Читать книгу Position, Navigation, and Timing Technologies in the 21st Century - Группа авторов - Страница 104

Assume that the residual carrier phase and Doppler frequency are negligible, that is, Δϕ ≈ 0 and Δf_D ≈ 0. Therefore, a coherent baseband discriminator may be used in the DLL. Figure 38.41 represents the structure of a coherent DLL that is used for tracking the code phase [55]. In what follows, the ranging precision of the DLL shown in Figure 38.41 is evaluated.

In the DLL, the received signal is first correlated with the early and late locally generated replicas of the SSS. The resulting early and late correlations are given respectively by

where T_c is the chip interval, t_eml is the correlator spacing (early‐minus‐late), and is the estimated TOA. The signal components of the early and late correlations, and , respectively, are given by

where is the propagation time estimation error, and R(·) is the autocorrelation function of s_code(t), given by

It can be shown that the noise components of the early and late correlations, and , respectively, are zero mean with the following statistics:

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 38.42 for t_eml = {0.25, 0.5, 1, 1.5, 2}.

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

(38.26)

where k_SSS is the slope of the discriminator function at Δτ_k = 0, which is obtained by

The mean and variance of D_k 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 B_{n, DLL} is the DLL noise‐equivalent bandwidth, and K_L is the loop gain. To achieve the desired loop noise‐equivalent bandwidth, K_L 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, , the variance time update is given by

(38.29)

At steady state, var{Δτ} = var {Δτ_{k + 1}} = var {Δτ_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 38.43 shows 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_eml must 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 38.44 shows the pseudorange error of a coherent DLL as a function of C/N₀, 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].