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I have a question about Sampling Theorem. Sampling Theorem states that a "band-limited" signal can be exactly reproduced if it is sampled at a frequency F, where F is greater than twice the maximum frequency in the signal.

But mathematically, a signal can never be truly band-limited. A law of Fourier transformations says that if a signal is finite in time, its spectrum extends to infinite frequency, and if its bandwidth is finite, its duration is infinite in time. Clearly we cannot have a time-domain signal of infinite duration, so we can never have a truly band-limited signal.

My question is that, how do we calculate the sampling rate for a real signal, that's not not band-limited?

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    \$\begingroup\$ But mathematically, a signal can never be truly band-limited NOT true A repeating waveform can have a limited BW as it is know until infinite time (as it repeats). A Sinewave function has only one frequency component and is thus bandwidth limited. \$\endgroup\$ – Bimpelrekkie May 4 '17 at 10:08
  • \$\begingroup\$ Signals contain information. To preserve that information, a signal-to-noise ratio is defined (all measurement systems and data recovery systems contain noise), and signal energy not benefiting the SNR can be filtered out. \$\endgroup\$ – analogsystemsrf May 4 '17 at 15:12
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The power of signals and noise below the Nyquist frequency is properly recovered after sampling. The power of the remainder of the signal (and noise) above the Nyquist frequency is folded into the base band and is usually regarded as interference (more noise): -

enter image description here

So, after sampling you have a signal to noise ratio and that SNR can be improved by using an anti-alias filter. What remains to be asked is: -

  • Is the signal below Nyquist sufficiently representative of what you want so that it is usable?
  • Is the noise sufficiently low for your purposes?
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  • \$\begingroup\$ Still I don't understand how we choose the sampling rate? \$\endgroup\$ – Shady May 4 '17 at 15:27
  • \$\begingroup\$ Well, you have to ask the questions posed in the final part of my answer and position the sampling rate to achieve enough intelligence left in the signal so that the sampled version is useful and that any noise (or discarded signal) folded down does not degrade the signal you have retained through the sampling process. It can be tricky to get right but using even a simple RC anti alias filter makes life easier. \$\endgroup\$ – Andy aka May 4 '17 at 18:14
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The higher the bandwidth, the higher the required sampling rate to avoid aliasing. For a non band-limited signal the bandwidth can be theoretically infinite, so the sample rate also should be infinitely high.

In real life: you look how much bandwidth you need for your information to pass through (audio/video/digital data) and filter only the needed part.

With some consideration sampling rate can also be lower than the highest frequency of the signal.

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  • \$\begingroup\$ The first paragraph only repeats the question, and the second forgets that (real) filters are never perfect... \$\endgroup\$ – pipe May 4 '17 at 10:35
  • \$\begingroup\$ @pipe So what's your solution? \$\endgroup\$ – Shady May 4 '17 at 10:53
  • \$\begingroup\$ @Shady The correct answer has been posted. \$\endgroup\$ – pipe May 4 '17 at 11:07
  • \$\begingroup\$ @Pipe It says " The power of signals and noise below the Nyquist frequency... ", I don't see where it explains how to choose this Nyquist frequency (sampling rate), when our signal is not band-limited. \$\endgroup\$ – Shady May 4 '17 at 15:43
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The fact that the band-pass filter does not remove all the frequencies above the Nyquist limit results in some aliasing and therefore an unwanted contribution in your sampled signal. In real systems you decide how small you need to make that contribution and design your filter and sampling rate accordingly. As a simple approximation we might treat the aliased signal as noise when looking to determine error rates in a discrete system.

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