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

The normalized signal component of the discriminator function is shown in Figure 38.18for t eml= {0.25, 0.5, 1, 1.5, 2}.

It can be seen from Figure 38.18that for small values of , the discriminator function can be approximated by a linear function given by

where α is the slope of the discriminator function at Δ t k= 0 [57], which is obtained by

Since R c( τ ) is symmetric,

and the linearized discriminator output becomes

(38.12)

It is worth noting that R c( τ ) and R c′ ( τ ) are obtained by numerically computing the autocorrelation function of the pulse‐shaped short code. Since the FIR of the pulse shaping filter h [ k ] is defined over only 48 values of k , the autocorrelation function R c( τ ) will be defined over 95 values of τ . However, interpolation may be used to evaluate R c( τ ) and R c′ ( τ ) at any τ . The mean and variance of D kcan be obtained from Eq. (38.12), and are given by

(38.13)

(38.14)

Now that the discriminator statistics are known, the closed‐loop pseudorange error is characterized next.

In order to achieve the desired loop noise‐equivalent bandwidth, K in Eq. (38.11)must be normalized according to

Figure 38.18 Output of the coherent baseband discriminator function for the CDMA short code with different correlator spacings (Khalife et al. [12]).

Source: Reproduced with permission of IEEE.

(38.15)

In cellular CDMA systems, for a t emlof 1.2, the loop filter gain becomes K ≈ 4 B n, DLL; hence, the choice of K in Section 38.5.2.3. Assuming a zero‐mean tracking error, that is, , the variance of the code start time error is given by

At steady state, var{Δ t k + 1} becomes

(38.16)

where Δ t is the steady‐state code start time error. Combining Eqs. (38.15)– (38.16)yields

(38.17)

The pseudorange can hence be expressed as

where v ( k ) is a zero‐mean random variable with variance σ 2= c 2· var {Δ t }. Figure 38.19shows a plot of the standard deviation of Δ t , denoted by σ , as a function of the carrier‐to‐noise ratio C / N 0for t eml= 1.25 chips.

This section presents experimental results for navigation with cellular CDMA signals. These results did not suffer from the BTS sector clock discrepancy issue (discussed in Section 38.7), since signals from only one sector antenna in each BTS cell were used. Experimental results exhibiting the BTS sector clock discrepancy issue and mitigation approaches are studied in [23, 25]. Section 38.5.4.1analyzes the pseudorange obtained by the receiver discussed in Section 38.5.2. Sections 38.5.4.2and 38.5.4.3present navigation results with aerial and ground vehicles, respectively.

Figure 38.19 Plot of σ , the standard deviation of Δ t , as a function of the carrier‐to‐noise ratio for t eml= 1.25 chips and B n,DLL= {0.5 Hz, 0.05 Hz} (Khalife et al. [12]).

Source: Reproduced with permission of IEEE.

The variation in the pseudorange obtained by the receiver discussed in Section 38.5.2is compared to the variation in the true range between a mobile receiver and cellular CDMA BTSs. For this purpose, the receiver was mounted on two platforms: (i) an unmanned aerial vehicle (UAV) and (ii) a ground vehicle [12, 18, 25].