| I {dB } = 10 log 10 | ( | I ––– I 0 | ) |
Neither ear drums nor microphones can convert sound intensity. We hear the pressure variations.
When the ratio is between two field strength or voltage values (so that the power being transmitted is proportional to the square of this value), the formula is:
| V {dB } = 20 log 10 | ( | V 1 ––– V 0 | ) | = 10 log 10 | ( | V 1 2 ––––– V 0 2 | ) |
It can therefore be seen that a ratio expressed in decibels is independent of whether the measurements are made as field strength or power values.
The bel was too large for everyday use, so the decibel (dB), equal to 0.1 bel (B), became more commonly used. The bel is still used to represent noise power levels in hard drive specifications.
The Neper is a similar unit which uses the Natural logarithm. The Richter scale uses numbers expressed in bels as well, though this is implied by definition rather than explicitly stated. In spectrometry and optics, the absorbance unit used to measure Optical density is equivalent to −1 B. In astronomy, the Apparent magnitude measures the brightness of stars logarithmically, since just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness.
The reason for using the decibel is that the Ear is capable of detecting a very large range of sound pressures. The ratio of the sound pressure that causes permanent damage from short exposure to the limit that (undamaged) ears can hear is above a million. Because the power in a sound wave is proportional to the square of the pressure, the ratio of the maximum power to the minimum power is above one (short scale) Trillion. To deal with such a range, logarithmic units are useful: the log of a trillion is 12, so this ratio represents a difference of 120 dB.
Psychologists have found that our perception of Loudness is roughly logarithmic — see the Weber-Fechner law. In other words, you have to multiply the sound pressure by the same factor to have the same increase in loudness. This is why the numbers around the volume control dial on a typical audio amplifier are related not to the voltage amplification, but to its logarithm.
Various Frequency weightings are used to allow the result of an acoustical measurement to be expressed as a single sound level. The weightings approximate the changes in sensitivity of the ear to different frequencies at different levels. The two most commonly used weightings are the A and C weightings; other examples are the B and Z weightings.
Sound levels above 85 dB are considered harmful, while 120 dB is unsafe and 150 dB causes physical damage to the human Body. Windows break at about 163 dB. Jet airplanes cause A-weighted levels of about 133 dB at 33 m, or 100 dB at 170 m. Eardrums rupture at 190 dB to 198 dB. Shock waves and sonic booms cause levels of about 200 dB at 330 m. Sound levels of around 200 dB can cause Death to humans and are generated near Bomb explosions (e.g., 23 kg of TNT detonated 3 m away). The space shuttle generates levels of around 215 dB (or an A-weighted level of about 175 dB at a distance of 17 m). Even louder are nuclear bombs, earthquakes, tornadoes, hurricanes and volcanoes, all capable of exceeding 240 dB.
Some other values:
| dB(SPL)\ | Source (with distance) |
|---|---|
| 250 | Inside of Tornado |
| 180 | Rocket engine at 30 m; Blue Whale humming at 1 m; Krakatoa explosion at 100 miles (160 km) Level Chart.txt |
| 150 | Jet Engine at 30 m |
| 140 | Rifle being fired at 1 m |
| 130 | Threshold of Pain; Train horn at 10 m |
| 120 | Rock Concert; jet aircraft taking off at 100 m |
| 110 | Accelerating Motorcycle at 5 m; chainsaw at 1 m |
| 100 | Pneumatic hammer at 2 m; inside Disco |
| 90 | Loud Factory, heavy Truck at 1 m |
| 80 | Vacuum cleaner at 1 m, curbside of busy street |
| 70 | Busy Traffic at 5 m |
| 60 | Office or restaurant inside |
| 50 | Quiet Restaurant inside |
| 40 | Residential area at Night |
| 30 | Theatre, no talking |
| 10 | Human breathing at 3 m |
| 0 | Threshold of hearing (human with good ears) |
Note that the SPL emitted by an object changes with distance from the object. Commonly-quoted measurements of objects like jet engines or jackhammers are meaningless without distance information. The measurement is not of the object's noise, but of the noise at a point in space near that object. For instance, it is intuitively obvious that the noise level of a volcanic eruption will be much higher standing inside the crater than it would be measured from 5 kilometers away.
Measurements of ambient noise do not need a distance, since the noise level will be relatively constant at any point in the area (and are usually only rough approximations anyway).
Measurements that refer to the "threshold of pain" or the threshold at which ear damage occurs are measuring the SPL at a point near the ear itself.
Under controlled conditions, in an acoustical laboratory, the trained healthy human ear is able to discern changes in sound levels of 1 dB, when exposed to steady, single frequency ("pure tone") signals in the mid-frequency range. It is widely accepted that the average healthy ear, however, can barely perceive noise level changes of 3 dB.
On this scale, the normal range of human hearing extends from about 0 dB to about 140 dB. 0 dB is the Threshold of hearing in healthy, undamaged human ears; 0 dB is not an absence of sound, and it is possible for people with exceptionally good hearing to hear sounds at −10 dB. A 3 dB increase in the level of continuous noise doubles the sound power, however experimentation has determined that the frequency response of the human ear results in a perceived doubling of loudness with every 10 dB increase; a 5 dB increase is a readily noticeable change, while a 3 dB increase is barely noticeable to most people.
Sound pressure levels are applicable to the specific position at which they are measured. The levels change with the distance from the source of the sound; in general, the level decreases as the distance from the source increases. If the distance from the source is unknown, it is difficult to estimate the sound pressure level at the source.
Since the human ear is not equally sensitive to all the frequencies of sound within the entire spectrum, noise levels at maximum human sensitivity — middle A and its higher harmonics (between 2,000 and 4,000 Hertz) — are factored more heavily into sound descriptions using a process called frequency weighting.
The most widely used frequency weighting is the "A-weighting", which roughly corresponds to the inverse of the 40 dB (at 1 kHz) equal-loudness curve. Using this filter, the sound level meter is less sensitive to very high and very low frequencies. The A weighting parallels the sensitivity of the human ear when it is exposed to normal levels, and frequency weighting C is suitable for use when the ear is exposed to higher sound levels. Other defined frequency weightings, such as B and Z, are rarely used.
Frequency weighted sound levels are still expressed in decibels (with unit symbol dB), although it is common to see the incorrect unit symbols dBA or dB(A) used for A-weighted sound levels.
In Radio electronics, the decibel is used to describe the ratio between two measurements of electrical power. It can also be combined with a suffix to create an absolute unit of electrical power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". Zero dBm is one milliwatt, and 1 dBm is one decibel greater than 0 dBm, or about 1.259 mW.
Although decibels were originally used for power ratios, they are commonly used in electronics to describe voltage or current ratios. In a constant resistive load, power is proportional to the square of the voltage or current in the circuit. Therefore, the decibel ratio of two voltages V1 and V2 is defined as 20 log10(V1/V2), and similarly for current ratios. Thus, for example, a factor of 2.0 in voltage is equivalent to 6.02 dB (not 3.01 dB!).
This practice is fully consistent with power-based decibels, provided the circuit resistance remains constant. However, voltage-based decibels are frequently used to express such quantities as the voltage gain of an amplifier, where the two voltages are measured in different circuits which may have very different resistances. For example, a unity-gain Buffer amplifier with a high input resistance and a low output resistance may be said to have a "voltage gain of 0 dB", even though it is actually providing a considerable power gain when driving a low-resistance load.
In professional audio, a popular unit is the dBu (see below for all the units). The "u" stands for "unloaded", and was probably chosen to be similar to lowercase "v", as dBv was the older name for the same thing. It was changed to avoid confusion with dBV. This unit (dBu) is an RMS measurement of voltage which uses as its reference 0.775 VRMS. Chosen for historical reasons, it is the voltage level at which you get 1 mW of power in a 600 ohm resistor, which used to be the standard impedance in almost all professional audio circuits.
Since there may be many different bases for a measurement expressed in decibels, a dB value is meaningless unless the reference value (equivalent to 0 dB) is clearly stated. For example, the Gain of an antenna system can only be given with respect to a reference antenna (generally a perfect isotropic antenna); if the reference is not stated, the dB gain value is not usable.
Decibels are used to account for the gains and losses of a signal from a transmitter to a receiver through some medium (free space, wave guides, coax, fiber optics, etc.) using a Link Budget.
Now 6.25 and 6.4 are approximately equal to 6.3, so we don't care. Thus the round numbers between 1 and 10 are these:
Ratio 1 1.25 1.6 2 2.5 3.2 4 5 6.3 8 10
dB 0 1 2 3 4 5 6 7 8 9 10
This useful approximate table of logarithms is easily reconstructed or memorized.
Examples:
Examples:
Another common sequence is 1, 2, 5, 10, 20, 50 ... . These preferred numbers are very close to being equally spaced in terms of their logarithms. The actual values would be 1, 2.15, 4.64, 10 ... .
The conversion for decibels is often simplified to: "+3 dB means two times the power and 1.414 times the voltage", and "+6 dB means four times the power and two times the voltage ".
While this is accurate for many situations, it is not exact. As stated above, decibels are defined so that +10 dB means "ten times the power". From this, we calculate that +3 dB actually multiplies the power by 103/10. This is a power ratio of 1.9953 or about 0.25% different from the "times 2" power ratio that is sometimes assumed. A level difference of +6 dB is 3.9811, about 0.5% different from 4.
To contrive a more serious example, consider converting a large decibel figure into its linear ratio, for example 120 dB. The power ratio is correctly calculated as a ratio of 1012 or one trillion. But if we use the assumption that 3 dB means "times 2", we would calculate a power ratio of 2120/3 = 240 = 1.0995 × 1012, giving a 10% error.