One of the most difficult aspects of Software Defined Radios for most people to wrap their heads around is the concept of "IQ sampling". This permits, for example, the entire AM broadcast band to be received at once and saved in an ordinary file. Later, you can read back that file and freely tune the entire AM band from that file -- including every station within it --- just as if you were tuning and listening live. Some non-trivial math (including complex numbers) is involved in this process, and it's really quite fascinating.