Precise timing using GPS

Looking at https://gssc.esa.int/navipedia/index.php/GPS_Navigation_Message and other websites, in order for a satellite receiver (which doesn't have other means of getting precise GPS time) to determine precise GPS time, after it's determined its position, and subsequently the distances from the satellites, it may look at the HOW word, which contains information about GPS time. (We can also assume the receiver has decoded clock corrections, ephemerides, etc., but here I'd like to focus on something much more basic.)

The ("legacy") NAV message is transmitted at around 50 bps, and let's say that translates to the receiver obtaining this HOW word every 6 seconds – because each subframe is 300 bits long, that means $$\frac{300~b}{50~bps} = 6~s.$$

The HOW from what I understand is a counter of 1.5-second-long periods since the start of the week (actually truncated because it's only incremented every 6 seconds, meaning the last two bits aren't needed) and let's assume these satellites start broadcasting as soon as the week starts, meaning at 0 s of the week. (We can assume the receiver knows when each week starts.)

My question is twofold, since I can't find this information anywhere:

1. Do satellites indeed start broadcasting as soon as the week begins (according to their internal clocks), meaning the first subframe is broadcast at exactly 0.0000000000 s of the week, the second at 6.0000000000 s, etc.?

2. Do satellites always, without exception, transmit HOW words at exactly those intervals, i.e. 0.0000000000 s, then 6.0000000000 s, etc.? Or can the HOW sequence sometimes be (translated to seconds) 0, 4.5, 12, 18, 22.5, etc. (as opposed to 0, 6, 12, 18, 24, ...) because maybe that 4.5 was in fact 5.9997994899 s, and the 22.5 was actually 23.9999593899 s? Or maybe they always do this 0, 6, 12, 18, 24, but really they don't start at 0.0000000000 s of the week, but rather at, say 0.0000030004 s, which means their next subframe transmission time will be at 6.0000030004 s.

I can only imagine this GPS timing thing working if satellites are programmed to send these subframes at exactly x.0000000000-second-intervals, according to satellite-internal clocks, and that the receivers are to trust that is the case.

If satellites don't emit subframes at precisely x.0000000000 s, then in my view we cannot possibly determine precise GPS time, because this 1.5(or in reality 6)-second counter that the HOW represents has too low a resolution (i.e. it has a 1.5-second resolution, as opposed to, say, milli- or micro-second resolution). Maybe this happens, but because there are multiple satellites involved (4+), the receiver sort of figures out if something like 22.5 s is encountered, and the actual time is 23.9999593899 s, that that's too big an offset because three other satellites are showing 24 s.

Just a couple of things to note here: I'm aware that there's a discrepancy between GPS time and UTC time, and between receiver and satellite clocks; I know the clocks exhibit drift from GPS time, and that due to the satellites' velocity, they're adjusted to 'tick' at a different frequency from normal GPS atomic clocks, and that there are instrument errors; I know how receiver position is obtained; I know that there's extra clock-correcting information embedded in the NAV message, precisely due to the drift, etc., of satellite clocks.

• The atomic clocks are very precise, but there is a very small error of about 10^-13 seconds. That is less than a millionth of a microsecond. Much better than your x.000 000 000 0-second-intervals, that is only 10^-10 seconds.
– Uwe
Jan 27, 2021 at 0:35
• Didn't even think of that! I just used an arbitrary number of zeros to suggest/illustrate accuracy/precision. Jan 27, 2021 at 0:41
• The GPS tag in Electronics SE has 371 questions and I've had some luck there myself. If no answers are forthcoming here, that might be a helpful alternative (or source for answers here).
– uhoh
Jan 27, 2021 at 0:53

The atomic clocks used in GPS satellites are very precise, the clock error is smaller than 10$$^{-13}$$ seconds.

There are many active GPS satellites, newer and older designs, but they all should work together in the system of all satellites and all ground stations for control.

It is essential that all signals like HOW and NAV you mentioned send by different satellites are synchronized. There are high speed digital counters and logic circuits generating these signals in the same way clocked by each satellites own atomic clock. No satellite is a little bit faster or slower in generating these signals, a very little offset between two satellites should be constant, it is not allowed to increase or decrease slowly over time.

When all digital clock circuits in the different satellites are operating synchronous, we have only the analog clock delay in the satellites transmitter and the cable to the antenna. This delay is about some nanoseconds and does not cause a problem as long as this delay is constant. There may be one satellite with one nanosecond more delay and another sat with one nanosecond less than all other satellites.

We only need all these individual delays to be constant over time. Any change of these delays caused by temperature change or by aging should be so slow and small that the satellites remain synchronized as seen from a user with a GPS receiver.

A constant delay in the transmitter circuit and the antenna cable is compensated by the digital clock circuits doing the HOW change a little bit earlier. For the system user it looks like there is no delay at all.

• Does it generally hold that any GPS receiver may safely assume that whenever the HOW counter shows 6 seconds, the sending satellite sent the signal at what it "thought" at the time was exactly 6.000 000 000 000 0 seconds (from the start of the week), according to the satellite's own time, as opposed to at 7.499 999 999 999 9 seconds, or 6.000 000 000 000 1 seconds? Jan 27, 2021 at 12:06
• The speed of light is about 3E8 m/s in vacuum, a time of 1E.13 seconds is equivalent to a distance of only 30 µm. So these very small differences of time or distance are not relevant for any GPS receiver. Those small distance may be measured using other methods but not with GPS.
– Uwe
Jan 27, 2021 at 15:36