4.2 L5 Coarse Acquisition
4.2.3 Detection Performance
The detection performance of the four aforementioned correlation strategies are compared theoretically and empirically using modified ROC curves and Post-correlation
SNR (PSNR) coefficients respectively.
Modified ROC curves
In Figure 4-3, the probability of detection of the four acquisition strategies are plotted against the incoming C/N0 for a fixed probability of false alarmPFA =10−4. As suggested
in Bastide (2004), the probability of false alarm is set to 4
10− rather than 3
10− (as is typically used for C/A signal acquisition) to account for the fact that the L5 uncertainty region is approximately ten times bigger than that of GPS C/A. Note that the results displayed in Figure 4-3 account for the effects of frontend filtering but not for those of code delay and Doppler uncertainty. As expected, and already reported in Bastide et al (2002) and in Yang et al (2004), single and coherently combined channel acquisition offer the worst and best detection performance, respectively. The low performance of the single channel acquisition is a direct consequence of the fact it only uses half of the available power. Conversely, the good performance of the coherently combined acquisition results from the optimal use that this strategy makes of all the available power. However, it is important to point out that for low C/N0, where the relative
data/pilot sign recovery becomes unreliable, the performance of all data/pilot combining methods tends to merge.
As suggested in Bastide et al (2002) the L5 acquisition threshold is taken as the incoming signal C/N0 required to reach the probability of detection PD =0.9. With the parameters
used herein, the acquisition threshold can be approximated at 42 dB-Hz.
Figure 4-3 – Probability of Detection versus Total C/N0 for Various L5 Acquisition
Strategies Using 1 ms Coherent Integration
Since intra-system cross-correlation peaks are the main cause of false alarms during signal acquisition, a more reliable test statistic is desired. To this end, the common approach is to assume the presence of a cross-correlation peak due to a strong interfering satellite in the incorrect search space bins. Hegarty et al (2003) introduced two cross- correlation levels: one at 19 dB-Hz when the cross-correlation occurs on both channels simultaneously and another at 16 dB-Hz when it affects a unique channel. Although seldom encountered in real life, the worst case is considered here. Another common step in making the signal detection more reliable is to increase the total pre-detection
integration time using M summations of the test statistic T . Both effects are illustrated for the non-coherently combined strategy in Figure 4-4.
The resulting test statisticTNCM is a non-centralχ2 RV under H1 and H0, and the non-
centrality parameters can be approximated, respectively, as
( )
2 0 ~2 1 R MP ≈ λ (4-19)( )
2 0 ~2 0 R MPI ≈ λ (4-20)where P is the cross-correlation level. I
Figure 4-4 – Probability of Detection versus Total C/N0 in the Presence of Noise and
Cross-Correlation (CC) Using 1 ms Coherent Integration and Various Non- Coherent Summation Numbers
As expected, and already reported in Hegarty et al (2003), the use of non-coherent summations can help improve the detection performance of the non-coherently combined strategy. This improvement, however, tends to vanish when the C/N0 decreases since the
squaring losses increase. It is anticipated that similar trends would be observed for the other L5 coarse acquisition strategies.
Similarly, Figure 4-4 shows that, in the presence of cross-correlation peaks the incorrect search space cells, the detection performance of the non-coherently combined acquisition strategy degrades, but only marginally. However, it is important to recall that, as illustrated in Table 2.2, the L5 PRN codes were designed to reduce the occurrence of intra-system cross-correlation peaks or, when cross-correlations do occur, to limit their time duration (Spilker & Van Dierendonck 2001).
Empirical Detection Performance – PSNR coefficients
To confirm the theoretical detection performance of the four acquisition strategies presented above, their PSNR is also computed using (Shanmugam 2008)
( )
[
( )]
(
)
( )
[
]
≠ ≠ − = = θ θ θ θ θ θ ˆ var ˆ ˆ log 10 2 10 T T E T PSNR (4-21)where the values T
( )
θˆ=θ , E[
T( )
θˆ≠θ]
and var T[
( )
θˆ≠θ]
are estimated over successive search spaces by averaging, respectively, the maximum, mean and variance of the test statistic.O’Driscoll (2007) shows that, when applied to a mean-shifted Gaussian detection problem, the PSNR corresponds to the Deflection Coefficient (DC) and, therefore, can be
used to exactly characterize the detection performance of this problem (Kay 1993). This relationship, however, does not hold for non-Gaussian detection problems. In such cases, empirical ROC curves provide the most relevant insight in terms of detection performance. However, in cases where the true code delay and Doppler offset of the incoming signal cannot be straightforwardly determined (e.g. when using real or hardware simulated data), the estimation of the probabilities of detection and false alarm over a large number of samples can become very tedious. Under these circumstances, empirical PSNR values can be used as a good approximation (Shanmugam 2008). Empirical PSNR coefficients are shown in Figure 4-5. For each PSNR coefficient (i.e. for each acquisition strategy and each C/N0), the values for the maximum, mean and variance
of the test statistic (used in Equation 4.21) are averaged over 1,000 two-dimensional search spaces.
As expected from the theoretical ROC curves displayed in Figure 4-3, the data/pilot combined strategies clearly outperform the single channel acquisition for the entire range of C/N0. Amongst these combined implementations and at high C/N0, the differential
combining approach seems to provide the higher PSNR, followed by the coherent and non-coherent combining strategies. At lower C/N0, the performance of coherent and non-
coherent combining degrades and, as a result, the performance of the combined implementations tends to merge. While these trends do not follow those observed in terms of ROC, where the coherent combining strategy provides the highest detection performance, it is important to recall that, for non-Gaussian problems, the PSNR is not an exact approximation of the detection performance. In fact, the superiority of the differential combining strategy in terms of PSNR can be explained by the low variance of the differential detector. Also, the ROC curves shown in Figure 4-3 are derived under the assumption that only noise is present under H0. In reality, some cross- and auto-
correlation side peaks are present in all the cells of the search space and modify the distribution of the test statistic under H0. In particular, it can be expected that the effects
of cross- and auto-correlation side peaks on the coherent and differential combining will be magnifiedat low C/N0.