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Decibels are everywhere in audio, but what exactly is a decibel? Well, it's a ratio between two things, which means that we always need to know the reference when we're talking decibels. Furthermore, decibels for power and signal amplitude are different, but we'll get to that in a moment. One decibel is one tenth of a bel, and usually we use decibels instead of bels because they're more convenient. Let's look at the formula for decibels for power:

dB=10*log10( P 2 / P 1 )

In the equation above, 10 comes from the ten decibels in one bel. P1 is the reference value, and P2 is the value of interest. Let's say we have a device that uses 20 watts of power. We compare it to a device that uses 2000 watts. In decibels this would be:

10*log10( 2000 / 20 ) =10*log10( 100 ) =20dB

A hundred-fold increase in power is only 20 decibels! Decibels are most useful when discussing large differences. There is a trick to it, though, and that is the decibels of signal amplitude, which are calculated differently. To best understand what's going on, it's beneficial to remember that power and signal are often in a squared relationship. So just as in electricity, where power is relative to the square of the voltage: P=UI= U 2 R , (R being a constant doesn't really affect the relation), there is an analogy in audio where power is relative to the square of the amplitude. For example,

dB=10*log10 ( L 2 / L 1 ) 2 =10*2*log10 ( L 2 / L 1 ) = 20*log10 ( L 2 / L 1 )

As the decibel scale is logarithmic, it also changes basic operations. Logarithm changes multiplication and division into addition and subtraction, respectively. While this makes some calculations easier, it makes adding and comparing decibel values a bit confusing sometimes. For example, 85 and 90 dB signals, when summed together, certainly don't produce a 175 dB signal! To get the correct answer, we first need to undo the logarithm, add the values, and take the logarithm again, like thus:

20*log10( 10^ ( 85/20 ) + 10^( 90/20 ) ) ≈ 94dB

Calculating logarithms all the time can be slow and cumbersome, so it's good to remember a couple of handy shortcuts. 6 dB means the signal doubles, and 20 dB means it grows ten times. Of course for power these are halves, so 3 and 10 dB respectively. I could write much more about this, but maybe I'll keep this post short and simple, and if there's something that needs a deeper look, it'll deserve a post of it's own. Do note that it's the logarithmic nature of human hearing that makes decibels so useful, and also why to make an amplifier sound twice as loud, you need to make it ten times more powerful. Yes, a 100 watt amp sounds only 4 times as loud as a 1 watt amp!