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If I have a signal with frequencies up to 2000 Hz but I'm only interested in the frequency range up to 500 Hz, is it correct to sample at 1000 Hz $(f_\mathrm s \geq 2 \cdot f_\mathrm{max})$? Or do I have to apply an analog lowpass filter with cut-off frequency 500 Hz first?

I read this similar post, but didn't understand it well: Ignoring Nyquist

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    $\begingroup$ You have to do lowpass filtering first, otherwise highpass part of the signal (500 - 2000Hz) will alias down to the lowpass one. And the signal of interest will be lost. To avoid this filter out a part of signal spectrum you're not interested in. $\endgroup$ – Serj Aug 19 '14 at 2:30
  • $\begingroup$ Ignore is not the correct word here, you can violate the Nyquist criteria if you are ok with the signals that will be aliased (e.g. pre-filtered and highly attenuated). $\endgroup$ – Christopher Felton Aug 19 '14 at 13:14
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You can only "ignore Nyquist" if you a-priori know that there is no higher frequency content (above your desired or required noise floor) that will alias down (or up!) to your band of interest. But since your signal has content at and above half your sample rate, you will need to low pass filter that signal before sampling.

You will also need a sampling rate ABOVE 1000 Hz, if you are interested in a finite length of signal at 500 Hz.

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