[kayson]

I know what you're getting at, but your statement, as others have pointed out, is incorrect. Your sampling rate always always has to be twice the highest frequency of the signal you are sampling.

If you are sampling an RF-modulated signal with a center frequency of 1GHz and 100MHz of baseband bandwidth, then yes, you do need to sample at 2.2GHz+. And some applications do exactly that.

If you're taking the RF signal, mixing it down to baseband, and filtering it to bandlimit, then you have a signal with maximum frequency component of 100MHz, and in that case, yes, your sampling rate can be 200MHz+

[abstrakraft]

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.

[Sesse__]

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).

[abstrakraft]

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.

[kayson]

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")

[kayson]

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

[diydsp]

Actually, GP is correct. See Bandpass Sampling: https://en.wikipedia.org/wiki/Undersampling.

"In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass-filtered signal at a sample rate below its Nyquist rate (twice the upper cutoff frequency), but is still able to reconstruct the signal.

When one undersamples a bandpass signal, the samples are indistinguishable from the samples of a low-frequency alias of the high-frequency signal. Such sampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF-to-digital conversion."

[kayson]

Yes, but this only works if, as the page points out, the signal is bandpass filtered, which GP did not mention. It's not true in the general sense, nor is it practical for many (most?) RF systems, especially those with multiple channels.

[squeaky-clean]

> your bandwidth is 100 MHz centered at 1 GHz

Implies a bandlimited signal centered around 1ghz.

[kayson]

I can see why you might think that but consider that in RF systems, while the wanted signal is bandlimited, you also have a lot of unwanted "blockers" all over the spectrum that need to be dealt with before sampling.

[gct]

I'm afraid you're mistaken (source: worked as DSP engineer for 15 years). Often you apply your filter around the RF frequency you want and then sample at a lower rate. You're right that the signal will get aliased doing that, but the information is always preserved.

If you sample s.t. your folding frequencies are in an appropriate place, you can fold your desired region into the first nyquist region without needing to mix it down. This is especially desirable if you can avoid having to build an IQ mixer because they're hard to keep balanced.

The worst case doing this is that your signal spectrum is reversed in frequency, but you can correct that easily digitally.

[kayson]

I'm afraid I'm not mistaken (source: I design integrated RF transceivers) ;)

Yes, you can subsample if you have a suitably bandpass-limited signal. But that's not the general case, nor is it what the nyquist-shannon theorem proves, which is where "nyquist frequency" comes from.

Nyquist frequency by the original definition is 2X highest frequency, though some papers textbooks evidently have started using it to mean 2X bandwidth, enough so that wikipedia[1] actually mentions it.

In integrated circuits, IQ mixing isn't problematic as we can fairly easily do gain and phase calibration to correct for the mismatch.

[1] https://en.m.wikipedia.org/wiki/Nyquist_frequency#Other_mean...

[gct]

You have to have a band limited signal to sample anyways, where it's at in the spectrum doesn't matter. The first thing you'll do before feeding anything to an ADC is running it through a filter to make _sure_ it's band limited. Whether that filter's at DC or some Rf doesn't matter.

Here's the result from his original paper where he specifically says that it doesn't have to be at DC:

https://imgur.com/uSywML7

[kayson]

My point is that practically speaking, it does matter where the signal is, depending on how you filter it. If you lowpass filter an RF- (or, more realistically, IF-) centered signal, you can't just sample it at 2X bandwidth because you'll get aliases from the unwanted content between DC and the bottom frequency edge of the signal.

It may not be a common scenario anymore, but it was very common in the early GSM days when the signal wasn't mixed to DC but near-DC.

[gct]

Ah yes you're right that you have to be careful, it'll fold at multiples of the nyquist frequency and you want to make sure your SOI is entirely contained in one of those zones.

[detaro]

(use \* to escape an * and prevent it being parsed as an italics marker.)