The Dreaded Decibel - Part 1

There's probably more confusion about the decibel than about any other topic in sound recording. It's actually quite simple provided you get to grips with the principles behind it.

But unfortunately it's so widely misused and misquoted by almost everyone from equipment manufacturers to sound engineers that separating the fact from the fiction can be difficult. But here goes.

What is a decibel?

The decibel is a measure of power ratio. That's the basic fact to grasp on to. The reason we use a power ratio in audio is that human hearing responds like that. It's capable of handling an enormous range of sound levels - from the sound of a mosquito at arms' length to that of a pneumatic drill at the same distance. But in general you have to multiply the sound power by ten for the sound to be perceived as twice as loud - which is why rock bands need such massive amplifiers.

The original basic unit of power ratio was the bel - defined by Alexander Graham Bell of telephone fame in the 1920s. One bel was defined as a power ratio of ten. But it was found to be rather too large as a basic unit for practical use, so the decibel - one tenth of a bel - was introduced and the symbol dB selected to represent it. While it might seem odd to use one tenth of something as a basic unit, it's no different in principle from the use of centimetres (cm) instead of metres to measure small objects.

So an increase of ten decibels (one bel) means multiplying the power (of an amplifier, for example) by ten, and a decrease of ten decibels means dividing it by ten. Ten decibels or one bel is thus in mathematical terms the base10 logarithm of a power ratio of 10. As the decibel is a logarithm, the difference between two decibel values (subtraction or addition) expresses the ratio (multiplication or division) of two (sound or electrical) powers. Note that I'm not using "power" in the mathematical sense here - I'm using it strictly in the sense of "energy per unit time" as in engines and amplifiers.

Confusion dispelled

Now we come to the sources of confusion. Firstly, as the decibel represents a ratio, it has to be referred to something in order to have a meaning. Multiplying three by two means something - six. But multiplying something unspecified by two means nothing. So in order to make sense, you have to specify not only a value in decibels but also the reference - what you are multiplying by the specified ratio. Various standard references have been defined to suit different requirements and the reference in use is shown by a suffix to the symbol - e.g. sound level (dB SPL), amplifier gain (dBu), digital audio signal strength (dBFS). How they are specified depends on their context. But very often the reference suffix gets left off published specifications due to sloppy thinking, and this causes massive confusion. The difference between two decibel values with different references means absolutely nothing. So always look for a reference suffix when you encounter values specified in decibels. If it's missing, you may be able to infer it from the context, but you'll sometimes not be quite sure your guess is what the provider of the documentation meant.

The second source of confusion is forgetting that the decibel always represents a power ratio. In digital audio recording, we're interested in sound pressure (because that's what microphones respond to) and voltage (because that's what gets converted into the numbers that are stored in the audio file). But basic physics dictate that, all else being equal, power varies as the square of voltage or pressure - doubling the voltage yields four times the power; ten times the voltage yields one hundred times the power, and so on. Inverting the relationship, a given power ratio equates to a change in pressure or voltage that is the square root of that power ratio. A power ratio of plus ten decibels - ten times the power - results from a voltage ratio of 2√10 or roughly 3.16.

So, as there are ten decibels in one bel, when we're discussing power ratios we express the ratio in decibels as 10×log10(power1/power2) where power1 and power2 are the two powers we're comparing. But when discussing voltage ratios we express the ratio in decibels as 20×log10(power1/power2), because the power ratio is the square of the voltage ratio. Odd as this may seem before you think it through, it really makes perfect sense, because decibels are always a measure of power ratio.

Fortunately, most of the time we sound recordists are only interested in voltage and sound pressure, so we normally use the second formula. Thinking in terms of voltage or sound pressure, if you're given a value in decibels, divide it by 20 and take the antilog (the inverse log). The result will be the voltage or pressure ratio.

Useful reference levels

There are numerous references for measuring signals in different systems. The ones that are of primary interest for the digital sound recordist are:

Standard Reference Levels
dB SPLa measure of sound pressure referred to the threshold of human hearing (2×10-5 Pascals) - always a positive value
dB(A)a measure of sound pressure as above, but weighted according to the nominal frequency response of the human ear
dBua measure of voltage referred to 0.775 volts RMS
dBFSa measure of relative amplitude (in voltage or digital data values) referred to the maximum representable digital value of a system - always a negative value

Note that dB SPL, dB(A) and dBu are generally measured as averages (RMS), but dBFS is a measure of peak value as that's what we need to know in order not to overload the digital recording system.

The strange-seeming value of 0.775 volts for the dBu reference is the voltage required to dissipate one watt of power in a 600 ohm load - the audio standard until the 1970s. This reference has been in use for decades and has been retained as the reference for professional audio even though 200 ohms is now almost universally used as the standard load.

In Part 2 we'll look at some real-world values expressed in decibels.