[time-nuts] Re: GPS antenna question

[Jeremy Elson]

The question of how GPS receivers get the time is a fascinating one and it
turns out to be absolutely integral to how they determine position. That
is: it is impossible to solve for position without solving for time, and
vice-versa.

One key technical requirement is that all the satellites carry extremely
precise clocks are synchronized *with each other*. This allows each one to
send a signal to a receiver, essentially timestamped with the time the
signal was sent according to the GPS timescale. A receiver can then look at
the set of received signals along with their timestamps. The range from
each satellite is not initially known, because we do not yet know the
difference between our (receiver's) clock and the shared GPS clock.
However, because we know that all the GPS clocks are synchronized with each
other, we know that there's just a single time bias value to solve for, not
one for each received signal.

So, we can essentially ask: what clock bias would make all the measured
range values converge?

I drew a picture of this for a presentation I did some 20 years ago:
https://www.circlemud.org/jelson/writings/localization/sld020.htm . Let's
think about the problem in two dimensions first. The 3 dots in the picture
are satellites. If we somehow knew the exact range to each satellite, we
could draw a circle around it and know we were somewhere on that circle. If
we had two circles, they'd intersect at our location. Since we don't know
the range, but the range plus an unknown bias, we can add a third satellite
and then solve for the bias: the key insight is that only a single bias
values will make all the circles converge. In the picture, some
incorrectly-computed bias is shown as the dotted circles, you can see the
three dotted circles do not meet at a single point. There is just a single
correct bias value, shown as the solid circles, that causes the circles to
converge -- and thus we have solved for both our position (the point where
the circles intersect) and the time (the bias values that caused the
circles to intersect).

In three dimensions, the circles are actually spheres, and you
theoretically need four satellites instead of three to account for the
extra dimension.

Of course, because of errors, the imaginary spheres never actually
intersect in one place. More and more satellites let us get better and
better estimates of the true bias and true location because it lets us
average away non-systematic errors.

With this model in mind, you might also now see why "survey mode" works
well for timing receivers. If we eliminate the position as a variable, but
assume it is known, the system is even more overconstrained; we can use 3
more satellites to average away errors rather than to solve for position.
In fact, if we theoretically knew our position a priori, we could determine
the time with just a single satellite.

Taking this argument to its extreme, if we know our position *and* the
time, we need "0 satellites", i.e., we can determine the clock error on the
satellite itself! And this is how the clocks in the GPS constellation are
set -- a receiver that has canonical USNO time (e.g., because it's at the
observatory) and a surveyed position listens for transmissions from a
satellite, determines the time error, and sends back a message with a clock
correction.

-Jeremy N3UUO



On Thu, Jul 22, 2021 at 6:01 PM Robert DiRosario <ka3zyx at comcast.net> wrote:

> If I want to use GPS for time and frequency standards, just how solidly
> does the antenna need to be mounted?
> The easiest and least expensive way to mount a GPS antenna would be up
> on two 10' TV mast sections, but that
> would move around a bit in the wind.  Maybe two or three inches.  Or do
> I need to do better?  All of the "easy" or
> "good" spots in my yard already have amateur radio antennas.
>
> A second question, and it may very between different GPS receivers, how
> to they get the time?  Do they just take one signal
> with a good S/N number and correct for the distance from that satellite,
> or do something more complicated with several signals?
>
> Thanks
> Robert
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