User:Boute/Decibel draft

Defining bel (B) and decibel (dB) --- draft, version 0

Preliminary remarks The literature on this topic is highly nonuniform, and reconciling the strict usages covered by the standards with the plethora of variants in engineering practice is a balancing act. Therefore referencing to source material will by necessity be unusually detailed, and most formulas below will reflect the common structure rather than the different variable conventions tailored to specific application areas. Also, in the interest of readers used to avoiding mathematics, it must be noted that for understanding the decibel, even at only an intuitive level, elementary notions of exponents and logarithms at the same level are indispensable, since they essentially reflect the same principles. Although there is in general no royal road to mathematics, for exponents and logarithms some sources easily available via the web come very close. The explanations below proceed gradually.

Introduction: informal definition and ramifications

The International System of Units (SI), categorizes bel, decibel and neper "outside the SI", and contains a significant cautionary remark: "some scientists consider that they should not even be called units" ( p. 127, p. 35). Still, as explained in, they can be given unit-like properties.

This introduction reflects the common aspects of standards and of informal usage in typical engineering practice and textbooks, but for the sake of clarity the numerous abuses of notation are avoided by omission.

The bel and the decibel are best understood as notations for conveniently representing and handling (positive) numbers that can be very large and very small. Informally,

A positive number $y$ is represented by $(\lg {\text{ }}y){\text{ B}}$, usually written $(10{\text{ }}\lg {\text{ }}y){\text{ dB}}$.

Equivalently, $10^{x}$ (any real $x$) is represented by $x{\text{ B}}$, usually written $(10{\text{ }}x){\text{ dB}}$.

Examples: $30{\text{ dB}}$ and $3{\text{ B}}$ represent 1000 whereas ${-30}{\text{ dB}}$ and ${-3}{\text{ B}}$ represent 1/1000. The following table shows a few additional values (those with decimal point are approximate). To the negative dB-values correspond the inverse numbers, for instance, $-3{\text{ dB}}$ represents 1/2 (approximately).

dB-value	0	1	2	3	4	5	6	7	8	9	10	20	30
represents	1	1.26	1.59	2.00	2.51	3.16	3.98	5.01	6.31	7.94	10	100	1000

The most important aspect to understand is that a number represented in decibel grows exponentially with the number of decibels. For instance, every increase with 3 dB doubles the number, and every increase with 10 dB multiplies the number by 10. More generally, the number represented by $(x+x'){\text{ dB}}$ is the product of the numbers represented by $x{\text{ dB}}$ and $x'{\text{ dB}}$.

Remarks

• In general, the decibel representation is just a correspondence, nothing more. As explained in more detail later with reference to the standards, the correspondence can be expressed in formula form as

$L_{y}=(\lg {\text{ }}y){\text{ B}}=(10{\text{ }}\lg {\text{ }}y){\text{ dB}}$.

By the properties of the logarithm, $L_{yy'}=(10{\text{ }}\lg {\text{ }}y+10{\text{ }}\lg {\text{ }}y'){\text{ dB}}$.

• Typically, $y$ is a ratio of so-called power quantities ( p. 37) of the form $y:=P_{1}/P_{0}$. However, in engineering practice the use is far more extensive: $y$ may be a temperature ratio, a frequency ratio, or anything where widely varying magnitudes are expected and dB-representation is useful.

• For representing absolute physical quantities, a reference quantity can be used as the denominator in the ratio. Conventions for expressing this are given by the following examples borrowed from the IEC standard(, page 19), assuming a power quantity $P$.

Recommended: $L_{P/1W}=x{\text{ dB}}$ (an inconsistent alternative is omitted here).

Allowed: dB (1 W), the space after dB and the parentheses being obligatory.

To be avoided (according to the cited standard): dB(1 W), dB (W), dB(W), dBW. Yet, dBW is common in engineering.

• Most people have heard about decibel mainly in conjunction with sound. It may be disappointing that such a topic of general interest cannot be covered satisfactorily in a few words, the reason being that apart from physics (pressure, power) it involves physiological issues (human perception) that must be taken into account in defining suitable variants of the decibel. Still, one can find initial information about this specific application in the web, in Wikipedia under Sound pressure and also below.

• Insofar as the examples show, the decibel can be used without knowing exactly what it means (or what $L_{y}$ in the formula is). This important aspect, extensively discussed in, explains why common engineering practice (discussed later) rarely refers to or even uses a precise definition.

The decibel in the context of standards

The International System of Units (SI), presents a rigorous definition for B (bel) and dB (decibel) in a small footnote which, up to a choice of variables to reflect more general uses, amounts to

$L_{y}=x{\text{ B}}=(10{\text{ }}x){\text{ dB}}$ means $x=\lg {\text{ }}y$ or, equivalently, $y=10^{x}$. Here and in all further formulas, $y$ is a positive real number and $x$ any real number.

Examples: $L_{100}=2{\text{ B}}=20{\text{ dB}}$ and $L_{1/100}=-2{\text{ B}}={-20}{\text{ dB}}$.

This formulation may appear somewhat circuitous, but many equivalent definitions in more direct form are given by various clarifying documents and papers and in other standardization documents. Up to the choice of variables, these definitions have the common form

$L_{y}=(\lg {\text{ }}y){\text{ B}}=(10{\text{ }}\lg {\text{ }}y){\text{ dB}}$.

An explicit definition of B and dB can be given via an additional definition for $L$ (or vice versa).

The convention that gives B the properties of a unit in a coherent system is the following.

${\text{L}}_{y}=\lg {\text{ }}y$ and, correspondingly, $x{\text{ B}}=x=(10{\text{ }}x){\text{ dB}}$.

This expresses the usual interpretation of the decibel.

Remarks (Mills et al.)

• If defined as a coherent dimensionless unit, B equals 1 (and dB = 1/10). In that case B can in principle be omitted in $L_{y}=(\lg {\text{ }}y){\text{ B}}$, but such units are written as "reminder labels to help the reader". If this is baffling, it may help recalling that $\sin {\text{ }}x=\sin {\text{ }}(x{\text{ rad}})$. Of course, using different labels in this manner is a convention outside mathematics and cannot enforce any mathematical difference between dimensionless coherent units with value 1.

• Values other than 1 may be useful when considering root-power quantities and in interpreting the correspondence with the neper, both discussed below.

• As seen by the earlier examples, for actual use the numerical values of B and dB are unimportant and hence "rarely required".

Use of decibel in relation to power quantities

The "native" application of the decibel is when $y$ is a ratio of power quantities. The standard terminology is as follows, assuming that $P_{0}$ is a fixed reference power quantity and $P_{1}$ and $P_{2}$ are any power quantities of the same type (dimensionality) as $P_{0}$.

$L_{P_{1}/P_{0}}$ is called the power level of $P_{1}$ relative to $P_{0}$.

$L_{P_{2}/P_{1}}$ is called the power level difference between $P_{2}$ and $P_{1}$. Here the value of $P_{0}$ need not be specified because $L_{P_{2}/P_{1}}=L_{P_{2}/P_{0}}-L_{P_{1}/P_{0}}$ irrespective of $P_{0}$.

Note: one can say that a power level or power level difference $x$ equals $(10{\text{ }}x){\text{ dB}}$, but for a given power ratio $y$ one can only say that it corresponds to $(10{\text{ }}\lg {\text{ }}y){\text{ dB}}$ in the sense that $L_{y}=(10{\text{ }}\lg {\text{ }}y){\text{ dB}}$.

Boute (talk) 15:45, 13 March 2011 (UTC)

Intermezzo: definitional arbitrariness

The concept of level appears in standards for expository purposes, but is not part of any standard. Expressions of the form $L_{y}$ rarely appear in engineering practice or textbooks. In fact, ISO 80000-3 (, p. vii) observes (quoted literally up to choice of variables):

"Generally it is not the logarithmic quantity itself, such as $L_{y}$, which is of interest; it is only the argument of the logarithm that is of interest, i.e. $y$."

This word statement is meant to convey the observation that any pairwise definition of $L_{y}$ and $x{\text{ B}}$ that satisfies

$L_{y}=x{\text{ B}}$ if and only if $x=\lg {\text{ }}y$

expresses the same correspondence between $x$ and $y$, evidently $x=\lg {\text{ }}y$ or, equivalently, $y=10^{x}$.

In general, if $L_{y}=x{\text{ B}}$ and $L_{y'}=x'{\text{ B}}$, then $L_{yy'}=(x+x'){\text{ B}}$. However, $L$ has logarithm-like properties (such as $L_{yy'}=L_{y}+L_{y}'$) only if ${\text{B}}$ is linear.

Summarizing, a statement such as "the power level is 20 dB higher" always means that the power itself is 100 times as high, without having to define power level or dB.

If in a given context various definitions of power levels and decibel are considered or related, it is helpful in mathematical calculations to distinguish them by different symbols. For instance, in some discussions about standards in the scientific literature, primes are used, as in B' and dB'. However, such a convention is for expository purposes only, since the standards themselves prohibit "decorating" symbols for units by any additional information.

Finally, the formulation $L_{y}=(\lg {\text{ }}y){\text{ B}}=(10{\text{ }}\lg {\text{ }}y){\text{ dB}}$ is general and application-neutral.

Use in relation to root-power quantities

A secondary application of the decibel concerns root-power quantities, formerly called field quantities. As defined by the ISO, "a root-power quantity is a quantity, the square of which is proportional to power when it acts on a linear system". Examples are voltage, current and pressure.

For such quantities, usually the rms (Root Mean Square) values are of interest here. If $q$ is a ratio of root-power quantities under the same loading conditions, then $q^{2}$ is the corresponding power ratio. For instance, if $V_{0}$ and $V_{1}$ are rms voltages, applying them to a resistor with resistivity $R$ leads to power dissipations given by $P_{0}=(V_{0})^{2}/R$ and $P_{1}=(V_{1})^{2}/R$, hence $p=P_{1}/P_{0}=(V_{1}/V_{0})^{2}=q^{2}$.

By a historical idiosyncrasy due to the original use of the decibel in a very narrow context (telephone cables), yet now embedded in engineering practice and in all cited standards, for a root-power ratio $q$ the level and the decibel are defined to be those used for the corresponding power ratio $p$, in the definitions assumed to be linked by $p=q^{2}$ by convention (whether or not this formula holds exactly in the considered application).

For instance, both a power ratio of 100 and a root-power ratio of 10 are expressed as 20 dB.

In practice the distinction between the interpretations is made mentally as just illustrated, but in formulas it must be kept explicit to avoid ambiguity. This can be done in a direct way by defining B' and dB' as

$x{\text{ B}}'=(x/2){\text{ B}}$ and $x{\text{ dB}}'=(x/2){\text{ dB}}$

Hence $p=q^{2}$ if and only if the following are equivalent

$L_{p}=x{\text{ B}}=(10{\text{ }}x){\text{ dB}}$

$L_{q}=x{\text{ B}}'=(10{\text{ }}x){\text{ dB}}'$.

In this manner, for the same argument $x$ these formulas yield the corresponding power and root-power ratios. For instance, the interpretation of 20 dB for power ratios is expressed by $L_{100}=20{\text{ dB}}$ and for root-power ratios by $L_{10}=20{\text{ dB}}'$.

These definitions for B' and dB' (including the disambiguating prime for expository purposes) coincide with those derived from the neper (Np) in , as discussed next.

The bel and the decibel in conjunction with the neper

The neper (Np) is defined in the same style as the bel (B) by the formula (up to variable naming)

$K_{y}=(\ln {\text{ }}y){\text{ Np}}$.

As observed in , this formula can be used independently of the similar formula for B and dB (which use $\lg$ instead of $\ln$), without ever having to establish a mutual link. When considered separately as coherent units, B and Np independently have value 1, and the label dB or Np is just added to indicate which defining formula is used.

However, in all standards considered, the neper is the (only) basic concept, whereas bel and decibel are derived concepts. The derivation is as follows (from, up to the choice of variable names).

The "native" use of the neper concerns root-power quantities, and to express secondary use of Np for power quantities $L'_{p}$ is defined by requiring that $L'_{p}=K_{q}$ in case $p=q^{2}$, hence $L'_{p}=((\ln {\text{ }}p)/2){\text{ Np}}$. A definition for the bel (B') and hence the decibel (dB') as a derived quantity is based on changing from the natural to the decimal logarithm using $\ln {\text{ }}y=(\ln {\text{ }}10)(\lg {\text{ }}y)$:

$L'_{p}=((\ln {\text{ }}p)/2){\text{ Np}}=((\lg {\text{ }}p)(\ln {\text{ }}10)/2)){\text{ Np}}=(\lg {\text{ }}p){\text{ B}}'$,

provided $B'$ is defined by

$x{\text{ B}}'=(x{\text{ }}\ln {\text{ }}10)/2){\text{ Np}}$, hence

$1{\text{ B}}'=((\ln {\text{ }}10)/2){\text{ Np}}$ and hence, approximately,

$1{\text{ B}}'=1.151293{\text{ Np}}$,

$1{\text{ dB}}'=0.1151293{\text{ Np}}$.

Observe that linearity was not assumed. Tacitly assuming linearity (as for units), further observes that this definition of B' and dB' as derived from Np is in fact the one given in the standards, where the expository prime is of course absent.

This new definition for B' (via Np) coincides with the earlier one, namely via B by $x{\text{ B}}'=(x/2){\text{ B}}$, provided

$K_{y}=(\ln {\text{ }}y){\text{ Np}}=(\lg {\text{ }}y){\text{ B}}=L_{y}$.

This correspondence has the important advantage of being application-neutral (same pattern without anomalous factors of 2). Finally, observe that

$L'_{y}=x{\text{ B}}'$ if and only if $L_{y}=x{\text{ B}}$,

which once more illustrates definitional arbitrariness, and indicates that the "leanest" formulation consists in discarding $L'$ while retaining both B and B' for use in formulas, also remembering that B' (defined via Np) is the definition given without primes in the standards. Summarizing:

$L_{y}=(\ln {\text{ }}y){\text{ Np}}=(\lg {\text{ }}y){\text{ B}}=(10{\text{ }}\lg {\text{ }}y){\text{ dB}}=(20{\text{ }}\lg {\text{ }}y){\text{ dB}}'$

$1{\text{ dB}}'=(1/2){\text{ dB}}=0.1151293{\text{ Np}}$.

The decibel in engineering practice

As already observed at various occasions, the shape of the formulas and the resulting definitional arbitrariness explain why engineering practice is not critically dependent on precise definitions of decibel, and why standards are rarely consulted or cited.

An unfortunate side-effect is the wide variety of negligent, obscure and even inconsistent notations that have gone through separate histories by being repeatedly copied and and partially patched up in different ways by various authors in the course of time. It suffices looking in textbooks and sources on the web to find numerous examples. This situation has probably contributed to the reputation of the decibel as an elusive concept.

A positive side-effect is the creative use of the decibel far beyond its narrow original application domain (telephone lines), for instance, to quantities other than root-power or power. Some of these applications, and the corresponding decibel-based notations, are listed below. Most of these are not (yet) covered by the standards, but are widely used.

These additional uses, as well as some informal yet nearly universal engineering practices around the basic (dimensionless) decibel, can be formalized in a manner that streamlines their mathematical treatment.

The decibel and the neper as nonlinear scaling

In engineering practice (as reflected in specifications, textbooks and literature), one often replaces lengthy turns of phrase such as "corresponds to" by the more concise "is", for instance, saying "The gain is 30 dB" or, in formulas, "Gain = 30 dB" (gain typically being a power ratio or a root-power ratio). Whereas this constitutes an abuse of terminology in the traditional interpretation and according to the standards, it allows a straightforward formalization that favors both intuition and symbolic calculation.

ISO 80000-3 (, p. vii) notes that, in general, one is interested not in $L_{y}$ but in $y$ itself. Hence many of the preceding complications are bypassed by directly defining $L_{y}=y$, thus simplifying all the preceding formulas that do not a priori assume linear decibel. This exploits definitional arbitrariness in a manner that is dual to $x{\text{ B}}=x$ (linear) and makes B, dB and Np into nonlinear scaling functions. The main resulting formulas, directly obtained from the preceding table, are the following.

$y=(\ln {\text{ }}y){\text{ Np}}=(\lg {\text{ }}y){\text{ B}}=(10{\text{ }}\lg {\text{ }}y){\text{ dB}}=(20{\text{ }}\lg {\text{ }}y){\text{ dB}}'$

$x{\text{ Np}}={\text{e}}^{x}$

$x{\text{ B}}=10^{x}$ and $x{\text{ dB}}=10^{x/10}$ and $x{\text{ dB}}'=10^{x/20}$

$1{\text{ dB}}'=(1/2){\text{ dB}}=0.1151293{\text{ Np}}$.

For instance, 0 dB = 1 and 20 dB = 100 and 20 dB' = 10. The inverse functions can be denoted using square brackets, for instance $[{\text{dB}}]$, read "expressed in dB", so one can write $100{\text{ }}[{\text{dB}}]=20$. Furthermore, $x{\text{ dB}}=(1{\text{ dB}})^{x}$ and $(x+x'){\text{ dB}}=(x{\text{ dB}})(x'{\text{ dB}})$ instead of the linear $(x+x'){\text{ dB}}=x{\text{ dB}}+x'{\text{ dB}}$ and $x{\text{ dB}}=x(1{\text{ dB}})$, which is mere multiplication by a constant.

Finally, since the representations in dB and Np now directly stand for the represented numbers themselves instead of for their logarithm in some arbitrary base, they can directly replace the numerical values in expressions with units. For instance, 1 kW = (30 dB) W. One can also define, for instance, $x{\text{ dBW}}=(x{\text{ dB}}){\text{ W}}$ and $x{\text{ dBV}}=(x{\text{ dB}}'){\text{ V}}$. As compared to the standard convention $L_{P/(1{\text{ W}})}=x{\text{ dB}}$ where $L$ is logarithmic, defining $L_{y}=y$ instead essentially liberates the units in the argument of $L$ to the effect that $P/(1{\text{ W}})=x{\text{ dB}}$, and 1 W can now be moved to the right hand side to yield $P=(x{\text{ dB}}){\text{ W}}$. All rules of dimensional arithmetic have become applicable to expressions with dB.

Further ramifications are discussed in.

(This concludes the draft. As was the intention, it is limited to the definitional part). Boute (talk) 14:48, 14 March 2011 (UTC)

Proposal for how to proceed from here

The preceding text is a first draft (version 0), awaiting comments and suggestions from others.

Below is version 1 under construction (initially empty), taking into account the feedback. When all subsections in version 1 are complete, version 0 will be archived and version 1 will replace it for further discussion.

The purpose of these drafts is serving as a source of material from which parts that have become more or less stable (e.g., by obtaining some degree of acceptance) can be gradually placed on the existing decibel page. Needless to say, anyone who feels that some parts are ready for posting can take the initiative. Boute (talk) 15:29, 15 March 2011 (UTC)

Defining bel (B) and decibel (dB) --- draft, version 1

(Proposed alternative title: Defining decibel and related concepts)

In the scientific literature, the decibel has been called "bewildering" and "elusive", reflecting the fact that "to the general public, most journalists and even some scientists have difficulty with the decibel". Especially in acoustics, which is just one application area, some consider the resulting confusion worse than in other fields of technology.

To alleviate such difficulties in a balanced manner, the explanations below proceed gradually, with many examples, offering various viewpoints and detailed references. Some very elementary notions of exponents and logarithms are helpful.

The decibel introduced informally

Since its origins, summarized later, the decibel has evolved from its native application (signal power in telephone systems) to much wider use for other purposes. The decibel is often designated as a "unit", but The International System of Units (SI) cautions that some scientists do not consider this term appropriate ( p. 127, p. 35). The neutral term "notation" circumvents this issue.

Intuitively, the decibel is best understood from the manner in which it is used in common engineering practice: as just a notation for conveniently representing and handling (positive) numbers whose magnitude can be very large and very small. Although most textbooks (more refs to be added) attempt giving a "definition", as explained later these definitions are inadequate and as a result not really used; instead, the manner in which the dB symbol is actually introduced is informal, often by example, and can be described in the following equivalent ways (the first for reading expressions in decibel, the second for writing them).

The expression $x{\text{ dB}}$ (any real $x$) represents the number $10^{x/10}$ (always positive).

Equivalently, a positive number $y$ is represented by the expression $(10{\text{ }}\lg {\text{ }}y){\text{ dB}}$.

The following examples illustrate the sufficiency of this informal characterization for practical use. Making more precise what "representing" or "dB" means is unnecessary (and may even be confusing) at this stage.

Examples: $30{\text{ dB}}$ represents 1000 whereas ${-30}{\text{ dB}}$ represents 1/1000. The following table shows a few additional values (numbers with decimal point are approximate). Note that $-x{\text{ dB}}$ represents the inverse of the number represented by $x{\text{ dB}}$, for instance, $-3{\text{ dB}}$ represents 1/2 (approximately).

expression	$0{\text{ dB}}$	$1{\text{ dB}}$	$2{\text{ dB}}$	$3{\text{ dB}}$	$4{\text{ dB}}$	$5{\text{ dB}}$	$6{\text{ dB}}$	$7{\text{ dB}}$	$8{\text{ dB}}$	$9{\text{ dB}}$	$10{\text{ dB}}$	$20{\text{ dB}}$	$30{\text{ dB}}$
represents	1	1.26	1.59	2.00	2.51	3.16	3.98	5.01	6.31	7.94	10	100	1000

An essential property is that the number represented by $x{\text{ dB}}$ grows exponentially with $x$. For instance, every increase of $x$ by 3 doubles the represented number, and every increase by 20 multiplies the number by 100. Thus a relative magnitude of, for instance, $10^{20}$ is compressed to 200 dB. The usefulness for calculations is illustrated later.

Writing (and reading) expressions with decibels

The main source of confusion is to "use the decibel as if it were a physical unit itself". Proper conventions for using the decibel are as follows, mainly according to the standards, but also considering common nonstandard notations in engineering.

• By definition, $x{\text{ dB}}$ represents just a (positive) number. Originally this number was a ratio (e.g., $P_{1}/P_{2}$) of so-called power quantities ( p. 37), but gradually the use encompassed temperature ratios, frequency ratios, or anything where widely varying magnitudes are expected and dB-representation is useful.

• For using the decibel to represent an absolute physical quantity $Q_{1}$, a reference quantity $Q_{0}$ is taken as the denominator in the ratio $Q_{1}/Q_{0}$, which is represented in decibels. Obviously, this reference quantity must be specified explicitly: just writing dB is meaningless. Notational conventions are shown via the following examples from the IEC standard(, page 19), assuming a power quantity for which the SI unit is W (watt).

Recommended: $L_{P/(1{\text{ W}})}=30{\text{ dB}}$, expressing that $P/(1{\text{ W}})=1000$, hence $P$ = 1 kW.

Allowed: $L_{P/{\text{W}}}=30{\text{ dB}}$ (omitting the number 1).

Allowed short form: dB (1 W), the space after dB and the parentheses being obligatory.

To be avoided (according to IEC): dB(1 W), dB (W), dB(W), dBW. However, dBW is common in engineering.

• Due to historical circumstances, for root-power quantities ( p. 37), the decibel is not used in the same way as for other quantities. According to ISO, "a root-power quantity (formerly called a field quantity) is a quantity, the square of which is proportional to power when it acts on a linear system". Typical root-power quantities are voltage, current and (sound) pressure. For instance, the power $P$ dissipated in a resistance $R$ when applying a Root Mean Square voltage $V$ is given by $P=V^{2}/R$. By convention, a root-power ratio $q$ is represented in decibel by taking its square $q^{2}$ as a surrogate, hence

A root-power ratio $q$ is represented by $(20{\text{ }}\lg {\text{ }}q){\text{ dB}}$.

Example: a voltage ratio of 10 is represented by 20 dB, and 10 dB (1 µV), commonly written 10 dBµV, represents 200 µV.

Some caution is needed: "Knowing when to multiply by 10 or 20 in such calculations is an endless source of confusion to the neophyte" and in textbooks errors directly related to this convention have also been reported. Disambiguation can be achieved by adding a prime and writing dB' to indicate that the convention for root-power quantities applies, but this is just for expository purposes and rigor in calculations: in standards and common practice, only the symbol dB is used without markings. In fact, the standards do not allow attaching anything to the symbol for a unit (an example from : one may write $U_{\text{max}}=1000{\text{ V}}$ but not $U=1000{\text{ V}}_{\text{max}}$).

Calculating with decibels

Understanding the reasons for using decibels at all (rather than alternatives such as scientific notation) depends on observing the advantages in actual calculations by various examples, as given below.

• The purpose of the decibel from the very beginning is facilitating the numerical computation of products and powers of the represented numbers, in the manner of logarithms. The decibel provides the equivalent of a slide rule for mental arithmetic (head calculation) which, according to some computer scientists, is especially important in the computer age, also to acquire a feeling for numbers and for plausibility when using calculators. The calculation rules for dB follow directly from the given (informal) definitions:

Sum rule: $(x+x'){\text{ dB}}$ represents the product of the numbers represented by $x{\text{ dB}}$ and $x'{\text{ dB}}$.

Product rule: $(xx'){\text{ dB}}$ represents the $x$-th power of the number represented by $x'{\text{ dB}}$ and, equivalently, the $x'$-th power of the number represented by $x{\text{ dB}}$.

• Examples of approximate calculations The following table is used.

expression	$0{\text{ dB}}$	$1{\text{ dB}}$	$2{\text{ dB}}$	$3{\text{ dB}}$	$4{\text{ dB}}$	$5{\text{ dB}}$	$6{\text{ dB}}$	$7{\text{ dB}}$	$8{\text{ dB}}$	$9{\text{ dB}}$
represents	1	1.26	1.59	2.00	2.51	3.16	3.98	5.01	6.31	7.94

Conversion to normalized scientific notation is very direct, as shown by writing out some head computations (using the table).

$+136=6+130$ hence $+136{\text{ dB}}$ represents 3.98 × $10^{13}$.

$-136=4-140$ hence $-136{\text{ dB}}$ represents 2.51 × $10^{-14}$.

Exponentiation (and roots) can be approximated similarly by mental arithmetic (as explained later, this is important for transmission lines).

$27{\text{ dB}}$ represents 501 and $(27/3){\text{ dB}}$ represents 7.94, the 3rd root of 501 (approximately).

$21{\text{ dB}}$ represents 126 and $(21/7){\text{ dB}}$ represents 2.00, the 7th root of 128 (approximately).

• An estimate for the accuracy is obtained via the sum rule: an absolute deviation $d$ in the argument $x$ causes a relative deviation of $10^{d/10}-1$ in the number represented by $x{\text{ dB}}$.

absolute deviation in the argument of dB	1	0.5	0.1
relative deviation for the represented number	26 %	12 %	2.3 %

• Decibel notation is close to scientific notation, but more convenient for multiplication and exponentiation. Indeed, scientific notation has the form $a$ × $10^{n}$, but for mental arithmetic it requires treating $a$ and $n$ differently in multiplication (multiplying $a$s, adding $n$s, renormalizing) and does not facilitate exponentiation.

Early history of the definition of the decibel

Here only a summary is given, and more complete history of the decibel can be found in .

The original definition under the preliminary name of TU (Transmission Unit) can be traced back to , where its introduction is motivated by calculations regarding signal power in telephone systems.

More specifically (, p. 401), the power efficiency of a subsystem can be expressed as the ratio of output power to input power, which, under controlled conditions (linear behaviour, proper termination), is constant. The combined effect of several subsystems (amplifiers, cables) connected one after another can be expressed as the product of their respective power ratios. For $n$ identical parts, each with power ratio $p$, the power ratio for their cascade connection is $p^{n}$. Cables can be modeled in this manner by allowing $n$ to be any real number; also the ratios usually considered for cables are input power to output power in order to obtain numbers greater than 1 and hence positive logarithms (expressing losses). In any case, the resulting multiplications and exponentiations become easier when using logarithms.

For a cable characterized by a power ratio $r$ for 1 mile, the power ratio $P_{1}/P_{2}$ for $N$ miles satisfies $P_{1}/P_{2}=r^{N}$ and hence $\log {\text{ }}(P_{1}/P_{2})=N{\text{ }}\log {\text{ }}r$, regardless of the base of the logarithm. For this logarithmic scale, $\log {\text{ }}r$ acts as a unit (and even equals 1 if base $r$ is chosen). Originally the MSC (Mile of Standard Cable) was used as a unit for power ratios represented by their logarithm.

The value $r$ is frequency-dependent: based on (p. 404) for the MSC $r={\text{e}}^{2a}$ where $a=0.00386{\sqrt {f}}$ assuming $f$ (dimensions stripped) is the frequency in Hz. For $f=900$ one finds approximately $r=1.26=10^{0.1}$, and hence $P_{1}/P_{2}=10^{N(0.1)}$ (approximately, for 900 Hz).

To avoid dependency on physical characteristics (frequency, distortion), in 1924 the TU was introduced as a replacement for the MSC. The TU is defined as follows ( p. 401-402), preserving the conceptions regarding magnitudes associated with the MSC.

"Any two amounts of power differ by $N$ units when they are in the ratio of $10^{N(0.1)}$."

The number $N$ of transmission units corresponding to the ratio of any two powers $P_{1}$ and $P_{2}$ satisfies $N=10{\text{ }}\log _{10}{\text{ }}(P_{1}/P_{2})$.

In 1929 the TU was renamed decibel, with symbol db. In due course, the lowercase b changed to uppercase B.

The definition does not state what "differ" means (obviously not just $P_{1}-P_{2}$) nor what the value of the unit is. This is resolved in (p. 402) by rewriting $N=10{\text{ }}\log _{10}{\text{ }}(P_{1}/P_{2})$ as

$N=\log(P_{1}/P_{2})/\log {\text{ }}10^{0.1}$ (unspecified base)

and defining the numerical value of the TU (or dB) to be $\log {\text{ }}10^{0.1}$; correspondingly the "difference" considered is $\log(P_{1}/P_{2})$. The term "difference" is based on the logarithmic property that, by introducing a reference power quantity $P_{0}$ (for instance, 1 W), one has

$\log(P_{1}/P_{2})=\log(P_{1}/P_{0})-\log(P_{2}/P_{0})$ (regardless of the choice of $P_{0}$ and of the logarithm base).

Although the rules for determining the number of decibels for a given power ratio are clear, the value of the decibel itself (unnecessary for applications) has caused confusion. A quotation of a 1954 paper found in reads:

"The fundamental relations have been further confused by describing the decibel as on-tenth of a Bel when the truth is that its value is the tenth root of the value of a Bel."

In this is interpreted as taking the value of the decibel to be the power ratio $P_{1}/P_{2}$ for which $N=1$.

According to, the decibel made its way into acoustics because the Bell System was conducting active research in that area and "sound power changes just detectable by the human ear are close to those corresponding to a mile of standard cable".

The decibel in acoustics and sound measurements

Most people have heard about decibel mainly in connection with sound, making this application a topic of general interest. Yet it must be cautioned that

• Acoustics and sound are only one of many fields where decibels are used.
• Sound involves various aspects of physics (sound intensity, sound pressure, density of and speed of sound in the medium, distance, direction, propagation conditions) and physiology (perception). These must be taken into account in defining and interpreting suitable variants of the decibel.
• In acoustics, dB-values are often quoted without mentioning any of these factors or even the reference values. This practice has been criticized by several authors and considered sufficient reason for abandoning the decibel in acoustics altogether in favor of using just SI units. A table in shows that the SI notations 20 μPa (the reference sound pressure in air is 20 micropascal) and 20 kPa correspond to 0 dB (20 μPa) and 180 dB (20 μPa) respectively, which suffices to cover the entire range of values encountered in practice.

A representative example from illustrates some of these issues as well as the advantages of the decibel for quick mental arithmetic. The author refers to an article in The Economist mentioning "tests of a sonar that produces extremely loud low-frequency sound (it has a maximum output of 230 decibels, compared with 100 decibels for a jumbo jet)". The quoted text suggests a difference of 130 dB. Yet, the comparison is meaningless for the folowing reasons.

• The 230 dB for the sonar pertains to a source level at 1 m distance, but the 100 dB for the jet pertains to a received level at an unspecified distance of several hundreds of meters.
• The reference level for sound pressure in water is 1 μPa as compared to 20 μPa in air.

For a jet taking off at a specified distance of 500 m the source used in mentions 120 dB (20 μPa). The corrections (in dB) needed for a more meaningful comparison are the following (applied to the jet), taking into account that sound pressure is a root-power quantity.

• Using 120 dB (20 μPa) for the jet (at the specified distance).
• Distance correction: $10{\text{ }}\lg {\text{ }}500^{2}$ or (using the table) 2 × 27, yielding 54 dB.
• Reference level correction from 20 μPa to 1 μPa (a ratio of 20): $20{\text{ }}\lg {\text{ }}20$ or (using the table) 2 × 13, yielding 26 dB.

Hence the jet noise corresponds to (120 + 54 + 26) dB (1 μPa) at 1 m distance. The resulting 200 dB (1 μPa) at 1 m for the jet can now be compared to 230 dB (1 μPa) at 1 m for the sonar, yielding a difference of 30 dB instead of the 130 dB suggested by the quoted text. To conclude, the ratio of the sound pressures (sonar versus jet) is only about 32 instead of over 3 million (values for 30/2 dB and 130/2 dB). In addition to this, actual propagation conditions and physiological factors (annoyance, pain) must be taken into account for a meaningful comparison.

The decibel in engineering practice and in the technical literature

In modern engineering practice, the decibel is used routinely for complex systems, often without worrying about stating definitions, and essentially using the informal view presented in the introduction.

For instance, in a text about satellite communication, the decibel is directly introduced via a concrete situation (p. 135). After deriving the following formula for the uplink carrier-to-noise ratio (here quoted without the subscripts u for "uplink")

${\frac {C}{N}}=({\text{EIRP}}_{\text{sat}}){\left({\frac {c}{4\pi fd}}\right)}^{2}\left({\frac {G}{T}}\right)\left({\frac {1}{kB}}\right)BO_{\text{i}}^{-1}L^{-1}$

decibels are casually introduced as follows: (quote) "The uplink and downlink uplink carrier-to-noise ratios in decibels are 10 times the logarithm (base 10) of the corresponding quantities."; for the uplink (omitting subscript u):

${\frac {C}{N}}({\text{dB}})={\text{EIRP}}_{\text{sat}}({\text{dBW}}){\text{ }}-{\text{ }}20\lg \left({\frac {4\pi fd}{c}}\right){\text{ }}+{\text{ }}{\frac {G}{T}}({\text{dB}}/{\text{K}}){\text{ }}-{\text{ }}10\lg k{\text{ }}-{\text{ }}10\lg B{\text{ }}-{\text{ }}BO_{\text{i}}({\text{dB}}){\text{ }}-{\text{ }}L({\text{dB}})$

The legend for the symbols in the first formula is (simplifying technical terms):

• ${\text{EIRP}}_{\text{sat}}$: a maximum power, to be seen in conjunction with a "backoff" $BO_{\text{i}}$ (dimensionless reduction factor)
• $c$: speed of light, $f$: frequency, $d$: distance
• $G$: antenna gain, $T$: noise tempertature, $k$: Boltzmann’s constant, $B$: noise bandwidth
• $L$: loss due to antenna tracking and atmospheric attenuation.

Tacitly the same symbols are used in the second formula (using decibels) for pure numbers denoting the numerical values in a coherent unit system (by definition, a coherent unit system allows omitting units in a formula without introducing conversion factors). Similar examples without preserving a coherent system (e.g., distance in km instead of m) but scaled correspondingly are found in .

Such examples demonstrate the following aspects typical for engineering practice.

• Introducing and using decibels without much comment and without definition.
• Avoiding the association of decibels with power ratios, but using them for representing any kind of number (temperature, frequency, distance).
• Obviating the 10 log versus 20 log issue by referring to the formulas used: squares automatically lead to the factor 20 where it is appropriate.

Especially regarding the 10 log versus 20 log issue, other authors have advocated similar reference to the relevant formulas, in particular for root-power quantities (such as sound pressure) applying the basic 10 log rule to the square of those quantities in an explicit manner, which obviates the 20 log rule.

Another source of difficulties and confusion are attempts at formal definitions via notations that are at odds with common mathematical conventions, and as a result cannot properly convey the desired information, nor be used for using the decibel algebraically.

Many textbooks (references here – how many?) provide a definition of the form

${\text{dB}}=10{\text{ }}\lg(P_{1}/P_{2})$

As a definition, this is mathematical nonsense since the left hand side of the equality is a constant expression (without variables) and the right hand side depends on $P_{1}$ and $P_{2}$. (Aside: as an equation with $P_{1}$ anf $P_{2}$ as unknowns it has a meaningful interpretation, but in a different way that is not intended.)

(to be continued) Boute (talk) 09:47, 24 March 2011 (UTC)

Recent formalizations of the decibel

(here comes a short overview of decibel, neper according to the standards and as obtained by formalizing engineering practice) Boute (talk) 13:25, 23 March 2011 (UTC)

References and Notes

Boute (talk) 09:06, 24 March 2011 (UTC)