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Comment by abstrakraft | Hacker News Reader

From an information theoretic perspective (which is the perspective Nyquist was originally coming from, though it didn't yet have that name), you don't need to mix the signal down. Assuming it is truly band-limited, you can sample the signal directly at RF, and reproduce it from those samples. Additionally, you will need to modulate the reproduced signal into the original band, which means you need to know where that band is - perhaps this is the detail you're pointing out?

Another way of looking at it is that sampling inherently does the mixing down to baseband. Although it may not be exactly the baseband you want if the spectrum isn't cleanly symmetric about a multiple of the sample frequency.

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I've worked on ultrasound systems that definitely worked this way, not just in theory but also in practice. Bandpass filter 20–40 kHz, sample directly at 40 kHz (giving 20 kHz bandwidth). No mixer step involved, but your spectrum becomes inverted (e.g. if you do an FFT, a 22 kHz tone will be in the 18 kHz bin, not the 2 kHz bin as you would perhaps expect).

Aliasing makes more sense (to me, anyway) if you think about the spectrum of complex signals, in which signals of real samples are modeled as the sum of positive and negative frequencies.

In the sampling operation, all sinusoids are shifted down to the "natural baseband" by adding or subtracting some multiple of the sampling frequency that places the resulting frequency within +/- half of the sampling frequency. So for your example of 22kHz, that real frequency has two components: +22kHz that gets shifted down to -18kHz=22kHz-40kHz, and -22kHz that gets shifted up to +18kHz=-22kHz+40kHz.

Note that this "natural baseband" is an abstraction of our own invention. You can just as easily think of the spectrum as ranging from 0Hz to the sampling frequency f_s, rather than -f_s/2 to f_s/2. The fact that some prefer one over the other is precisely why fftshift exists.

To clarify: "band-limited" usually means X(w) = 0 for abs(w) > B for some B, where X is the frequency spectrum. And that's the definition Shannon used in the original proof, which is where the idea of Nyquist Frequency comes from.

If you add the additional constraint of the signal being "bandpass-limited" where, X(w) = 0 for A > abs(w) > B for some A, B, then yes, you can under sample.

And that's where the information-theory idea comes in where the amount of information contained in the band only "needs" 2X sampling rate to reconstruct perfectly.

You can think of aliasing being somewhat orthogonal to that in the sense that you need 2X bandwidth so you don't corrupt the signal, but 2X max frequency so you don't alias anything else into the signal. (I say this realizing that aliasing is what would cause the former signal corruption, hence "somewhat")

Looks like I bundled that second inequality. Band pass is X(w) = 0 for abs(w) < A or abs(w) > B

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