As anyone who has tried to research this topic will know, this simple question leads to a very complex answer. So a few warning comments to begin with will assist:
- This article will greatly simplify some very complex, but very important, issues.
- Consequently, we will try to eliminate the quite complex mathematics.
- We will try to have some fun along the way by introducing some interesting by-products of the investigation.
Starting at the beginning, a dB is the unit for a decibel, which is one tenth of a bel [B]. And a bel (and hence a decibel) is a unit that measures a ratio of two values.
So far, so good. I think we are all comfortable with the prefix “deci” meaning one tenth. But it always reminds me of the word “decimate”, which has come to mean “kill, destroy, or remove a large proportion of something”. But the origins of the word comes from the Latin of the Romans who, as you know, could be very nasty when they chose. When they dealt with mutinous troops, or barbarian peoples, they would decimate them – meaning that they would kill every tenth person in that group. Apparently it was a very effective method of control!
But back to our subject. The bel originated as a result of a need in the very early days of the telegraph and telephone to quantify the signal loss down lengths of telephone cable. Much of this early work was done by the Bell Telephone Laboratories in the US who came up with the name “Bel” in 1924 to honour of Alexander Graham Bell. In practice, the bel is seldom used – the decibel being the preferred unit.
More digressions. Both Alexander Graham Bell and the Bell Telephone Laboratories are very interesting in their own rights. Bell was born in Edinburgh, Scotland in 1847, later moved to Canada and then the USA. He claimed citizenship of all three countries! He was a prolific inventor/scientist and, among other things, invented the telephone (in 1876) and co-founded the Bell Telephone Company (in 1877). He died in Nova Scotia, Canada in 1922.
The Bell Telephone Company went through many changes of ownership and names – among others, AT&T Bell Laboratories (in 1984), and finally it was purchased by the Finnish company Nokia in 2016. These labs have been responsible for a truly amazing range of discoveries – e.g. radio astronomy, the transistor, the laser, and a range programming languages. In total 9 Nobel Prizes have been awarded for work at these laboratories!
But we digress. So, what is a dB and why do we want to make that so complicated?
The dB is a logarithmic way of describing a ratio of two numbers. This means that when it is used to give the sound level (for example) for a single sound rather than a ratio, a reference level must be chosen. Suffixes are attached to the basic dB unit to indicate this reference value. And we use the dB to measure so many things – e.g. acoustics, voltage, audio electronics, radar, antennas, etc. So there is a large range of reference levels and, therefore, suffixes. Some examples are dBA, dBV, dBm, dBZ, dBi and many, many more. It is this range that creates the complexity.
A reflection of this complexity is the refusal, in 2003, of the International Committee for Weights and Measures to include the dB in the International System of Units (SI)!
Before leaving the subject of suffixes, let’s briefly review some protocols:
- Both dBA and dB(A) are used to indicate acoustic levels filtered (weighted) to approximate the human ear’s response to sound
- However, dBa is something quite different – this one is used telephone circuitry
- dBL is the unfiltered measurement of acoustic levels. The Texcel GTM and ETM measure dBL. Since the measured frequencies range from 2 Hz to 250 Hz (by Australian Standard) and some of these frequencies are below the normal hearing range, we call this overpressure or, sometimes, infrasound.
Another diversion here. We have been using the term “logarithm” a lot here – so what is a logarithm?
The simple definition is: A logarithm (log) is the power to which a number must be raised in order to get some other number.
For example, Log10 100 = 2. In other words, 102 = 100.
Logarithms can be to any base (not necessarily 10 as above) and there is also a natural logarithm, but let’s not discuss them here.
Interestingly, the Babylonians sometime between 2000 and 1600 BCE, initiated the invention of the logarithm. But the logarithm as we know it was invented in about 1614 by an English mathematician. Using logarithms allows you to do multiplication by addition (and division by subtraction), and mathematicians spent many decades calculating tables of logarithms to more and more decimal places to increase the accuracy of these processes.
The slide rule, in common use by engineers up until the 1970s, multiplied and divided by using a graphical representation of logarithms. The slide rule was also invented in the 1600s (probably around 1630), although the modern version was created in 1859.
The importance of logarithms in science and engineering was expressed as recently as 1958 as follows:
“Probably no work has ever influenced science as a whole, and mathematics in particular, so profoundly as this modest little book (the Descriptio – the original paper written about logarithms in 1614). It opened the way for the abolition, once and for all, of the infinitely laborious, nay, nightmarish, processes of long division and multiplication, of finding the power and the root of numbers.”
(Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times (1958)
Back to our main subject.
The definition of a bel is easy:
If two signals differ by 1 bel, one is 10x bigger than the other, so the bel is a logarithmic unit defined as:
1 bel is the logarithm of a ratio of 10
I.e. Log10(10) = 1
But 10 is too big a ratio to be useful in a lot of situations – hence the decibel.
A decibel can be defined as the ratio whose logarithm is 0.1:
1 decibel is 0.1 x Log10(10) = Log10(100.1) = 1.2589
I.e. Log10(1.2589) = 0.1
As stated above, the dB is a logarithmic way of describing a ratio of two numbers. But dB is often used to measure an absolute value, as in dBA which measures a level of sound or noise in the air. To make a dB value measure an absolute value (so we can compare measurements), we must fix one of the two values in the ratio to a reference level. This is what we do with sound pressure levels below. Other reference levels are defined in Australian and International Standards.
Just for the record, microphones (e.g. those used by Texcel’s GTM and ETM) actually measure sound pressure levels – expressed in Pascals (Pa). We convert these pressure levels to dBL by using the formula 20 x Log10(Pa / PaRef) where Pa is the sound pressure level recorded by the microphone and PaRef is the reference level for sound energy (20 µPa).
In summary, despite all the digressions, hopefully you now know enough about a dB to at least answer some trivia questions. But seriously, if you would like more information, please reach out – we’d love to assist.
