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i've got a problem with the sampling theorem

Sampling theorem states that a signal can be reconstructed exactly from it's samples if the original signal has no frequencies above half the sampling frequency.

But what about frequencies exactly half the sampling frequency?? let's say i sample a sine (with an arbitrary phase and amplitude) with a frequency exactly double the sine frequency. I will be unable to reconstruct the phase and the amplitude of the sine because i don't know how the phase shifted the sine in relation to my samples (for example, if i happen to sample exactly on the zero-crossings of the sine, my samples will all be zero).

what's the solution to that problem?

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The only signal that's bandlimited to frequency = half of sampling frequency and is zero at the sample points is a sine/cosine signal (appropriately phase-shifted), so you can reconstruct the original signal in this case too. –  Alok Singhal Feb 10 '10 at 22:18
This is a programming related question. In fact programming is the only context in which it makes any sense. Please do NOT vote to close questions just because you do not understand them. Programming is a much larger field than that. –  RBarryYoung Feb 11 '10 at 6:48
The people who closed this question have clearly never done any DSP or signal processing SOFTWARE. Sorry it's not a web, database, or cool language feature, but it most certainly IS programming. –  phkahler Feb 11 '10 at 12:30
@GregS - The FAQ says, not "software" but, that questions "should be about programming. You know, with a computer." So the D in DSP means what exactly? Or do you do the binary calculation by hand? This question is so much more worthy for SO than the multitude of questions about spheres colliding and basic trig that get answered here all the time, that the rule really seems to be "only trivial questions about math are allowed." –  tom10 Feb 12 '10 at 22:04
I replied to your argument here because you made it here, which seems the right place for it since it's a discussion about whether THIS question should be closed. You should feel free though to go on meta for a discussion if you like, but please don't tell me to. –  tom10 Feb 13 '10 at 7:16
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closed as off topic by pavium, Peter, Jason Punyon, dmckee, YOU Feb 11 '10 at 4:25


2 Answers

How about:

This sufficient condition can be weakened, as discussed at Sampling of non-baseband signals below.


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Check this: http://en.wikipedia.org/wiki/Nyquist_rate#Nyquist_rate_relative_to_sampling It's clearly stated that the sampling rate should exceed the Nyquist rate, which is double the highest frequency component.

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and exceed it by a good margin –  pavium Feb 10 '10 at 22:18
@martiert, a better reconstruction? if you sample at more than twice the maximum frequency contained in the signal, you can reconstruct the signal exactly from the samples... doesn't get any better than that :). –  vicatcu Feb 10 '10 at 22:27
@vicatcu No you can't reconstruct it exactly. Not even close. All you can do is avoid aliasing. –  phkahler Feb 11 '10 at 1:30
@phkahler - No, vicatcu (and others) are correct. The point of Nyquist is that you can exactly reconstruct the signal if you sample above the Nyquist rate. Just google it and read the theorem (it probably won't say "exactly" as reconstruct means exactly reconstruct). –  tom10 Feb 12 '10 at 21:55
Just to clarify: en.wikipedia.org/wiki/File:CriticalFrequencyAliasing.svg that's at the nyquist rate, so how exactly do you reconstruct "the right one"? –  phkahler Apr 15 '10 at 20:37
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