The-clock-synchronization-problem.md

 Satellites send a signal and users’ receptor receives it. But in order to read the signal properly, both clocks (the satellites and the receptor’s) must have the same time scale.
 
-**1st synchronization problem:**
+# **1st synchronization problem:**
 
-Indeed, if we calculate the time a signal makes to go from the satellites to the receptor, in an ideal world, we only would have to use the fact that ![f11]. As time is multiplied by the speed of light, distance is very sensible to time mistake:
-![f1]
+Indeed, if we calculate the time a signal makes to go from the satellites to the receptor, in an ideal world, we only would have to use the fact that ![f11](http://chart.apis.google.com/chart?cht=tx&chl=d=ct). As time is multiplied by the speed of light, distance is very sensible to time mistake: ![f1](http://chart.apis.google.com/chart?cht=tx&chl=dD=%5Cfrac%7Bdt%7D%7Bt%7D)
 
- Without any other mistake, time can be written  ![f2]  with ![f3] the propagation time, ![f4] the bias of the receptor, ![f5] the bias of the satellite.
+Without any other mistake, time can be written ![f2](http://chart.apis.google.com/chart?cht=tx&chl=t=t\_%7Bi%7D+t\_%7Bb,r%7D-t\_%7Bb,s%7D) with ![f3](http://chart.apis.google.com/chart?cht=tx&chl=t\_%7Bi%7D) the propagation time, ![f4](http://chart.apis.google.com/chart?cht=tx&chl=t\_%7Bb,r%7D) the bias of the receptor, ![f5](http://chart.apis.google.com/chart?cht=tx&chl=t\_%7Bb,s%7D) the bias of the satellite.
 
-And the pseudo-distance between the satellite and the receptor ![f6] with ![f7]. The satellites clock is very precise and permanently controlled by the “ground segment”. We know its bias with the accuracy of less than a nanosecond. Concerning the receptors clocks, they are much less accurate. Phones clock are often quartz clock, and their accuracy is unknown.
+And the pseudo-distance between the satellite and the receptor ![f6](http://chart.apis.google.com/chart?cht=tx&chl=p=D\_%7Bi%7D+c(t\_%7Bb,r%7D-t\_%7Bb,s%7D)) with ![f7](http://chart.apis.google.com/chart?cht=tx&chl=D\_%7Bi%7D=ct\_%7Bi%7D). The satellites clock is very precise and permanently controlled by the “ground segment”. We know its bias with the accuracy of less than a nanosecond. Concerning the receptors clocks, they are much less accurate. Phones clock are often quartz clock, and their accuracy is unknown.
 
 Therefore, the localisation problem has 4 unknown parameters: 3 geometrical parameters x, y and z and time t.
 
-**2nd synchronization problem:**
+# **2nd synchronization problem:**
 
 The same problem affects the measurements made thanks to the Doppler Effect. The clock drift is a phenomenon that describes the precision of a clock through time. This drift depends on the quality of the clock : the drift of an atomic clock will be more predictable and smaller than the drift of a quartz clock.
 
-We get from the derivation of the previous equality that : 
-![f8]
+We get from the derivation of the previous equality that : ![f8](http://chart.apis.google.com/chart?cht=tx&chl=%5Cfrac%7Bdp%7D%7Bdt%7D=%5Cfrac%7BdD\_%7Bi%7D%7D%7Bdt%7D+c%5Cbigg(%5Cfrac%7Bdt\_%7Bb,r%7D%7D%7Bdt%7D-%5Cfrac%7Bdt\_%7Bb,s%7D%7D%7Bdt%7D%5Cbigg))
 
-with ![f9] the clock drift of the receptor, ![f10] the clock drift of the satellite.
+with ![f9](http://chart.apis.google.com/chart?cht=tx&chl=%5Cfrac%7Bdt\_%7Bb,r%7D%7D%7Bdt%7D) the clock drift of the receptor, ![f10](http://chart.apis.google.com/chart?cht=tx&chl=%5Cfrac%7Bdt\_%7Bb,s%7D%7D%7Bdt%7D) the clock drift of the satellite.
 
-![f10] is controlled and considered insignificant for the measurements.
+![f10](http://chart.apis.google.com/chart?cht=tx&chl=%5Cfrac%7Bdt\_%7Bb,s%7D%7D%7Bdt%7D) is controlled and considered insignificant for the measurements.
 
-Therefore, the localisation problem has 4 unknown parameters : the speed following the 3 axis of the reference frame, and the clock drift of the receptor.
-
-The question that arises is: how to make sure those clocks are synchronized?
-
-[f1]: http://chart.apis.google.com/chart?cht=tx&chl=dD=\frac{dt}{t}
-[f2]: http://chart.apis.google.com/chart?cht=tx&chl=t=t_{i}+t_{b,r}-t_{b,s}
-[f3]: http://chart.apis.google.com/chart?cht=tx&chl=t_{i}
-[f4]: http://chart.apis.google.com/chart?cht=tx&chl=t_{b,r}
-[f5]: http://chart.apis.google.com/chart?cht=tx&chl=t_{b,s}
-[f6]: http://chart.apis.google.com/chart?cht=tx&chl=p=D_{i}+c(t_{b,r}-t_{b,s})
-[f7]: http://chart.apis.google.com/chart?cht=tx&chl=D_{i}=ct_{i}
-[f8]: http://chart.apis.google.com/chart?cht=tx&chl=\frac{dp}{dt}=\frac{dD_{i}}{dt}+c\bigg(\frac{dt_{b,r}}{dt}-\frac{dt_{b,s}}{dt}\bigg)
-[f9]: http://chart.apis.google.com/chart?cht=tx&chl=\frac{dt_{b,r}}{dt}
-[f10]: http://chart.apis.google.com/chart?cht=tx&chl=\frac{dt_{b,s}}{dt}
-[f11]: http://chart.apis.google.com/chart?cht=tx&chl=d=ct
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+Therefore, the localisation problem has 4 unknown parameters : the speed following the 3 axis of the reference frame, and the clock drift of the receptor. The question that arises is: **how to make sure those clocks are synchronized?**
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