I tried to calculate the shift as a joke and came to the conclusion that it's about 1/38038th of a second IF both objects move at the greatest relative speed to each other (17600mph - disregarding triangulation and relative angles). This is on the order of 26000 nanoseconds of shift (7867 metres per second, 0.19m is the wavelength of the GPS L1 signal@1575MHz, so about 41405 wavelenghts to 7867 metres, or 1/38038th of a second - same calculation if speed of light is used).

However, apparently general relativity plays a bigger role here (https://www.avionicswest.com/Articles/howGPSworks.html) and due to it the faster moving clocks on the satellites need to be slowed down, and by a relatively substantial amount too:

"... The physics of general relativity states that space-time is warped in the presence of massive bodies (the earth) with the result that clocks run slower as they are brought closer to that body. Satellite clocks at 4 earth radii are influenced less than the same (atomic) clock running on the surface of the earth. So the atomic clocks on the satellites must be slowed down to stay in sync with clocks in your GPS receiver. The frequency is reduced from 10.23 MHz by 4.547 milli-Hertz (a half a part in one billion) to stay in sync with earth clocks. While small, the error if not corrected would accumulate to a 38 ms advance per day, or a distance error of 10 km."

The Doppler shift in the frequency from a satellite transceiver is measurable and needs to be taken into account. In ham radio use this means adjusting the frequency on the ground transceiver during the satellite pass. Commercial systems will automate such corrections.

https://n1aae.com/visualizing-satellite-doppler-shift/

https://upcommons.upc.edu/handle/2117/123510

What will help is something in the signal that the receiver can "lock on" to in order to tolerate slight mistuning. With analogue modulation that's a "carrier" frequency which has no data but provides a frequency reference the receiver can "lock on" to. AM, FM, and analogue TV include such a carrier. By contrast single sideband with no carrier, popular in ham radio, will give the receiver an audible pitch shift if they are not precisely tuned. For digital modulation things are more complicated but there are still ways to handle that mistune, and indeed devices don't give the user control over the precise frequency.

https://en.wikipedia.org/wiki/Carrier_wave

I would also add that Earth-Moon-Earth propagation can also cause Doppler shift to a meaningful degree.

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It happens with satellites all the time.

On the Huygens probe into Titan, it was realized that they had forgotten to account for the Doppler effect properly, and that the signal from Huygens would not be received. They figured out a spacecraft geometry that reduced the Doppler shift enough to avoid the problem. https://www.thespacereview.com/article/306/1

And of course that's how a radar gun measures how fast your car is going.

But yeah, radio communications with airplanes don't have to worry about it, as it's too small to matter and they can tweak the frequencies if they need to to stay locked on to each other.

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