The Heisenberg uncertainty principle in quantum mechanics comes about precisely because position and momentum are a Fourier transform pair.

In that vein, I've always wondered whether the fact that we live in a fundamentally quantum universe is just a mathematically consistent side effect of the Fourier transform and the Universe's limited extent in space-time. Is the Planck time just the point at which we can't determine the difference in frequency of two signals because the wavelength of their beat frequency has to fit inside the Universe? It's fun to think about.

And you can essentially "observe" the Heisenberg principle when looking at a moving object. If you are observing a plane for example to more accurately know its velocity you will need to observe over a longer time, but if you do this you loose accuracy about its position. This does affect radar systems, who can either send short pulses to accurately pin down the position or long pulses to measure the speed.