Hi Lazer, I couldn’t reply to your actual comment, so I’ll reply to S. McBride and hope you see it!

The time difference is computed by autocorrelation. Very basically,
The first bit from signal one is multiplied by the first bit of signal two. For example, if the first bits from the two signals both have values −1, then the result is (−1) × (−1) = +1. Similarly, if both bits have values +1, then the result is +1. On the other hand, if the two bits disagree, the result is (+1) × (−1) = −1. This process is repeated for the second pair of bits, and so on. The result can be written as a sequence of +1 (where the bits agree) and -1 (where the bits disagree). This sequence is then summed, and divided by the total number of bits in each signal. For example, if signal A can be written (+1, −1, −1, +1, −1), and signal B can be written (+1, +1, −1, −1, +1), then multiplication gives (+1, −1, +1, −1, −1); the sum of which gives −1; then dividing by the number of bits (5) gives −0.2.

When the two signals are not properly matched in time, the result of autocorrelation gives an answer close to zero; if the signals are matched in time, the result is close to +1 (but not exactly, since a real signal also has noise, so some bits are incorrect). One can see that the larger the number of bits that are compared, the better the resolution. This is because the random bits will average to zero better, the more bits we compare.

Hi, thanks for your explanation. If I understand correctly, the receiver knows what the satellite’s time was when the message was sent by the satellite, as well as what its own time is when it receives the message. But if his time and the satellite’s aren’t synchronized, it can’t just compute dt = my_time – satellite_time.

I thought devices regularly kept themselves synchronized with the satellites’ time. If not, can you explain how dt is computed ?

see my response to Alex earlier today.

One misleading thing I saw is the assertion that we know the distance to each satellite. We don’t. We do know the satellite’s location from its broadcast data, but don’t yet know our location (or what time our clock read when each satellite sent its timestamp). All we can tell from the raw data is the differential distance to the satellites in use. With four or more satellites, we can solve for our location and could perhaps synchronize our clock to GPS time. See my response to Alex earlier today.

Good question and one that at least most websites do not appear to address correctly.
If we assume that the receiver’s clock is not synchronized to the satellites, then the data you start with are time difference dt and satellite location Sxyz from each satellite.
The system of vector equations to solve to find your location Lxyz is |Sxyz – Lxyz| – D0 = c*dt, where D0 is the unknown distance to the arbitrary satellite assigned dt=0 and c is the speed of light. That has four unknowns Lxyz and D0, so at least four satellites are needed.

You just stated the reason. Could you explain it?