Theory: Sampling Theorem & Nyquist Frequency

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+6 Q:

Theory: Sampling Theorem & Nyquist Frequency

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?

+6 A:

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.

ahmadabdolkader 2010-02-10 22:17:08

and exceed it by a good margin

pavium 2010-02-10 22:18:45

The Sampling theorem only states that it should exceed the Nyquist rate, not with a good margin. Of course, you will get a better reconstruction of the image with a higher sampling rate, but the sampling theorem is about avoiding aliasing, which happens if your sampling rate exceeds the Nyquist rate.

martiert 2010-02-10 22:23:23

@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 2010-02-10 22:27:06

@vicatcu No you can't reconstruct it exactly. Not even close. All you can do is avoid aliasing.

phkahler 2010-02-11 01:30:10

@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 2010-02-12 21:55:20

Just to clarify: http://en.wikipedia.org/wiki/File:CriticalFrequencyAliasing.svg that's at the nyquist rate, so how exactly do you reconstruct "the right one"?

phkahler 2010-04-15 20:37:45

How about:

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

More recent statements of the theorem are sometimes careful to exclude the equality condition; that is, the condition is if x(t) contains no frequencies higher than or equal to B; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency B.

Dieter 2010-02-10 22:23:01

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Theory: Sampling Theorem & Nyquist Frequency

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