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Nyquist Frequency

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Re: Nyquist Frequency

#81

Post by gct »

kayson wrote: post #77
gct wrote: post #74

Earlier quoted context omitted.

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 t…

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 2…

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

Re: Nyquist Frequency

#82

Post by Chinjut »

Chinjut wrote: post #61
EarthIsHome wrote: post #3

One misconception that many make regarding the Nyquist frequency is thinking that the sampling rate needs to be twice the highest frequency. Your sampling should really really be twice the bandwidth . e.g. your bandwidth is 100 MHz centered at 1 GHz (it needs to actually be bandlimited to 100 MHz**). You do not need to sample at 2.2 GHz. You sample at 200 MSPS (really, you should sample a little more than that, say 2…

Consider a signal whose value at x seconds is f(2x) - 2 f(3x) + f(4x), where f(x) = sin(2πx)/x. Considering that the absolute frequencies of f(x) are uniformly distributed from 0 to 1 Hz, the absolute frequencies of this total signal should be constrained to between 2 and 4 Hz. Thus, a bandwidth of 2 Hz. But if we sample at 6 Hz (three times the bandwidth!) including x = 0, we'll get all zeros. Granted, we might say…

That is, it's not quite as simple as saying you just need to sample at any frequency at least twice the bandwidth. Rather, it's the more complicated behavior described by this graph: https://en.wikipedia.org/wiki/Undersampling#/media/File:Samp.... That is, the general rule is that the ratio of the highest frequency in the signal to half the sample rate, and the ratio of the lowest frequency in the signal to half the sample rate, have to lie within an interval of consecutive natural numbers.

When the lowest frequency is zero, this is the familiar rule that the sample rate has to be at least twice the highest frequency in the signal. But more generally, it's more complicated.

Re: Nyquist Frequency

#83

Post by kayson »

gct wrote: post #81
kayson wrote: post #77

Earlier quoted context omitted.

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 2…

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

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.

Re: Nyquist Frequency

#84

Post by gct »

kayson wrote: post #83
gct wrote: post #81

Earlier quoted context omitted.

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

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 w…

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.

Re: Nyquist Frequency

#85

Post by Sesse__ »

lumb63 wrote: post #47

Signals and systems was a tough course for me. It was what crushed my 4.0 GPA. Nyquist frequency was a concept I could not wrap my head around. I’ve improved, but it still doesn’t click as I’d like it to. When I took the course, it made no sense to me that you could sample at twice the frequency of the signal and reconstruct it. Consider a sine wave at 1 Hz. If you sample at 2 Hz, you’d get readings of 0, 1, 0, -1, e…

What you are saying is generally correct, but: If you sample a 1 Hz sine at 2 Hz, you wouldn't get readings of 0, 1, 0, -1, etc.; you would get readings of 1, -1, 1, -1, etc., or if you're very unlucky, 0, 0, 0, 0, …! The _exact_ case is of Fs/2 is, well, an edge case.

Re: Nyquist Frequency

#86

Post by Chinjut »

Chinjut wrote: post #61
EarthIsHome wrote: post #3

One misconception that many make regarding the Nyquist frequency is thinking that the sampling rate needs to be twice the highest frequency. Your sampling should really really be twice the bandwidth . e.g. your bandwidth is 100 MHz centered at 1 GHz (it needs to actually be bandlimited to 100 MHz**). You do not need to sample at 2.2 GHz. You sample at 200 MSPS (really, you should sample a little more than that, say 2…

Consider a signal whose value at x seconds is f(2x) - 2 f(3x) + f(4x), where f(x) = sin(2πx)/x. Considering that the absolute frequencies of f(x) are uniformly distributed from 0 to 1 Hz, the absolute frequencies of this total signal should be constrained to between 2 and 4 Hz. Thus, a bandwidth of 2 Hz. But if we sample at 6 Hz (three times the bandwidth!) including x = 0, we'll get all zeros. Granted, we might say…

Whoops, I should've pulled the division by x out of the definition of f. The example I had in mind was [sin(4πx) - 2 sin(6πx) + sin(8πx)]/x. [Another good example is [sin(6πx) - 2 sin(8πx) + sin(10πx)]/x, whose frequencies are between 3 Hz and 5 Hz, thus a bandwidth of 2 Hz, but sampling at 4 Hz or even 8Hz gets all zeroes.]

Anyway, the details on that example don't matter, the Wikipedia graph and article makes things more clear.

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