Analytical derivation of the False Alarm Probability and Detection Probability for GNSS NLOS Detection using a Dual Frequency Receiver

Kurzfassung

These days, the global positing system (GPS) is the most common position system in the world. To determine a position, at least 4 satellites are necessary. Therefore it measures the signal propagation time between the mobile terminal and the satellites. However, in indoor or urban area, the direct path may be blocked. Thus, the satellite signals propagate an additional distance due to reflection and diffraction. This effect, where no direct path is available, is called non-line-of-sight (NLOS) propagation and adds a positive bias to the geometric LOS (GLOS) propagation time. This can cause an estimation error in the order of hundreds of meters, cf. [1], [2]. Therefore to obtain better positioning results it is necessary to identify NLOS signals and mitigate this additional bias by reconstructing the LOS path. If enough measurements to identify the position are available, it is possible to discard the NLOS signals or weight the signal with a lower priority. However it poses a great challenge to mitigate the NLOS impact through processing at the receiver. This paper discusses a novel analytical derivation of the false alarm probability (FAP) and detection probability (DP) for GNSS NLOS detection using a dual frequency receiver. Generally, dual frequency receivers are used to obtain better positioning results by correcting for example the ionosphere errors. We derive in this paper the FAP and DP for power–scaled detectors based on the multipath signals for detecting NLOS signals. The concept of this paper bases on the idea of [3]. The authors in [3] present a derivation of the FAP and DP of detecting pilot bursts of W-CDMA systems. Instead of considering an antenna array where the long-term channel statistics are identical, we consider receiver for simultaneous reception of two GNSS frequency bands. The derivation considers frequency–selective fading channels taking sub–chip multipath interference, channel dynamics and initial frequency offsets into account. Each antenna element consists of a correlator bank with Ns correlators which partitions the received symbol sequence in M non–overlapping subsequences of length Nc. The last part of the receiver normalizes the power and searches for the maximum of all correlator outputs. If the output is above a threshold the signal is considered as LOS or NLOS if blow. Thus the FAP defines the probability that the detection is beyond the threshold whereas no signal is on the GLOS path, which is caused by noise or by strong multipaths. The DP states that the detection is beyond the threshold and the received signal is LOS. To determine DP, we compute either the threshold corresponding to a defined FAP or vice versa and calculate afterwards the corresponding DP. To obtain these probabilities and since the received signal is zero-mean complex Gaussian distributed, the correlator outputs are also zero-mean Gaussian distributed. Therefore the test characteristics can be characterized by its covariance matrix for each frequency. We calculate the cumulative density function (CDF) of the FAP and DP by the eigenvalues of the covariance matrix for each delay and each partitioned sequence. Contrary to [3], the eigenvalues for the FAP are not equivalent. Hence we have to use the lemma of Gil–Pelaez (cf. [4]) to calculate the CDF. We can evaluate this CDF numerically for the FAP to obtain either the FAP for a defined threshold or the threshold for a defined FAP. Afterwards, we can calculate with these values the DP in the same way, also with the lemma of Gil–Pelaez. Especially in this paper we have to calculate both probabilities with the lemma of Gil–Pelaez, because of different eigenvalues. If all eigenvalues of the covariance matrix are identical, the lemma of Gil–Pelaez reduces to the chi–square distribution [3], where the probability computation simplifies. To verify the derived results, we consider the L1 and L5 frequencies of GPS with a chip rate of 1024 kHz which is scrambled by a Gold–code. We consider rectangle pulse–shaped symbols and a frequency selective Rayleigh fading channel. The simulations consider a receiver velocity of 100 km/h and are done for each single frequency and both frequencies together which use the coherent channel knowledge. The simulation results show clearly the advantages of dual frequency receivers. Additionally we see the influence of sub–chip interference to the FAP and DP. Especially it shows also the trade–off between temporal non–coherent versus coherent averaging. Due to the high receiver mobility of 100 km/h, the symbol partitioning into more blocks performs best, since the number of chips of coherent correlation is smallest. To conclude, the results of this paper show a simple way to calculate the DP versus FAP. Simulation results verify the analytical probabilities and show the trade–off between temporal non–coherent versus coherent averaging. Clearly, the optimum noise averaging depends on the length of the symbol sequence and channel dynamics. Additionally, the simulations show the improved DP of dual frequency receiver. Furthermore we see the influence of sub–chip interference in the simulation results. Thus the final paper will present the derivation of the FAP and the DP in more detail. Especially we state the covariance matrix and the derivation of the eigenvalues where the influence of the sub–chip interference and the dual frequencies are obvious. Furthermore we will provide additional simulation results which show the FAP/DP versus SIR and like in [3] the optimal partitioning of the symbol sequence depending on the velocity. [1] J. a. Figueiras and S. Frattasi, Mobile positioning and tracking: from conventional to cooperative techniques, ser. New ecologies for the twenty-first century. Wiley, 2010. [2] K. Yu, I. Sharp, and Guo. (2009) Groundbasedwireless positioning. [3] L. Schmitt, V. Simon, T. Grundler, C. Schreyoegg, and H. Meyr, “Initial Synchronization of W-CDMA Systems Using a Power-Scaled Detector with Antenna Diversity in Frequency-Selective Rayleigh Fading Channels,” in Proceedings of the IEEE Global Communications Conference (GLOBECOM 2003), San Francisco, USA, December 2003. [4] J. P. Imhof, “Computing the distribution of quadratic forms in normal variables,” Biometrika, vol. 48, no. 3/4, pp. 419–426, 1961.