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Nyquist Frequency

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Re: Nyquist Frequency

#21

Post by GeompMankle »

elromulous wrote: post #13

To add another misconception, the Nyquist frequency is a lower bound, below which you necessarily get aliasing. It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is. E.g. sampling a 1hz signal at 2hz still doesn't tell you if the signal was a 1hz sin or a 1hz sawtooth (depending on how lucky or unlucky you are).

What is the point of adding a misconception?

The conception of the theorem is that if the signal being sampled is sufficiently integrable AND bandlimited AND the signal is uniformly sampled at at least the Nyquist rate over all time/space THEN then reconstruction of the bandlimited signal is exactly possible using the sinc interpolator. The proof is covered in "Shannon's original proof" in the Wikipedia article and most books on signal analysis such as Gaskill's Linear System book. Most EE people will have to do the proof as an intro course assignment in the first month of a DSP class.

OTOH, if you are not able to sample the function over all space or time AND the function happens to be periodic outside the interval you did sample THEN reconstruction of the bandlimited periodic signal is possible using the Dirchlet kernel.

If you are not able to sample the function overall space (from the first) AND that function is not periodic, you have small problems which occasionally become big problems if you are no careful. Most DSP books have a chapter about windowing discrete data and dealing with this conundrum. Basically, exact reconstruction is not guaranteed and context-specific techniques need to be employed to ensure desirable fidelity.

Re: Nyquist Frequency

#22

Post by duped »

elromulous wrote: post #13

To add another misconception, the Nyquist frequency is a lower bound, below which you necessarily get aliasing. It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is. E.g. sampling a 1hz signal at 2hz still doesn't tell you if the signal was a 1hz sin or a 1hz sawtooth (depending on how lucky or unlucky you are).

> It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is.

Formally, the Shannon-Nyquist theorem states that if you sample a band limited signal at twice its bandwidth, an ideal reconstruction filter can be used to perfectly reconstruct the input signal. There's some wiggle room over ideal sampling/filtering, but the point is that it tells you exactly what the input was, provided it was band limited.

The misconception I think you're having is that band width is not the period of a signal.

Re: Nyquist Frequency

#23

Post by PaulDavisThe1st »

Evidlo wrote: post #15
elromulous wrote: post #13

To add another misconception, the Nyquist frequency is a lower bound, below which you necessarily get aliasing. It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is. E.g. sampling a 1hz signal at 2hz still doesn't tell you if the signal was a 1hz sin or a 1hz sawtooth (depending on how lucky or unlucky you are).

A 1Hz sawtooth contains frequencies above 1Hz. It actually has frequency components that go out to infinity, so its impossible to perfectly reconstruct a sawtooth without knowing beforehand that its a sawtooth. This is true for any signal with discontinuities (i.e. not "band-limited").

This is incorrect, though subtly, and for several different reasons:

1) It is completely possible to create a sawtooth wave that contains only a single frequency. However, you could also consider the wave to be an (infinite) sum of sinusoids at different frequencies. Both views are "correct", and which is more appropriate depends on the context.

2) Related to (1): natural (acoustic) sounds are almost always best considered as a sine series. While there are such sounds which are most easily described as a sawtooth, when you consider the physical/mechanical process by which they are formed, the sine series is a more obvious approach.

3) A digital 1Hz sinusoid can trivially contain no harmonics at all. However, the moment you attempt to convert this into an acoustic pressure wave, the nature of the physical world essentially guarantees that the acoustic pressure wave will have a series of harmonics going out far beyond the base frequency. Once you start actually moving things (like magnetic coils, speaker cones and air), it's more or less impossible to avoid generating harmonics. But since the original signal was genuinely a pure sine tone, it becomes a little tricky to decide what the correct way to describe this is.

Re: Nyquist Frequency

#24

Post by stagger87 »

PaulDavisThe1st wrote: post #23
Evidlo wrote: post #15

Earlier quoted context omitted.

A 1Hz sawtooth contains frequencies above 1Hz. It actually has frequency components that go out to infinity, so its impossible to perfectly reconstruct a sawtooth without knowing beforehand that its a sawtooth. This is true for any signal with discontinuities (i.e. not "band-limited").

This is incorrect, though subtly, and for several different reasons: 1) It is completely possible to create a sawtooth wave that contains only a single frequency. However, you could also consider the wave to be an (infinite) sum of sinusoids at different frequencies. Both views are "correct", and which is more appropriate depends on the context. 2) Related to (1): natural (acoustic) sounds are almost always best cons…

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 about sounds and pressure waves that I don't think had the effect you intended. I think you lost the plot somewhere along the way.

Re: Nyquist Frequency

#25

Post by mhh__ »

wittenbunk wrote: post #12
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…

Only true for continuous RF sources. For transient signals you need at least Nyquist frequency.

In that sense isn't the bandwidth 0-Max anyway though?

Re: Nyquist Frequency

#26

Post by diydsp »

kayson wrote: post #18
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…

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 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."

Re: Nyquist Frequency

#27

Post by Sesse__ »

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…

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).

Re: Nyquist Frequency

#28

Post by ska »

elromulous wrote: post #13

To add another misconception, the Nyquist frequency is a lower bound, below which you necessarily get aliasing. It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is. E.g. sampling a 1hz signal at 2hz still doesn't tell you if the signal was a 1hz sin or a 1hz sawtooth (depending on how lucky or unlucky you are).

That isn't really what is going on. If the signal doesn't contain any higher frequency information, the Nyquist limit establishes what you need to exactly reconstruct the signal. It is therefore sufficient for any use.

So your case, a 1hz sin doesn't contain any higher frequencies, and will be reconstructed perfectly. A 1hz sawtooth contains higher frequencies, and so is not.

I think what you are really getting into is that a signal with periodicity of, say 1hz, does not mean that the Nyquist limit is 1hz. Square waves and sawtooths are particularly obvious examples of this, because the sharp edges cannot be achieved without (very many) high frequency contributions.

Now you can avoid this by creating a different set of component functions and a different sense of "frequency" but that just pushes the problem around. Also, since you are doing non-standard things you need to explain it, especially if what you are using doens't form a proper basis.

Finally, of course this is all in the idea mathematical setting, in real world noise etc. also has to be taken into effect.

Re: Nyquist Frequency

#29

Post by klodolph »

mikepavone wrote: post #17
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…

Is this assuming you have some analog hardware that's demodulating the signal in front of your ADC? How do you demodulate a signal from a 1GHz carrier with 200 MSPS?

No, this assumption is incorrect. You can ADC first and then demodulate afterwards. The spectrum of your high-frequency (near 1 GHz) signal will be aliased at frequencies below the Nyquist frequency, but it’s easy to calculate the original frequency, if you know that the signal is band-limited.

Re: Nyquist Frequency

#30

Post by Sesse__ »

elromulous wrote: post #13

To add another misconception, the Nyquist frequency is a lower bound, below which you necessarily get aliasing. It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is. E.g. sampling a 1hz signal at 2hz still doesn't tell you if the signal was a 1hz sin or a 1hz sawtooth (depending on how lucky or unlucky you are).

> To add another misconception, the Nyquist frequency is a lower bound, below which you necessarily get aliasing. It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is.

Yes, it does. The Nyquist criterion gives exactly the (minimum) sampling frequency you need for perfect reconstruction of a bandlimited signal.

> E.g. sampling a 1hz signal at 2hz still doesn't tell you if the signal was a 1hz sin or a 1hz sawtooth (depending on how lucky or unlucky you are).

A 1 Hz sawtooth is not a bandlimited signal, so the Nyquist theorem does not apply.

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