Recommendation B.12[1])
A.1.1
The bel (symbol B) expresses the ratio of two powers by the
decimal logarithm of this ratio. This unit is not often used, having been
replaced by the decibel (symbol dB) which is one-tenth of a bel.
A.1.2 The decibel may be used to express the ratio of two field
quantities, such as voltage, current, sound pressure, electric field,
charge velocity or density, the square of which in linear systems is
proportional to power. To obtain the same numerical value as a power ratio, the
logarithm of the field quantity ratio is multiplied by the factor 20, assuming
that the impedances are equal.
The relationship between a current or voltage ratio and that of the
corresponding power ratio is impedance dependent. Use of the decibel when the
impedances are not equal is not appropriate unless adequate information is
given concerning the impedances involved.
For example, if P1 and P2 are two powers,
their ratio expressed in decibels is:
10 lg
If P1 and P2 represent the powers
dissipated by currents I1 and I2 in
resistances R1 and R2:
10 lg = 10 lg = 20 lg + 10 lg
A.1.3 The decibel may also be used to express the ratio of two values of a
quantity connected with power by a well-defined relationship. In this case, the
logarithm of this ratio must be multiplied by a factor representing the
relationship which connects the quantity with a power, and a term representing
a multiplying factor may be added to it.
The corresponding formulae, together with an example, are given in Appendix I,
§ I.2.
The neper (symbol Np) expresses the ratio of two field
quantities such as voltage or current, the square of which is proportional to
power by the natural logarithm of this ratio. The value of a power ratio in
nepers is one half of the natural logarithm of the power ratio. The values in
nepers of the ratio of two field quantities and of the corresponding powers are
equal only if the impedances are equal.
One neper corresponds to the value of e of a field quantity ratio and
to the value e2 of a power quantity ratio.
Sub-multiples such as the decineper (dNp) are also used.
In some disciplines, nepers may be used to express the logarithm of a power
ratio without the factor ½. An example is optical depth or attenuation in
radiometry. Such usage is deprecated in telecommunications in order to avoid
ambiguity. Under this definition, the neper would in fact be equal to 4.34 dB,
instead of 8.68 dB as is traditionally the case.
Countries can continue to use either the neper or the decibel for measurement
purposes within their own territory and, to avoid conversion of values,
countries which prefer to do so may continue to use the neper between
themselves by bilateral agreement.
For the international exchange of information concerning transmission
measurement and related values and for the international specification of
limits for such values, the only logarithmic expression to be used is the
decibel.
For theoretical or scientific calculations, where ratios are expressed in
terms of naperian logarithms, the neper will always be used, implicitly or
explicitly.
As a result of some calculations on complex quantities, a real part in nepers
and an imaginary part in radians are obtained. Factors may be applied for
converting to decibels or degrees.
The conversion values between the neper and the decibel are as follows:
1 Np = (20 lg e)dB 8.686 dB
1 dB = (0.05 ln 10)Np 0.1151 Np
Concerning the symbols that include the symbol dB, the following rules should be used as far as possible:
The symbol dB without additional indication should be used to indicate a difference between two power levels or a ratio of two powers, two power densities, or two other quantities clearly connected with power.
The symbol dB followed by additional information within parentheses should be used to express an absolute level of power, power flux density or any other quantity clearly connected with power, in relation to a reference value within the parentheses. In some cases, however, common use may give rise to simplified symbols such as dBm instead of dB(mW).
The symbol dB followed by additional information without parenthesis should be used to express by convention, special conditions such as measurements through specified filters or at a specified point of a circuit.
The attenuation or loss is a decrease between two points of an
electric, electromagnetic or acoustic power. The attenuation is also the
quantitative expression of a power decrease, generally in decibels; this
decrease is expressed by the ratio of the values at two points of a power or of
a quantity related to power in a well-defined manner.
The gain is the increase between two points of an electric,
electromagnetic or acoustic power. The gain is also the quantitative expression
of a power increase, generally in decibels; this increase is expressed by the
ratio of the values at two points of a power or of a quantity related to power
in a well-defined manner.
The exact designation of the loss or gain in question must be given (e.g.
image-attenuation coefficient, insertion loss, antenna gain) which in fact
refers to the precise definitions of the ratio in question (terminal
impedances, reference conditions, etc.).
A.5.1 Transmission loss (Refs. Recommendation 341, CCIR Volume V and
Recommendation 573, term A43, CCIR Volume XIII)
This is the ratio, expressed in decibels, of the transmitted power
(Pt) to the received power (Pr):
In many cases, the comparison of a quantity, here called x, with a
specified reference quantity of the same kind (and dimension),
xref is expressed by the logarithm of the ratio
x/xref. This logarithmic expression is often called "the level
of x (with respect to xref)" or "the x-level (with
respect to xref)". With the general letter symbol for
level L, the level of the quantity x may be written
Lx.
Other names and other symbols exist and can be used, x may in itself be
a single quantity, e.g. power P, or a ratio, e.g. P/A,
where A is area, xref is here supposed to have a fixed
value, e.g. 1 mW, 1 W, 1 µW/m2, 20 µPa, 1 µV/m.
The level representing the quantity x with reference quantity
xref may be indicated by the quantity symbol:
Lx (with respect to xref), and may be
expressed in decibels, when the reference quantity is a power, or a quantity
linked to power, in a well defined way.
Example:
The statement that the level of a certain power, P, is 15 dB above the
level corresponding to 1 W can be written:
LP (with respect to 1 W) = 15 dB, which means 10 lg
(P/1 W) = 15[3])
The "absolute power level" corresponds to the ratio of P and a reference
power, e.g. 1 W.
If P = 100 W and the reference power 1 W, we obtain:
LP = 10 lg (P/1 W) dB
= 10 lg (100 W/1 W) dB
= 20 dB
with the condensed notation 20 dB(1 W) or 20 dBW, dBW being the abbreviation
for: dB(1 W). With the reference power 1 mW and P = 100 W we obtain 50
dB(1 mW), or with the special notation mentioned earlier. 50 dBm, being the
abbreviation for: dB(1 mW). The notations dBW and dBm are currently used in the
CCIR and the CCITT. See § A.8 below.
The logarithmic expression corresponds to the ratio of P/[Delta]f
(where [Delta]f denotes a bandwidth) and a reference quantity, e.g. 1
mW/kHz. P may be a noise power. The logarithm will in this case, as in
all other cases, be taken of a pure number.
An example with a condensed notation is 7 dB(mW/kHz) or that which is the same
thing: 7 dB(W/MHz) or 7 dB(µW/Hz).
The logarithmic expression corresponds to the ratio of P/A,
where A is area, and a reference power density, e.g. 1 W/m2.
A notation in a certain case can be:
-40 dB(W/m2)
or -10 dB(mW/m2).
The logarithmic expression corresponds to the ratio of P/T,
where T is temperature, and a reference power density, e.g. 1 mW/K,
where K is kelvin.
An example is: 45 dB(mW/K)
or: 15 dB(W/K).
The logarithmic expression corresponds to the ratio of P/(A .
[Delta]f) and a reference density e.g. 1 W/(m2 . Hz).
An example is: -18 dB(W/(m2 · Hz))
or: -18 dB(W · m-2 · Hz-1).
A variant sometimes used is, dB(W/(m2 · 4 kHz)).
The strength of an electromagnetic field can be expressed by a power
flux-density (P/A), by an electric field strength E or by
a magnetic field-strength H. The field-strength level
LE is the logarithm of the ratio of E and a reference
field-strength, usually 1 µV/m.
An example with a condensed notation is:
LE = 5 dB(µV/m).
As the power carried by an electromagnetic field is linked to the square of
the field strength, this notation means:
20 lg E(µV/m) = 5.
The level corresponds to the ratio of sound pressure and a reference pressure,
often 20 µPa.
Example: 15 dB(20 µPa).
As acoustic power is linked to the square of sound pressure, this means:
20 lg (p/20 µPa) = 15[4])
This is either the ratio of the signal power (Ps) to the
noise power (P0), or the ratio of the signal voltage
(Us) to the noise voltage (Un) measured at
a given point with specified conditions. It is, expressed in decibels:
This is either the ratio of the wanted signal power (Pu) to
the maximum permissible interfering signal power (Pi), or the
ratio of the wanted signal field-strength (Eu) to the maximum
permissible interfering signal field-strength (Ei). It is
expressed in decibels:
This is the ratio Pc/(Pn/[Delta]f) -
where Pc is the carrier power, Pn the noise
power, [Delta]f the corresponding frequency bandwidth. This ratio has a
dimension of frequency, it cannot be expressed without caution in terms of
decibels, for power is not linked with frequency on a well-defined basis.
This ratio could be expressed in relation with a reference quantity such as 1
W/(W/Hz) which clearly indicates the origin of the result.
For example, with Pc = 2 W, Pn = 20 mW,
and [Delta]f = 1 MHz, for the logarithmic expression corresponding to
C/N0 we have:
10 lg = 50 dB (W/(W/kHz))
This expression is abbreviated to read 50 dB(kHz) which should however be
avoided if it is liable to give rise to any misunderstanding.
The figure of merit (M) characterizing a receiving radio station is a
logarithmic expression which is related to the antenna gain G (in
decibels) and the overall noise temperature T (in kelvins) in the
following way:
M = dB (W/(W . K))
The decibel notation may be abbreviated to read dB(K-1) which
should however be avoided if it is liable to give rise to misunderstanding.
Examples of special notations, the use of which may be continued are given
below. These notations are often made in addition to other notations.
For absolute power level (see Appendix I, § I.1.1)
dBW: absolute power level with respect to 1 watt, expressed in decibels;
dBm: absolute power level with respect to 1 milliwatt, expressed in
decibels;
dBm0: absolute power level with respect to 1 milliwatt, expressed in decibels,
referred to a point of zero relative level;
dBm0p: absolute psophometric power level (weighted for telephony) with respect
to 1 milliwatt, expressed in decibels, referred to a point of zero relative
level;
dBm0s: absolute power level with respect to 1 milliwatt, expressed in decibels,
referred to a point of zero relative level in sound programme transmission;
dBm0ps: absolute psophometric power level (weighted for sound-programme
transmission) with respect to 1 milliwatt, expressed in decibels, referred to a
point of zero relative level in sound programme transmission.
For absolute level of an electromagnetic field (see Appendix I, §
I.2.1):
dBµ or dBu: absolute level of the electromagnetic field with respect to
1µV/m, expressed in decibels.
For absolute voltage level including the audio-frequency noise level
(see Appendix I, §§ I.2.2 and I.2.3):
dBu: absolute voltage level with respect to 0.775 V, expressed in decibels;
dBu0: absolute voltage level with respect to 0.775 V, referred to a point of
zero relative level;
dBu0s: absolute voltage level with respect to 0.775 V, referred to a point of
zero relative level in sound-programme transmission;
dBqps: absolute weighted voltage level measured according to Recommendation
468, CCIR Volume X-1, in sound-programme transmission;
dBq0ps: absolute weighted voltage level measured according to Recommendation
468, CCIR Volume X-1, referred to a point of zero relative level in
sound-programme transmission;
dBq0s: absolute unweighted voltage level measured according to Recommendation
468, CCIR Volume X-1, in sound-programme transmission with respect to 0.775 V
referred to a point of zero relative level.
For relative power level (see Appendix I, § I.1.2):
dBr: decibels (relative);
For relative voltage level in sound-programme transmission (see
Appendix I, § I.2.4):
dBrs: relative power level expressed in decibels, referred to another point in
sound-programme transmission.
For absolute acoustic pressure level:
dBA (or dBB, dBC): weighted acoustic pressure level with respect to 20
µPa, mentioning the weighting curve used (curves A, B or C, see IEC
Publication 123).
For antenna gain in relation to an isotropic antenna:
dBi.
For antenna gain in relation to a half-wave dipole:
dBd.
Note 1 - In the case of the ratio "energy per bit to spectral noise
density", E/N0, which is used in digital transmission,
the ratio is made between two quantities homogeneous with spectral power
density, and this ratio may normally be expressed in decibels, like power
ratios (see § A.1 above). However, it is necessary to ensure that the
units used for the expression of both terms in the ratio are equivalent: for
example, joule (J) for energy and watts per hertz (W/Hz) for spectral noise
density.
Note 2 - Appendix I gives the principles for the use of the term
decibel in telecommunication.
The examples given in the present Recommendation are illustrations of these
principles.
Note 3 - In Appendix II is given the principle of the notation
recommended by the IEC for expressing the level of a quantity with respect to a
specified reference. The notations used in the present Recommendation are
applications of this principle.
The absolute power level is the ratio, generally expressed in decibels, between
the power of a signal at a point in a transmission channel and a specified
reference power.
It should be specified in every case whether the power is real or apparent.
It is necessary for the reference power to be indicated by a symbol:
- when the reference power is one watt, the absolute power level is expressed
in "decibels relative to one watt" and the symbol "dBW" is used;
- when the reference power is one milliwatt, the absolute power level is
expressed in "decibels relative to one milliwatt" and the symbol "dBm" is used.
The relative power level is the ratio, generally expressed in decibels, between
the power of a signal at a point in a transmission channel and the same power
at another point in the channel chosen as reference point, generally at the
origin of the channel.
It should be specified in every case whether the power is real or apparent.
Unless otherwise specified, the relative power level is the ratio of the power
of a sinusoidal test signal (at 800 or 1000 Hz) at a point in the channel to
the power of that signal at the transmission reference point.
In the old transmission plan, the CCITT had defined "the zero relative-level
point" as being the two-wire origin of a long distance circuit (point 0 of
Figure I-1/B.12).
In the presently recommended transmission plan the relative level should be
-3.5 dBr at the virtual switching point on the sending side of a four-wire
international circuit (point V of Figure I-2/B.12). The "transmission reference
point" or "zero relative level point" (point T of Figure I-2/B.12) is a virtual
two-wire point which would be connected to V through a hybrid transformer
having a loss of 3.5 dB. The conventional load used for the computation of
noise on multi-channel carrier systems corresponds to an absolute mean power
level of -15 dBm at point T.
If a measuring signal with an absolute power level LM (in
dBm) is applied at point T, the absolute power level of signal appearing at a
point X, where the relative level is LXR (in dBr), will be
LM + LXR (in dBm).
Conversely, if a signal at X has an absolute power level LXA
(in dBm), it is often convenient to "refer it to zero relative level point" by
computing L0 (in dBm0) by the formula:
Definition: Quotient of a power by another quantity, for example, an
area, a bandwidth, a temperature.
Note 1 - The quotient of a power by an area is called "power
flux-density" ("puissance surfacique") and is commonly expressed in "watts
per square metre" (symbol: W . m-2 or W/m2).
The quotient of a power by a frequency bandwidth is called "power spectral
density" and can be expressed in "watts per hertz" (symbol: W .
Hz-1 or W/Hz). It can also be expressed with a unit involving a
bandwidth characteristic of the technique concerned, for example, 1 kHz or 4
kHz in analogue telephony, 1 MHz in digital transmission or in television; the
power spectral density is then expressed in "watts per kilohertz" (W/kHz) or in
"watts per 4 kHz" (W/4 kHz) or even in "watts per megahertz" (W/MHz).
The quotient of a power by a thermo-dynamic temperature, used particularly in
the case of noise powers, has no specific name. It is usually expressed as
"watts per kelvin" (symbol: W . K-1 or W/K).
Note 2 - In some cases a combination of several types of power
densities can be used, for example a "spectral power flux-density" which is
expressed as "watts per square metre and per hertz" (symbol: W . m-2
. Hz-1 or W/(m2 . Hz)).
Definition: Expression in logarithmic form, usually in decibels, of the
ratio between the power density at a given point and a reference power
density.
Note - For example, if one watt per square metre is chosen as the
reference power flux-density, the absolute power flux-density levels are
expressed as "decibels with respect to one watt per square metre" (symbol:
dB(W/m2)).
Similarly, if one watt per hertz is chosen as the spectral reference power
density, the absolute spectral power density levels are expressed as "decibels
with respect to one watt per hertz" (symbol: dB(W/Hz)).
If one watt per kelvin is chosen as the reference for power density per unit
temperature, the absolute power density levels per temperature unit are
expressed as "decibels with respect to one watt per kelvin" (symbol:
dB(W/K)).
This notation can easily be extended to combined densities. For example, the
absolute spectral density levels of the flux-density are expressed as "decibels
with respect to one watt per square metre and per hertz" for which the symbol
is: dB(W/(m2 . Hz)).
Current practice has led to an extension of the use of the term decibel to
ratios of quantities which are only indirectly connected with power or which
are linked to it through the medium of a third quantity. In these various
cases, the decibel should be used with the utmost precaution and should always
be accompanied by a note indicating the conventions adopted and the sphere of
validity of this usage.
A case extremely common in practice, is where the ratio of two powers
P1 and P2 depends solely on the ratio of the
values X1 and X2 of another quantity
X by an equation in the form:
The electromagnetic field set up by a transmitter is of concern to some
services. At considerable distances from the antenna this field is generally
defined by its electric component E, for which it is often convenient to
use a logarithmic scale.
For a non-guided wave propagated in a vacuum, or in practice in the
atmosphere, there is a clearly defined relationship between the electric
field E and the power flux-density p:
The absolute voltage level is the ratio, generally expressed in decibels, of
the voltage of a signal at a point in a transmission channel to a specified
reference voltage.
The nature of the voltage in question, e.g. r.m.s. value, should be specified
in every case.
A reference voltage with an r.m.s. of 0.775 volts is generally adopted which
corresponds to a 1 milliwatt power dissipated in a resistance of 600 ohms,
since 600 ohms represents a rough approximation to the characteristic impedance
of certain balanced telephone lines.
I.2.2.1 If the impedance at the terminals of which the voltage
U1 is measured, is in fact 600 ohms, the absolute voltage level
thus defined, corresponds to the absolute power level with respect to 1
milliwatt, and so the number N exactly represents the level in decibels
with respect to 1 milliwatt (dBm).
I.2.2.2 If the impedance at the terminals of which the voltage
U1 is measured, is R ohms, N equals the number of
dBm increased by the quantity 10 log (R/600).
Measurement of audio-frequency noise in broadcasting, sound recording or
sound-programme transmission is made, normally through a weighting network and
by following the quasi-peak value method of Recommendation 468 using a
reference voltage of 0.775 volt at 1 kHz and a nominal impedance of 600 ohms
and expressing the results normally in dBqp.
Note - The two notations in "dBq" and "dBm" should not be used
interchangeably. In sound-programme transmission the notation "dBq" is
restricted to level measurements of noise with single or multiple tone bursts
whereas the notation "dBm" only applies to sinusoidal signals used for lining
up the circuit.
The relative voltage level at a point in a sound-programme transmission chain is the ratio, expressed in dB, of the voltage level of a signal at that point relative to the voltage level of the same signal at the reference point. This ratio is expressed in "dBrs", the "r" indicating "relative level" and "s" indicating that the ratio refers to levels in a "sound-programme" system. At the reference point (the point of zero relative level, 0 dBrs) a test signal at the alignment level (see Recommendation 645, CCIR Volume X-1, has a level of 0 dBu. Note that in some broadcasting chains, there may be no point of zero relative level. However, points of measurements and interconnection may still be given a level (in dBrs) relative to hypothetical reference point.
In certain spheres such as audio frequencies, the concept of voltage is
sometimes more important than that of power. This is the case, for example,
when low output- and high input-impedance two-port networks are associated in
tandem. In this way a deliberate departure is made from the impedance matching
conditions in order to simplify the formation of these networks. When this is
done, only the voltage ratios at different points in the link need to be taken
into consideration.
It is then convenient to express these voltage ratios in a logarithmic scale,
e.g. to the base 10, by defining the number N of corresponding units by
means of the equation:
N = K lg
In this equation the coefficient K is a priori arbitrary.
However, by analogy with the operation:
N = 20 lg
which expresses in decibels the ratio of the I2R loss
as in two equal resistances at the terminals of which the voltages
U1 and U2 respectively, are applied, one is
led to adopt the value 20 for the coefficient K. The number N
then expresses in decibels the power ratios which would correspond to the
voltage ratios, if the latter were applied to equal resistances, although in
practice this is not generally the case.
If the impedance at the terminals of which the voltage is measured is not
specified, the corresponding power level cannot be calculated. However, a
number N can be defined conventionally in accordance with § I.3.1
with respect to a reference voltage and can be expressed in decibels. To avoid
any confusion, it is essential to specify that an absolute voltage level is
concerned and the symbol dBu must be used. The symbol dBu appears to create no
confusion with the use defined in § I.2.1 as the absolute level of the
electromagnetic field referred to 1 microvolt per metre. If, however, there is
any risk of confusion, the expression dB (775 mV) must be written, at least the
first time.
A similar text will be submitted to the
CCIR as a revision of Recommendation 574-2.
[2)] In this Recommendation, the notation
letter lg is used for the decimal logarithm in accordance with ISO 31 (Part XI)
and usage within the IEC (Publication 27-3). The notation log10 is
also used within ISO and the IEC.
[3)] In the ratio (P/1 W), it is evident
that both powers must be expressed in the same units.
[4)] In the ratio (p/20 µPa), it is
evident that both sound pressures must be expressed in the same units.
[5)] The omission of the reference level,
permitted by the IEC, is not permitted in CCIR and CCITT texts.