Position, Navigation, and Timing Technologies in the 21st Century

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

Figure 38.44 Coherent baseband discriminator noise performance as a function of C / N 0for different t emlvalues. 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 emlis the correlator spacing (early‐minus‐late); is the propagation time estimation error; and are the estimated and the true TOA, respectively; and R (Δ τ ) ≈ sinc ( W SSSΔ τ ) 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 DiscriminatorThe dot‐product discriminator function is defined as

where S kis the signal component of the dot‐product discriminator given by

and N kis 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 Δ τ / 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 Δ τ k, the discriminator function can be approximated by a linear function according to

(38.33)

where and is given by

(38.34)

The mean and variance of D kare calculated to be

(38.35)

(38.36)

Early‐Power‐Minus‐Late‐Power DiscriminatorThe 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 kcan be shown to be

and N kis 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}.