Thursday, October 25, 2012

How Many Bits in Infinity?

Why are 24-bit digital recordings the benchmark for capturing live sound?

It has to do with our limited ability to measure physical phenomena. Experiencing and recording sound is essentially measuring pressure over time.

Regardless of whether you are measuring continuously or sampling at intervals, at any instant you are trying to capture the magnitude of a phenomenon, or signal, that exists as a real number — arbitrarily scaled to lie in the closed interval [0,1] — and its direction. Real numbers cannot (all) be represented exactly.

The set of real numbers has infinite size. The set is so big, the real numbers are not even countably infinite, like the integers. While there are an infinite number of integers, there are a finite number between any two of them, so that the set of integers in a closed interval has a known size N. A set of size N can be represented using b bits, where b is the smallest whole number such that N ≤ 2b. Between any two real numbers are an infinite number of real numbers. How many bits in infinity?

Thus, our ability to record physical phenomena is defined by our ability to quantize it. Quantization maps infinite sets of real values to single values, to create a finite set of approximation sample values ("words") large enough both to have high dynamic range (DR), the maximum decibel (dB) level of a signal minus the aveage noise level, and have an acceptable error factor (inversely proportional to the size of the quantized set).

How big is an accurate quantization set and how many bits are needed to represent its members? Each bit added to the sample word accomplishes three important things:
  1. Doubles the size of the quantized value set it can represent;
  2. Cuts quantization error by half;
  3. Adds 6 dB of DR.
There is a well-known term for this — exponential growth. Not only does the sample set expand as you add bits, the rate of that expansion accelerates. 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?

Since the threshold of hearing is near 0 dB SPL, and since the "threshold of pain" is often defined as 120 dB SPL, it is said that the DR of human hearing is approximately 120 dB. Thus, 24 bits is the first natural computer word size (divisible by 8) that offers a DR geater than that of human hearing. Larger word sizes, while greater precision for other kinds of measurements, don't add meaningful information for sound. 24 bits is the right combination of precision and practicality.

Anything less than 24-bit digital audio has been a compromise. 16-bit samples were chosen for CD audio due both to the requirement to store more than 70 minutes of audio on a disc and to the limited space offered by optical disc technology in the 1970's. Lossy compressed audio was a concession to the covenience of being able to store a lot of songs on the low-capacity flash memory devices that were the first generation of portable digital music players. Now there is sufficient storage space and wireless bandwidth inside our homes to make 24-bit studio master recordings the de facto standard for digital music acquisiton and playback.

The reproduction is never going to be the original perfomance; or, as Alfred Korzybski said, "The map is not the territory." But a richly detailed map is better than the back of a napkin drawing. To paraphrase one of my design heroes Edward Tufte, summaries can emerge from high-information sources, but there is nowhere to go if we begin with a low-information source. A 24-bit studio master recording is a richly detailed map. Why settle for a summary?

             Vinyl-to-Digital Restoration #9            

Title: Desperado
Artist: Eagles
Genre: Rock
Year: 1973
When I bought my first CD player c. 1985, I already had an extensive album collection. Never the record labels' dream consumer, I rarely re-purchased on CD material I had on vinyl. For years, my turntable and CD player peacefully co-existed. So my Eagles albums went silent when I retired the turntable in the mid-90s (before the vinyl revival) for lack of space and a general frustration that an artist shuffle is not possible when the material is spread over seven LPs. It's nice to have the band back together after converting all that physcial media to weightless 24-bit digital. Hell Freezes Over, anyone?

© 2012 Thomas G. Dennehy. All rights reserved.

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