GPS satellites transmit time values regularly, but if we put in consideration the time delay between the satellite and the receiver, the time value received wouldn't be accurate insofar as I know GPS receivers aren't able to determine their position until they've got accurate time. Our devices don't have any atomic clocks synchronized with GPS.
How do our GPS receivers calculate time from the signals of GPS?
"...the time value received wouldn't be accurate insofar as I know GPS receivers aren't able to determine their position until they've got accurate time."
The time value isn't used to tell the receiver what time it is (at least not directly, although that is helpful later). It's used so that the receiver can tell relatively what the distance is to each satellite.
If you hear Sat A say that the time is 0.00000 and Sat B says the time is 0.00010, then if they are in sync, you must be closer to B than to A. You can tell exactly how much closer you are by the specific time difference.
Repeat the calculations with a few other satellites and you will find that there is only one place (and time) that the receiver can be located.
The GPS receiver computes a solution that simultaneously provides Position, Velocity, and Time (PVT). It's not that one is calculated first, then the other is. They all fall out simultaneously.
To give a big picture view of how the GPS solution is determined, consider the following equation:
$\rho_i = \sqrt{(x_i-x_u)^2+(y_i-y_u)^2+(z_i-z_u)^2} +c\Delta t$
where $\rho$ is essentially a range/distance from the user to the GPS satellite, $x,y,z$ are position coordinates, the subscript $i$ indicates the particular satellite, $c$ is the speed of light, and $\Delta t$ is a time delay.
Assuming that you have knowledge of the GPS space vehicle (SV), the $x_i,y_i,z_i$ values are known from the satellite ephemeris (this can be obtained from publicly available data, and more accurate ephemeris can be obtained via more secure methods). There are now 4 unknowns, implying that we need 4 GPS SVs to solve for the user location $(x_u,y_u,z_u)$ and time delay. More SVs can be observed, and an over-determined solution can be found from various numerical methods (e.g., a least squares solution), or a best-4 SV solution can be employed.
The time delay is essentially in the $\Delta t$ term. Various errors can be accounted for by augmenting the system of equations to include, but in no way limited to, ionospheric & tropospheric delays, relativity effects, and clock errors present in the receiver.
A multitude of simple and complex differential methods exist to essentially exploit similar delays between measurements and remove them without even solving for them (e.g., differential GPS and real time kinematics
Here is a short paper that discusses the observation equations and, more specifically, the GPS signal and code-generation.
The GPS receiver's time has to be synchronized with atomic clocks located in GPS satellites. It is kinda chicken or the egg problem. The receiver needs precise time to calculate precise distance and precise distance to synchronize time (to calculate the time difference).
It is the fourth satellite that gives your receiver the precise time.
But this is the perfect scenario - you've got 4 precise distances and they perfectly fit into single point. But precise measurement of distances requires synchronized time on your receiver.
So what if your receiver's time is not synchronized with satellites? In that case the fourth measurement - the distance to satellite D - will be way off. It won't align with neither of the two points, but it will be notably closer to one of them. Let's call the distance between the point and the sphere around satellite D the error.
At this point the receiver solves the problem backwards. We know that if the receiver has precise time and 4 distances it should yield precise position. So by adjusting the unsynchronized time of your receiver and recalculating the error with the adjusted time, the error increases or decreases.
The process of synchronizing time is reduced to the problem of minimizing the error by adjusting local time. Once the error is minimized, the precision of receiver's time is maximized.
Imagine you have 4 distances. Pick any 3 distances of them and if their spheres overlap, they will create two points where all of them intersect. No matter how imprecise the 3 distances are, as long as their spheres overlap they will form two perfectly defined points. But that does not mean the points relate to the real world.
The fourth measurement is the one that cannot be loose, in order to the 4 spheres overlap in single point.
In theory, 3 satellites can give you enough data to reduce your possible location to two points. And one of the points usually will be in absurd distance or moving absurd speed so it could be ruled out.
So the fourth satellite is not required to rule out one of the two points but really to synchronize the time of your receiver. Because without that the two points produced by overlapping only three spheres are not reliable at all.
Yes.
Let's say the time on your receiver was way off, let's say off by 10 minutes.
So, the delay for the signal from each of 4 satellites would appear to be 10 minutes plus some small fraction of a second.
The signal from the satellite with the shortest delay would be the satellite that is closest to you.
Subtracting that delay from the delays of the other 3 satellites will let you calculate the difference of the delay between each of the satellites.
Some other information you know as facts:
With just this information from these 4 satellites it should be possible to fairly accurately calculate your latitude and longitude. Adding data from additional satellites would allow improved accuracy including calculation of your altitude on earth.
TL;DR
The clock on each GPS satellite is calibrated/synchronized to the same time. Each transmits a signal representing the time of emission from that satellite, as well as ephemeris data describing satellite positions and orbital parameters.
A GPS receiver collects this data to build its own copy of the satellite ephemeris and acquires what is essentially a set of relative differences in time-of-flight when comparing satellite signals one to another. If we consider four satellites we'll call A,B,C, and D, the receiver determines time-of-flight differences between A and B, between A and C, and between A and D. There will be only one point in space for which those difference values will be observed. A mathematical technique called trilateration is employed to derive that point. Once that point has been computed, the device can determine the exact GPS time by using the time from any of the satellites and the computed distance between that satellite and the receiver.
Depending on the positions of the receiver and actual satellites observed, there will be some uncertainty in the computed position. Additional satellites are used to reduce that uncertainty.
I'm pretty sure many of the answers above are at least partially off. You don't need 4 GPS satellites for a solution, you need 3. As stated above, 1 gets you a sphere, 2 a circle, and 3 one or most likely two points on the circle. One of those points is far in space or inside the Earth. The GPS receiver uses the mean surface distance of the Earth, the geode, to determine the correct (closer) one. Whichever is the closest to the geode wins. More than 3 satellites just allow better reduction of errors for timing and atmospheric issues, system faults, etc.
In terms of time, I'm not 100% sure, but I would imagine that the actual time of day is not of great importance, but just listening to ephemeris data should get sub-second accuracy for that. To get the receiver to have near-atomic accuracy I believe the system just makes an initial guess based on the data it has (perhaps an average of all the times it has heard from the satellites + a few MS for the mean distance of receivers to sats? Just speculation there) and then computers the error in the position. If the clock is fast, the error is big in one direction; slow produces a large error in the opposite direction. The receiver's clock is continually sped up or slowed down via a type of Phase-Locked Loop (PLL) until the error is as close to zero as possible; this continues for the entire time the device is running. If the clock starts to drift again, the PLL will catch it and push it back on track. By doing this, the clock is kept to nearly the same accuracy as those on the satellite, and thus accurate enough for navigation purposes or other precision timing purposes (NTP, radio transmission synchronization, etc). I suspect that's also (at least partly) why you see a larger error when you first turn on a GPS receiver that gets progressively smaller as the clock gets progressively more in sync.
PLL's have been around for a long time and are used everywhere. A VCXO is commonly used with a PLL to provide a cheap, variable, high frequency source based on the output of a lower speed, high quality fixed oscillator in radios and other communications gear, along with computers and other electronics. Hence where clock multiplier comes from with computing, a slow speed, fixed XO drives a faster bus via PLL to get your FSB speed, your CPU then using another multiplier and PLL for its own speed. PLL's also have been used for clock synchronization and recovery for communications for a very long time in communications networks.
Wikipedia has a decent article on it. It provides an analogy of cars racing around a track, each one under their own control trying to obtain the fastest safe/practical speed. When a pace car comes out, they all must get in line and follow the car in front of them, without passing. The pace car drives a fixed speed based on their speedometer. Each driver behind it tries to keep a similar distance to it or the car in front of them; they speed up to close gaps or slow down to lengthen them. Thus they all end up nearly evenly spaced and traveling at almost exactly the same speed. Remove the reference (pace car), and they're all off at random again.
