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I read this: To reproduce a given frequency, the sample rate must be at least twice that frequency. Why is this so?

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This is called the Nyquist criterion. Here's the wikipedia page on that subject: – Saaru Lindestøkke Jan 19 '13 at 21:39
up vote 6 down vote accepted

That's Nyquist-Shannon sampling theorem. Think about it in the other way: imagine you are given a set of samples (the black dots in the image).

The most intuitive way to recover the original analog signal would be by joining these samples. But you can easily notice that there is not a unique way of joining them. In fact, there are infinite possible signals that would contain the whole set of samples. Obviously, given the sample set, you would join them the easiest way, i.e. the blue signal.

Now, what about sampling? In the image you also have a red signal (which frequency is 1 Hz) that has been sampled too slowly, resulting in the black dot sample set. To have a clue about the real signal, we should have sampled it at, at least, 2 Hz. That way, we also would have samples (dots) at 0.5, 1.5, 2.5, etc seconds. And, as you can draw in Paint, the easiest path to join all the samples would be, again, the red signal. So we would have recovered our original signal successfully.

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