Position, Navigation, and Timing Technologies in the 21st Century. Группа авторов

(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].

      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 = kT_sub 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(t_eml), which is shown as a function of t_eml (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 t_eml values. Solid and dashed lines represent the results for B_{n, DLL} = 0.05 Hz and B_{n, DLL} = 0.005 Hz, respectively (Shamaei et al. [73]).

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

Figure 38.45 General structure of the DLL to track the code phase (Shamaei et al. [74]).

      Source: Reproduced with permission of IEEE.

      Assuming the receiver’s signal acquisition stage to provide a reasonably accurate estimate of f_D, the in‐phase and quadrature components of the early, prompt, and late correlations can be written as

      where x is e, p, or l and κ is –1, 0, or 1 for early, prompt, and late correlations, respectively; t_eml is the correlator spacing (early‐minus‐late); is the propagation time estimation error; and are the estimated and the true TOA, respectively; and R(Δτ) ≈ sinc (W_SSSΔτ) is the autocorrelation function of s_code(t).

      It can be shown that the noise components and of the correlations have (i) uncorrelated in‐phase and quadrature samples, (ii) uncorrelated samples at different time, (iii) zero‐mean, and (iv) the following statistics:

      (38.31)

      (38.32)

      where x′ is e or l.

      Open‐Loop Analysis: The open‐loop statistics of the code phase error using dot‐product and early‐power‐minus‐late‐power discriminators are analyzed next.

       Dot‐Product Discriminator The dot‐product discriminator function is defined as

      where S_k is the signal component of the dot‐product discriminator given by

and N_k is the noise component of the discriminator function, which has zero mean. Figure 38.46(a) shows the normalized S_k/C for t_eml = {0.25, 0.5, 1, 1.5, 2}. It can be seen that the signal component of the discriminator function is nonzero for Δτ/T_c > (1 + t_eml/2), which is in contrast to being zero for GPS C/A code with infinite bandwidth. This is due to the sinc autocorrelation function of the SSS versus the triangular autocorrelation function of the GPS C/A code.

      For small values of Δτ_k, the discriminator function can be approximated by a linear function according to

(38.33)

      where