Can compressed sensing make a better oscilloscope?
The STM32 has a 1Msps 12-bit ADC, and there are oscilloscope projects using it. But a decent oscilloscope has at least 20MHz of bandwidth, and the Miniscope has 461kHz, so it’s about 2.3% of a decent oscilloscope.
Actually though the STM32F103C8T6 used in that project and in the Blue Pill has two such ADCs. If you apply them both to the same input, though, you won’t get any more information because (IIRC) they sample in sync. This is ideal if you’re trying to measure the voltage-current characteristics of some device but suboptimal if you want to measure a single signal faster. You could perhaps put an analog filter on one of the inputs in order to phase-shift some signal components.
But what if you can fire the ADCs, or an external sample-and-hold circuit feeding them, at effectively random times? Then you could sample the signal in a time-domain basis that’s incoherent with respect to its frequency-domain content. Then maybe you could use a standard ℓ₁-minimizing basis-pursuit algorithm to look for a sparse frequency-domain signal that explains what you saw?
It seems like that might be enough to get you an effective 20MHz or so of bandwidth, though of course only for signals that really are sparse.