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signal-processing capabilities increased, such time difference estimations and measurements became possible. Note that synchronization at the mobile receiver s end is thus avoided as long as time differences are carried out. The basic idea was to obtain two such differences in order to allow the calculation of the intersection point of the resulting two hyperbolae (Fig. 1.13). This approach leads to a theoretical single point in a two-dimensional space. The rst system that used this technique was the Decca,8 which came into operation at the end of World War II. It worked within a frequency band of 70 128 kHz, allowing for approximately 450 km of operational range. The resulting accuracy was in the range of a few hundred meters, depending on propagation conditions. The new era of radio electric signals allowed for a rigorous evaluation of accuracy a very important parameter. The current e-Loran9 is also a hyperbolic system, but which added new features concerning the modulation scheme, based on pulse trains forwarded by each master and slave station.10 This approach allowed for a more precise positioning, together with an ef cient way of identifying the various stations. The frequency band used was rst in the 1750 1950 kHz range for LORAN-A, and is currently in the 90 110 kHz range for eLoran (which is the latest development of LORAN-C). This frequency involves the use of very large antennas (Fig. 1.14) in order to achieve long range and high power transmission.

The same shape tween, without (top) and with (bottom) shape hints, respectively. Shape hints contain letters (a through z) for identifying which points correspond in the starting and ending shape. You can use up to 26 shape hints. Shape hints are yellow in a starting keyframe, green in an ending keyframe, and red when not on a curve. For best results when tweening shapes, follow these guidelines:

The problem of synchronization has to be clearly distinguished from those of precision or stability The satellite-based time, driven by atomic clocks for each constellation, is very accurate One could have imagined some receivers also to be equipped with atomic clocks The time ow would have been very precise too However, synchronization is related to the fact that all the system components deal with a common timescale; this is of primary importance when carrying out time measurements in the GNSS elds One has to keep in mind the fact that 1 ns is equivalent to 30 cm23 It is then necessary to know the satellites time drift and the synchronization of the receiver must be very accurate! The problems of satellite synchronization and receiver synchronization are very different, as are the techniques implemented to deal with these problems.

At the satellite end, we have already mentioned that atomic clocks are used, but in order to achieve synchronization, the ground segment predicts both the satellite s position and clock at future times and estimates the accuracy of these predictions Speci c computations are carried out (Kalman ltering) in order to improve the accuracy of this extracted data, used to update the navigation message In such a way, the ground segment monitors the satellites clocks and uploads the corresponding parameters to each satellite Thus, the synchronization of satellites clocks to GPS time is obtained from the navigation message The problem of receiver synchronization is even more crucial as it must be carried out on a frequent time base, the receiver not being equipped with atomic clocks.

The solution to this fundamental question is achieved in a clever way: the bias of the receiver s clock is considered as being an unknown variable in the positioning problem of the receiver Indeed, the solution vector of three-dimensional positioning is made up of three spatial variables, typically X, Y, and Z, or longitude, latitude, and altitude, plus one temporal variable, the receiver s clock bias with respect to the GPS time Of course, an additional unknown means that an additional measurement will be required,24 but this also allows the constraints on the receiver clock to be loosened This feature was required to open the way to affordable mass market receivers..

To calculate the probability of decoding error, we need to know the probability that conditionally independent sequences are jointly typical. Let S1 , S2 , and S3 be three subsets of {X (1) , X (2) , . . . , X (k) }. If S1 and S2 are conditionally independent given S3 but otherwise share the same pairwise marginals of (S1 , S2 , S3 ), we have the following probability of joint typicality.