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It is well known that no time-limited signal is frequency-limited. So the frequencies of a time-limited signal extend upto infinity - may be decaying slowly or fastly in frequency domain. But we take signal's frequency content to be zero beyond some frequency and take that as the maximum frequency of the signal. My two questions are:

  1. Is there a solid fail-proof rule by which we can fix the maximum frequency?
  2. When the max freq is fixed and sampling is done, how do we guarantee that the infinite foldings that occur in frequency spectrum due to sampling, do not reinforce each other, at some particular frequency within folding frequency and distort the spectrum of the sampled signal?
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  • $\begingroup$ For 1 - the time goes to infinity $\endgroup$ – Moti Mar 12 '16 at 4:26
  • $\begingroup$ For 2 - the sampling frequency is twice the max frequendy $\endgroup$ – Moti Mar 12 '16 at 4:27
  • $\begingroup$ and you can LPF data to fix the max frequency. $\endgroup$ – robert bristow-johnson Mar 12 '16 at 4:47
  • $\begingroup$ @ Moti: That is begging the question, my question is about how do you fix the 'max frequency'? $\endgroup$ – Seetha Rama Raju Sanapala Mar 12 '16 at 5:00
  • $\begingroup$ @ robert bristow-johnson: No LPF is ideal! Complete attenuation is not possible in a practical filter. $\endgroup$ – Seetha Rama Raju Sanapala Mar 12 '16 at 5:01
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In the real world, one must a-priori know or assume something about your time-limited signal, such as that all the stuff in the stop-band of your signal, system, and chosen anti-aliasing filter is below your desired noise floor at or above half your chosen sample rate.

If there's more out-of-band noise than you expect, say a nuclear warhead EMP goes off next to your microphone or 20 kHz audio low pass filter, and melts it, your assumptions may have failed.

Thus, nothing is fool proof. You might be able to statistically characterize your signal to a distribution that fits within some bounds with sufficient likelihood.

Also, one doesn't necessarily assume anything to be zero. There's usually some noise floor, even if you set it below preceptability. Thus, you might be able to adjust your max frequency by requiring/allowing a different/higher noise floor for a given signal.

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  • $\begingroup$ :Thanks for the clarification. EMP is extraneous. But the signal content upto infinity is inherent. I am just wondering if any investigations have been done on the circumstances in which the infinite content of the signal spectrum and the resulting foldovers, do not distort the signal in band due to the finite sampling rate. This has been nagging me for decades but of course I did not do any research on this. $\endgroup$ – Seetha Rama Raju Sanapala Mar 14 '16 at 3:52

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