# Digital Audio - Frequency and Time Domain Requirements for Audio Reproduction

Taking a relatively low resolution signal such as Redbook audio and carefully interpolating samples between the existing samples to increase the effective resolution is the kind of “distortion” I can get behind. I imagine a crude drawing on graph paper with medium sized blocks transposed onto another piece of graph paper with much smaller blocks.

44.1/16-bit is in no way “low resolution.” It 100% perfectly represents a signal up to half the sampling rate. Meaning that the EXACT same original analog signal can be recreated from the digital data within the bandwidth limits. That is a mathematical fact. See the Nyquist–Shannon sampling theorem (wiki it). If up-sampling is improving the sound, it is not because of curve smoothing (“medium sized blocks transposed onto another piece of graph paper with much smaller blocks”). The idea that curve smoothing is needed is based on some poor technical explanations of digital signal theory and on deceptive marketing. The graph paper plots of the digital data are a gross simplification to make complex digital theory more understandable. In reality, it is not a jagged curve unless that was the original signal. Any sound difference is most likely due to how your DAC handles the up-sampled data.Its filter may work better with 192K (for example) than with 44.1. It is arguable if this is audible, but there may be cases where it is.

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Poor Nyquist abused again. The Nyquist theorem says that such a perfect reconstruction can be done in theory, and that you cannot do better (the last part is very important). It doesn’t say that perfect reconstruction can be done in practice. Perfect reconstruction requires processing infinite amounts of data, which is awkward because it requires infinite amounts of time and power and money. (Wikipedia says, in its more polite way, “neither method is numerically practical”.)

Any real world DAC will do worse than Nyquist.

Upsampling is not a way to get past Nyquist, it is a practical engineering solution to manage the distortion into more benign form.

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Not within the bandwidth limit.

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I don’t want to bog down this discussion with a debate about sampling, so I’ll point to a very good article by a digital music expert on the topic:

https://xiph.org/~xiphmont/demo/neil-young.html#toc_1bv2b

The relevant part starts about a third in. Look for “Sampling fallacies and misconceptions.”

He explains the stair-step pattern we often see in articles on digital. He also discusses upsampling. One quote:

All signals with content entirely below the Nyquist frequency (half the sampling rate) are captured perfectly and completely by sampling; an infinite sampling rate is not required. Sampling doesn’t affect frequency response or phase. The analog signal can be reconstructed losslessly, smoothly, and with the exact timing of the original analog signal.

So the math is ideal, but what of real world complications? The most notorious is the band-limiting requirement. Signals with content over the Nyquist frequency must be lowpassed before sampling to avoid aliasing distortion; this analog lowpass is the infamous antialiasing filter. Antialiasing can’t be ideal in practice, but modern techniques bring it very close. …and with that we come to oversampling.

Perfection does require infinite processing, regardless of bandwidth limit, because the convolution function needs to be infinitely wide.

Shannon-Nyquist is a mathematical theorem, not an engineering one. It does not claim that such a device can be built, can be physically realized.

And I’m not just being picky about some theory/practice distinction: Shannon-Nyquist does not take a position on how close you can get with a real-world device. Maybe you can get “pretty close” but S-N doesn’t say so. It does say you can’t do better.

Every single authoritative source I’ve read clearly says that an 48K sample rate can perfectly represent a signal of up to one half of that rate. I’m going to trust the author of Ogg Vorbis (and other audio codecs), who I quoted above. I could add a dozen other authoritative sources here, but I have a hunch that I will not be able to convince you.

Shannon (of Nyquist-Shannon fame) wrote:

If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.

A sufficient sample-rate is therefore 2B samples/second, or anything larger. Equivalently, for a given sample rate fs, perfect reconstruction is guaranteed possible for a bandlimit B < fs/2.

@ Nick, @AndersVinberg
Isn’t it true that both you and the people you’re debating could be right?

What I’m getting from this discussion is that 48K can perfectly represent the signal, but it is difficult to “perfectly” to convert that 48k signal back to analog without unwanted artefacts. One thing that is also objectively true is that there are almost no DAC’s that work with 16 44.1k data without upsampling it. So that suggests that both of those points are correct.

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DACs upsample for the benefit of their filter algorithms, not due to any inherent flaw in representing data up to 22Khz in 16/44.1.

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It depends on the definition of “can be”. I argue that the theorem says that it “can be” in theory but not in practice.

Nyquist certainly doesn’t say that every DAC ever built is perfect. (My 1982 CD player is not.) And he doesn’t say that every DAC that is built today is perfect: the theorem doesn’t mention 2015. It doesn’t even say that there exists a single DAC today that is perfect.

Does he say that a perfect DAC can actually be built, by somebody, at some time? I don’t see any reference to engineering in the theorem. It is pure math.

So the attribution of perfect engineering to Nyquist seems difficult to support.

Of course, it could be true even if Nyquist doesn’t say it. If you want to move from mathematical possibility to an engineering practicality claim you would have to prove it. But I argue not only that it is unstated by Nyquist, but that it is in fact not possible.

Wikipedia (the ultimate authority ) says:

It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.

Sounds good. But later, it says:

The mathematically ideal way to interpolate the sequence involves the use of sinc functions. Each sample in the sequence is replaced by a sinc function, centered on the time axis at the original location of the sample, nT, with the amplitude of the sinc function scaled to the sample value, x[n]. Subsequently, the sinc functions are summed into a continuous function. A mathematically equivalent method is [bla bla]. Neither method is numerically practical. Instead, some type of approximation of the sinc functions, finite in length, is used. The imperfections attributable to the approximation are known as interpolation error.

There is the practicality gap. The sinc function is infinite in size, we can’t process that, we approximate, less than perfect.

How big is that imperfection? Nyquist takes no position on that. We see that the industry has been learning how to do better, and will doubtless continue learning. That’s what the species does.

But we also oversell…

Anders- I think we have both made our points clearly, so how about if we drop this and let the topic continue? Feel free please to have the last word.

I argue that the theorem says that it “can be” in theory but not in practice.

You are clearly a smart person, but I think you got this one wrong, as the sources I’ve cited show. You can’t “argue” the math.

I don’t see any reference to engineering in the theorem. It is pure math.

Math is of course the basis of all engineering.

Nyquist certainly doesn’t say that every DAC ever built is perfect

Agreed. Pretty much all DACs show differences when measured. The differences come from their filter implementations, internal timing, suppression of incoming jitter, suppression of internal and external noise, plus all of the common analog domain problems.

The original point I was making was that 44.1/16 is not “low resolution” for data under 22 KHz. It is a perfect representation. Secondly, that upsampling that data will not smooth the data and make it convert to analog more accurately, but it might improve the digital filter’s performance in some DACs. It also might introduce intermodulation distortion, so it is not automatically a good thing in every case with every DAC.

You are absolutely correct that DACs can introduce interpolation error. However, it is my understanding that this error is in the time domain, not the frequency domain. As far as I know, upsampling does not solve this.

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Just wanted to say thankyou to all contributors for the good natured debate. I have seen threads on other fora which have become heated for some reason. This hasn’t and it is very refreshing to read.

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Anders I kind of disagree with you. Rob Watts (Chord WTA filter designer) states that near perfect reconstruction would require a 1 million tap FIR filter. If he is right, you must be wrong since one million is on equal to infinite. Watts also stated that a 1 million tap filter is currently not feasible. Or we could say that we agree far more dan we do with Nick

BTW how can people persist in stating the Shannon/Nyquist is telling us that digital sounds perfect. Don’t these people listen? Only those designers that think about filtering problems, like those at Meridian, Chord, PS Audio and dCS, bring us results I find really good sounding.

Right, 1 million is not the same as infinite.
But “near perfect” is not the same as perfect, either.

I hope my cynical English was good enough to make clear that I did and do agree with your😜

You are misunderstanding. No one in this discussion said that digital sounds perfect. I said that Nyquist tells us that 16/44.1 is a perfect representation of the original auido signal if it is under 22Khz. No data is lost in the encoding, meaning that one does not need to upsample for the purpose of restoring lost data. (The are sometimes other reasons to upsample.) How it sounds, depends on many other factors in the design of your DAC and, perhaps, the rest of your playback chain.

As an aside, I don’t think analog ever sounds “perfect” either. It will depend on how one defines perfect in a practical sense, as Anders hinted above.

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You are right, no one said that. But if you say that math is the basis of all engineering, you imply that when the equation works out, it should sound good. Why otherwise would you say that. Shannon (and not Nyquist) published the theorem on sampling but it would take a long time - until the late 70s, before we could more or less use it for audio. Also the anti-aliasing filter introduces distortion so your “No data is lost in the encoding” is incorrect. See the recent work of Craven and Steward. The same goes for the reconstruction filter - where Nyquist’s theorem comes into account. And that’s what I meant by my remark that people - in general and not explicitly meaning you - can’t be listening and stating that both theorems work out in practice.Any trained listener hears the time smearing, damaged transients and - often but not always - the coloration caused by the filters. My sympathies are with Anders points of view.

You have taken what I said out of context. Further, I don’t follow your logic that leads to that claimed implication. It seems nonsensical to me. Are you saying that math is not the basis of engineering? Should we really waste our time debating that?

I think we have talked this to death already and that I’m not likely to change your opinion.

That happens in reconstruction, not the sampling.

symantecs. The anti aliasing filter by nature causes time smearing and so on. You could say that the filter isn’t the sampling process but according to Shannon the band limiting is an integral part of the sampling. And the anti-aliasing filter is only present in the adc, not in reconstruction.

Anti-aliasing only occurs in sampling when the frequency exceeds the Nyquist bandwidth limit. That is why I have specified over and over in this discussion a limit of 22Khz for 16/44. It is not really a problem in audio unless you try to feed say a 96Khz signal to an 16/44 ADC, for example.

Perhaps I’m now the one claiming specious implications, but I get the distinct impression from the way you try to twist my words that you are trolling me and trying to pick a fight. For that reason and because I really think this discussion has run it course, I will not be responding further.