If you want to know who's broadcasting anywhere in the range from 1MHz to 10GHz (example only -- I don't know the real range for this device), that's a bandwidth of 9.999 GHz.
You could scan freqencies sequentially, but it's slow and you might miss something. You can't afford to build a 20 GHz A-to-D converter, and even if you could, you don't want to build a computer big enough to process the resulting 80 GB/s data stream (assuming a 40 GHz sample rate with 16-bit samples).
The compressive sampling process is what allows you to sample at a lower rate (as long as you can assume that the signal you're looking for doesn't use too much of the total bandwidth). The convex optimization is apparently used to decide how to do the sampling in a way that's guaranteed to recover your signal.