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Nyquist Frequency

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Re: Nyquist Frequency

#51

Post by cushychicken »

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…

The folks who are telling you you’re wrong don’t understand Nyquist’s criterion very well. Curse those undergrad courses for only effectively teaching about Nyquist at baseband frequencies.

You can sample 100MHz of bandwidth at 1GHz just as you describe at 210MSPS. You’ll get everything in the 950-1050MHz band.

Trouble is, without an antialiasing filter, you’ll get every other band that’s a multiple of that sampling rate. The Nyquist criterion works at every multiple of the sampling frequency.

Bandpass filter your analog input appropriately from 950-1050MHz and you’re golden.

This is the way nearly every commodity Wi-Fi chip downsamples 2.4/5GHz raw RF. Sigma-delta ADCs are cheap, fast, and space efficient for die area using this method.

Re: Nyquist Frequency

#52

Post by labcomputer »

mikepavone wrote: post #49
labcomputer wrote: post #38

Earlier quoted context omitted.

As the sibling comment mentioned, you don’t need to demodulate first, because that is actually what the sampling process of your ADC does. You can think of it as multiplying the original signal by a comb (in the time domain) of delta functions, which folds everything (in the frequency domain) back into the nyquist frequency of your ADC. Each delta function corresponds to one sample. If your original signal was truly…

Thanks for the explanation! Between your comment and the Undersampling wiki page diydsp linked to I think I am on the path to enlightenment. > If your original signal was truly band-limited to 100MHz In practice, this means you need to band pass before the ADC, right? i.e. "signal" in this case is the entire input to the ADC and not just the particular modulated signal you care about

> In practice, this means you need to band pass before the ADC, right? i.e. "signal" in this case is the entire input to the ADC and not just the particular modulated signal you care about

Right and right.

And, you’d normally want that to be a contiguous 100 MHz band of frequencies (you could in principle have multiple discontiguous bands that add up to 100 MHz if they are spaced right (they don’t fold down to the same base frequencies), but that would be quite an unusual application).

Re: Nyquist Frequency

#53

Post by kimburgess »

InitialLastName wrote: post #48
Sesse__ wrote: post #46

Earlier quoted context omitted.

I'd guess the “why” was “why on earth did they not have an antialiasing filter”, not “why did they downsample”. A good lowpass filter is easy to design, cheap to apply, and protects you from this kind of stuff.

I was working off the quote, but I can see some reasons that someone would decide not to AA filter. Depending on the context it might be reasonable to assume that the signal is band-limited anyway (talk-oriented radio especially is often low-pass filtered) and it's easy to miss that some point in the system can introduce an (inaudible to most humans) artifact. Those assumptions, along with the desire to avoid complex…

> I'd also emphasize how little most of the people involved in these systems care about the quality of the archive. If it's good enough to a) confirm there was signal on the channel and b) understand the voices involved, it's good enough to not worry about further.

This is uncomfortably accurate. I work with the capture side of these system and people in that space care deeply about the integrity of the signal, but have little concern for what it contains. Archival is the inverse: the information content of the signal is what's important, not the signal itself.

Re: Nyquist Frequency

#54

Post by jancsika »

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…

I've had an open GSoC project for some years to create a library that makes a handful of these audio misconceptions true. So the student would design an oscillator or oscillator bank where the closer you get to Nyquist, the more some "bad thing" happens to the corresponding output. Morphing into a triangle would be one way to do it.

Re: Nyquist Frequency

#55

Post by kayson »

diydsp wrote: post #26
kayson wrote: post #18

Earlier quoted context omitted.

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,…

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 sam…

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.

Re: Nyquist Frequency

#56

Post by kayson »

abstrakraft wrote: post #20
kayson wrote: post #18

Earlier quoted context omitted.

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,…

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…

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")

Re: Nyquist Frequency

#58

Post by squeaky-clean »

kayson wrote: post #55
diydsp wrote: post #26

Earlier quoted context omitted.

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 sam…

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.

> your bandwidth is 100 MHz centered at 1 GHz

Implies a bandlimited signal centered around 1ghz.

Re: Nyquist Frequency

#59

Post by squeaky-clean »

PaulDavisThe1st wrote: post #31
stagger87 wrote: post #24

Earlier quoted context omitted.

At the "textbook"/"theory" level, the person you are replying to is not wrong. A sawtooth waveform has infinite harmonics. If you were going to be nitpicky (which your response was in that spirit), the best thing to have said (IMO) was that the high frequency harmonics are going to drop off and be below any sort of "noise floor" or sensitivity of the system and not matter anyways. Instead you wrote a bunch of stuff a…

> A sawtooth waveform has infinite harmonics This is only true if you consider the waveform to be a sine series. As I indicated, this is a perfectly legitimate way to think about a sawtooth (and indeed, it appears to be fundamentally how the human ear works too). But a sawtooth waveform is also nothing more than a very sharp rise/drop in air pressure followed by a longer drop/rise, repeated over and over again. If yo…

> However, digital synthesis does not require this sort of conception at all, and can be constructed without any summing of a harmonic series.

A naive sawtooth algorithm (linear rise from -1.0 to 1.0) will create aliasing and not be a true saw. You cannot filter out this aliasing unless you do this with extreme oversampling. Otherwise you need to synthesize the waveform in an alias-free (or alias minimizing) method.

There's quite a few ways to digitally synthesize a sawtooth, all with some compromise, but they're all based on sine summation theory.

One of the more common ways is to precalculate a table of single-cycle bandlimited waveforms for every 1/3 octave or so, and choose the nearest table index for a given note-frequency being played, and interpolate as needed. (It's essentially mipmapping).

Re: Nyquist Frequency

#60

Post by femto »

cushychicken wrote: post #51
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…

The folks who are telling you you’re wrong don’t understand Nyquist’s criterion very well. Curse those undergrad courses for only effectively teaching about Nyquist at baseband frequencies. You can sample 100MHz of bandwidth at 1GHz just as you describe at 210MSPS. You’ll get everything in the 950-1050MHz band. Trouble is, without an antialiasing filter, you’ll get every other band that’s a multiple of that sampling…

The most fiendish application of this effect that I've seen is polyphase filtering. I can't remember the details, but at the time I can remember the wonder of understanding (in a lecture by fred harris) how most the logic was running at a low sampling rate yet the input was at a high rate. The mixing was done by aliasing.

Details here:

https://www.dsprelated.com/thread/7758/understanding-the-con...

https://s3.amazonaws.com/embeddedrelated/user/124841/fbmc_bo...

https://s3.amazonaws.com/embeddedrelated/user/124841/fbmc_ch...

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