Calculating Time-to-First-Fix
November 1, 2011 By: Nicolas Couronneau, Peter J. Duffett-Smith, Alexander Mitelman GPS World
Flow-graph representation of the acquisition process for one channel. FA is the false-alarm state and D the correct detection of the signal from this satellite. H1 and H0 represent respectively states in which the signal is and is not present. PFA|H1 is the probability of false alarm in a window where the signal is present and PFA|H0 the probability of false alarm in a window where the signal is not present. P D is the probability of detection, and PMD the probability of missed detection.
Cell-phone users are often more concerned about the speed of positioning than the accuracy, making time-to-first-fix the most important factor in a GNSS mass-market receiver’s perceived performance. However, TTFF is generally difficult to characterize and optimize because of the need to encompass a wide range of environments, including indoors.
One method of characterizing the time-to-first-fix (TTFF) is to measure it directly, using a signal generator and a real receiver. This method avoids the approximations of analytical solutions, but it is usually time consuming and it does not provide much insight into the factors affecting the TTFF since it is gen erally not possible to change the receiver’s architecture. Another approach is to use Monte Carlo simulations and a model of the acquisition process. This approach is more flexible than direct measurement, but again it can take a long time to simulate weak-signal environments.
We have developed a third approach based on analytical methods but regulated by measurements of the signal-to-noise ratio in target environments. Using this approach, one can quickly calculate the probability distribution of the TTFF for different signal strengths and acquisition parameters.
To illustrate this method, we consider a model of an assisted-GPS receiver combined with experimental measurements of the GPS L1 C/A signal taken indoors. The results are presented in Figure 1, where the probability of the TTFF (horizontal axis) is plotted as a function of the time after the beginning of the data series at which the acquisition process started (vertical axis), calculated using a 400-second GPS data series measured indoors. The strength of our approach is that we can quickly calculate the TTFF probability for any given confidence level and it is quite general so that it can be extended to other types of receivers.
Figure 1. The probability of the TTFF (horizontal axis) as a function of the time after the beginning of the data series at which the acquisition process started (vertical axis), calculated using a 400-second GPS data series measured indoors. Note that the colored scale is not linear.
Modeling the Acquisition Process
A GPS receiver must first acquire signals from a sufficient number of satellites before it is able to calculate a position. This search is often the major contributor to the TTFF.
GPS Acquisition Architecture. The acquisition can be represented as the search for a specific, yet unknown, combination of three parameters in a larger search space. These are:
- the Gold-code number used to generate the pseudo-random noise (PRN) sequence,
- the code phase, and
- the carrier frequency offset.
The last of these has contributions from the frequency offset caused by the relative motion of the satellite and receiver (the Doppler effect) and the frequency bias of the receiver’s local oscillator.
In general, signal detection is performed by correlating incoming signals with a local satellite signal replica for every combination of parameters in the search space. The correlated signal is then integrated and a “hit” is declared if the integrated value crosses a pre-determined threshold. The time required to test for the presence of a satellite signal for each combination of parameters is called the dwell time. We suppose here that this is approximately equal to the integration time.
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