To make things simpler think of a single sine wave. A low frequency sine wave will consist of hundreds of samples so even if you connected them with straight lines it wouldn't look too bad. This is the mistake that everyone who doesn't understand digital audio makes.

The samples are not connected by straight lines or in a staircase like fashion, but there has to be a continuous sine wave with some frequency X that has to match the sample values.

As you increase frequency, connecting samples with straight lines will break down even visually. But you can still fit only one sine wave in there since we have to have more than 2 samples per cycle.

Exactly two samples doesn't work because think what would happen if the sample value was taken exactly at the zero-crossing of the sine wave - all sample values would be zero which is silence and not a tone.

If you have less than 2 samples per cycle you end up with aliasing. That is frequencies above half the sampling rate will be "mirrored back" to below the sampling rate.

That is why the low-pass filtering is absolutely necessary. In the image above a low pass filter would attenuate the blue sine wave because its frequency is greater than the sampling rate.

Edited by xnor - 6/10/13 at 2:39am