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Mix Magazine

This installment of The Bitstream column appeared in the March 2003 issue of Mix Magazine.

The Bitstream

This column discusses single versus double precision arithmetic in digital signal processing…

So, Are You Single?

If you own a digital desk, either in hardware or software, you routinely throw away from 2 to 8 bits of your hard earned 24. As I asked last month, should you care? Not really, since you can’t do anything about it but, you can work smarter within the constraints imposed by necessity. This time around we’re gonna take a look at one of the most basic questions a budding audio novice asks; “Mommy, why do we need 24 bits when our hearing’s only 20?”

Let’s start with young Elizabeth’s question. Her mom has told her time and again that her hearing has about 120 decibels from the quietest perceptible sound to the “threshold of pain,” the loudest sound pressure tolerable before instantaneous hearing damage occurs. Even though, when considering the dynamic range of our hearing, we are talking about sound pressure level, we can still use the same ratio measuring system, the decibel or dB, to measure the dynamic range represented by AES data. 20 bits worth of dynamic range is about 120 dB since each PCM bit represents about 6 dB of amplitude. So, Young Liz has nailed the number and, to answer her question, we need to look at gain staging, that juggling of gain through a complex system to maximize resulting dynamic range and minimize accumulated noise.

OK, think about this for a moment…take two signals: acoustic, analog or digital and add them together. The resultant mix is usually louder, though there are exceptions. Take that concept and apply it not only to mixing but EQ’ing and processing and you can see that, without a bit of forethought on the part of a system designer or engineer, the final gain will either not take advantage of the dynamic range available in the system or will quickly overload some bus. Too little or too much, neither is good and usually the problem leans toward too much signal and not enough dynamic range in which to fit it.

So, how does a designer deal with this dilemma? Well, remember earlier I said that a PCM bit encodes about 6 dB of dynamic range. Looking at that from another angle, you can think of dynamic range as being expressed as word length. A word length of 16 bits encodes roughly 16 times 6 dB or 96 dB of dynamic range and again, a 20 bit word encodes 120 dB. The AES/EBU protocol can carry a 24 bit essence, enough for a solid 20 bits of signal ( thank you, young Elizabeth) plus 4 additional low level bits generated by processing.

Now that we’re past the basics, let’s look at one of the most overlooked features of better quality digital audio geegaws and the whole point of this month’s column. A bonus for the discriminating buyer and boon to those who value retention of low amplitude detail, double precision calculations are one of the keys to better bits. Double precision is akin to the American Way: if some is good, more is better. Most products use single precision arithmetic, though, if you dig a bit (yuk yuk), you find a few pearls in among the swine. Double precision fixed point would employ 2 time 24 or a 48 bit word length and double precision floating point would use a 2 times 32 or 64 bit mantissa. Twice as many bits, which allows low amplitude details to be retained rather than lost to rounding or truncation. Low amplitude signal retention is one area where a DSP programmer distinguishes him or herself, since I feel this is a key differentiator between truly professional products.

So, how do twice as many bits help retain subtle details in your material? Here’s an example…Take two numbers, representing the instantaneous amplitude of signal and some amount of gain change being applied to that signal:

0.92 [original signal amplitude] x
0.707 [negative gain] =
0.65044 [resultant signal amplitude after gain change]

Notice that the result of multiplying those numbers produces a new number with many more digits than when we started. Think of how many gain changes, which includes panning, EQ and mixing, that occur during the course of a typical project and you’ll get some idea of how 24 fixed or 32 floating bits can get “filled up” rather quickly. Those low order bits, say the 22nd through 24th bit, carry low amplitude detail in the data.

Another example, basic gain staging in a digital mixer, should drive the point home. When two or more signals are summed or mixed, the result is usually louder than either source. Below is a chart of the amount of “padding,” as it was called in the analog days, or negative gain needed versus the number of channels being mixed in order to prevent overload in the downstream mix bus.

No. of Channels To Mix
Gain Reduction in dB per Channel
Delta: Increased Number of Channels
4
2
-
8
3
4
16
4
8
24
5
8
32
5
8
48
6
16
64
6
16
80
7
16
112
7
32
144
8
32
176
8
32
240
8
64
304
9
64

Table 1 — Gain Staging for Digital Mixers

Looking at those numbers, it’s plain to see that, with 144 channels to mix, a whopping 8 bits of gain reduction must be applied to each input channel prior to mixing. Those 8 lower order bits, the quietest stuff, must be handled in one of four ways:

1) truncated or tossed away,
the cheese ball approach

2) rounded up or down to the nearest integer,
a better method

3) re–dithered and word length reduced,
better still or

4) stored as double precision

Storing the data as double precision retains most all of the low amplitude detail and postpones the inevitable word length reduction until the end of the production cycle. However, realize that there is no standard for double precision data interchange between products, so files and AES/EBU output both end up as single precision between processing islands in the production chain. Still, a few host–based DAWs, such as the AudioCube and Sonar XL, have double precision modes while SonicStudio HD, a hardware–based DAW, is fully double precision throughout. The Cakewalk folks at Twelve Tone Systems advertise their Sonar XL as having “all the capabilities of SONAR plus two 64-bit, fully automatable DirectX 8 mastering effects…” Notice the mention of “64-bit,” which indicates that the plugins are double precision floating point. My guess is that more gear will joint this elite list as vendors attempt to differentiate their offerings based on quality rather than price.

I hope that this month’s rant hasn’t caused your head to rotate at too high a speed and helped you to understand why some gear sounds like crap at any setting except 11. Hopefully, it’ll also aid you in making more informed decisions when you need to choose or expand your technology base in days to come.

Bio

OMas, in his search to get you the freshest geek info, pounds show floors ’till he canno’ take it no more. This column was written while under the influence of the original quadraphonic version on SACD of Mike Oldfield’s Tubular Bells .