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

and is given by

(38.34)

      The mean and variance of D_k are calculated to be

      (38.35)

(38.36)

       Early‐Power‐Minus‐Late‐Power Discriminator The early‐power‐minus‐late‐power discriminator function is defined as

Figure 38.46 Normalized signal component of non‐coherent discriminator functions: (a) dot‐product and (b) early‐power‐minus‐late‐power for different correlator spacings (Shamaei et al. [74]).

      Source: Reproduced with permission of IEEE.

      where S_k can be shown to be

      and N_k is the noise component of the discriminator function, which has zero mean. Figure 38.46(b) shows the normalized S_k/C of the early‐power‐minus‐late‐power discriminator function for t_eml = {0.25, 0.5, 1, 1.5, 2}.

The discriminator function can be approximated by a linear function for small values of Δτ_k (cf. Eq. (38.33)) with

(38.37)

      The mean and variance of D_k are calculated to be

      (38.38)

(38.39)

Closed‐Loop Analysis: An FLL‐assisted PLL produces a reasonably accurate pseudorange rate estimate, making first‐order DLLs sufficient. At steady state, var{Δτ} = var {Δτ_{k + 1}} = var {Δτ_k} and using Eq. (38.29) yields

(38.40)

      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 g_α(t_eml) for 0 ≤ t_eml ≤ 2. It can be seen that g_α(t_eml) is a nonlinear function and increases significantly faster for t_eml > 1. Figure 38.48 shows the standard deviation of the pseudorange error for a dot‐product DLL as a function of C/N₀ with t_eml = 1 and B_{n, 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 g_β(t_eml) for 0 ≤ t_eml ≤ 2. It can be seen that g_β(t_eml) is significantly larger than g_α(t_eml). To reduce the ranging error due to g_β(t_eml), t_eml 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 B_{n, DLL} = {0.05, 0.005} Hz and t_eml = 1. It can be seen that decreasing the loop bandwidth decreases the standard deviation of the pseudorange error. However, very small values of B_{n, 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 g_α(t_eml) shown in (a), while for an early‐power‐minus‐late‐power discriminator it is related through g_α(t_eml) and g_β(t_eml) shown in (b) (Shamaei et al. [74]).

      Source: