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38.6.3.2 Non‐Coherent DLL Tracking
ОглавлениеIn a typical DLL, the correlation of the received signal with the early, prompt, and late locally generated signals at time t = kTsub 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(teml), which is shown as a function of teml (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 teml values. Solid and dashed lines represent the results for Bn, DLL = 0.05 Hz and Bn, 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 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); is the propagation time estimation error; and are the estimated and the true TOA, respectively; and R(Δτ) ≈ sinc (WSSSΔτ) is the autocorrelation function of scode(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 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 and is given by
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 using Eq. (38.29) yields
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 gα(teml) for 0 ≤ teml ≤ 2. It can be seen that gα(teml) is a nonlinear function and increases significantly faster for teml > 1. Figure 38.48 shows the standard deviation of the pseudorange error for a dot‐product DLL as a function of C/N0 with teml = 1 and Bn, 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 gβ(teml) for 0 ≤ teml ≤ 2. It can be seen that gβ(teml) is significantly larger than gα(teml). To reduce the ranging error due to gβ(teml), teml 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 Bn, DLL = {0.05, 0.005} Hz and teml = 1. It can be seen that decreasing the loop bandwidth decreases the standard deviation of the pseudorange error. However, very small values of Bn, 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 gα(teml) shown in (a), while for an early‐power‐minus‐late‐power discriminator it is related through gα(teml) and gβ(teml) shown in (b) (Shamaei et al. [74]).
Source: Reproduced with permission of IEEE.
Figure 38.48 DLL performance as a function of C/N0 for non‐coherent discriminators: dot‐product discriminator (solid line) and early‐power‐minus‐late‐power discriminator (dashed line), Bn, DLL = {0.05, 0.005} Hz, and teml = 1 (Shamaei et al. [74]).
Source: Reproduced with permission of IEEE.
