GPS Position Can Be Computed without the Navigation Data

This paper presents an algorithm for computing a GPS receiver's position and the current time from C/A-code phase measurements to at least six satellites and a set of valid satellite ephemerides. Reference position and reference time are not necessary. The process uses a cost function having numerous local minima in addition to the global minimum. An exhaustive search over the 5dimensional time-position-bias search space will reliably find the global minimizer, and thus solve the approximate position and time. Introducing a reference time within one minute from the true time resulted in worst-case computation time of a few seconds. If the current pseudorange correction parameters are available, the accuracy of this method is comparable to that of the conventional pseudo-range methods. As a conclusion, an accurate GPS position can be solved in a reasonable time even without the navigation data or precise system time. INTRODUCTION The method presented in this paper is intended for GPS positioning in weak signal situations where the satellite signals are so noisy that C/A codes can be tracked but the navigation messages are beyond recognition [1,2,3]. The ephemeris and pseudo-range correction parameters could be already in the receiver's memory, or they could be acquired via e.g. cellular network along with a delayed reference time [1]. A local positioning algorithm for this situation that requires initial time and position approximations has already been presented in [4]. This paper describes a global extension to the local algorithm. BACKGROUND: THE LOCAL ALGORITHM Use the following symbols: t, r, b system time, 3D user position and user clock bias to be solved, φi C/A code phase measurement to ith satellite, εi approximate ranging error compensation, si (t) position of ith satellite at system time instant t, and τi approximate time-of-travel from ith satellite to user. The range fit to the ith satellite is defined by (1) ( ) ( ) ( ) , , frac i i i i i q t b b t φ ε τ Λ = + + − − − r s r where 300 km Λ ≈ is the length of the C/A code sequence and the lambda-fraction operator is defined by (2) frac round x x x Λ − Λ Λ . Note that the user clock bias b can be restricted between 1 2 − Λ and 1 2 Λ since the fraction operator eliminates integer multiples of Λ.