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Decibel

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Decibel
This article is about the unit of level. For other uses, see
Decibel (disambiguation).The
decibel
(
dB
) is a logarithmic unit used to expressthe ratio between two values of a physical quantity, of-ten power or intensity. One of these quantities is often a
reference value, and in this case the decibel can be usedto express the absolute level of the physical quantity, asin the case of sound pressure. The number of decibelsis ten times the logarithm to base 10 of the ratio of twopower quantities,
[1]
or of the ratio of the squares of twoﬁeldamplitudequantities. Onedecibelisonetenthofone
bel
, named in honor of Alexander Graham Bell. The belis seldom used without the
deci-
preﬁx.The deﬁnition of the decibel is based on the measure-ment practices in telephony of the early 20th century inthe Bell System in the United States. Today, the unit isused for a wide variety of measurements in science andengineering, most prominently in acoustics, electronics,
and control theory. In electronics, the gains of ampliﬁers,
attenuation of signals, and signal-to-noise ratios are often
expressed in decibels. The decibel confers a number ofadvantages, such as the ability to conveniently representvery large or small numbers, and the ability to carry outmultiplication of ratios by simple addition and subtrac-tion.A change in power by a factor of 10 corresponds to a 10dB change in level. A change in power by a factor of twoapproximately corresponds to a 3 dB change. A changein voltage by a factor of 10 results in a change in powerby a factor of 100 and corresponds to a 20 dB change. Achange in voltage ratio by a factor of two approximatelycorresponds to a 6 dB change.The decibel symbol is often qualiﬁed with a suﬃx that in-dicates which reference quantity has been used or someother property of the quantity being measured. For ex-ample,
dBm
indicates a reference level of one milliwatt,while
dBu
is referenced to approximately 0.775 voltsRMS.
[2]
In the International System of Quantities, the decibel isdeﬁned as a unit of level or of level diﬀerence, equal toone-tenth of a bel. The bel is then deﬁned in terms ofthe neper, an alternative unit of level of root-power quan-tities, applicable when the natural logarithm (base
e
) isused to deﬁne the level.
[3]
1 History
The decibel srcinates from methods used to quantifysignal losses in telephone circuits. These losses weresrcinally measured in units of
Miles of Standard Cable
(MSC), where 1 MSC corresponded to the loss of powerover a 1 mile (approximately 1.6 km) length of standardtelephonecableatafrequencyof5000radianspersecond
(795.8 Hz), and roughly matched the smallest attenuationdetectable to the average listener. Standard telephone ca-ble was deﬁned as “a cable having uniformly distributedresistance of 88 ohms per loop mile and uniformly dis-tributed shunt capacitance of .054 microfarad per mile”(approximately 19 gauge).
[4]
The
transmission unit
(TU) was devised by engineers ofthe Bell Telephone Laboratories in the 1920s to replacetheMSC.1TUwasdeﬁnedastentimesthebase-10loga-rithmoftheratioofmeasuredpowertoareferencepowerlevel.
[5]
The deﬁnitions were conveniently chosen suchthat 1 TU approximately equaled 1 MSC (speciﬁcally,1.056 TU = 1 MSC).The threshold of hearing is 25 dB
[6]
In 1928, the Bell system renamed the TU the decibel,
[7]
being one tenth of a newly deﬁned unit for the base-10logarithm of the power ratio. It was named the
bel
, inhonor of their founder and telecommunications pioneerAlexander Graham Bell.
[8]
The bel is seldom used, as thedecibel was the proposed working unit.
[9]
The naming and early deﬁnition of the decibel is de-scribed in the NBS Standard’s Yearbook of 1931:
[10]
Since the earliest days of the telephone,the need for a unit in which to measure thetransmission eﬃciency of telephone facilitieshas been recognized. The introduction of ca-ble in 1896 aﬀorded a stable basis for a con-venient unit and the “mile of standard” cablecame into general use shortly thereafter. Thisunit was employed up to 1923 when a new unitwas adopted as being more suitable for moderntelephone work. The new transmission unit iswidely used among the foreign telephone orga-nizations and recently it was termed the “deci-bel”atthesuggestionoftheInternationalAdvi-sory Committee on Long Distance Telephony.The decibel may be deﬁned by the state-ment that two amounts of power diﬀer by 1decibel when they are in the ratio of 10
0.1
andany two amounts of power diﬀer by N deci-bels when they are in the ratio of 10
N(0.1)
. The1
2
2 DEFINITION
numberoftransmissionunitsexpressingthera-tio of any two powers is therefore ten times thecommon logarithm of that ratio. This methodof designating the gain or loss of power in tele-phone circuits permits direct addition or sub-traction of the units expressing the eﬃciencyof diﬀerent parts of the circuit...
1.1 Standards
In April 2003, the International Committee for Weightsand Measures (CIPM) considered a recommendation forthe decibel’s inclusion in the International System ofUnits (SI), but decided not to adopt the decibel as an SIunit.
[11]
However, the decibel is recognized by other in-ternational bodies such as the International Electrotech-nical Commission (IEC) and International Organizationfor Standardization (ISO).
[12]
The IEC permits the useof the decibel with ﬁeld quantities as well as power andthis recommendation is followed by many national stan-dards bodies, such as NIST, which justiﬁes the use of thedecibel for voltage ratios.
[13]
The term
ﬁeld quantity
isdeprecated by ISO, which favors root-power. In spite oftheir widespread use, suﬃxes (as in dBA or dBV) are notrecognized by the IEC or ISO.
2 Deﬁnition
The decibel (dB) is one tenth of the bel (B): 1B = 10dB.The bel is (1/2) ln(10) nepers.The bel represents a ratio between two power quanti-ties of 10:1, and a ratio between two ﬁeld quantitiesof √10:1.
[14]
A
ﬁeld quantity
is a quantity such as volt-age, current, pressure, electric ﬁeld strength, velocity, orcharge density, the square of which in linear systems isproportional to power.
[15]
A
power quantity
is a poweror a quantity directly proportional to power, e.g., energydensity, acoustic intensity and luminous intensity.The method of calculation of a ratio in decibels dependson whether the measured property is a
power quantity
ora
ﬁeld quantity
.Two signals that diﬀer by one decibel have a power ratioof
10
110
which is approximately 1.25892, and an ampli-tude (ﬁeld) ratio of
√
10
110
(1.12202).
[16][17]
Although permissible, the bel is rarely used with other SIunit preﬁxes than
deci
. It is preferred to use
hundredths of a decibel
rather than
millibels
.
[18]
2.1 Conversions
ThebelisdeﬁnedbyISOStandard80000-3:2006as(1/2)ln(10) nepers (Np), where ln denotes the natural loga-
rithm. Because the decibel is one tenth of a bel, it followsthat 1 dB = (1/20) ln(10) Np. The same standard deﬁnes1 Np as equal to 1 (thereby relating all of the units asnondimensional natural log of ﬁeld-quantity ratios, 1 dB= 0.11513..., 1 B = 1.1513...). Since logarithm diﬀer-encesmeasuredintheseunitsareused to representpowerratios and ﬁeld ratios, the values of the ratios representedby each unit are also included in the table.
2.2 Power quantities
When referring to measurements of
power
or
intensity
, aratio can be expressed in decibels by evaluating ten timesthe base-10 logarithm of the ratio of the measured quan-titytothereferencelevel. Thus,theratioofapowervalue
P
1
to another power value
P
0
is represented by
L
B, thatratio expressed in decibels,
[19]
which is calculated usingthe formula:
L
dB
= 10
log
10
P
1
P
0
Thebase-10logarithmoftheratioofthetwopowerlevelsisthenumberofbels. Thenumberofdecibelsistentimesthenumberofbels(equivalently, a decibelisone-tenthofabel).
P
1
and
P
0
mustmeasurethesametypeofquantity,andhavethesameunitsbeforecalculatingtheratio. If
P
1
=
P
0
in the above equation, then
L
B = 0. If
P
1
is greaterthan
P
0
then
L
Bispositive; if
P
1
islessthan
P
0
then
L
Bis negative.Rearranging the above equation gives the following for-mula for
P
1
in terms of
P
0
and
L
B:
P
1
= 10
L
dB
10
P
0
2.3 Field quantities
When referring to measurements of ﬁeld
amplitude ,
it isusual to consider the ratio of the squares of
A
1
(measuredamplitude)and
A
0
(referenceamplitude). Thisisbecausein most applications power is proportional to the squareof amplitude, and it is desirable for the two decibel for-mulations to give the same result in such typical cases.Thus, the following deﬁnition is used:
L
dB
= 10
log
10
A
21
A
20
= 20
log
10
A
1
A
0
.
The formula may be rearranged to give
A
1
= 10
L
dB
20
A
0
Similarly, in electrical circuits, dissipated power is typi-callyproportionaltothesquareofvoltageorcurrentwhen
3the impedance is held constant. Taking voltage as an ex-ample, this leads to the equation:
G
dB
= 20
log
10
V
1
V
0
where
V
1
is the voltage being measured,
V
0
is a speciﬁedreference voltage, and
G
B is the power gain expressed indecibels. A similar formula holds for current.The term
root-power quantity
is introduced by ISO Stan-dard 80000-1:2009 as a substitute of
ﬁeld quantity
. Theterm
ﬁeld quantity
is deprecated by that standard.
2.4 Examples
All of these examples yield dimensionless answers in dBbecausetheyarerelativeratiosexpressedindecibels. Theunit dBW is often used to denote a ratio for which thereferenceis1W,andsimilarlydBmfora1mWreferencepoint.
ã
Calculating the ratio of 1 kW (one kilowatt, or 1000watts) to 1 W in decibels yields:
G
dB
= 10
log
10
1000
W
1
W
≡
30
dB
ã
The ratio of
√
1000
V
≈
31
.
62
V to
1
V in decibelsis
G
dB
= 20
log
10
31
.
62
V
1
V
≡
30
dB
(31
.
62
V
/1
V
)
2
≈
1
kW
/1
W , illustrating the conse-quence from the deﬁnitions above that
G
dB
has the samevalue,
30
dB , regardless of whether it is obtained frompowers or from amplitudes, provided that in the speciﬁcsystem being considered power ratios are equal to ampli-tude ratios squared.
ã
The ratio of 1 mW (one milliwatt) to 10 W in deci-bels is obtained with the formula
G
dB
= 10
log
10
0
.
001
W
10
W
≡−
40
dB
ã
The power ratio corresponding to a 3 dB change inlevel is given by
G
= 10
310
×
1 = 1
.
99526
...
≈
2
A change in power ratio by a factor of 10 is a change of10 dB. A change in power ratio by a factor of two is ap-proximately a change of 3 dB. More precisely, the factoris 10
3/10
, or 1.9953, about 0.24% diﬀerent from exactly2. Similarly, an increase of 3 dB implies an increase involtagebyafactorofapproximately
√
2
,orabout1.41,anincrease of 6 dB corresponds to approximately four timesthepowerandtwicethevoltage, andsoon. Inexacttermsthe power ratio is 10
6/10
, or about 3.9811, a relative errorof about 0.5%.
3 Properties
The decibel has the following properties:
ã
The logarithmic scale nature of the decibel meansthat a very large range of ratios can be representedby a convenient number, in a similar manner toscientiﬁc notation. This allows one to clearly visu-alize huge changes of some quantity. See Bode plotand semi-log plot. For example, 120 dB SPL maybe clearer than a “a trillion times more intense thanthe threshold of hearing”, or easier to interpret than“20 pascals of sound pressure”.
ã
Levelvaluesindecibelscanbeaddedinsteadofmul-tiplying the underlying power values, which meansthat the overall gain of a multi-component system,suchasaseriesofampliﬁerstages, canbecalculatedby summing the gains in decibels of the individualcomponents, rather than multiply the ampliﬁcationfactors; that is, log(A × B × C) = log(A) + log(B)+ log(C). Practically, this means that, armed onlywith the knowledge that 1 dB is approximately 26%power gain, 3 dB is approximately 2× power gain,and 10 dB is 10× power gain, it is possible to de-termine the power ratio of a system from the gainin dB with only simple addition and multiplication.For example:A system consists of 3 ampliﬁers inseries, with gains (ratio of powerout to in) of 10 dB, 8 dB, and 7dB respectively, for a total gain of25dB.Brokenintocombinationsof10, 3, and 1 dB, this is:25 dB = 10 dB + 10 dB +3 dB + 1 dB + 1 dBWith an input of 1 watt, the outputis approximately1 W x 10 x 10 x 2 x 1.26x 1.26 = ~317.5 WCalculated exactly, the output is 1W x 10
25/10
= 316.2 W. The ap-proximatevaluehasanerrorofonly+0.4% with respect to the actualvalue which is negligible given theprecision of the values supplied andthe accuracy of most measurementinstrumentation.
4
5 USES
4 Advantages and disadvantages
4.1 Advantages
ã
According to Mitschke,
[20]
“The advantage of usingalogarithmicmeasureisthatinatransmissionchain,there are many elements concatenated, and each hasitsowngainorattenuation. Toobtainthetotal,addi-tion of decibel values is much more convenient thanmultiplication of the individual factors.”
ã
The human perception of the intensity of, for exam-ple, sound or light, is more nearly linearly related tothe logarithm of intensity than to the intensity itself,per the Weber–Fechner law, so the dB scale can beuseful to describe perceptual levels or level diﬀer-ences.
4.2 Disadvantages
According to several articles published in
Electrical En- gineering
[21]
and the
Journal of the Acoustical Society of America
,
[22][23][24]
the decibel suﬀers from the followingdisadvantages:
ã
The decibel creates confusion.
ã
The logarithmic form obscures reasoning.
ã
Decibels are more related to the era of slide rulesthan that of modern digital processing.
ã
Decibels are cumbersome and diﬃcult to interpret.
ã
Representing the equivalent of zero watts is not pos-sible, causing problems in conversions.Hickling concludes “Decibels are a useless aﬀectation,which is impeding the development of noise control asan engineering discipline”.
[23]
Anotherdisadvantageisthatquantitiesin decibelsarenotnecessarily additive,
[25][26]
thus being “of unacceptableform for use in dimensional analysis .
[27]
For the same reason that decibels excel at multiplicativeoperations (e.g., antenna gain), they are awkward whendealing with additive operations. Peters (2013, p. 13)
[28]
provides several examples:
ã
“iftwomachineseachindividuallyproducea[soundpressure] level of, say, 90 dB at a certain point, thenwhen both are operating together we should expectthe combined sound pressure level to increase to 93dB, but certaintly not to 180 dB!
ã
“supposedthatthenoisefromamachineismeasured(including the contribution of background noise)and found to be 87 dBA but when the machine isswitchedoﬀthebackgroundnoisealoneismeasuredas 83 dBA. ... the machine noise [level (alone)] maybe obtained by 'subtracting' the 83 dBA backgroundnoise from the combined level of 87 dBA; i.e., 84.8dBA.”
ã
“in order to ﬁnd a representative value of the soundlevel in a room a number of measurements are takenat diﬀerent positions within the room, and an aver-age value is calculated. (...) Compare the logarith-mic and arithmetic averages of ... 70 dB and 90 dB:logarithmic average = 87 dB; arithmetic average =
80 dB.”
5 Uses
5.1 Acoustics
The decibel is commonly used in acoustics as a unitof sound pressure level, for a reference pressure of 20micropascals in air
[29]
and 1 micropascal in water. Thereference pressure in air is set at the typical threshold ofperception of an average human and there are commoncomparisons used to illustrate diﬀerent levels of soundpressure. Sound pressure is a ﬁeld quantity, therefore the
ﬁeld version of the unit deﬁnition is used:
L
p
= 20
log
10
p
rms
p
ref
dBwhere
p
ᵣₑ is equal to the standard referencesound pressure level of 20 micropascals in airor 1 micropascal in water.The human ear has a large dynamic range in audio recep-tion. The ratio of the sound intensity that causes perma-nent damage during short exposure to the quietest soundthat the ear can hear is greater than or equal to 1 tril-lion (10
12
).
[30]
Such large measurement ranges are con-veniently expressed in logarithmic scale: the base-10 log-arithm of 10
12
is 12, which is expressed as a sound pres-sure level of 120 dB re 20 micropascals. Since the hu-man ear is not equally sensitive to all sound frequencies,noise levels at maximum human sensitivity, somewherebetween 2 and 4 kHz, are factored more heavily intosomemeasurementsusingfrequencyweighting. (SeealsoStevens’ power law.)Further information: Examples of sound pressure andsound pressure levels
5.2 Electronics
In electronics, the decibel is often used to express poweroramplituderatios(gains), inpreferencetoarithmeticra-
tios or percentages. One advantage is that the total deci-bel gain of a series of components (such as ampliﬁersandattenuators)canbecalculatedsimplybysummingthe

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