GPS receivers listening to 44W transmitters from 12,000 miles away can be heard saying hold my beer.

AND with postage stamp sized antennas. That one blows my mind regularly.

From under the noise floor. If anything that is the impressive feat.

I heard this in a podcast recently, imagine working at Bell Labs mid 20th century and calling a meeting with Nyquist, Hartley and Shannon.

Even if they didn’t show the invite would be a keepsake.

It would be hard to tell the names apart from the various tech bits and pieces named after the participants in the meeting. Shannon: "But what about the Nyquist frequency? Nyquist: "My trembling is much reduced, thank you." ...

Nothing special about the noise floor, it's just the point where you can only transmit around one bit per second per hertz

From an information-theoretic POV, yes, you can squeeze information from less, but the definition I found is based on signal levels not transmission capacity, and it seems to have other notable consequences.

AFAICT it's around this point that you can't tell whether there is a transmission, unless you know what it looks like; tuning requires decoding and/or fancy math, not a spectrometer; communication works just fine (with proper transmission modes), but there are nontrivial practical consequences as you approach or go below the noise floor.

I wouldn't expect to see analog equipment operating below the noise floor.

Link to the proof? Or at least, does the theorem has a well-known name?

The Shannon–Hartley theorem. It assumes additive white Gaussian noise (which is a good model for most kinds of thermal-ish noise), and provides a bound on the channel capacity that practical codes closely approach.

https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theore...

That's actually true for most electromagnetic telecoms. Ten orders of magnitude between the transmitted and received power is usually the easy case. With more careful techniques you can do even twenty or more orders of magnitude.

And yes, it baffles me as well.

There's a reason the word 'magic' is invoked around RF more than just about any technology.

My favorite is GPS. Not only below the noise floor, but significantly below it.. and all the satellites are on the same frequency. Gold codes are insane.

Power laws are powerful things.

This comment perfectly encapsulates what makes ham radio is so much fun. I'd argue that WSPR is even more fun than this.

If you enjoy WSPR, check out JS8Call. Its Slow mode has 30 second transmissions with extremely narrow bandwidth- very powerful even with low wattage, and it's a conversation mode and a beacon network mode :)