I suppose that this is the right forum for this topic, so here goes.

I'm not sure how many of the Outlaw brethren are acousticians, engineers, or signal processing types - I suspect most of us are not - so I decided to write a quick little primer on the subject of decibels, but with the primary focus on decibels as they pertain to sound - and what we perceive.

To start with, sound is really nothing more than a very small dynamic perturbation in pressure, superimposed on a very large static pressure. That is, sound is a dynamic signal, and this is what we perceive, which 'rides' on the static pressure (barometric).

For those comfortable with circuitry / electronics, a good analogy of this concept is that of putting two power supplies in series; one of these is a DC power supply set to some voltage, and the other is a function generator, and for now, let's assume that said function generator is cranking out a 100 Hz sine wave.

If you were to look at the net voltage on something like an oscilloscope, you would see that the comparatively small AC signal revolves around the DC signal. So, suppose we had 100V DC and we had 20 mV of the AC signal - when you look on the oscilloscope, if it is set to show the DC signal (i.e. DC coupling), all you would see on the 'scope would be a horizontal line, with a barely perceptible 'wiggle' superimposed on the DC voltage. Were you to change the coupling on the 'scope to AC, then it would ignore the DC signal, and you would be able to see the comparatively small AC signal.

Sound is very much like this. In the analogy above, sound is the AC signal, and barometric is the DC portion.

Think about it - we walk around under a sea of air, exerting roughly 14 psi on all that we see, but we perceive sound.

As far as where sound pressure level (SPL) number comes from, to answer that we have to do a little math.

Sound Pressure Level (SPL) is defined as twenty times the base 10 logarithm of the measured RMS pressure divided by the reference pressure. I know...that's a lot to digest in verbal form. So, alternately, here is the forumal that governs this quantity:

(1): SPL = 20 * Log10 (Pmeas/Pref)

Where Pmeas is the time-varying pressure pertubation, and Pref is equal to 20 micro Pascal.

Now, we all have heard / encountered THE DECIBEL in our lives - whether you have heard about it as a community noise standard, the level of a typical rock concert, or whatever, you have heard of this concept. However, most of us just don't have a feel for it and what it really means.

So let's revisit the expression above. First, let's look at what's in the parenthesis - we have a ratio there. Now, what if the measured pressure (Pmeas) is equal to the reference pressure (Pref)? In that case, what's inside of the parenthesis becomes equal to 1.0. We know that the base-10 logarithm (Log10) of 1.0 equals 0.

So, in this particular case (when the measured pressure equals the reference pressure), the equation looks like this:

SPL = 20 * Log10 (1) ... which equals
SPL = 20* (0) ... which equals
SPL = 0 dB

Thus, the measured sound pressure level in this case is 0 dB; this is what is often referred to as the absolute threshold of hearing (in a young, healthy, not-having-been-exposed-to-elevated-levels-of-noise ear).

OK, so now we know where the lower limit of hearing is, and how we got there (mathematically). When you think about 'typical' rock concert levels (i.e. 110 - 120 dB) we are talking about something very, very loud. But...what is the measured pressure associated with (for example) 100 dB?

You can rearrange the expression above to solve for the pressure. That is, if we just shuffle some terms about and invoke the properties of logarithms, we can produce the following equation to solve for the measured pressure (Pmeas):

(I'll do this in steps):

SPL = 20 * Log10 (Pmeas/Pref) ...now let's take the first step by dividing both sides of the equation by 20. When we do this we get:

(2): SPL/20 = Log10(Pmeas/Pref) ...OK, so now we're closer to getting what we want to know (Pmeas) by itself. By properties of logs, we can take the next step in rearranging the equation, which yields:

(3): 10^(SPL/20) = (Pmeas/Pref) ... where the carat ( ^ ) symbol means 'raised to the power'. We're almost there - all that's left is a little algebra. So, let's now multiply both sides of the expression by Pref (the reference pressure). When we do that, we get:

(4): 10^(SPL/20) * Pref = Pmeas

So, if you wanted to know the pressure that equates to 100 dB, you now just have to substitute the value into the expression (4)above. By doing so we get:

(5): Pmeas = 10^(100/20) * 20E-6 ... which yields...

(6): Pmeas = 10^(5.0) * 20E-6 ... which yields...

(7): Pmeas = 100,000 * 0.00002 ...which yields...

(8): Pmeas = 2.0 Pascal

OK, so now what?

Well, the more useful relationship to keep in your head is what follows, as it is useful, and will likely clear up a lot of confusion you may have about interpreting SPL. Here goes...

"For every change of 10 dB SPL, we perceive roughly a two-fold increase in intensity."

Practically speaking, this means that for a given sound, if it is presented to you at 100 dB, it will seem twice as loud as the same sound presented to you at 90 dB. Likewise, the same sound presented to you at 80 dB will seem 1/2 as loud as the sound at 90 dB, and logically, 1/4 as loud as the sound at 100 dB.

The heart of this relationship lies in the logarithm; you would think (logically enough) that if SPL describes how loud we perceive something, that by changing the SPL by a factor of two, we would perceive a two-fold change...but...this is not the case. Again, blame it on the logarithm (as opposed to the Bossa Nova) for this, but it's good to keep in mind.

Remember...something that's 50 dB is roughly half as loud as something that's 60 dB, one-fourth as loud as something at 70 dB and so on. So this would apply to the difference between 62 and 72 dB, 27 and 37 dB, and so on.

This is a bit of an oversimplification, because what's in the signal as well as its temporal aspects (how / if it changes over time) can play a huge role in the perceived intensity of the sound.

So, how is this useful to the Home Theater scene?

Well, there are a number of ways.

First of all, keep in mind that the SPL produced by a speaker (operating in its linear range and the linear range of the associated amplifier) is proportional to VOLTAGE, not to the power delivered.

The volume controls on the 990 et al are all calibrated in decibels with respect to full-scale (0 dB), but the volume control controls the output VOLTAGE - not the actual SPL (though it is possible to take this into account (as it affects the change in SPL in dB), but that's another post). Nevertheless, within that statement lies some useful information, which goes like this...

Since the volume control is in dB...
...and since the thing we are controlling is the VOLTAGE out of the processor...
...and since we know that SPL from a speaker is proportional to VOLTAGE...
...then we know, that each time you change the volume control (on your 990, 950, 2150...or whatever) by 10 dB from its current position, you will effectively halve or double the perceived intensity of the sound.

Example...

You're playing your favorite piece of music. The volume control shows - 32dB. If you turn the volume control up to - 22 dB you will perceive the sound as being roughly twice as loud as it was at the - 32 dB setting. You decide that it's not loud enough, and you think "this would sound even better were it twice as loud". Thus, turning the volume control up to - 12 dB will result in the music being four times as loud as it was at the - 32 dB setting, and twice as loud as it was at the - 22 dB setting. This also applies to things like streaming music servers connected to your gear via the ANALOG outputs (the digital ones are fixed, not variable). So, if your processor was set to - 32 dB on its volume control, and your streamer was set to - 17 dB on its volume control, then by changing the volume on the streamer to - 7 dB (and not touching the gain on your processor), you would likewise create a two-fold increase in perceived intensity of the sound. This is because the gain control on all devices controls the voltage.

For grins...also keep in mind that differences in SPL of 1 dB are pretty much imperceptible. Under highly controlled listening tests, and depending upon the signal to which you are listening, it may be a bit easier to distinguish this difference.

However, a good rule of thumb is to remember that a 3 dB change in SPL is generally viewed as 'just noticeable'.

Remember, the volume control is looking at dB relative to full-scale (but controls the voltage)...but remember... the SPL from a speaker is proportional to the voltage delivered.

Thus, if you were using a sound level meter, and (for instance) playing some test material (maybe random noise, a warble, whatever), then if at a given gain (volume) setting on your processor you measure the test signal as being 83 dB, turning the gain up on your processor by 3 dB (i.e. 3 dB more than its current position) will yield a 3 dB increase in measured SPL (86 dB SPL).

So, decibels can indeed be tricky, and in fact, nowadays, many people in the noise control / signal processing industry and community use a linear scale for loudness, with units = sone.

That is, a 10-sone loud is twice as loud as a 5-sone sound; likewise, a 20-sone sound is twice as loud as the 10-sone sound, and four times as loud as the 5-sone sound.

I may write another primer on dB as they apply to voltage and power if there is any interest, but in the mean time, I hope that for those of you who shy away from things like the decibel, that this makes it a bit friendlier, and a bit more understandable to you.

Thanks,

Mark
PS: If you want to know more about weightings (i.e. dB(A), dB(C), and so on), there are many good links on the web to such materials and I can scare some up and post them if there is any interest.