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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: en.wikipedia.org/wiki/Nyquist_frequency – 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).

http://i.stack.imgur.com/ld5VI.png

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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If the sampling rate is s, an input frequency of f will yield precisely the same sequence of samples as f+ks for any positive or negative integer k. If input frequencies are allowed to go to zero and if if the input signal may have any arbitrary phrase, then any input frequency higher than s/2 would be indistinguishable from a frequency within that range.

Note that the Nyquist criterion is only relevant in cases where inputs are allowed to go zero. If the input will only contain a narrow range of frquencies, it may be safely sampled at a much lower rate without loss of information. In some cases, such sampling may be be useful to shift a frequency which is too high to measure into a range where measurement is possible. For example, if one has a very precise 10.000000MHz reference and wants to measure a frequency which is known to be between 100.01Mhz and 100.02Mhz, sampling the input signal at a rate of 10.00Mhz will yield a signal which is between 10,000 and 20,000Hz which may in some cases be more easily measured than the direct 100+MHz signal. Note, however, that signals near any multiple of 10.00Mhz will be indistinguishable from those of any other.

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