Sampling: What Nyquist didn't say, and what to do about it (2018) [pdf]

https://www.wescottdesign.com/articles/Sampling/sampling.pdf

by luu • 2 years ago

192 74 2 years ago

diydsp2• 2 years ago

Many brilliant points. Particularly about Bessel filters instead of Butterworth. And also about lowering Fc and increasing Fs, rather than increasing order. And also about considering time domain as well as frequency domain.

The dense interview question I use to assess this area of knowledge: "How do you choose the stopband attenuation of a filter?" You can assess a lot from the interviewee's response. Stopband attenuation, with respective to the input signal magnitude at the stopband frequency, or start of the stopband, is the most relevant term for determining the magnitude of noise in the sampled signal. And that is the upper bound on the performance of the downstream algorithm.

patrakov• 2 years ago

My response would be: "to validate the chosen stopband attenuation, compare the unwanted signal or interference that passes through the filter with the noise that you already have or with other factors that would make this unwanted signal or interference unnoticeable".

diydsp2• 2 years ago

That's a good way of looking at it. The "unnoticeable" part is where more can be teased out:

That is, to expect a specification for the desired S/N (signal-to-noise) ratio and bandwidth, and/or bit rate at which the downstream algorithm is expected to work. This doesn't have to be a perfect estimate/evaluation/measurement, but it helps to get the ballpark amount of computation power, delay, etc. needed to actually make the system work. All too often I see people react to a noisy signal, by "putting a filter on it," without knowing how much attenuation/gain/phase shift, etc. are actually needed to make the system work acceptably.

E.g. You don't want to spend too long designing the perfect filter... And you want to know if there's any hope at all cleaning up the signal to make use of it!

Good luck with your filtering projects!

f5ve• 2 years ago

If you're thinking "The highest rate I need my signal to be able to replicate is X, so I should set my sampling rate to 2X," then you're wrong and this article gives several reasons why.

As far as I can tell, though, it doesn't mention what may be the most important reason (especially to the folks here at hackernews): resampling and processing.

This is why professional grade audio processing operates at a sample rate many multiples higher than human hearing. It's not because of the quality difference between, say, 192 and 96 kHz, but rather if you're resampling or iterating a process dozens of times at those rates, eventually artifacts will form and make their way into the range of human hearing (20 kHz).

gumby• 2 years ago

Yeah, a lot of people think “Nyquist” is a synonym for 2 and stop thinking further.

Espressosaurus• 2 years ago

Heh. Then they don't actually understand what it implies.

Which makes sense I suppose.

rcxdude6• 2 years ago

Not just eventually: many effects, such as basically any non-linear mapping like a distortion, will create overtones that will immediately alias down if you are not oversampling. You either need to use some DSP tricks or oversample (usually a mix of both) to avoid this happening, which often happens in just one step of an effects chain.

markkitti• 2 years ago

Even the term "oversampling" implies that sampling beyond Nyquist rate is excessive. I think you would agree that one is not being excessive. It is necessary to sample well beyond accepted "Nyquist rate" in order to reconstruct the signal.

scns• 2 years ago

A great explanation by Dan Worrall:

https://www.youtube.com/watch?v=-jCwIsT0X8M

shampto3• 2 years ago

You’re right, but I fear this idea has become prevalent in audiophile communities where they only want to listen to files that are 96kHz or higher.

In my opinion, having a high sample rate only really matters during the production phase and does not have a noticeable effect on the final form factor. If the producer uses high sample rate during the creation process, I see no reason why the listener would care if the file they’re listening to is higher than even 44.1kHz unless they are planning on using it for their own production.

hunter2_3• 2 years ago

The same could be said about bit depth: 24 bits offers far less quantization artifacts than 16 bits, and those artifacts can readily show up during production processes such as dynamic range compression, but they are extremely well hidden by dithering with noise shaping which gets applied during mastering so ultimately listeners are fine either way.

However, any type of subsequent processing in the digital domain, even just a volume change by the listener if it's applied digitally in the 16 bit realm (i.e., without first upscaling to 24 bits), completely destroys the benefit of dithering. For that reason, we might say that additional processing isn't confined to the recording studio and can happen at the end user level.

I'm unsure whether this same logic applies to sampling frequency, but probably? I guess post-mastering processing of amplitude is far more common than time-based changes, but maybe DJs doing beat matching?

duped2• 2 years ago

People should prefer 48k over 44.1 but not for fidelity. It would just make the world a better place if 44.1k audio files died out. The reasons it was chosen are invalid today and we're stuck with it, and now every audio stack needs to be able to convert between 44.1/88.2 and 48/96 which is a solved problem, but has a tradeoff between fidelity and performance that makes resampling algorithms a critical design feature of those stacks.

All because Sony and Philips wanted 80 minutes of stereo audio on CDs decades ago.

charcircuit• 2 years ago

Another factor which I don't see mentioned is that the tech audio signal is not always directly going to your ear. Is it possible for the sound bouncing around the room to break such assumptions?

throwaway06652• 2 years ago

This is all about representing signals inside a computer. Audio played from a speaker (or as it exists in the physical domain) is continuous and your ear doesn't have a sample rate. So there's no concept of a Nyquist limit or aliasing with physical sound.

flyinghamster• 2 years ago

Also, when the sampling rates get extreme (software-defined radio), it is well worth moving to complex samples. Doing so allows you to use a sampling rate equal to your theoretical maximum bandwidth, instead of 2x. That's not such a big deal at audio bandwidth, but when your Airspy is slinging a 6 MHz chunk of spectrum, it becomes a huge deal.

jmsgwd• 2 years ago

I think you're mixing up the effects of _sample rate_ and _bit depth_ here!

Everything you said about sample rate applies more to bit depth. Higher bit depth (bits per sample) results in a lower noise floor. When audio is digitally processed or resampled, a small amount of noise ("quantization distortion") is added, which accumulates with further processing. This can be mitigated by working at higher bit depths - which is why professional grade audio processing routinely uses 24 bit formats (for storage) and 32-bit or 64-bit floating point internally (for processing), even if the final delivery format is only 16 bit.

Sample rate, on the other hand, affects bandwidth. A higher sample rate recording will contain higher frequencies. It doesn't have any direct effect on the noise floor or level of distortion introduced by resampling, as I understand. (It could have an indirect effect - for example, if certain hardware or plugins work better at particular sample rates.)

A survey of ~2,000 professional audio engineers done in May 2023 showed that 75% of those working in music use 41.1 kHz or 48 kHz, while 93% of those working in post production use 41.1 kHz or 48 kHz.[1] These are the basic CD-derived and video-derived sample rate standards.

From this it's clear that even in professional audio, higher sample rates are a minority pursuit. Furthermore, the differences are extremely subjective. Some audio engineers swear by higher sample rates, while others say it's a waste of time unless you're recording for bats. It's very rare (and practically, quite difficult) to do proper tests to eliminate confirmation bias.

[1] https://www.production-expert.com/production-expert-1/sample...

EDIT: add link to survey.

rightbyte2• 2 years ago

Higher sampling rate makes it easier to identify non-sound disturbances. Like vibrations or electrical, that can show up in multiple orders of some frequency.

anonymous-panda6• 2 years ago

It’s interesting to me how frequently people will talk about sampling performance profilers and not know about Nyquist.

Given a CPU runs at many GHz but SW sampling profilers run at ~1 or even maximum 10khz, it’s really hard to write software if you’re targeting processing at MHz rates.

rcxdude6• 2 years ago

I'm not sure what you mean here, but I would say that sampling profilers are a good example of Nyquist applied correctly for the most part: your signal (the distribution of where your software is spending its time) is likely to have an extremely low bandwidth (as in, it's basically static), so why try to sample super rapidly? It's much more of a question of whether you get enough samples to represent that distribution accurately, not how quickly you get those samples.

anonymous-panda6• 2 years ago

You’re talking about profiling a normal running program. If you’re profiling a benchmark that’s always executing the hot path, where exactly the hot path contribution lies becomes less clear. This is because you run into aliasing with the samples being collected at unhelpful points. Maybe if you run long enough you start to see a picture but at 1khz you’re going to have to run for a very long time. The other way aliasing comes into effect is that it becomes to see the impact of “individually cheap” but often executed pieces of code (eg a simple bounds check may not show up even if it’s responsible for a 20% slowdown because the probability of your sampler hitting it is small when that bounds check takes nanoseconds to execute vs your ms profiler sample rate.

Basically a 1ms sampler can pick out a signal that’s 2ms or longer in periodicity if sampled once (all faster signals will get aliased). To get to 1ghz (once a nanosecond) would require capturing 1 million times more samples and you’re still dealing with aliasing screwing up the picture you’re getting (although maybe with stack sampling you get disambiguation to combat aliasing? Not sure).

gct• 2 years ago

Fun fact static is very broadband

rcxdude6• 2 years ago

I have referred so many people to this paper over the years. It does a great job of dispelling a lot of misunderstandings people have about Nyquist's theorem.

nofu7ur3• 2 years ago

...there is no point to distributing music in 24-bit/192kHz format. Its playback fidelity is slightly inferior to 16/44.1 or 16/48....

from "24/192 Music Downloads ...and why they make no sense" (2012) [1]

[1] https://people.xiph.org/~xiphmont/demo/neil-young.html

hamstergene2• 2 years ago

This could be a useful disclaimer, to say "we are talking about recording and processing and NOT about playback".

Audiophiles demand streaming services to provide 192/24 because they see the music being originally mastered at high sample rates, and from that conclude that listening to 48/16 is a loss of "original recorded quality".

I can totally envision audiophiles picking up the Wescott's article and using it as "scientific" argument to distribute more Hi-Res music, and some half-technical manager buying that. They won't even read it.

shin_lao• 2 years ago

High sample rate isn't useful for playback, but 24-bit is great for nuance in low intensity sound.

KeplerBoy3• 2 years ago

so for stuff which didn't properly use the dynamic range of the standard codecs?

Are human ears even capable of hearing 24 bits of dynamic range?

throwaway06652• 2 years ago

The noise floor of 16-bit music is imperceptible so for mastered music at least 24-bits is useless. Both a higher sample rate and bit depth can be useful for recording purposes - provided the room is perfectly treated and there are no external sources of noise.

salty_biscuits• 2 years ago

The biggest one I have run into is that the bandwith doesn't need to be contiguous. If you know the the important bits of the signal are contained in certain frequency bands then you can get much lower sampling rates. Basically why things like l1 reconstruction work.

mwbajor• 2 years ago

Bandpass sampling is a regular technique used in the RF world to mix down a signal to its final IF or baseband frequency, digitally. In this case, you are undersampling but using a well defined signal that you have apriori knowledge of where it is before and after sampling.

https://en.wikipedia.org/wiki/Undersampling#:~:text=In%20sig....

greesil3• 2 years ago

I ran into this problem with a codec chip. We had to run the output at 48 khz, and the input and output clock had to be the same. Well we didn't have enough CPU to process the input at 48 khz, we only cared about 8 khz bandwidth for human speech. Boxcar averaging then decimating produced way too much aliasing, so much that the ML classifiers wouldn't work. The solution was putting a honking large FIR anti aliasing filter on the codec, like 100 taps, because -that- chip had oodles of CPU to spare as it turned out.

Why FIR? It had a super sharp cutoff and of course linear phase. If I'd known about Bessel filters at the time I'd have tried those out. Live and learn and ship it.

anthomtb• 2 years ago

To clarify - you were able to run the FIR filter on the codec chip because the main CPU didn't have enough <somthing - clock rate?> to handle the 48kHz input?

Nifty story, just wanted to validate my understanding.

greesil3• 2 years ago

Yep that's it. The main cpu was a Cortex M3 or something like this. It's been awhile so my recollection is a little hazy.

rightbyte2• 2 years ago

Did you try a simple iir filter and then decimation?

greesil3• 2 years ago

I could have done that and moved the cutoff frequency back to, oh let's say 4khz but then I'm adding attenuation at the higher end of the band and nonlinear group delay. That's maybe fine if that's how we processed the training data, but I don't want us to retrain the classifier every time I change the anti aliasing filter.

iamcreasy3• 2 years ago

So, we first need to find the highest motion frequency a sensor might experience in the experiment, and then make sure the sampling rate it at least twice that to avoid aliasing?

Meaning, if an IMU sensor is mounted on a very slow moving, such as 2cm/s, RC vehicle then the sampling rate can be very slow. But if the sensor is on a fast moving drone, we need to estimate the highest frequency of the motion and make sure our sampling rate is at least double that?

KeplerBoy3• 2 years ago

Yes, pretty much, but it depends. Your 2 cm/s RC vehicle might still experience high-frequency vibrations (in the range of the motor rpm) and you might want to sample that.

iamcreasy3• 2 years ago

If we are not trying to capture motor rpm vibration - can't we just ignore it?

jeanl2• 2 years ago

For a mini-surprise, check the prime factors of 44100.

Dylan16807• 2 years ago

It's a multiple of 7?

It doesn't have big factors?

I don't know what I'm supposed to be surprised about.

mppm• 2 years ago

44100 = 210^2 = (2 * 3 * 5 * 7)^2

_kb• 2 years ago

For anyone interested in an interactive tool for playing with the concepts noted, here’s something I put together a while back for demonstrating to colleagues: https://www.desmos.com/calculator/pma1jpcuv0.

sramsay• 2 years ago

This is the most lucid explanation of aliasing I think I've ever read.

I mostly deal with this stuff in the realm of audio where the matter of analog vs digital rages regularly. The reason one hears "Nyquist says" so often from people defending digital audio is that the pro-analog people are imagining stair-stepped signals in which the information that is "lost" results in a noticeable degradation in sound quality. This (11-year-old) video is the gold standard for addressing this concern:

https://www.youtube.com/watch?v=cIQ9IXSUzuM

Curiously, no one ever seems to mention the actual problems (like aliasing).

bgnn• 2 years ago

That depends on the profession I think. If you are designing A/D or D/A converters, you know sampling and all the artefacts by heart. Nyquist criterion is extremely important, but it is simplistic. You need to consider a lot of other stuff like out-of-band noise and harmonics, reconstruction filters etc etc.

We have periodic steady state analysis to simulate sampled systems and harmonic transfer functions by the way. It is extremely useful especially to stimulate noise aliasing.

Sampling: What Nyquist didn't say, and what to do about it (2018) [pdf]

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