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### Abstract:

Reliable, fast and low-complexity signal acquisition plays a key role in satellite navigation. Prior to tracking or data demodulation, the user must achieve coarse synchronization with the respective signal’s code-phase and Doppler frequency. For navigation signals of the current generation (e.g. E1 OS, L1C), this task is cumbersome in terms of computation time and complexity due to their long spreading code, which is why coarse/acquisition (C/A) signals with short codes and low data rates are having a renaissance in signal design for future global navigation satellite systems (GNSS). Due to their potentially short time to acquisition and low demand on hardware resources, they are an attractive option both from a mass-market and high-performance user’s perspective. However, short spreading codes cannot be assumed to be mutually orthogonal. When multiple satellite signals are received, multiple access interference (MAI) will act as an additional source of noise. From a signal design point of view, a tradeoff between shorter and shorter codes and acquisition reliability needs to be achieved. Acquisition reliability is measured in terms of the receiver operating characteristic (ROC): the ROC curve is obtained by plotting the global detection probability versus the global false alarm probability at various detection thresholds. In [1], this has been done on a per-bin basis for individual search cells, but not yet for the complete search space of code-phase bins and Doppler bins. The contribution of this work is twofold. Firstly, we derive the global ROC by extending our previous work form [1] to the full search space. Secondly, the new ROC model is used to minimize the spreading code length under a reliability constraint. The first part of the work deals with the global ROC model. When considering the complete search space, one needs to account for the fact that C/A signals act as very strong MAI on some few Doppler bins, but as very weak MAI on most Doppler bins. This phenomenon has sometimes been referred to as “Doppler-crossings”, and is due to the fact that a C/A signal repeats a short code many times before a new data bit is transmitted. Repeating a code many times leads to a signal power spectral density that exhibits spectral lines; these spectral may align when the difference between the search bin’s Doppler frequency and the interferer’s Doppler frequency is an integer multiple of the code rate. It has been demonstrated that the conventional spectral separation coefficient (SSC) is not suited to model Doppler-crossings, but modified versions have been proposed [2], [3]. Using this constellation-dependent SSC for each interfering satellite, we derive a probability density function for the MAI’s effective noise floor I_0 (in units of Watts per Hertz) affecting each search bin. Finally, the ROC is derived as follows. Since MAI is modelled in terms of its white noise equivalent, we can assume that each search bin output follows a chi-square distribution when conditioned on the MAI noise floor I_0. This conditional ROC is then marginalized over the distribution of I_0 to obtain the actual ROC. In the second part of the work, the actual tradeoff between spreading code length and reliability is described. We aim to minimize the spreading code length, while ensuring that the acquisition performance is not eaten up by the increase of MAI; therefore, we perform a constrained minimization. The reliability constraint is that the ROC contain operating points with acceptable false alarm/detection performance. Starting from a reliable initial spreading code length (e.g. 1023 chips), we calculate the ROC with the previously described method. While the constraint is satisfied, the spreading code length is decreased by a fixed stepsize and the ROC is calculated again. This process terminates as soon as the reliability constraint is violated. The last spreading code length which satisfied the constraint is the optimal short code length. The proposed methodology offers a systematic way for acquisition optimization from a GNSS signal design point of view. Preliminary simulation results suggest that reasonable short code lengths are on the order of 300, depending on the application scenarios. Such a signal can help to dramatically improve the time to acquisition and acquisition complexity. [1] C. Enneking, F. Antreich, M. Appel, A. L. F. de Almeida, „Pure Pilot Signals for GNSS: How Short Can We Choose Spreading Codes?”, Proceedings of the 2019 International Technical Meeting of The Institute of Navigation, Reston, Virginia, January 2019, pp. 925-935. [2] C. Enneking, F. Antreich, L. Krieger and A. L. F. de Almeida, „Gaussian Approximations for Intra- and Intersystem Interference in RNSS“, IEEE Communications Letters (Early Access), Oct. 2018. [3] C. J. Hegarty, “A Simple Model for C/A Code Self-Interference”, Proceedings of the 27th International Technical Meeting of the ION Satellite Division, ION GNSS+ 2014, Tampa, Florida, September 8-12, 2014.