[OnlineGladiator]

I was actually wondering about this the other day. So what equation(s) would you use to determine the variance in distance estimation relative to velocity estimation? And it sounds like you're strongly implying the distance variation is immeasurably small while accurately estimating velocity - is this correct? I'm not sure the macro point makes sense, since you could have a large object with only one point measuring it (or more realistically a dozen points, but still far from what people mean when they say macro). But I'm curious to learn more if you can provide the math.

[PhaseLockk]

I'm pretty sure the effect you are discussing has to do with the uncertainty relationship inherent to the Fourier Transform [0]. This is very closely related to the Heisenberg uncertainty principle, and states you cannot simultaneously constrain time and frequency, which are the values you need to measure for position and velocity, respectively. In the context of signal processing applications, I don't think the particle nature of light is typically considered, which is why it may not be exactly correct to refer to it as the Heisenberg uncertainty principle in this context. This is a bit outside my domain though, so take it with a grain of salt.

[0] https://en.wikipedia.org/wiki/Uncertainty_principle#Signal_p...

[kyzyl]

So your're correct that there is a Fourier Transform analogy for the uncertainty principle, but in the context of FMCW lidars (which brought up the question of velocity vs position uncertainty), the measurement of frequency actually determines both the position and the velocity. It's actually a problem for most FMCW lidars because you just get 1-2 frequency measurements and somehow need to disentangle what the range frequency is, as well as what the doppler (velocity) frequency is. A massive amount of effort has been put into developing lidar methods and architectures that solve this problem well.

But in summary, the uncertainty principle as encountered in quantum mechanics has ~nothing to do with a trade off between range accuracy and range uncertainty. It's possible that it could come into play in a very detailed treatment of FMCW lidar SNR, in the context of counting return photons, but also not generally necessary there. The time-frequency uncertainty plays a role in that the range and velocity resolution both get better the longer you stare at a signal. So for a given amount of reflected light, at a given range/velocity, there is a fundamental lower bound to how long you must integrate to a) get a signal at all and b) achieve a desired precision.

[mattkrause]

It's not just an analogy--the underlying math is the same. These course notes have a nice little summary + a proof: http://www.its.caltech.edu/~matilde/GaborLocalization.pdf

[OnlineGladiator]

Thank you for this! This is exactly what I was looking for.

[mattkrause]

It seems to be an extract from "Foundations of Field Computation" by Bruce MacLennan, if you want to read the whole thing: http://web.eecs.utk.edu/~bmaclenn/FFC.pdf

He and Dr. Marcolli have a bunch of interesting stuff on their websites if you like this sort of stuff.