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

where T sub= 10 ms is the subaccumulation period, which is chosen to be one frame length. The transfer function of the frequency loop filter is given by

(38.23)

where ω n, fis the undamped natural frequency of the frequency loop, which can be related to the FLL noise‐equivalent bandwidth [54]. The output of the frequency loop filter is the rate of change of the angular frequency , expressed in rad/s 2. It is therefore integrated and added to the output of the phase loop filter. The frequency loop filter transfer function in Eq. (38.23)is discretized and realized in state space.

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

where Γ is a normalization constant given by

where and are the early and late correlations, respectively, is the chip interval, W SSS= 63 × 15 = 945 kHz is the SSS bandwidth, is the expectation operator, and is the interference‐plus‐noise variance. The calculation of the overall noise level including interference and channel noise is discussed in [65].

The DLL loop filter is chosen to be similar to Eq. (38.23), with a noise‐equivalent bandwidth B n, DLL(in hertz). The output of the DLL loop filter v DLL(in s/s) is the rate of change of the SSS code phase. Assuming low‐side mixing, the code start time is updated according to

The SSS code start time estimate is used to reconstruct the transmitted frame. Figure 38.38shows the block diagram of the tracking loops, where ω c= 2 πf cand f cis the carrier frequency (in hertz). Finally, the pseudorange estimate ρ can be deduced by multiplying the code start time by the speed of light c (cf. Eq. (38.10)).

Figure 38.38 LTE SSS signal tracking block diagram (Shamaei et al. [64, 65]).

Source: Reproduced with permission of Institute of Navigation, IEEE.

Figure 38.39 LTE SSS tracking results with a stationary receiver (Shamaei et al. [64, 65]).

Source: Reproduced with permission of Institute of Navigation, IEEE.

Figure 38.39shows tracking results with real LTE signals. Here, the PLL, FLL, and DLL noise‐equivalent bandwidths were set to 4, 0.2, and 0.001 Hz, respectively. To calculate the interference‐plus‐noise variance, the received signal was correlated with an orthogonal sequence that is not transmitted by any of the eNodeBs in the environment. Then, the average of the squared‐magnitude of the correlation was assumed to be the interference‐plus‐noise variance. Since the receiver was stationary and its clock was driven by a GPS‐disciplined oscillator (GPSDO), the Doppler frequency was stable around zero. Note that the aiding term τ is computed in the Timing Information Extraction block to improve SSS tracking. The term τ gets added to in the SSS generator block. The calculation of τ is discussed next.

In LTE systems, the PSS and SSS are transmitted with the lowest possible bandwidth. The ranging precision and accuracy of the SSS is analyzed in [73, 74], which shows that the SSS can provide very precise ranging resolution using conventional DLLs in an environment without multipath. However, because of its relatively low bandwidth, the SSS is extremely susceptible to multipath. To achieve more precise localization using LTE signals, the CRS can be exploited. The ranging precision of SSS and CRS in a semi‐urban environment with multipath was studied experimentally in [63], which showed that CRS is more robust to multipath.

In the timing information extraction stage of the receiver, the TOA can be estimated by detecting the first peak of the channel impulse response (CIR). The CIR can be computed from the received signal model in the i‐ th symbol, given in Eq. (38.19). The subscript i will be dropped in the sequel for simplicity of notation. The estimated CFR of the u ‐th eNodeB is given by

(38.24)

where . Equation 38.24is obtained using the fact that . The CIR estimate is obtained by taking the IFFT of the estimated CFR, yielding

(38.25)