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 fD, 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; teml is the correlator spacing (early‐minus‐late);
It can be shown that the noise components
(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 Sk is the signal component of the dot‐product discriminator given by
and Nk is the noise component of the discriminator function, which has zero mean. Figure 38.46(a) shows the normalized Sk/C for teml = {0.25, 0.5, 1, 1.5, 2}. It can be seen that the signal component of the discriminator function is nonzero for Δτ/Tc > (1 + teml/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 Δτk, the discriminator function can be approximated by a linear function according to
where
The mean and variance of Dk are calculated to be
(38.35)
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 Sk can be shown to be
and Nk is the noise component of the discriminator function, which has zero mean. Figure 38.46(b) shows the normalized Sk/C of the early‐power‐minus‐late‐power discriminator function for teml = {0.25, 0.5, 1, 1.5, 2}.
The discriminator function can be approximated by a linear function for small values of Δτk (cf. Eq. (38.33)) with
The mean and variance of Dk are calculated to be
(38.38)
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 {Δτk + 1} = var {Δτk} and
