The decibel is widely known as a measure of sound pressure level, but is also used for a wide variety of other measurements in science and engineering, most prominently in acoustics, electronics, and control theory. In electronics, the gain of amplifiers, attenuation of signals, and signal to noise ratios are often expressed in decibels. It confers a number of advantages, such as the ability to conveniently represent very large or small numbers, a logarithmic scaling that roughly corresponds to the human perception of sound and light, and the ability to carry out multiplication of ratios by simple addition and subtraction.
The decibel symbol is often qualified with a suffix, that indicates which reference quantity or frequency weighting function has been used. For example, dBm indicates that the reference quantity is one milliwatt, while dBu is referenced to 0.775 volts RMS.[2] and dBμV/m referenced to microvolts per meter for radio frequency signal strength.
The definitions of the decibel and bel use logarithms to base 10. The neper, used in electronics, uses natural logarithm to base (e).
History
The decibel originates from methods used to quantify reductions in audio levels in telephone circuits. These losses were originally measured in units of Miles of Standard Cable (MSC), where 1 MSC corresponded to the loss of power over a 1 mile (approximately 1.6 km) length of standard telephone cable at a frequency of 5000 radians per second (795.8 Hz), and roughly matched the smallest attenuation detectable to an average listener. Standard telephone cable was defined as "a cable having uniformly distributed resistances of 88 ohms per loop mile and uniformly distributed shunt capacitance of .054 microfarad per mile" (approximately 19 gauge).The transmission unit (TU) was devised by engineers of the Bell Telephone Laboratories in the 1920s to replace the MSC. 1 TU was defined as ten times the base-10 logarithm of the ratio of measured power to a reference power level.[3] The definitions were conveniently chosen such that 1 TU approximately equaled 1 MSC (specifically, 1.056 TU = 1 MSC).[4] Eventually, international standards bodies adopted the base-10 logarithm of the power ratio as a standard unit, named the bel in honor of the Bell System's founder and telecommunications pioneer Alexander Graham Bell.[5] The bel was larger by a factor of ten than the TU, such that 1 TU equaled 1 decibel.[6] For many measurements, the bel proved inconveniently large, giving way to the decibel becoming the common unit of choice.
In April 2003, the International Committee for Weights and Measures (CIPM) considered a recommendation for the decibel's inclusion in the International System of Units (SI), but decided not to adopt the decibel as an SI unit.[7] However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission (IEC).[8] The IEC permits the use of the decibel with field quantities as well as power and this recommendation is followed by many national standards bodies, such as NIST, which justifies the use of the decibel for voltage ratios.[9]
Definition
A decibel (dB) is one tenth of a bel (B), i.e. 1B = 10dB. The bel is the logarithm of the ratio of two power quantities of 10:1, and for two field quantities in the ratio
.[10] A field quantity is a quantity such as voltage, current, sound pressure, electric field strength, velocity and charge density, the square of which in linear systems is proportional to power. A power quantity is a power or a quantity directly proportional to power, e.g. energy density, acoustic intensity and luminous intensity.The calculation of the ratio in decibels varies depending on whether the quantity being measured is a power quantity or a field quantity.
Power quantities
When referring to measurements of power or intensity, a ratio can be expressed in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference level. Thus, the ratio of a power value P1 to another power value P0 is represented by LdB, that ratio expressed in decibels, which is calculated using the formula:
Rearranging the above equation gives the following formula for P1 in terms of P0 and LdB:
.
.
Field quantities
When referring to measurements of field amplitude it is usual to consider the ratio of the squares of A1 (measured amplitude) and A0 (reference amplitude). This is because in most applications power is proportional to the square of amplitude, and it is desirable for the two decibel formulations to give the same result in such typical cases. Thus the following definition is used:
and 
is one of the standard properties of logarithms.The formula may be rearranged to give
Examples
- To calculate the ratio of 1 kW (one kilowatt, or 1000 watts) to 1 W in decibels, use the formula
- To calculate the ratio of
to
in decibels, use the formula
, illustrating the consequence from the definitions above that GdB has the same value, 
, regardless of whether it is obtained with the 10-log or 20-log rules; provided that in the specific system being considered power ratios are equal to amplitude ratios squared.- To calculate the ratio of 1 mW (one milliwatt) to 10 W in decibels, use the formula
- To find the power ratio corresponding to a 3 dB change in level, use the formula
A change in power ratio by a factor of 10 is a 10 dB change. A change in power ratio by a factor of two is approximately a 3 dB change. More precisely, the factor is 103/10, or 1.9953, about 0.24% different from exactly 2. Similarly, an increase of 3 dB implies an increase in voltage by a factor of approximately
, or about 1.41, an increase of 6 dB corresponds to approximately four times the power and twice the voltage, and so on. In exact terms the power ratio is 106/10, or about 3.9811, a relative error of about 0.5%.Please see the pic at right hand side :An example scale showing x and 10 log x. It is easier to grasp and compare 2 or 3 digit numbers than to compare up to 10 digits.
Common reference levels and corresponding units
Although decibel measurements are always relative to a reference level, if the numerical value of that reference is explicitly and exactly stated, then the decibel measurement is called an "absolute" measurement, in the sense that the exact value of the measured quantity can be recovered using the formula given earlier. For example, since dBm indicates power measurement relative to 1 milliwatt,
- 0 dBm means no change from 1 mW. Thus, 0 dBm is the power level corresponding to a power of exactly 1 mW.
- 3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power level corresponding to 103/10 × 1 mW, or approximately 2 mW.
- −6 dBm means 6 dB less than 0 dBm. Thus, −6 dBm is the power level corresponding to 10−6/10 × 1 mW, or approximately 250 μW (0.25 mW).
[edit] Electric power
dBm or dBmW- dB(1 mW) – power measurement relative to 1 milliwatt. XdBm = XdBW + 30.
- dB(1 W) – similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm; XdBW = XdBm − 30.
Decibel (dB) and dB relative to a milliwatt (dBm) represent two different but related concepts.
A dB is a shorthand way to express the ratio of two values. As a unit for the strength of a signal, dB expresses the ratio between two power levels. To be exact, dB = log (P1/P2).
Using the decibel allows us to contrast greatly differing power levels (a common predicament in radio link design) with a simple two- or three-digit number instead of a more burdensome nine- or 10-digit one.
For instance, instead of characterizing the difference in two power levels as 1,000,000,000 to 1, it's much simpler to use the decibel representation as 10*log (1,000,000,000/1), or 90 dB. The same goes for very small numbers: The ratio of 0.000000001 to 1 can be characterized as -90 dB. This makes keeping track of signal levels much simpler.
The unit dBm denotes an absolute power level measured in decibels and referenced to 1 milliwatt (mW). To convert from absolute power "P" (in watts) to dBm, use the formula dBm = 10*log (P/1 mW). This equation looks almost the same as that for the dB. However, now the power level "P" has been referenced to 1 mW. It turns out that in the practical radio world, 1 mW is a convenient reference point from which to measure power.
Use dB when expressing the ratio between two power values. Use dBm when expressing an absolute value of power.
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Working with Decibels
If you want to communicate effectively with EMC engineers, it’s important to get comfortable with decibels (dB). Decibel notation is a convenient way of expressing ratios of quantities that may or may not span many orders of magnitude. It is also used to express the amplitude of various signal parameters such as voltage or current relative to a given reference level.
A power ratio, P2:P1, in dB is simply calculated as,
(1)For example, if we are comparing a 10-watt received power to a 5-watt specification, we could say that the received power exceeded the specification by,
(2)If the impedance associated with two power levels is constant, then the power is proportional to the voltage (or current) squared. In this case, we can also express voltage (or current) ratios in dB,
(3)or,
(4)Decibels can also be used to express ratios of power densities or electromagnetic field strengths. For example, if the electric field strength incident on a composite surface is 3 V/m and the reflected field strength is 1 V/m, the ratio of incident to reflected field strengths is,
(5)Antenna or amplifier gains are usually reported in dB. So are cable or filter losses. An amplifier that receives a 1-watt signal and produces a 100-watt signal has a gain of,
(6)A cable whose input signal has an amplitude of 3.0 volts and whose output signal has an amplitude of 2.8 volts exhibits a gain of,
(7)or a loss of,
(8)Note that the inverse of any ratio is expressed by changing its sign in dB. A ratio of 1 is 0 dB. Phase or negative values cannot be expressed in dB.
Quiz Question:A signal traveling one kilometer in a coaxial cable loses one-half its voltage. Express the, a.) input-to-output voltage ratio b.) input-to-output power ratio c.) input-to-output voltage ratio in dB d.) input-to-output power ratio in dB. |
Of course, the input-to-output voltage ratio is 2:1, while the input-to-output power ratio is (2)2:(1)2=4:1. The voltage ratio expressed in dB is 20 log (2/1) = 6 dB. The power ratio is 10 log (4/1) = 6 dB. This illustrates one of the primary advantages to expressing gains or losses in dB. As long as the impedance is constant, it is not necessary to specify whether a ratio is power or voltage when it is expressed in dB. A 6-dB gain unambiguously means the power has quadrupled whether the original measurement was voltage, current or power. On the other hand, if we were simply to say that one signal was twice as strong as another, it would not be clear whether it had twice the power or twice the amplitude.
Example 1-1: Specifying ratios in dB
Specify the following ratios in dB:
| 200 μV/m : 100 μV/m |  |
| 300 mV : 100 mV |  |
| 400 mA : 100 mA |  |
| 500 μA/m : 100 μA/m |  |
| 2 μW : 1 μW |  |
| 3 mW : 1 mW |  |
Expressing Signal Amplitudes in dB
Signal amplitudes can also be expressed in decibels as a ratio of the amplitude to a specified reference. For example, a 100-μvolt signal amplitude can also be expressed as,
(9) Quiz Question:Express the following signal or field amplitudes in their normal units, a.) 6 dB(μV) b.) 20 dB(μA) c.) 20 dB(A) d.) 100 dB(μV/m) e.) 100 dB(μW) |
The units in parentheses following the "dB" indicate that the quantity being expressed is an amplitude.
Each of the quantities above is simply converted as follows: | (10) |
 | (11) |
 | (12) |
 | (13) |
 | (14) |
Using Decibels
Why bother expressing signal amplitudes in dB? After all, there's never any ambiguity concerning whether a quantity is a power or voltage when the amplitude and its units are provided. The real power of working in dB is calculating ratios.
Previously, we mentioned comparing a 10-watt receiver to a 5-watt specification. Using Equation (2), we showed that the receiver was 3 dB over the specification. In this case, if the powers had been expressed in dB(W),
(15)
(16)we could have calculated the ratio as,
(17)Rather than dividing amplitudes to determine the ratio, we can simply subtract amplitudes expressed in dB(·). Again, as long as the impedance is constant, it won't matter whether we are working with units of power, voltage or current.
Example 1-2: Specifying ratios in dB
Specify the following ratios in dB:
| 46 dBμV/m) : 40 dB(μV/m) | -> | 46 dB(μV/m) - 40 dB(μV/m) = 6 dB |
| 50 dB(mV)) : 40 dB(mV) | -> | 50 dB(mV) - 40 dB(mV) = 10 dB |
| 52 dB(mA) : 40 dB(mA) | -> | 52 dB(mA) - 40 dB(mA) = 12 dB |
| 54 dB(μA/m) : 40 dB(μA/m) | -> | 54 dB(μA/m) - 40 dB(μA/m) = 14 dB |
| 3 dB(μW) : 0 dB(μW) | ->> | 3 dB(μW) - 0 dB(μW) = 3 dB |
| 7 dB(mW) : 0 dB(mW) | -> | 7 dB(mW) - 0 dB(mW) = 7 dB |
dBm
One of the most common units expressed in decibels is dB(mW) or dB relative to 1 milliwatt. This is almost always written in the abbreviated form, dBm (i.e. without the "W" and without the parentheses).Many oscilloscopes and spectrum analyzers optionally display voltage amplitudes in dBm. Since dBm is a unit of power, we must know the impedance of the measurement in order to convert dBm to volts. For example, a voltage expressed as 0 dBm on a 50-ohm spectrum analyzer is,
(18)Example 1-3: Specifying voltages in dBm
Specify the following voltages in dBm assuming they were measured with a 50-ohm oscilloscope:
| 1 μV |  |
| 2 μV |  |
| 10 μV |  |
| 1 V |  |
| 2 V |  |
| 10 V |  |
In this example, we can see that doubling the voltage adds 6 dB (e.g. 13 dBm + 6 dB = 19 dBm) and increasing a voltage by a factor of 10 adds 20 dB. This is true no matter what units of voltage are being used and is an example of why it is often convenient to work with decibels.
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