The "chip rate" of GPS L1 C/A (the main one) is 1023 kchips/sec. So you end up with a signal that is over 1 MHz wide to encode 50 bits/sec. Nyquist-Shannon theorem says* you thus need over 1 Msamp/sec (using complex numbers), probably more like 2 Msps because of Doppler, to capture that. GPS is pretty forgiving and 4 bits/sample is plenty (2 bits is usable), but that would still be 1 MB/s of high entropy data. Note the system linked in the parent comment only records in 12 ms bursts. That captures enough info to find position offline, but only if you add in the historical orbit information that normally takes 30 sec to download off of the GPS signal. Streaming 1 MBps of data is doable, but I think would draw much more power than solving on device. Just recording to an SD card is far less.* The Nyquist-Shannon theorem actually says the converse, but for anything you'll encounter outside the recesses of a math department, it's still the optimal solution.
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