Tim Wescott wrote:
Mike Andrews wrote:

I thought about the reverse-GPS approach, but couldn't figure out how
to determine absolute position. The most I could come up with was that
you'd know times-of-arrival at the various receivers, and that would
give you deltas from the earliest time-of-arrival. But until you know
the distance of the transmitter from any one of the receivers, you
can't determine position w.r.t. _any_ of them. As soon as you have
distance from one of the receivers and N deltas, you have a fix in
(min(N-1,3)) dimensions -- assuming that the processor knows where all
the receivers (or antennas, at least) is in that space.

OK, maybe reverse LORAN. If you know the difference in the times of
arrival between two stations you can plot the hyperbolic surface where
your transmitter must lie. With four stations you should have six
different surfaces. The intersections won't agree, but you can get a
maximum likelihood estimation of the transmitter's position in
three-dimensional space.

Being a mathematician by trade would make this easier, and more fun...

While I do computer science now, rather than math, my degree is the
5-year Bachelor's in math, for what _that's_ worth. Every now and
again I get to actually use a bit of real math at work, generally to
the amazement of the in-juh-nears here at WeBuildHighways.

My point here is definitely not to wave my degree, as I'm quite sure
that others here have degrees more advanced than mine, or do math for
a living instead of as a hobby, etc., but to point out that having
a math background didn't make it any easier for me. It's still fun,
though.

Actually three receivers would do it unambiguously most of the time, but
four would be more accurate at the cost of a bunch more math.

Seems to me that N+1 receivers gives you an unambiguous fix in
(min(N-1,3))-space: 2 receivers locate the transmitter on a line, 3
locate it on a plane, and 4 locate it in 3 dimensions. Since we only
get to sense 3 spatial dimensions, more than 4 receivers are useful
only to provide an overdetermined solution, which may permit more
precision.

Of course, the "closer" the receivers are to one another as seen by
the transmitter (think of the solid angle that the receiver array
subtends from the transmitter), the more ill-conditioned the matrix of
coefficients that one uses to determine the position.

An interesting variation on the problem would be one in which the
receivers also received or derived some precise time signal, such as
GPS time, and the transmitter to be located transmitted a signal which
contained a precise time referenced to the same standard.

This turns out to provide a good location for the transmitter, I
believe.

--
Mike Andrews

Tired old sysadmin