Decibel

The decibel is a relative unit of measurement equal to one tenth of a bel. It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a power ratio of 10^1/10 or root-power ratio of 10^1/20.
The strict original usage above only expresses a relative change. However, the word decibel has since also been used for expressing an absolute value that is relative to some fixed reference value, in which case the dB symbol is often suffixed with letter codes that indicate the reference value. For example, for the reference value of 1 volt, a common suffix is "V".
As it originated from a need to express power ratios, two principal types of scaling of the decibel are used to provide consistency depending on whether the scaling refers to ratios of power quantities or root-power quantities. When expressing a power ratio, the corresponding change in decibels is defined as ten times the logarithm with base 10 of that ratio. That is, a change in power by a factor of 10 corresponds to a 10 dB change in level. When expressing root-power ratios, a change in amplitude by a factor of 10 corresponds to a 20 dB change in level. The decibel scales differ by a factor of two, so that the related power and root-power levels change by the same value in linear systems, where power is proportional to the square of amplitude.
The definition of the decibel originated in the measurement of transmission loss and power in telephony of the early 20th century in the Bell System in the United States. The bel was named in honor of Alexander Graham Bell, but the bel is seldom used. Instead, the decibel is used for a wide variety of measurements in science and engineering, most prominently for sound power in acoustics, in electronics and control theory. In electronics, the gains of amplifiers, attenuation of signals, and signal-to-noise ratios are often expressed in decibels.

History

The decibel originates from methods used to quantify signal loss in telegraph and telephone circuits. Until the mid-1920s, the unit for loss was miles of standard cable. 1 MSC corresponded to the loss of power over one mile of standard telephone cable at a frequency of radians per second, and matched closely the smallest attenuation detectable to a listener. A standard telephone cable was "a cable having uniformly distributed resistance of 88 ohms per loop-mile and uniformly distributed shunt capacitance of 0.054 microfarads per mile".
In 1924, Bell Telephone Laboratories received a favorable response to a new unit definition among members of the International Advisory Committee on Long Distance Telephony in Europe and replaced the MSC with the Transmission Unit. 1 TU was defined such that the number of TUs was ten times the base-10 logarithm of the ratio of measured power to a reference power. The definition was conveniently chosen such that 1 TU approximated 1 MSC; specifically, 1 MSC was 1.056 TU. In 1928, the Bell system renamed the TU as the decibel, being one tenth of a newly defined unit for the base-10 logarithm of the power ratio. It was named the bel, in honor of the telecommunications pioneer Alexander Graham Bell. The bel is seldom used, as the decibel was the proposed working unit.
The naming and early definition of the decibel is described in the NBS Standard's Yearbook of 1931:
The word decibel was soon misused to refer to absolute quantities and to ratios other than power. Some proposals attempted to address the resulting confusion. In 1954, J. W. Horton considered that 10 be treated as an elementary ratio and proposed the word logit as "a standard ratio which has the numerical value 10 and which combines by multiplication with similar ratios of the same value", so one would describe a 10 ratio of units of mass as "a mass logit". This contrasts with the word unit which would be reserved for magnitudes which combine by addition and reserves the word decibel specifically for unit transmission loss. The decilog was another proposal to express a division of the logarithmic scale corresponding to a ratio of 10.
In April 2003, the International Committee for Weights and Measures considered a recommendation for the inclusion of the decibel in the International System of Units, but decided against the proposal. However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission and International Organization for Standardization . The IEC permits the use of the decibel with root-power 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. In spite of their widespread use, [|suffixes] are not recognized by the IEC or ISO.

Definition

The IEC Standard 60027-3:2002 defines the following quantities. The decibel is one-tenth of a bel: 1 dB = 0.1 B. The bel is ln nepers: 1 B = ln Np. The neper is the change in the level of a root-power quantity when the root-power quantity changes by a factor of e, that is, thereby relating all of the units as nondimensional natural log of root-power-quantity ratios, = =. Finally, the level of a quantity is the logarithm of the ratio of the value of that quantity to a reference value of the same kind of quantity.
Therefore, the bel represents the logarithm of a ratio between two power quantities of 10:1, or the logarithm of a ratio between two root-power quantities of :1.
Two signals whose levels differ by one decibel have a power ratio of 10^1/10, which is approximately, and an amplitude ratio of 10^1/20.
The bel is rarely used either without a prefix or with SI unit prefixes other than deci; it is customary, for example, to use hundredths of a decibel rather than millibels. Thus, five one-thousandths of a bel would normally be written 0.05 dB, and not 5 mB.
The method of expressing a ratio as a level in decibels depends on whether the measured property is a power quantity or a root-power quantity; see Power, root-power, and field quantities for details.

Power quantities

When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to reference value. Thus, the ratio of P to P₀ is represented by L_P, that ratio expressed in decibels, which is calculated using the formula:
The base-10 logarithm of the ratio of the two power quantities is the number of bels. The number of decibels is ten times the number of bels. P and P₀ must measure the same type of quantity, and have the same units before calculating the ratio. If P₀ in the above equation, then L_P = 0. If P is greater than P₀ then L_P is positive; if P is less than P₀ then L_P is negative.
Rearranging the above equation gives the following formula for P in terms of P₀ and L_P :

Root-power (field) quantities

When referring to measurements of root-power quantities, it is usual to consider the ratio of the squares of F and F₀. This is because the definitions were originally formulated to give the same value for relative ratios for both power and root-power quantities. Thus, the following definition is used:
The formula may be rearranged to give
Similarly, in electrical circuits, dissipated power is typically proportional to the square of voltage or current when the impedance is constant. Taking voltage as an example, this leads to the equation for power gain level L_G:
where V_out is the root-mean-square output voltage, V_in is the rms input voltage. A similar formula holds for current.
The term root-power quantity is introduced by ISO Standard 80000-1:2009 as a substitute of field quantity. The term field quantity is deprecated by that standard and root-power is used throughout this article.

Relationship between power and root-power levels

Although power and root-power quantities are different quantities, their respective levels are historically expressed in the same units, typically decibels. A factor of 2 is introduced to make changes in the respective levels match under restricted conditions such as when the medium is linear and the same waveform is under consideration with changes in amplitude, or the medium impedance is linear and independent of both frequency and time. This relies on the relationship
holding. In a nonlinear system, this relationship does not hold by the definition of linearity. However, even in a linear system in which the power quantity is the product of two linearly related quantities, if the impedance is frequency- or time-dependent, this relationship does not hold in general, for example if the energy spectrum of the waveform changes.
For differences in level, the required relationship is relaxed from that above to one of proportionality, or equivalently,
must hold to allow the power level difference to be equal to the root-power level difference from power P and F to P and F. An example might be an amplifier with unity voltage gain independent of load and frequency driving a load with a frequency-dependent impedance: the relative voltage gain of the amplifier is always 0 dB, but the power gain depends on the changing spectral composition of the waveform being amplified. Frequency-dependent impedances may be analyzed by considering the quantities power spectral density and the associated root-power quantities via the Fourier transform, which allows elimination of the frequency dependence in the analysis by analyzing the system at each frequency independently.