# Science and Math for Audio Humans–Ringing Decibels

12/28/2011 1:01:53 PM
By

Yes, the title is a wisecrack. I don't think you can blame a guy for that too much.

My disclaimer, though? That's not a wisecrack. Take a look: Everything that I set before you should be read with the idea that “this is how I've come to understand it.” If somebody catches something that's just flat-out wrong, or if you just think that an idea is debatable, please take the time to start a discussion via the comments.

The last part of this series primed you (via a discussion of logarithms) for understanding how decibels work. At this point, we're ready to get into the real “grit and grunge” of what decibels mean for audio folks.

So–what in tarnation is a decibel anyway?

If you're familiar with SI units, you know that a “deci[unit]” is one tenth of the basic unit. For instance, a decimeter is 1/10 of a meter. In the same way, a decibel is 1/10 of a “bel,” which was a unit name dreamed up by–wait for it–Bell Telephone Laboratories. The decibel is a convenient shorthand unit for describing the ratio of measured power to a reference power level. The convenience comes from the decibel using a logarithmic scale to effectively compress large ratios into a manageable numerical value. For the decibel, the chosen logarithmic base is 10 (also called the “common log”). What this results in is that 10 times more power gives you 10 more decibels.

This chosen behavior for the decibel is provided by the mathematical term shown in Figure 1.

 Figure 1

If we decide to use a reference power of one watt, we can get a handful of useful examples.
 Figure 2 Figure 3 Figure 4

 Figure 5

Of particular note is the 2:1 ratio giving 3 dB greater output. Depending on who you ask, humans perceive “twice as loud” when they get 6-10 dB more out of an audio source. The problem is that “twice as loud” is actually just “twice the power,” or 3 dB. If you have 500 watts flowing continuously through an audio system, and then want 3 dB more output, you're going to have to find a way to get 1,000 watts continuous to flow through that rig. Worse, if people really want to feel like that rig is twice as loud, you will probably have to find a way to get the equivalent output of 2,000-5,000 watts. (I say “equivalent” because simply increasing the power output into a given set of drivers is not always a smart idea. In fact, just buying bigger amplifiers and not getting any more drivers involved is a prime way to cook an audio system.)

Now, let's take a moment to delve into one of the sticky bits, which is a piece of the puzzle that has thrown both green and seasoned audio folks for a loop. The thing about decibels is that they really are meaningless without knowing what the reference is. If you don't know what is being referenced when speaking in decibels, you are wide open to all manner of misunderstandings. For instance, I can clearly remember the case of an experienced but confused audio technician being very upset about his new, digital mixing console. The console had meters that would not read beyond 0 dB. The tech in question, being seasoned in the world of analog audio, “knew” that a good, solid level was 0 dB. Having set his levels to be averaging about 0 dB on his meters, he was quite angry at how terribly distorted his signals sounded. What had to be pointed out to him was that his new console used a meter reference where 0 dB was 0 dBFS–decibels referenced to Full Scale. In other words, when you reached 0, your signal was just about to clip. What he had wrongly assumed was that the scale was the same as his analog consoles, where the 0 dB reference was up to 24 dB below the point of a signal clipping.

I myself get a bit lazy when using decibels to refer to sound pressure level. I can have a meaningful conversation with folks at my regular gig by simply saying things like, “Yeah, those guys were pretty mellow. We came out of the gate at only about 100.” The reason, though, that the conversation is meaningful is that we all have experience in what that sloppy reference to “100” actually means. To be really correct, though, I would have to be much more specific: “Yeah, those guys were pretty mellow. We came out of the gate at only about 100 decibels SPL (Sound Pressure Level), Z-weighted (actually unweighted), averaged over 1 second.” That sentence is rather a fuss to pronounce, but it would be the only way to give a complete stranger a correct picture of what 100 dB meant in our context.

So–how about a handy list of some common decibel reference points?

Well, I hate to hit you with a cliffhanger, but now is not quite the time for that list. The reason is because a lot of the references we use in audio are voltage and not power. In order to show you how voltage-based references shake out when using a unit meant for power ratios, I'll need to first explain Ohm's Law.

...and that's for another installment.