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## Decibel Games

#### Interconnect

A thorough explanation of the 'decibel' (dB) and its applications in recording.

How do I convert dBm to dBV, and how many millivolts is that? 'Zero level' isn't attained by switching off the mixer — so how come? And how many dB do I need to double the level — was it 3, 6 or 10? Ben Duncan presents a readable investigation into the decibel's foibles, plus some handy tables to hang on the studio wall.

Let's forget all about dBm, dBu and dBV for the moment. Decibels (dB) are fundamentally a means of expressing ratios, so we'll begin from that angle.

A ratio is a numeric abstraction, used to compare the sizes of two objects, signal levels or whatever, without referring to their actual size. For an example, if an amplifier's output is 100 times bigger than the input, the output to input ratio is 100:1, though this says nothing about the actual signal level - it could be millivolts or megavolts. We can also express this as x 100 amplification. In decibel language though, we say the output is +40dB or "40dB up", referred to the input. But why invoke a set of fancy 'dB' numbers just so that we can express 100:1, or an amplification of 100:1? The answer is, believe it or not, to make the engineer's sound life simple; decibels are a valiant attempt to banish nasty maths (ugh!) from the control room.

In audio circuits and acoustics, we need to handle a very wide range of signal magnitudes. For example, in studio electronics, the lowest noise levels and highest amplifier outputs span a range of 1000 million! But by using decibels, however, we can simplify a monster ratio such as 1 to 2371373.7, to a cool -127.5dB, which I'm sure you'll agree is less of a mouthful.

In fact, with decibels, we need never go above 3 figures - the entire range of physical magnitudes we need to cover is encompassed by 180 decibels. And because the numbers are simpler, mistakes are less likely; if you slip a decimal place on 120dB, you'll get 12.0dB, which is all too obviously an error. But that monster number (now was it 237137.37 or 23713773.7?) is all too easy to confuse.

#### Signal Chain

Figure 1 shows a simple signal chain in a studio (by adding a few processors, we could make it considerably more complex). Each stage of amplification, from the microphone through to the monitor amp multiplies the original signal coming out of the mic, whereas the faders and front-end pad attenuate, or divide down the level.

The problem is to discover the overall amplification (or attenuation) between the mic input and the amp's output. As you can see, any amount of complexity results in a fairly horrendous equation if we use ordinary numerical ratios. Figure 2, meanwhile, presents the same situation, but with the amplification and attenuation expressed in decibels. Figure 2. The same studio chain - but expressed in decibels.

Because dB are a logarithmic quantity, there's no multiplication or division required - we simply add up the figures; it's like adding up your groceries and subtracting a few 10p discount coupons. The only snag is that the answer (+54dB) may not mean very much - at least until you learn to think in dB.

In the meantime, use the table (see Figure 3) to get an idea of the ballpark numeric ratio. The overall amplifiation is actually x501, and fanatical readers can check this by tapping (54 / 20 x Antilog) into a scientific calculator (Figure 4). With this knowledge, we can then arrive at the mic's output level (Vin), knowing perhaps from the metering that the voltage at the amplifier's output is 22 volts, viz: 22/501 = 43mV. But more on this later, when we look at decibel scales.

#### Manipulating Decibels

Learning to think in decibels is made easier (we hope) by the tables in Figure 3. Table A concentrates on the key decibel numbers - their ratios are easy to remember with a little practice. Once you've assimilated these, you can go on to attack Table B, which fills in the gaps. The main scale from 10 to 130dB is particularly easy to remember, because it has a distinct pattern. You'll also find it helpful to note the following:

1) Adding 10 to any decibel number multiplies it by 10.

2) Subtracting 10 from any decibel number divides it by 10.

3) A negative (-dB) number, which represents signal attenuation, can be represented as a division (eg. -6dB = / 2), but often, it's more convenient to multiply the reciprocal, so -6dB can be equally presented as 'x0.5' - Table A gives some further examples.

Thus far, decibels are easy sums you can tot up in your head, or on the back of a Marlbro packet (cough, choke), which is good news for the practical engineer doing a rough check on levels in his studio. But for more accurate work, you need only dig out your scientific calculator and wipe the dust off those special notation keys you last used in the dim past. Figure 4 shows you how, and offers some actual examples.

#### Decibel Scales

So far, we've looked at decibels in their abstact, ratiometric format. But they're even more useful if we give them the absolute qualities of a unit, like volts, watts or pascals (sound pressure), so they can become part of a scale.

To do this, we need to decide that so many decibels equals so many volts (or whatever). Now because dB can be positive or negative, and extend infinitely in either direction, the most sensible reference point is halfway, at 0dB ("zero dB"). And quite simply, this is where the words 'zero level' originate from - it's any voltage (or wattage) that we decide to make equal to 0dB for the purposes of establishing a scale. All other levels refer back to the zero level according to their respective ratios: smaller levels are minus (-dB), bigger ones plus (+dB).

For example, if we make 1 volt = 0dB, 2 volts will be +6dB, whereas ½ volt will be -6dB. Trouble is, how do we remember next week that +6dB = 2 volts, and not 17mV or 10kV?? And more to the point, how does the rest of the cosmos get to know about this wondrous, yet entirely arbitrary definition?

The answer is to use the letter V (for 1 volt) - so the dB scale built upwards and downwards from 1 volt is christened dBV. And looking back to Figure 3, Table B, dBV are readily sussed; just read off the ratios as a voltage. Thus 2dBV = 1.26 volts, +30dBV is around 31 volts, and -60dBV = 0.001 volts (1 millivolt).

We can now tackle the mysteries of dBu and dBm, which are more widespread scales than dBV - in fact, they're the standard scale for most pro-audio work.

History being full of human mistakes, which once set in concrete can't readily be amended, both dBu and dBm have the bizarre distinction of equalling 0.775 volts (or 775 millivolts) at their zero level, which is hardly the world's most convenient number, because (groan!) a whole new set of figures have to be learned. In fact, growing awareness of this setback led to a proposal to make more use of dBV (it makes life easier for newcomers), but the dBm/dBu scale dates back over sixty years, and equipment manufacturers can be quaintly conservative over matters like this, despite handling the leading edge of electronics technology! In other words, dBm and dBu will be with us for the foreseeable future, so we may as well join in. The table in Figure 5 displays dBu/dBm versus actual signal voltages. Meanwhile, a range of levels at key points in the signal chain are given alongside.

Next question: "If 0dBu and 0dBm both equal 775mV, what's the difference between the two?"

Well, dBm was originally defined in terms of power (rated in watts), ie. 0dBm = 1 milliwatt, but is also tied in with the archaic interconnections which had 600 ohm termination resistors. And it's no coincidence that the voltage needed to dissipate 1 mW (0.001 watts) across a 600 ohm resistor is 775mV. So that's where this silly number came from...!

In 1984, though, we've progressed beyond dumping power into 600 ohm resistors (see previous HSR articles on voltage matching, Mating Microphones, July 1984), and load impedances are, as a rule, much higher. More important, the actual impedance is indeterminate - it varies from unit to unit, and from studio to studio. And this is where dBu comes in: 0dBu equals 775mV regardless of whether it's measured across a 250 ohm mic, a 4 ohm speaker, a line input with a 100k input impedance, or an open circuit. 0dBm, on the other hand, strictly equals 775mV ONLY when it's measured across 600 ohms. Today, though, this formality is dispensed with, and dBm is routinely used like dBu - they've both ended up being synonymous, except in reactionary circles.

But what about zero level? Though the term was originally derived from zero dB, today zero level is NOT necessarily 0dBu or 0dBm - or even 0dBV. Instead, various sectors of the audio business have voted for their own, idiosyncratic zero level standard eg. -10dBu (Oriental home studio equipment), +4dBu (Recording studios) and +8dBu (Broadcast studios).

#### Sound & Power Levels

The voltage in any electrical circuit is analagous to pressure and so sound pressure (which corresponds to perceived loudness) is the acoustic equivalent of voltage. Sound Pressure Levels (SPLs) can be measured in pounds per square inch, but for sound, we use much smaller metric units, namely pascals.

Like the levels in electronic circuits, sound pressures also encompass a wide range, so decibels are once again a handy tool to handle routine calculations and comparisons. In fact, because our perception of sound intensity is logarithmic, dB are a particularly apt scale, and even make intuitive sense, to the extent that aged politicians and country vicars (gulp) know what 'decibels' (of the sonic variety) are, and that 120dB is reprehensible(!) whereas 88dB are decent and legal.

But that's jumping the gun. First, we have to establish a zero level which equals so many pascals. This scale is known as dBSPL, and the main difference is that the zero level was originally set not at a convenient, easy to remember level (like 1 pascal), or even at an inconvenient one (775 millipascals!), but at the lowest audible level. As a result, sound decibel levels are usually positive, so there's no need to use + and - signs. I say, usually, because -dBSPL are possible.

After the zero level was established, psychoacoustic researchers discovered that at midband frequencies, our hearing is more sensitive than was originally envisaged. So the threshold of sound perception is actually a negative figure (around -10dBSPL) at 3kHz, under ideal conditions. Incidentally, 0dBSPL = 0.0002 pascals but there's no need for a conversion table, because most readers will already be familiar with sound levels expressed in decibels alone; only acoustic researchers and microphone designers need to convert pascals into dBSPL, and vice versa.

Turning now to power levels, watts have different dimensions to volts or pascals. The crucial thing to remember is that an XdB rise (let's call it 6dB, for example) in volts or SPL also spells a 6dB rise in power, but the numeric ratio is different - it's the square of the voltage ratio.

So if we wind up a fader by 10dB, the mixing desk's output voltage, the amplifier's power output and the SPL in front of the speaker will ALL increase by 10dB. But in numeric terms, while the SPL and voltage have increased by 3.16 times, the power has increased (x3.16)2 = 10 times. Conversely, a power reduction of -10dB effectively divides the power by ten. That one's easy to remember! The table in Figure 6 illustrates this. But note that unless a dB ratio is specifically identified as a power ratio, we can normally assume that voltage ratios are being discussed.

#### Decibel Facts

How many dB double the level? This is a boring question, and you should fire back with another, "Which level do you refer to?". To double a signal voltage, or the acoustic SPL (as read out on an SPL 'decibel' meter), we add 6dB. To double power output (watts), we add 3dB, whilst to double the perceived acoustic level (ie. what we actually hear), we need to add an extra 10dB.

Although plain decibel ratios were originally confined to comparing signal levels - including noise - or describing amplification and attenuation along the signal chain, today they're increasingly used for equipment specifications.

For example, you may see the distortion of a mixer quoted as -80dB. In the more familiar percentage terms, this works out as (0.0001 x 100)%, that's 0.01%. Or you may see some other parameter, such as 'slew rate' described as being 6dB better, meaning twice as good. Because decibels align with the natural logarithmic relationships between a paper specification and what we actually hear, all sorts of audio specifications can be usefully described and more readily assessed when they're presented in decibel format.

When talking levels in a studio, it's common practice to leave the dBu (or dBm) out of a sentence, so long as the 'plus' and 'neg' is left intact. If someone says "The limiter output is plus 3", the word 'plus' makes it obvious they're referring to a decibel scale (cf. a decibel ratio), and the fact that it's the dBm or dBu scale may usually be assumed.

You can also boost your credibility as a sound engineer by writing the decibels abbreviation correctly. In prosaic adverts, or Nipponese instruction manuals, you'll see DB, Db and db in print, but the one written by educated people is emphatically dB - small d, big B. This might seem stuffy, but we are talking about a deci-Bel, ie. a tenth of a Bel, and in this context, writing DB is just as naughty as saying your console's zero level is 100MV, (100 megavolts!) when you meant 100 millivolts.

Figure 3
 Table A Table B Approximate ratio Exact ratio db -dB +dB db -dB +dB 0 ÷1 x1 0 x1 x1 1 x0.89 x1.12 2 x0.66 x1.26 3 ÷1½ (x0.7) x1½ 3 x0.66 x1.41 4 x0.60 x1.58 5 x0.56 x1.78 6 ÷2 (x0.5) x2 6 x0.5 x2.0 7 x0.45 x2.24 8 x0.40 x2.51 9 x0.35 x2.82 10 ÷3 (x3.0) x3 10 x0.316 x3.16 20 x0.1 x10 30 ÷31 (x0.03) x31 30 x0.032 x31.6 40 ÷100 (x0.01) x100 40 x0.01 x100 50 x0.0032 x316 60 ÷1000 (x0.001) x1000 60 x0.001 x1.000 70 x0.00032 x3,160 80 x0.0001 x10,000 90 x0.000032 x31,600 100 (etc) x100,000 110 x316,000 120 x1000,000 130 x3160,000

Figure 4. Decibel Conversions
A) To convert any ratio (or number)'N' into dB:

1) Enter N x Log x 20
2) Add minus if required for 'divide by'
3) eg: 72 x Log = 1.8573,
(1.8573 x 20) = 37dB, whilst ÷ 72 would be -37dB

B) To convert 'N' dB to a numerical ratio:

1) Enter (N ÷ 20) x Antilog
2) Add ÷ sign if -dB
3) eg: -23dB ÷ 20 = 1.15,
(1.15 x Antilog) = ÷ 14.1 (or x0.07)

Figure 5. dBu and dBm.
dBu/dBm Voltage Typical studio levels contrasted
DM CM DI LL ML
+40 77V
+30 24V
+25 13.8V
+20 7.7V
+15 4.4V
+10 2.4V
+5 1.4V
+2.2 1 volt
0 775mV
-5 550mV
-10 250mV
-15 140mV
-20 77mV
-25 44mV
-30 25mV
-35 14mV
-40 8mV
-45 4.5mV
-50 2.5mV
-55 1.4mV
-60 775μV
-70 250μV
-80 77μV
-90 25μV
-100 8μV

Note: To convert dBu/dBm to dBV, subtract 2.2 dB (+2.2dBu = 0dBV)
To convert any dBV level to dBu or dBm, add 2.2dB (0dBu = —2.2dBV)

Key to levels
DM = Dynamic mic
CM = Capacitor mic
DI = DI feed
LL = Line
ML = Monitor

Figure 6. Power Ratios
dB Ratio
0 x1
1 x1.26
3 x2
6 x4
10 x10
20 x100
30 x1000
40 x10,000

Sub Woofers

#### Home & Studio Recording - Copyright: Music Maker Publications (UK), Future Publishing. Home & Studio Recording - Nov 1984

Donated & scanned by: Mike Gorman

Topic:

#### Sound Fundamentals

Feature by Ben Duncan

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