Sensitivity of GPS Acquisition to Initial Data Uncertainties

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Abstract
To acquire a GPS satellite signal, it is necessary to estimate the Doppler frequency shift due to the motion of the satellite relative to the receiver. This estimate is sensitive to uncertainties in the receiver position and in the GPS “Time-of-Week,” which is used to compute the satellite's position and motion. Knowing this sensitivity and the uncertainties in the initial data, the receiver can determine the number of frequency bins that must be searched, which directly impacts the “Time-to-First-Fix.” Ignoring this sensitivity and searching the wrong number of bins can result in missing the signal on the one hand or excessive delays on the other. Since the Doppler shift is proportional to the range rate, its sensitivity to position errors is maximized by taking the gradient of range rate, and its sensitivity to time errors is given by the time derivative. Somewhat surprisingly, the Doppler estimate is most sensitive to both kinds of initial errors when the satellite is directly overhead, implying that the common practice of acquiring the highest satellite first needs to be reexamined. Also, the elevation angle alone does not determine sensitivities, since it takes two angles to specify the orientation of the satellite's orbit to the receiver—even when earth rate is neglected. In addition, numerical results for position and time sensitivities are tabulated for the entire spectrum of receiver/satellite orientations. These show that in the worst case, an error of one second in time causes nearly as large an error in estimated Doppler as does an error of one kilometer in position, roughly one hertz. One implication of this result is that a continuous, battery-powered “Real-Time-Clock” can significantly improve the Doppler estimate and reduce the acquisition time.

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