain. To achieve the desired loop noise‐equivalent bandwidth, K Lmust 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, , the variance time update is given by

(38.29)

At steady state, var{Δ τ } = var {Δ τ k + 1} = var {Δ τ k}; hence,

(38.30)

From Eq. (38.30), it can be seen that the standard deviation of the ranging error is related to the correlator spacing through g ( t eml). Figure 38.43shows g ( t eml) for 0 ≤ t eml≤ 2. It can be seen that g ( t eml) is not a linear function, and it increases significantly faster when t eml> 1. Therefore, to achieve a relatively high ranging precision, t emlmust be set to be less than 1. It is worth mentioning that for the GPS C/A code with an infinite bandwidth, g ( t eml) = t eml.

Figure 38.44shows the pseudorange error of a coherent DLL as a function of C / N 0, with B n, DLL= {0.005, 0.05} Hz and t eml= {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].

In a typical DLL, the correlation of the received signal with the early, prompt, and late locally generated signals at time t = kT subare calculated according to

where x can be either e , p , or l representing early, prompt, or late correlations, respectively. Figure 38.45represents 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 Δ τ is related to the correlator spacing through g ( t eml), which is shown as a function of t eml(Shamaei et al. [73]).