# 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
Commented Oct 12, 2019 at 15:00
• Ideal analog filters, I forgot to add that. I meant what would happen purely theoretically. Commented Oct 12, 2019 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. Commented Oct 12, 2019 at 17:23

Yes you can bandpass filter an adequate portion of a sampled (ideal impulse modulated) signal spectrum and still retain the same information of the lowpass filtered version.

As you have stated, the sampled signal has a spectrum which includes shifted and weighted copies of the original (possibly baseband) signal. Assuming no aliasing occured during the sampling operation, then any one of those shifted copies includes all the necessary and sufficient information to reconstruct the original continuous-time signal. So you can use it, but be careful, the signal originally was baseband, so you have to correctly shift the bandass spectrum into baseband.

Not sure what you mean by "wrong".

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.