The Heisenberg uncertainty principle is just one specific example of a much more general, relatable, non-quantum phenomenon.
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For more on quantum mechanical wave functions, I highly recommend this video by udiprod:
Minute physics on special relativity:
Main video on the Fourier transform
Louis de Broglie thesis:
More on Doppler radar:
Radar basics: https://www.eetimes.com/document.asp?doc_id=1278808
There's a key way in which the description I gave of the trade-off in Doppler radar differs from reality. Since the speed of light is so drastically greater than the speed of things being detected, the Fourier representation for pulse echoes of different objects would almost certainly overlap unless it was played for a very long time. In effect, this is what happens, since one does not send out a single pulse, but a whole bunch of evenly spaced pulses as some pulse repetition frequency (or PRF).
This means the Fourier representation of all those pulses together can actually be quite sharp. Assuming a large number of such pulses, it will look like several vertical lines spaced out by the PRF. As long as the pulses are far enough apart that the echoes of multiple objects on the field from different targets don't overlap, it's not a problem for position determinations that the full sequence of pulses occupies such a long duration. However, the trade-off now comes in choosing the right PRF. See the above article for more information.
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