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After obtaining an initial coarse estimate of the code start time and Doppler frequency , the receiver refines and maintains these estimates via tracking loops. A phase‐locked loop (PLL) or a frequency‐locked loop (FLL) can be employed to track the carrier phase, and a carrier‐aided delay‐locked loop (DLL) can be used to track the code phase. FLLs are generally more robust than PLLs, are useful when transitioning from acquisition to tracking, and can track in more challenging environments [54, 55]. Figure 38.12 depicts a block diagram of a PLL‐aided DLL tracking loop [12, 18]. The PLL and DLL are discussed in detail next.

Figure 38.11 Cellular CDMA signal acquisition front panel showing |Z_k|² along with , PN offset, and C/N₀ for a particular BTS (Khalife et al. [18]).

Source: Reproduced with permission of IEEE.

Figure 38.12 Tracking loops in a navigation cellular CDMA receiver. Thick lines represent complex quantities (Khalife et al. [18]).

Source: Reproduced with permission of IEEE.

PLL: The PLL consists of a phase discriminator, a loop filter, and a numerically controlled oscillator (NCO). Since the receiver is tracking the data‐less pilot channel, an atan2 discriminator can be used, given by

where is the prompt correlation. The atan2 discriminator remains linear over the full input error range of ±π and could be used without the risk of introducing phase ambiguities. In contrast, a GPS receiver cannot use this discriminator unless the transmitted data bit values of the navigation message are known [54]. Furthermore, while GPS receivers require second‐ or higher‐order PLLs due to the high dynamics of GPS SVs, lower‐order PLLs could be used in cellular CDMA navigation receivers. It was found that the receiver could easily track the carrier phase with a second‐order PLL with a loop filter transfer function given by

(38.9)

where is the damping ratio, and ω_n is the undamped natural frequency, which can be related to the PLL noise‐equivalent bandwidth B_n,PLL by [55]. The output of the loop filter v_{PLL, k} is the rate of change of the carrier phase error, expressed in rad/s. The Doppler frequency is deduced by dividing v_{PLL, k} by 2π. The loop filter transfer function in Eq. (38.9) is discretized and realized in state space. The noise‐equivalent bandwidth is chosen to range between 4 and 8 Hz.

DLL: The carrier‐aided DLL employs a non‐coherent dot‐product discriminator given by

where Λ is a normalization constant given by Λ = T_c/2C; C is the carrier power, which can be estimated from the prompt correlation; and , and are the prompt, early, and late correlations, respectively. The prompt correlation was described in Section 38.5.2.1. The early and late correlations are calculated by correlating the received signal with an early and a delayed version of the prompt PN sequence, respectively. The time shift between and is defined by an early‐minus‐late time t_eml, expressed in chips. Since the autocorrelation function of the transmitted cellular CDMA pulses is not triangular as in the case of GPS, a wider t_eml is preferable in order to have a significant difference between , and . Figure 38.13 shows the autocorrelation function of the cellular CDMA PN code as specified by the cdma2000 standard and that of the C/A code in GPS. It can be seen from Figure 38.13 that for t_eml ≤ 0.5 chips, R_c(τ) in the cdma2000 standard has an approximately constant value, which is not desirable for precise tracking. A good rule of thumb is to choose 1 ≤ t_eml ≤ 1.2 chips.

The DLL loop filter is a simple gain K, with a noise‐equivalent bandwidth Hz. The output of the DLL loop filter v_{DLL, k} is the rate of change of the code phase, expressed in s/s. Assuming low‐side mixing, the code start time is updated according to

Figure 38.13 Autocorrelation function of GPS C/A code and cellular CDMA PN sequence according to the cdma2000 standard (Khalife et al. [12]).

Source: Reproduced with permission of IEEE.

In a GPS receiver, the pseudorange is calculated based on the time a navigation message subframe begins, which eliminates ambiguities due to the relative distance between GPS SVs [55]. This necessitates decoding the navigation message in order to detect the start of a subframe. These ambiguities do not exist in a cellular CDMA system. This follows from the fact that a PN offset of one translates to a distance greater than 15 km between BTSs, which is beyond the size of a typical cell [56].

Finally, the pseudorange estimate ρ can be deduced by multiplying the code start time by the speed of light c; that is,

(38.10)

Figure 38.14 shows the intermediate signals produced within the tracking loops of the cellular CDMA navigation receiver: code error; phase error; Doppler frequency; early, prompt, and late correlations; pseudorange; and in‐phase and quadrature components of the correlation.