# Passing a sampled signal through a filter

I was wondering why is it wrong to use a band-pass filter on a sampled signal?

If the signal we want to sample has frequencies up to fmax, we sample it with frequency fs = 2fmax (so that Nyquist theorem condition is fulfilled), and the spectrum of the sampled signal will have scaled "copies" of the original spectrum centered around frequencies k * fs (k = +-1, +-2 ...).

What would happen if we filtered that sampled signal with a band-pass filter [fs-fmax, fs+fmax], and why is it wrong? Wouldn't we end up with the signal we started with?

• how would you construct such a filter? analog? digital? ideal filters? – user28715 Oct 12 '19 at 15:00
• Ideal analog filters, I forgot to add that. I meant what would happen purely theoretically. – Dan D. Walsh Oct 12 '19 at 16:50
• To fulfill Nyquist in finite time, you need to sample above 2*Fmax. Nyquist at 2*F only applies to sampled signals longer than the existence of the universe. – hotpaw2 Oct 12 '19 at 17:23

The Nyquist criteria simply requires you to have "two samples per Hz of bandwidth". It doesn't have to be $$[-f_{max},-f_{max}]$$, it can be any frequency range that includes at least $$2 \cdot f_{max}$$ of bandwidth. However, for real signals you need to figure out what to do with the negative frequencies.