## Thursday, January 19, 2012

### We'll Always Have Paris. Or a 24-bit Approximation of Paris.

 Quantization maps infinite sets of real values to single values.
When it comes to digitally recording music from analog source material (LP, tape) the best approach is, "Sample often, sample well."

In a previous post we introduced non-subjective numerical methods to establish 96 KHz (ninety-six thousand samples per second) as an information-rich practical sampling rate. That takes care of "sampling often."

But what about "sampling well?" Can near 100% accuracy be achieved for the samples being collected? 24-bit word size is the closest practical approximation. Why? Read on.

Digital sound samples are real numbers in the closed interval [-1, +1] and can take on any of an uncountably infinite number of values within that interval. The sound capture process therefore must quantize the infinite set of real sample values into a set of approximating values that has a finite size yet is still information-rich. How big should that set be and how many bits are needed to represent its members?

With b bits available, a value set of size 2^b (2 raised to the power of b) can be encoded. At one bit, the sign bit, there are two values available (2^1 = 2) and we can only express whether a sample is greater than 0 or less than 0. Arbitrarily assign the value +0.5 to any positive sample (bit value 0) and -0.5 to any negative sample (bit value 1). The maximum quantization error ε with one bit is 50% (± 0.5). Clearly we need more bits.

Each bit added to the sample word doubles the size of the quantized value set it can represent and cuts quantization error by half. There is a well-known term for thisexponential growth. Not only does the sample set expand as you add bits, the rate of that expansion accelerates.

At 8 bits, 256 values can be represented and ε is just under 0.5%. Not good. At 16 bits, the set has more than 65,000 values and ε around 0.002%. Better, but still room for improvement. At 24 bits, the set has more than 16 million values and ε around 0.000006%. Every added bit is more significant than all the bits that came before it, up to a point. When does adding bits stop adding information?

A copy can't contain more information than the original. Most digital studio recording is done at 24-bit, so adding bits beyond that in the home audio capture process adds no new information on digital-to-analog source material (e.g. new releases on vinyl). Using 24-bit sampling to record analog-to-analog material (old vinyl) mimics current industry best practices. And 24-bit is a sample size supported by more and more computer sound interface devices, so it's likely you can find one at a price you're willing to pay.

Therefore my next recommendation for creating numerically accurate digital copies of your LPs is to set the sample size to 24-bit when recording. What do you think? Leave a comment.

Vinyl-to-Digital Restoration #11

Title: Waiting For Columbus
Artist: Little Feat
Genre: Rock
Year: 1978
Double albums present special challenges in the digital transfer process. Live albums present still other challenges. Double live albums are a double whammy. If a multi-disc set comprises a single work, you must normalize the output sound level across the entire set (not disc by disc) to prevent odd changes in volume if you shuffle the tracks. Deciding exactly where to split between tracks on a live album (during applause) is an art, not a science. You'll be happy to have a 24-bit/96KHz recording on which you can zoom way in to make a precise cut.