**Measuring noise, part 2**

If the Prince of Denmark had been an acoustic or electronic engineer, he would have had no need to ask his famous question; he would have known that the decibel, or dB for short (and pronounced dee-bee, hence the pun in the title of this blog), is a really neat idea. I may not be able to convince you how great a concept the dB is, but at least I will try in this blog to give you a practical understanding of the idea.

It all starts with the unit which is known as the “bel”, abbreviated to B. This unit relates to the ratio of the power of one signal to another (expressed as a quotient), given by the expression:

log_{10}(P2/P1), where (P2/P1) is the ratio of the two signal powers.

So now you know why I devoted all of my previous blog (2 Jun) to logarithms.

In practice nobody uses the bel: the unit that is used is the decibel or dB, which is one-tenth of a bel. So the power ratio in decibels is given by the expression:

10log_{10}(P2/P1).

This term may also be used to define a measured power level (P_{m}), by setting P_{1} to be some reference level P_{ref}. If we do this, the level P_{ref} expressed in dB is 10log_{10}(P_{ref}/P_{ref}).

Since I determined in my previous blog (2 Jun) that log_{10}(1) = 0, the sound level of P_{ref} is 0 dB. So P_{ref }becomes the zero point of our decibel scale, just like the freezing point of water is the zero point on the Celsius temperature scale.

Since the decibel is defined by the base-10 logarithmic function it is a non-linear measuring scale. So whereas on a linear scale zero and the units 1, 2, 3, 4, etc are equally spaced along the scale, as on a ruler, using the decibel results in the decades, 0, 1, 10, 100, etc, being evenly spaced.

The Richter scale, used to rate earthquakes, is another example of a base-10 logarithmic scale.

It is convenient to use dB as the measurement of sound levels (and other physical quantities such as power in electronic circuits) for three reasons.

Firstly, decibels appear to confuse people and so give engineers an advantage and make them feel superior to those not in the know.

Secondly, it is easy to manipulate decibels in situations where power levels vary due to changes in the power emanated at the source, or changes in the path through which the sound has to travel. For example, as a rough rule of thumb sound power increases by four times every time you halve the distance from the sound source. Calculating this is a doddle using decibels since if (P2/P1) is 4 then 10log_{10}(P2/P1) is 6 dB. So all you have to do is add 6 dB to the signal power, avoiding the need to use fractions, or multiplication (remember logarithms allow you to add instead of multiplying).

Thirdly, there is a very wide range wide between the quietest sound that can be detected by the human ear and the largest that can be tolerated. The non-linear logarithmic scale of the decibel conveniently compresses this wide variation into a small range of numbers. This makes it much easier to draw graphs, for example. Conveniently, and hard to believe I know, the decibel scale also approximates to the human perception of the way that the intensity of a sound varies with its actual power. So all round it’s a good way of representing sound power.

Now one last thing that I want to cover in this blog is something that gives engineers the occasional chuckle and feeling of superiority. Question – if there are two sound sources each of 90 dB above our reference level, what is the combined power level? The answer is surely 180 dB above our reference level. No you are wrong (sounds off of engineers laughing).

So how do we work that out? If we assume that a single noise source has a power P_{m}, then the power in decibels is 10log_{10}(P_{m}/P_{ref}). If we add a second source with the same power then the resulting combined power is 10log_{10}(2P_{m}/P_{ref}).

Now we have a multiplication in that term, 2P_{m. }and we are taking the logarithm of this multiplication. From what we found out in the previous blog (2 Jun) log_{10}(A.B) = log_{10}(A) + log_{10}(B).

So the term for the combined power can be written 10log_{10}(P_{m}/P_{ref}) + 10log_{10}(2). By whipping out my *Chambers’s Four Figure Mathematical Tables* again, I can tell you that 10log_{10}(2) is 3.010, or 3 dB near enough. So combining two sound sources of 90 dB above our reference level gives a combined power of 93 dB above our reference level. Now who would have thought that? You see you can expect weird things like that to happen when you use a non-linear scale.

In the unlikely event that this blog has made you completely enthralled with the possibilities of the decibel, the Wikipedia article on the subject (here) will lead you to the heights of ecstasy.