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Digital Audio: Theory And Reality

Looking into the promise of perfect audio and the Nyquist Theorem.

By Nigel Redmon September 12, 2018

Most people who’ve looked at digital audio before know about the Nyquist theorem.

If you sample an analog signal at a rate of at least twice its highest frequency component, you can convert it back to analog, passing through a low-pass filter, and get back the same thing you put in. Exactly. Perfectly.

The Real World

In the real world, though, many people argue that analog “sounds better.” How can this be, if digital audio is perfect?

For one thing, we’ve grown to like some of the deficiencies of analog recording. Just as tube amplifiers give a more pleasant distortion and compression to musical signals than transistors, analog tape similarly warms up and fattens the sound.

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Of course, this alone isn’t a reason to forsake digital’s many conveniences. We can always use other means, such as tube compressors, to fatten the sound if needed. The real problems lie with the real-world problems Nyquist didn’t warn us about.

First, there is no such thing as the perfect low-pass filter required by Nyquist’s theorem.

This article is provided by EarLevel Engineering.

A real filter has a finite slope, so we need to set its cut-off a little lower than theory. Also, a steep filter has a lot of phase shift near and above the cutoff. And some aliasing is bound to leak through at the very high end. A technique called oversampling has been developed to reduce these problems.

Another big problem is finite word length effects—we’re using 16-bit samples, not the pure numbers of the Nyquist theorem, so we have to compromise the sample values.

To start, 16 bits is not as great as it seems. Yes, it translates into 96 dB dynamic range, but that’s an absolute ceiling—you can’t go any higher. So, the average music level must be much lower in order to allow headroom for peaks. And at the low amplitudes, noise floor can become a problem.

On top of this, any gain change (from mixing tracks or changing volumes) causes individual samples to be rounded to the nearest bit level, adding distortion. Fortunately, a technique called dithering relieves these problems.

Clock jitter is another problem.

If the sample clock timing is not perfect, it creates another kind of distortion. For a self-contained unit, the solution is simply more accurate timing; reducing timing errors reduces the distortion to a negligible level. When digitally interfacing with other units, though, the issue becomes a little more complex, but is not a problem when handled correctly.

Finally, an often overlooked detail in digital audio discussion is that Nyquist’s samples are instantaneous values—impulses.

Our digital systems generally output stairsteps to the converter and low-pass filter, holding the current sample level until the next. This causes a frequency droop and loss of highs—impulses carry more high-frequency energy than stairsteps. The solution is not to produce impulses—which are impossible to produce perfectly—but to simply adjust the frequency response with filtering.

Fortunately, it’s trivial to add this adjustment to an oversampling filter.

Read and comment on the original article, as well as view the additional links and definitions of terms mentioned, here.

Nigel Redmon is a musician, electrical and software engineer, and independent developer, specializing in digital audio signal processing applications. He has developed products for Line 6, Equator Audio, Alesis, Oberheim, and others. He shares his research at EarLevel Engineering. View more of his articles here.

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Dr.Frederick Ampel says

While useful this is Digital 101 - for the umpteenth time- not sure why it needs to be published again- all of this has been covered before in many many many other articles a, books, and white papers.

Erik Veach, MS INCE says

Very well explained. You could even look at it from the other angle, too, and recognize that what we call "analog" (typically vinyl or tape media) signals are themselves actually discrete signals at some level as well, similar to digital signals. Is the variation in amplitude in those LP grooves truly continuous? Not really. There's a fundamental physical limit to how continuous the signal is, even if that limit is defined at the molecular level. So, in a sense, even analog signals do still act like digital signals in the real world; and with today's technology it should in fact be possible to record a digital signal with sufficient bit-depth and high enough sampling rate to match, or even exceed, analog source audio sample limitations imposed by fundamental physical properties of the media. Likely people would still say "analog sounds better" - further proof that they like the way the analog media storage "distorts" the sound, which has nothing to do with "superior quality" but is instead simply a measure of subjective taste. Great article, Nigel - thanks!

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