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dB stands for "decibel", which is literally one tenth of a Bel, a unit named for telephone pioneer Alexander Graham Bell. A Bel is the logarithm (base 10) of the ratio between two power levels, but the whole Bel is never used... only decibels. So since there are 10 decibels in a Bel:
dB = 10 * log10(P1 / P0)
where P1 is the power of the signal being measured and P0 is some reference power it is being compared to. The reference power is either stated, or implied by a particular standard.
For example, when talking about frequency response of a system, it is common to refer to the upper and lower -3 dB frequencies (or simply the 3 dB points, since the minus sign is understood). These are the frequencies where the power is reduced to half of what it is in some middle frequency range which is used as a reference. Since the log of 1/2 is -0.30103, multiplying by 10 gives about -3 dB.
In most cases power is proportional to the square of voltage, such as when the voltage is driving a constant load. In this case a voltage-based formula may be used, but note that the dB value is still a power ratio due to the squares:
dB = 10 * log10(V1^2 / V0^2)
which is simplified to:
dB = 20 * log10(V1 / V0)
Note that dB is a signed value, but the sign is often implied by the method of expression. For example, you could say that one power is 3 dB lower than another, or you could say that it is -3 dB relative to the other.
You might want to think of dB as a nonlinear way to express "percent" for power measurements; for example, "-3 dB" is just another way of saying "50%", where it is always assumed you are talking about power (not voltage). It doesn't matter whether the power is microwatts or megawatts, half is always -3 dB.
Why not just stick with percentages? Consider that the ratio between the largest and smallest powers that a system deals with may be billions to one; who would want to deal with numbers like 0.0000000001 percent? Rather than counting the leading zeros, we can just use -120 dB.
Although the dB system may seem awkward at first, it is much more intuitive than percentages in actual use. Since our senses (and other systems) respond in a logarithmic fashion, the same number of dB gives roughly the same change in sensation at different parts of the range. In other words, a change of (say) 2 dB sounds like the same relative change whether it is going from 73 dB above your threshold of hearing to +71 dB, or from +56 dB to +54 dB. You can easily see that there is a 2 dB difference in both cases, but try that with percentages: 1995262315% is to 1258925412% as 39810717% is to 25118864%, but it sure isn't obvious!
Another point to keep in mind is that, like a percentage, a dB value is always relative to something. You can talk about a change in a single value, like "a 7 dB reduction", or you can talk about comparing different readings, like "this harmonic is 13 dB below the fundamental". In these cases it should be pretty clear what is relative to what. In other cases the reference must be explicitly specified. The common convention is to use something like "dB re: 1 kW", here specifying that the 0 dB reference is a power of 1 kW. Daqarta labels the Y-log axis as "dB re:FS" to denote that a 0 dB signal is 100% of the Full-Scale input level on the current ADC range. There are also notations like "dB SPL" where the 0 dB reference is a specified standard value, making these absolute dB readings. (See Absolute dB (SPL, etc.).)
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