The-clock-synchronization-problem.md

@@ -4,23 +4,34 @@ Satellites send a signal and users’ receptor receives it. But in order to read
 
 # **1st synchronization problem: the distance calculation is very sensitive to clock errors**
 
-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)
+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 ![lagrida_latex_editor__72_](uploads/e6e11d7df889518818c8123f954700d7/lagrida_latex_editor__72_.png).
+With :
+- ![lagrida_latex_editor__69_](uploads/5c926f7e6aac01d7485d65355efeb0b6/lagrida_latex_editor__69_.png) the distance between the satellite and the user,
+- ![lagrida_latex_editor__70_](uploads/f347b61d1739c4711ddf9b5d8e232a3a/lagrida_latex_editor__70_.png) the speed of light,
+- ![lagrida_latex_editor__71_](uploads/f778811e0c38f1979de5e9219e485bec/lagrida_latex_editor__71_.png) the time.
 
-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.
+As time is multiplied by the speed of light, distance is very sensible to time mistake.
 
-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.
+Without any other mistake, time can be written ![lagrida_latex_editor__73_](uploads/1c3f0af394ac5ae992313930157c854a/lagrida_latex_editor__73_.png) with ![lagrida_latex_editor__74_](uploads/7ad505e13cf61943281e9f62418e7e26/lagrida_latex_editor__74_.png) the propagation time, ![lagrida_latex_editor__83_](uploads/9cc25f573f404c68b5001c37c22023f8/lagrida_latex_editor__83_.png) the bias of the receptor, ![lagrida_latex_editor__76_](uploads/a4480fac0d264f77733a7bb0458e0ca1/lagrida_latex_editor__76_.png) the bias of the satellite.
 
-Therefore, the localisation problem has 4 unknown parameters: 3 geometrical parameters x, y and z and time t.
+And the pseudo-distance between the satellite and the receptor ![lagrida_latex_editor__87_](uploads/cee25cb5eb25a1e4c0cc7d59f536673f/lagrida_latex_editor__87_.png)
+
+With :
+- ![lagrida_latex_editor__78_](uploads/4cdb333547b229ed0c6d41b51ed302eb/lagrida_latex_editor__78_.png)
+
+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 ![lagrida_latex_editor__79_](uploads/71eaa8a8f5fe7768fa1bf8028e8ee623/lagrida_latex_editor__79_.png), ![lagrida_latex_editor__80_](uploads/5eb2b052f97cb5b7ae0afa42edcec878/lagrida_latex_editor__80_.png) and ![lagrida_latex_editor__81_](uploads/1652090369310750579599ddac8fbe1c/lagrida_latex_editor__81_.png) and time ![lagrida_latex_editor__82_](uploads/3ad9ddfa58eb14b73f44a58c4fde2af8/lagrida_latex_editor__82_.png).
 
 # **2nd synchronization problem: the problem with the frequency**
 
 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](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))
+We get from the derivation of the previous equality that : ![lagrida_latex_editor__86_](uploads/4b27654bc7469881a814e5c19b3a75ec/lagrida_latex_editor__86_.png)
 
-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.
+with ![lagrida_latex_editor__88_](uploads/36716b7a0e028ae59b0c0910b02dd90f/lagrida_latex_editor__88_.png) the clock drift of the receptor, ![lagrida_latex_editor__89_](uploads/66e79711f996767b54b8a2dd782916ea/lagrida_latex_editor__89_.png) the clock drift of the satellite.
 
-![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.
+![lagrida_latex_editor__89_](uploads/834c1ea6350f3d584837a695495615cc/lagrida_latex_editor__89_.png) 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?**