Tuesday, January 17, 2012

100% Fidelity is Possible as We Approach Infinity. Are We There Yet?

A back-of-a-napkin capture of the Riemann Integral. 
The first step in ephemeralizing your LPs is to record them digitally, creating permanent archival copies. This is a step you only have to take once for each of your titles, but if you do it right, once is enough.

Recall that an information-rich map is always better than a back-of-a-napkin drawing. Create rich 24-bit/96KHz copies of your source material for archiving. Why? Read on.

Because we know we can't capture infinity digitally, the recording process is to sample the original; that is, to take a reading of the original sound wave at regular intervals Δi. The question is, how do we design a sampling process that will produce an optimal copy? Many people will tell you that "optimal" is a completely subjective characterization—not so (not "completely").

While virtually any scientific debate can turn subjective when opinion and evidence clash, math is uniquely impervious to opinion. "a(b + c) = ab + ac. Politicize that, bi***es." (Randall Munroe)

Without a priori knowledge about the sound being recorded, it is impossible to know if it is being captured accurately. But there is a proxy calculation that we can examine objectively.

A sound wave is a function F(t) of pressure v. time. Once a sample F(ti) is taken at time ti, the recorded sound value remains constant for Δi seconds until the next sample can be taken. Any Calculus Hero will recognize that, as we are sampling a sound wave, we are simulatneously calculating the Riemann Integral (approximate area below the curve) for it. Dude.

The width Δi and height F(ti) of a rectangle in the Riemann Integral determine the accuracy of the approximation. Both dimensions are under your control. Let's concentrate here on getting the proper width via high frequency sampling. Next time we'll look at getting the proper height by taking the best possible samples.

The Riemann Integral aproaches 100% accuracy as Δi 0. Thus you get progressively better approximations the more samples you can take in a closed interval. That's not my opinion, that's not even consensus opinion. That's math.

Because sound sampling is a real-time process, the total number of samples is less important than the number of samples you can take per second. This is your sampling rate, expressed in samples/sec or Hertz (Hz). Riemann says, the higher your sampling rate, the more accurate your source recording.

So what is a good practical sampling rate? Common sampling rates are:
  • 44.1 KHz CD Audio (CDDA)
  • 48 KHz DVD Audio (DVD-A)
  • 88.2 KHz 2x CDDA Rate
  • 96 KHz 2x DVD-A Rate
96KHz make sense for several reasons. First and foremost, it's higher that the other candidate rates (more is better). Second, 96 KHz is a common sampling rate supported by a wide variety of computer sound interface devices, so it's likely you can find one at a price you're willing to pay. Finally, because sampling theory says you can capture any frequency by sampling at twice that frequency 2f, 96KHz sampling captures frequencies up to 48KHz, well into the range inaudible to humans and above the upper range limit of even audiophile-class speakers.

Therefore my first recommendation for creating numerically accurate digital copies of your LPs is to set the sampling rate to 96 KHz. What do you think? Leave a comment.

              Vinyl-to-Digital Restoration #10              

Title: Gallery
Artist: Gallery
Contributing Artists: David Samuels; Michael DiPasqua; Paul McCandless; David Darling; Ratzo Harris
Genre: Jazz
Year: 1982
Despite the personnel involved having a pretty damn good pedigree, this album is lost to history. ECM Records never released it on CD, and even today has no entry for it on the ECM discography. Long ago I bought Gallery on vinyl for its link between the Paul Winter Consort and Orgeon. One of the best reasons for shepherding your analog past into the digital future is that your memory can be jogged. The record labels' memory can't.

© 2012 Thomas G. Dennehy. All rights reserved.


  1. Totally agree with the 96 kHz recommendation. Just have to take an issue with your statement that math is uniquely impervious to opinion. Yes, a(b + c) = ab + ac. But the simple fact that you support your argument with the use of math shows that you are an elitist, liberal know-it-all, who think you are better than others just because you have some fancy-schmanzy university degree.

  2. By using math to back up his arguments, he is proving he's better than the rest of us.