Talk:Decibel/Archive 5

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Size of 1dB versus number of number of dB in a given quantity

Dear catslash editor

About your correction to my change, I understand that this issue is confused. I have analyzed previous discussions and I propose now to use the recoomenations of Stephen Meigs (I copy below) Please let me know if you agree

To be perfectly clear, say either "the decibel is a unit whose value is ten times the logarithm to base 10 of the ratio of two power quantities" or say "the decibel is a unit defined as 1/10 the unit whose value is the logarithm to base 10 of the ratio of two power quantities". How about the following as something one could agree on? "The decibel is a unit whose value is ten times the logarithm to base 10 of the ratio of two power quantities. The actual base 10 logarithm of the ratio itself is called a bel. Though less simply defined, the smaller decibel unit has proven to be more popular than the bel unit. Since the value of a measurement using decibels is 10 times that of a measurement using bels, the decibel (as a unit) equals 1/10 bel, whence the "deci" prefix."Stephen A. Meigs (talk) 16:42, 11 August 2010 (UTC)

Regards OscarJuan --OscarJuan (talk) 18:41, 3 December 2011 (UTC)

The statement "the value of a measurement using decibels is 10 times that of a measurement using bels" is correct.

The statement "the decibel is a unit whose value is ten times the logarithm to base 10 of the ratio of two power quantities" is incorrect.

Dondervogel 2 (talk) 13:34, 5 December 2011 (UTC)

About Dondervogel comment

Yes, I understand the subtle difference that the word "unit" has in this context. But It do not changes my point: the definition of this page is not good enough. The proof of it is that this discussion page is full of comments between editors. As you know, this situation is not usual at technical Wiki pages (Politics is other world...) My opinion is as follows: a) The definition of the decibel unit as: 1 dB = 0,1 B is not clear at all... altough it is semantically correct b) The definition of this page looks like a paradox. Then we must add a line to explain why this is that way and not an error If the principal editors of this page are in agreement with this improvement, then we can work on it. Best regards --OscarJuan (talk) 01:00, 7 December 2011 (UTC)

What looks like a paradox? And what change would you propose to improve it? The present scheme seems clear and correct to me. Dicklyon (talk) 02:45, 7 December 2011 (UTC)

Yes, it is quite clear, it is the recent attempted edits that are confused. A lengthy talk page does not mean the article has problems - that is not logical - the talk page will be just as long after the problems are fixed. On the contrary, it means the article has been thoroughly discussed and improved. SpinningSpark 10:13, 10 December 2011 (UTC)

Dear editors: The paradox in my opinion is the unclear definition. I copy the definition that you have in this page, at the beginning and under DEFINITION. At the beginning of the article you said: A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities. Proposition-1)A decibel is one tenth of a bel and we can read below (under DEFINITION) Proposition-2) A decibel (dB) is one tenth of a bel (B), i.e. 1B = 10dB. Proposition-3 )The bel is the logarithm of the ratio between two power quantities of 10:1, Then we understand, from proposition 2) dB is 0,1 Bel; from proposition-3) then dB is 0,1 Bel = 0,1 log (p/pref) But it is not the correct definition as we know from the fist cited statement. Please note that from the begining of this page this was a central dilemma between editors. Yes I know that is is very common in most text books. Then I propose tu give under DEFINITION title the following statement: Decibel is defined as ten times the logarithm to base 10 of the ratio of the power quantity to be measured divided by a reference power quantity. From a historical point of view the decibel is a magnitude submultiple of the bel (then the name deciBEL),defined as: logarithm to base 10 of two power quantities. Then a decibel (dB) is one tenth of a bel, in the same sense as decimeter is one tenth of a meter It is my proposition Regards --OscarJuan (talk) 23:58, 10 December 2011 (UTC)

I don't think it makes sense to say "Decibel is defined as ten times the logarithm..."; the current wording "A ratio in decibels is ten times the logarithm..." makes more sense; or maybe better would be something like "A ratio can be expressed in decibels as ten times the logarithm...". If you want to define a decible, probably 1/10 of a bel is the best thing; but if you want to say how to represent a ratio as a number of decibels, then the other kind of statement is sensible; we should have both, but not the vague and confusing "Decibel is defined as...". Dicklyon (talk) 04:11, 11 December 2011 (UTC)

Dear editor: Please note that this page is about decibel and we must define it Your proposition about "a ratio in debibeles" is only a indirect definition because you are defining "a ratio in decibeles" but this is not what we must define because our issue is "decibel" not "ratio en decibeles" Other matter is that it is importand to have into definition the usual concept of "measured magnitude"/ "reference magnitude" that the definiton you are using now do not have Best regards--OscarJuan (talk) 00:25, 18 December 2011 (UTC)

I agree with OscarJuan's assertion that an article entitled "Decibel" ought to define the term "decibel". Or at least it should strive to do so, and the event of not achieving this should explain the reason for the omission. Dondervogel 2 (talk) 00:38, 18 December 2011 (UTC)

Sure, I agree. I think we have been striving to define it clearly and correctly. You have a suggested improvement? Dicklyon (talk) 01:02, 18 December 2011 (UTC)

What it says is "A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities.[1] A decibel is one tenth of a bel, a seldom-used unit." These statements are correct but they do not contain a definition of "decibel" (except indirectly through the bel). I'm sure I could find an explicit definition if one is wanted, but I have a feeling of deja vu here. The problem that I foresee is that explicit definitions (of the decibel) are notoriously difficult to follow. Perhaps it could say "the formal definition is <new (complicated) explicit definition> and what this means in practice is <existing implicit definition>". I'll see what I can find. Dondervogel 2 (talk) 01:39, 18 December 2011 (UTC)

It would be true to say: A decibel is a unit measuring gain or loss of electrical or mechanical power. One decibel is one tenth of a bel, and one bel is a ten-fold increase in power. A decibel is thus an increase in power by a factor of 10^1/10. Truth though is insufficient, it must also be verifiable by consulting books. This is a problem as the vast majority of books either state dB = 10log₁₀(P₁/P₂) which is clearly false, or else are ambiguous or self-contradictory. Here however is one text that gives the above definition[1] (though first it says Gain in dB = 10log₁₀(P₁/P₂)), it may be possible to find more such sources. --catslash (talk) 01:50, 18 December 2011 (UTC)

(edit conflict) Under Taking Stock above (10-11 March 2011) I found (from the ANSI Acoustical Terminology Standard)

"decibel. Unit of level when the base of the logarithm is the tenth root of ten, and the quantities concerned are proportional to power. Unit symbol, dB."

"level. In acoustics, logarithm of the ratio of a quantity to a reference quantity of the same kind. The base of the logarithm, the reference quantity, and the kind of level shall be specified."

This explains the deja vu at least. What happened to the proposal Boute was working on? Dondervogel 2 (talk) 01:56, 18 December 2011 (UTC)

Well, it's a definition all right, but rather indirect, vague, and incomplete; it doesn't even mention power, which is what it needs to be for that specified log base. Dicklyon (talk) 06:07, 18 December 2011 (UTC)

I agree there's room for improvement (especially with regard to the need to mention power), but IMO the fact that is this the definition of the American National Standards Institute gives it sufficient weight to justify quoting it verbatim along the lines of "According to American National Standard Acoustical Terminology S1.1-1994 (ASA 111-1994) the decibel is defined as ...". I will keep my eyes open for something better. Dondervogel 2 (talk) 10:40, 18 December 2011 (UTC)

ISO standard 80000-8 includes 3 definitions for the bel (B), depending on whether it is used for sound pressure level, sound power level or sound exposure level. The definition for sound power level is “1 B is the sound power level when P/P0 = 10”, accompanied by the remark “In practical applications and consistent with the definition of sound power level, the submultiple decibel, dB, is used instead of the bel, B.”

The definition given for sound power level is “LW = 10 lgP/P0 dB where P is sound power … and the reference value is P0 = 1 pW”, followed by the remark “For a general definition of the level of a power quantity, see also ISO 80000-3:2006, item 3-22.” I do not have a copy of ISO 80000-3:2006 to hand. Dondervogel 2 (talk) 11:08, 18 December 2011 (UTC)

I found Boute's proposal as well. It includes a comprehensive bibliography. Dondervogel 2 (talk) 14:06, 18 December 2011 (UTC)

I am happy to see the constructive work of several editors. I will add some personal comments, as follows; ***** ONE Since Bel is an unused magnitude (only created by an American private company, 83 years ago, as tribute to Graham Bell, founder of the company and inventor) I do not think that we must define the decibel unit as a fraction of an old magnitude, never used, and created only as an tribute... Mi proposition is that after the definition as: 10 times logarithm to base 10 of (Power / Reference power) we could add: "From a historical point of view the decibel is a submultiple of the bel (deci-BEL),defined in 1928 as: logarithm to base 10 of two power quantities, as a tribute to Alexander Graham Bell" ******* SECOND As a comment; the WEBSTER Dictionary, 3rd edition, defines decibel without mention of the bel unit Regards --OscarJuan (talk) 16:28, 18 December 2011 (UTC)

Something like that would be useful; it's better not to leave out mention of Bel altogether, as it is the concept that clarifies "deci", and why a smaller unit has a bigger factor in front in its calculation. As for Boute's proposal, there's some good stuff there, but it seems to largely idiosyncratic, defining "x dB" as representing "a number" rather than a power ratio, and calling the common formula "dB = ..." "nonsense" rather than explaining it as simply a notationally strange way to define a ratio as a number of dB. Dicklyon (talk) 16:48, 18 December 2011 (UTC)

Regardless of whether one approves of Boute's choice of words, he did some valuable groundwork and provides a comprehensive list of source references. By the way, I've just noticed that the ANSI definition *does* mention power. I still think it could be quoted verbatim somewhere - perhaps in the Acoustics section. Dondervogel 2 (talk) 20:47, 18 December 2011 (UTC)

First of all, my best wishes for 2012 to everyone interested in the decibel topic. I apologize for letting the work overload concerning other topics keep me from contributing during the past year (and for several months to come). The problem with the decibel is that there is no "common definition" or common defining formula whatsoever, and that attempts to make the concept more precise has led to at least half a dozen different interpretations over the past 80 years, fortunately all equivalent with respect to practical applications. Let me say, though, that the "common" formulas (where?) starting with "dB = ..." are likely to be unsound. Also, the ratio of quantities with the same dimension, in particular power, is always a pure (i.e, dimensionless) number. On the other hand, in the interpretation of one of the early authors (not the ones who wrote the original papers), dB is meant to be written after a pure number to indicate that the number was obtained by taking 10 times the logarithm of a power ratio (and no other kinds of quantities). This is analogous to writing m (meter) after a pure number obtained by taking the ratio of two lengths, one being a fixed standard length. The analogy would be complete only if one of the powers were a fixed quantity (as in dBm, milliwatt), which was not the intention of the original authors (namely, an approximate equivalent for the attenuation in the "standard mile of cable"). The analogy was further destroyed by the tendency, heavily deplored by some of the early authors (see my earlier bibliography list), to extend the notion to voltage ratios (which are not powers), resulting in errors (e.g., if impedances differ). However, by then the genie was out of the bottle. Engineers in satellite communication now freely use dB for other kinds of ratios (temperatures, distances), and they do so without errors, because they always refer to quantities linked by a formula, and ratios of quantities with the same dimension are recognized as pure numbers. The only (yet very serious) mess remaining from earlier decisions is that factors 10 and 20 are embedded in the definition of two kinds of decibels (one for power, one for root power quantities), instead of having a single definition for decibel and letting the introduction of a factor 2 (or other factor) be governed by the presence of a square (or other exponent) in the formula relating the quantities. The latter option would be clean, clear, and fully general. Boute (talk) —Preceding undated comment added 21:16, 1 January 2012 (UTC).

Correctly dated Boute (talk) 21:21, 1 January 2012 (UTC)

Can Wikipedia get this right?

The main question

The amount of discussion regarding the decibel on these Wikipedia pages and the diversity of fundamentally incompatible formulations in the literature at large show that there is a serious problem.

Wikipedia's prime duty is providing clear and reliable information to the community. An overly narrow interpretation of the Wikipedia rules (e.g., WP:Verifiability) is counterproductive. In particular, for "verifiability" of statements with an essential yet simple mathematical content, reasoning (verifiable in situ by any interested reader) should be the prime criterion.

Even then, before Wikipedia can arrive at a simple and reliable information about the decibel, consensus must be reached about the rationale regarding the introductory definition. Should this definition be burdened by the historical complications and inconsistencies, or should it rather reflect current practices, which (perhaps thanks to evolution) can be captured by a simple and consistent unifying definition?

This is the main question to be answered before meaningful progress can be made.

Technical motivation: the decibel confusion

Judging by the literature in the first few decades after 1924, the decibel falls woefully short of Boileau's principle "Ce que l'on conçoit bien s'énonce clairement" (what is well-conceived is clearly stated). One can rightly conclude that the original ideas underlying the decibel (a replacement for the MSC - Mile of Standard Cable) were not well-conceived, and that later attempts at restricting it exclusively to power ratios have only aggravated the situation. Such attempts were futile to start with, since the constructive elements of the idea are jailbroken time and again, as is fitting in any technology. The confusion is well-documented in The Decibel Report by Richard Katz.

In brief, the earliest 1924 papers (Hartley, Martin; see the Alcatel-Lucent site) "define" (!) the TU (later called decibel) in a very roundabout way. The only light-shedding part is Hartley's passing reference to the term level used by Clark (1923 A.I.E.E. paper) in discussing power level charts, nowadays called link budgets. Thirty years later, Horton notes that the decibel is still "bewildering", and adds to the confusion by emphasizing that the decibel is not a dimensionless unit, but a concept as fundamental as other physical units. By contrast, comments by designers of the current standards (see my earlier references) explain that dB simply stands for the number 1, just like rad (radian). Etcetera, ad nauseam.

All older references and the comments just mentioned share the unspoken convention (easily inferred in retrospect) that the decibel is not a measure for a power ratio, but for a (relative) power level. The exponentiation (or its inverse) takes place outside the decibel, and the decibel itself is technically trivial. Here are some implications

(a) The levels interpretation does not allow saying "3 dB equals 2", but only "3 dB corresponds to 2".

(a) What everyone has (correctly) learned about units, namely that one cannot add meters to seconds, becomes invalid. Indeed, with the level interpretation, in link budgets one happily writes 10 dBm - 30 dB + 6 dB but, then again, 10 dBm - 30 dBm + 6 dB is nonsense!

Meanwhile, over the past 50 years, throughout the engineering literature and in data sheets,

(a) writing an equality sign as in 3 dB = 2 has become customary;

(b) the decibel has been found convenient for about anything: dBHz for frequencies, dBK (K for kelvin) for temperatures etc.

While this would be "sloppy abuse of notation" by the levels interpretation, this evolutionary common practice is internally fully consistent, and conceptually much simpler:

(a) for dimensionless quantities (including power ratios)

x B = 10^x and, correspondingly, x dB = (x/10) B. The obvious rule is (x + y) dB = (x dB)(y dB), showing that this decibel is technically nontrivial, which is precisely what makes it useful.

(b) for dimensioned quantities, e.g. power, the units are simply written after the number x dB, as usual:

x dBW = ((x/10) B) W. The rules for combining various dimensions are the familiar ones, independently of the decibel.

So one simply writes 30 dBW = 1000 W = 1 kW. Problems occur only if contamination with the older conventions is allowed to occur. For instance:

By the level interpretation conventions, cable losses are expressed in dB/m, say, x dB/m. The losses over 100 m are then (100 x) dB.

By the actual conventions, noise is often expressed in dBW/Hz say, x dBW/Hz. For 100 Hz bandwidth, the intended result is not (100 x) dBW (fortunately!) but only 100(x dBW) or, equivalently, (20 + x) dBW.

Summarizing: the described definitions are not a radical new convention, but merely a recognition of what many engineers have been doing in practice since many decades.

Some random examples from the literature:

McClaning and Vito, Radio Receiver Design, p. 7: 0 dBW = 1 W; p. 11: 0 dBHz = 1 Hz

Pratt and Bostian, Satellite communications, p. 143: kTB = -141.4 dBW (for specific T and B)

More generally, Googling dBm/Hz/K yields many sources mentioning that, for Boltzmann's constant k = -198.6 dBm/Hz/K

This leads back to the question: do we reflect current practice, which (perhaps by evolution) implicitly uses a simple and consistent mathematical basis, or do we perpetuate confusion, just for historical reasons? Boute (talk) 15:19, 19 March 2012 (UTC)

More examples of current practice

A more thorough search (systematic rather than random) reveals that the current notational practice outlined above is quite universal in link budget calculations. In the following references:

Kamilo Feher, Digital Communications (1981), p. 43, Table 1.5

Pratt and Bostian, Satellite Communications (1986), p. 133, Table 4.6

Tri T. Ha, Digital Satellite Communications (1986), p. 137, Table 4.1

Dharma Raj Cheruku, Satellite Communication (2009), p. 51 and following

Maral, Bousquet, Sun, Satellite Communications Systems (2010), p. 223 and following

the following notations consistently appear:

Satellite G/T in dB/K

Noise bandwidth in dBHz

Boltzmann's constant k = -228.6 dBW/Hz/K

Bit rate in dBHz

The only issue that mars the picture in these books is the very sloppy, inconsistent and ad hoc calculation style, clearly due to contamination from the old level interpretation. Indeed, calculation examples are written in the form 10 dBW - 30 dB + 6 dB = -14 dBW. The consistent form is (10 dBW)(-30 dB)(6 dB) = (10 - 30 + 6) dBW = -14 dBW. More generally, the consistent form gets and the decibel scalings and the dimensional analysis correct in the calculations. Clearly, the tweaks to make the sloppy calculations consistent are very minor. Interestingly, the referenced link budget calculations in tabular form are correct under the consistent interpretation and need no tweaking (simply because the addition or multiplication sign is not written). Boute (talk) 13:42, 21 March 2012 (UTC)

Replies

We can agree there is confusion. The correct course is:

• not to reproduce confused accounts
• nor to prescribe our own reformed interpretation

but instead:

• describe current practice (tell it like it is), elucidating any inherent inconsistencies
• if there's a source that uses words such as confused, contradictory inconsistent etc., then report it
• if there's an acceptable source that proposes a reformed interpretation, then report that too.

Incidentally, the dB is still commonly used as unit of gain or loss, in the sense of the original mile of standard cable (MSC), in which circumstances there are few problems: 2 MSC + 3 MSC = 5 MSC as with any other unit. The possible sources of confusion are:

• The common implicit definition 10log₁₀(P_out/P_in). So we should say most authors define the dB implicitly by...which means that 1 dB is an increase in power by a factor of 10^1/10 ~ 1.25893.
• The implicit definition again, making readers think that 1 B should be one tenth of a dB. So give both implicit and explicit definitions of the Bel.
• The application of the of the above formula to cases where P_out is not a fixed multiple of P_in such as S/N and in particular the case where P_in is a fixed reference level (the main problem). So we should say by extension the above formula is used to define a logarithmic measure of the power P_out, by setting P_in equal to some arbitrary reference level, such as 1 milliwatt. In this case the reference level is usually appended as a suffix to the dB symbol as in dBmW. Although written as unit, these absolute decibels cannot be meaningfully added or subtracted as can the relative decibels used to measure gain and loss.

--catslash (talk) 00:15, 22 March 2012 (UTC)

Many people already did ample reporting on sources exposing ambiguities and inconsistencies regarding the decibel. We need not repeat all that. More importantly, for issues with mathematical content, sources are of secondary relevance: simple logic and some mathematical insight are more effective in noticing and correcting inconsistencies, and any devoted reader can verify it on the spot. The conventions tacitly used in the past 40 years by communications engineers (see the earlier examples) reveal an underlying rationale that is entirely consistent and thereby superior to the earlier conventions for which there has never been any uniformity or agreement in sight during the past 85 years. Hence the best way out of the morass is making the aforementioned rationale explicit, as was done in the above explanation and examples. These certainly described current practice exactly as it is, verifiable down to the page numbers. The unsurprising fact that the old inconsistent interpretations are also still found in the literature is not a good reason for perpetuating them. Boute (talk) 22:38, 23 March 2012 (UTC)

I cannot see anything wrong with the common practice of adding dB as a level to dB as a ratio. For instance,

$10\mathrm {\ dBm} +20\mathrm {\ dB} =30\mathrm {\ dBm}$

is a perfectly correct result. It does not matter that this does not fit with the usual rules of dimensional analysis, it is equivalent to,

$10\mathrm {\ mW} \times 100=1000\mathrm {\ mW}$

No one complains that the "100" is in different "units". I appreciate that all of this does not fit in with the SI scheme of how units should behave (even the suffix "m" would not be allowed - affixing suffixes to units is forbidden) and is probably the reason it has never been adopted by the SI. With dB, it is not adding apples to oranges that causes a problem, but adding apples to apples;

$10\mathrm {\ dBm} +10\mathrm {\ dBm} \neq 20\mathrm {\ dBm}$

13 dBm would be a more correct answer. SpinningSpark 07:47, 22 March 2012 (UTC)

You clearly saw things wrong with the addition example: it does not fit with the usual rules of dimensional analysis. Why conclude that it does not matter? We should care, in the interest of the readers.

Regarding your example, simply consider writing

$10\mathrm {\ mW} \times 100=1000\mathrm {\ mW}$

with decibels as

$10\mathrm {\ dBmW} \times 20\mathrm {\ dB} =30\mathrm {\ dBmW}$

The two equations are factorwise identical, and dimensional analysis goes unhampered. Similarly for more complicated equations involving dBW/Hz/K and so on. Note: recognizing that dBm stands for "dBmilliwhat?", we prefer writing dBmW instead.

Similarly,

$10\mathrm {\ dBmW} +10\mathrm {\ dBmW} =2\times (10\mathrm {\ dBmW} )=13\mathrm {\ dBmW}$

which is exactly what you wanted. Boute (talk) 22:38, 23 March 2012 (UTC)

I will consider doing it that way when the average textbook follows that scheme. Until then it is an enormous confusion (and against policy) to describe that addition in anything like that way in the article. Multiplying decibels? never heard of it. SpinningSpark 23:18, 23 March 2012 (UTC)

No one must feel obliged doing it that way --- unless he/she values consistency: then there is no other way. The real cause of all confusion (admittedly enormous for the decibel) are inconsistent notational conventions and interpretations. What is a reasonable time estimate for the average textbook using any consistent scheme for the decibel if the vicious circle is not broken? What policy proscribes adding a formulation that cleans everything up? Anyway, no one is multiplying decibels, but just standard physical quantities or numbers (that happen to be "expressed in decibels"). Boute (talk) 06:36, 24 March 2012 (UTC)

I think the point you are missing is that it is not the purpose of Wikipedia to "break the vicious circle" or establish a "consistent scheme". We summarise what is out there in reliable sources, not what could be or should be. If it is not in RS then the policies involved could be WP:V, WP:OR, and WP:NOT. Has any RS ever written 10 dBmW x 20 dB = 30 dBmW? Using the times symbol is a multiplication in my book. Using it to mean something else is certainly OR. SpinningSpark 10:13, 24 March 2012 (UTC)

As made clear repeatedly, with examples: many reliable sources (such as the textbooks listed above) do establish a consistent scheme. They use expressions with dBW/Hz/K and dBHz and, for this example, effectively multiply them so the Hz cancels. The only "smudge" is that some still express this with a plus symbol in formulas due to contamination by older (inconsistent) decibel conventions, so the units and the decibels cannot possibly combine correctly. Hence the problem is just the opposite: abusing plus. Studying the above formulas and explanations more carefully shows that the times symbol does mean multiplication as we all know it. Most importantly, an issue that most of the comments I am reading in these talk pages are systematically hedging, is that the purpose of all policy rules such as WP:V, WP:OR, WP:NOT, is not in themselves, but in ensuring reliable information to the readers. For a mathematical issue such as characterizing the decibel, the only sensible criterion for judging whether something is a RS is consistency, not tradition. In fact, WP:OR explicitly exempts Routine Calculations from the OR designation. It is clear that routine logic (i.e., verifiable by any diligent reader) deserves the same dispensation. Boute (talk) 17:45, 24 March 2012 (UTC)

Would you mind pointing out to me where exactly in the Decibel Report or whatever source you are referring to that it sanctions dBmW x dB, because I am not seeing it. I don't even see any mention of dBm. As for WP:V, it is one of our core policies. Not only that, it is a founding principle. The fact that 2.5 + 7.3 = 9.8 is a "routine calculation" and does not require a citation. But if one wanted to express that in a different way, say because it better fitted with the conventional representation of a function, as Щ(2.5, 7.3) = 9.8 then a citation would certainly be needed. SpinningSpark 21:56, 24 March 2012 (UTC)

For the answer to your first questions, I reset the indentation below. Here I just note that dBmW x dB, without variables or concrete numbers, will indeed not be found anywhere. The idea that Щ(2.5, 7.3) = 9.8 better fits with the conventional representation of a function is very strange since, for nearly all concrete 2-argument functions, infix notation is the conventional one. For the decibel, it is the contamination with old inconsistent convention that deviates from common mathematical practice, and the many quoted more recent sources are effectively restoring common mathematical practice.

(Returning to the left margin) Of course, one has to look on the indicated pages. Also, not everything is found together in one place, since the authors clearly take the viewpoint that the conventions are "obvious from usage" (and they are, as seen next).

Starting with a basic example: in McClaning and Vito, page 9, example 1.3, equation 1.28, one finds 24 mW = 133.8 dBf (meaning dBfW, femtowatt). The obvious convention is that (a) x dB = 10^(x/10) and hence, unavoidably, (x dB) (y dB) = (x + y) dB and (b) and that units are treated in the normal way (without the traditional inconsistencies around dB). This is more evident in more advanced examples, as shown next.

For the more advanced examples (insofar as basic arithmetic is advanced), all quoted sources are similar. Taking Pratt and Bostian, Satellite Communications as an example, page 133, table 4.6,

Satellite transponder system noise temperature (500 K): 27 dBK

Channel RF bandwidth, 30 kHZ: 44.8 dBHz

Boltzmann's constant k: -228.6 dB/Hz/K

Hence: Noise power at transponder input: -156.8 dBW

The cancellation of the units reflects multiplication with units as usual, which combines with decibels via (x dB) (y dB) = (x + y) dB, which is multiplication with numbers as usual. So all that is used is basic multiplication with dimensioned quantities as taught in elementary school.

A systematic account can be found in The decibel done right: a matter of engineering the math which, being peer-reviewed publication, falls in the category of the most reliable sources according to WP:IRS. In retrospect, the unavoidable emphasis in the abstract on novelty (as needed for publishing research papers!) requires some mitigation: more recently I found out that it exactly synthesizes current practice as reflected in many textbooks, such as the ones referenced.

If any inconsistencies are left in those (relatively) more recent sources, and inconsistencies still are there, they are clearly due to contamination with the older, inconsistent conventions. However, the common underlying idea is both consistent and undeniable. Below we argue that this is the presentation of the decibel most suited for Wikipedia. Boute (talk) 08:57, 25 March 2012 (UTC)

Back to the original policy question

The question was: do we present the decibel in the simple, consistent way reflected by current practice, or do we burden the readers with the old inconsistencies just for historical reasons?

Here is my view. Wikipedia's first obligation is towards its readers. Trying to meet the "verifiability" criterion by the "reliable sources" approach is absurd, since most sources are internally and mutually inconsistent. Hence, which one should we choose? Given the fundamental differences, using any single source would violate the WP:NPOV rule. More importantly, the examples given for the various Wikipedia policy rules without exception are about historical, sociological etc. issues, where obviously publications are the main (even only) source of information. They totally ignore that, for mathematical issues, logic and consistency is more in the interest of the readers than any other source. Maybe this huge (hopefully temporary) gap in the policy rules is what Wikipedia policy makers recognized by issuing WP:Ignore All Rules. However, it seems that not the rules themselves, but an extremely narrow interpretation of the rules, is what prevents us from making progress towards a clear article about the decibel.

One final remark: in founding principles one reads (concerning these principles) "People who strongly disagree with them are nonetheless expected to either respect them while collaborating on the site or turn to another site", which is among the most totalitarian statements I have read in a long time. The danger is that enforcing rules (or a specific interpretation thereof) may become a hobby. This hampers reaching consensus about how to best serve the readers. Hence, before continuing the technical discussion, we must agree on basic policy. Boute (talk) 08:57, 25 March 2012 (UTC)

This page is already TLDR and is probably dissuading other editors from taking part. It is absolutely not the right place to debate policy. Please do not continue to argue for a change, or reinterpretion, of policy here. If you want that debate, take it to a more suitable page such as WP:VPP. When you have succeeded in getting policy changed, come back and let us know. SpinningSpark 10:25, 25 March 2012 (UTC)

I agree with your TLDR observation, but refrain from blaming anyone. Also, I did not argue for change of policy, but against very narrow interpretations that represent a disservice to the readers. If I wanted a broader debate, I would evidently take it to the suitable pages. Here we should stick to the specific question: do we present the decibel in the simple, consistent way reflected by current practice, or do we burden the readers with the old inconsistencies just for historical reasons? That can only be debated here. Boute (talk) 11:17, 25 March 2012 (UTC)

I think I was entitled to interpret some of your comments such as "which is among the most totalitarian statements I have read in a long time" as arguments aqainst current policy. What exactly is it that you are proposing to add or remove from the article? If it it is any form of dBm x dB then I absolutely oppose. You have presented no RS which explicitly use this and your interpretation that it is implicit in some sources is WP:SYNTH. I have not relooked to see if this is in your own publication, but that is going to have WP:COI and WP:N issues. SpinningSpark 23:55, 25 March 2012 (UTC)

The various sources quoted show that decibel has evolved considerably from the original narrow MSC idea. The difficulty seems reflecting this in the Wikipedia article. Clearly WP:SYNTH is meant to avoid that one synthesis among many other plausible ones would be presented as the only one and may even be wrong or misleading, as illustrated by the examples in the WP:SYNTH guidelines --- all from history or sociology. The issue here is mathematical, and making explicit the calculation rules that are used implicitly in these various sources is not really synthesis at all but Routine Calculation in Wikipedia terminology. Indeed, the explicitation given is the only consistent (in some real sense, inevitable) one; it is clearly correct and has no chance of being misleading. Anyway, why would (10 dB) x (20 dBW) = 30 dBW be more objectionable than 10 x 100 W = 1000 W which (in the consistent interpretation) is exactly the same?

WP:COI concerns conflict between personal interests of an editor and those of Wikipedia. Here I have no personal interests (not even presumable ones), and Wikipedia (meaning: its readers) would benefit from a clear and simple explanation of the decibel, reflecting the current state of the art. The WP:N refers to a topic deserving an article, but the decibel article de facto exists and no one has objected thus far.

In view of the TLDR specter and other work, I let this matter rest for some time, while remaining very interested in further comments. More reflection/discussion is clearly needed, but quality work cannot be hurried. Boute (talk) 07:09, 26 March 2012 (UTC)

A third view

(Please try and respond to each issue separately.) Fgnievinski (talk) 23:48, 26 March 2012 (UTC)

Boute: I appreciate all your effort in trying to clarify the existing confusion in doing calculations with decibels. But you have to understand that Wikipedia has to be somewhat conservative with regard to new approaches. The reason is that the average reader -- and editor -- is not sufficiently expert to be able to follow the argument. We have to rely on verifiability, not corrrectedness, as a criterion for inclusion in Wikipedia. See, e.g., the news coverage in Talk:Haymarket_affair. Fgnievinski (talk) 23:48, 26 March 2012 (UTC)

That being said, I think Boute's approach is not original research nor synthesis, since it's already been published in IEEE. Maybe one day most textbooks will adopt the multiplication sign as he advocates, but until then, one has to admit that it's still a fringe theory. And as such I think it deserves mention in the article, although without undue weight, of course. There could be a potential conflict of interest if Boute himself were to add a citation to his own IEEE article, so if other editors agree on the wording, someone else can make the addition. Fgnievinski (talk) 23:48, 26 March 2012 (UTC)

I would be against covering Boute's paper in the aticle unless it can be shown that it has been referenced in other peer reviewed works or discussed in RS books. From WP:FRINGE "Fringe theories may be excluded from articles about scientific topics when the scientific community has ignored the ideas", and "Fringe theories may be mentioned in the text of other articles only if independent reliable sources connect the topics in a serious and prominent way." Many scientific papers and patents are published every year, but we do not cover anywhere near all of them in Wikipedia. The test of whether to include novel ideas is notability, WP:N - how widely it is discussed in RS, and the test of whether a novel idea is mainstream or fringe is how widely it is discussed in scholarly papers. SpinningSpark 01:45, 27 March 2012 (UTC)

I think the extensive references collected in User:Boute/Decibel_draft are very valuable and could be integrated in the main article without much contention. Let me offer these suggestions: (1) a paragraph stating that decibels is often seen as confusing; (2) a sub-section for under Decibel#Suffixes and reference levels listing Satellite communication units, e.g., dB (re 1 Hz), dB (re 1 W/(Hz*K)), dB (re 1 m^-2), etc. Fgnievinski (talk) 23:48, 26 March 2012 (UTC)

I don't support saying decibels are confusing unless RS specifically state what the confusions are and how they are resolved. I would be fine with pointing out that decibels do not fit into the usual scheme of units and how they are different. Whether or not that causes confusion is POV - personally I find it quite self-evident how dB calculations should be handled regardless of the apparent contradictions. I'm fine with covering satcomm units, that's completely uncontroversial. SpinningSpark 01:45, 27 March 2012 (UTC)

Whether or not the concept of a decibel is confusing might be considered POV. But I think it is reasonable to claim that the way the decibel is used in *practice* is confusing. On request I can supply several examples where this point is made by serious scientists. Dondervogel 2 (talk) 15:59, 27 March 2012 (UTC)

Yes, please. The only way the article can sensibly discuss "confusion" and ways to reduce it is via what others have said in WP:RSs. Dicklyon (talk) 17:21, 27 March 2012 (UTC)

Here's a good example that mentions confusion on every other page, and originally published in 1998 in the journal Canadian Acoustics. I will look out for others. Dondervogel 2 (talk) 10:31, 28 March 2012 (UTC)

... and here are some more Dondervogel 2 (talk) 10:44, 28 March 2012 (UTC)

Thanks for these. What exactly are the claimed confusions? Can we agree on a specific wording? Then we can agree on where to place it in the article.Fgnievinski (talk) 18:35, 28 March 2012 (UTC)

Restricting attention to the Chapman & Ellis article, multiple confusions are claimed, due to: multiple uses (different physical quantities expressed in same units); used frequently without stating a reference value (without which it is "useless"); different reference values used in air and water for same physical quantity; unclear what weighting applied (or not) to correct for human (or other) hearing. Their conclusions include the statement

Some sensible acousticians have advocated abandoning the use of the decibel — which is partly to blame for our woes — in favor of the good old SI (i.e., metric) units for sound pressure, acoustic intensity, power, etc. Until that happy day dawns, let us include reference values with our decibels, so we don’t end up with fruit salad dBs.

I leave the choice of words (how to phrase all this) to others. Dondervogel 2 (talk) 16:42, 29 March 2012 (UTC)

I'd be in favor of quoting literally, which however lengthy, at least we won't be accused of synthesis. Let's say, at least one sentence, at most one paragraph, per reference. I'll start a new talk section to work on this text. Fgnievinski (talk) 02:36, 30 March 2012 (UTC)

Coming full circle, the handling of units in decibel calculations remains unstated in the article. I think it's better to acknowledge the omission having a stub section instead of silently ignoring the issue. Initially we could say as little as "units might require a more careful combination than the numerical values are handled.", and tagging it as a section-stub. Then later we'd discus how best to explain it. Fgnievinski (talk) 23:48, 26 March 2012 (UTC)

Since Boute's solution may be seen as overkill, can't we just say that "In decibel calculations, numerical values are added directly, but the original units are implicitly multiplied in the result." For example, in link budget we do: power spatial density in dB (re 1 W/m^2) equals transmitted power in dB (re 1 W) minus propagation distance squared in dB (re 1 m^2) minus 11 dB, where 11 dB = 10*log10(4*pi) is the spherical spreading factor. Fgnievinski (talk) 23:48, 26 March 2012 (UTC)

This is merely stating that a multiplication in a logarithm's arguments is an addition of the logarithms. The point is made in the lede of the article and also in the "Merits" section, but the article could benefit from an addition to the examples section. SpinningSpark 01:45, 27 March 2012 (UTC)

SpinningSpark, you are underestimating the importance of handling units correctly in decibel calculations. What the article states -- that the log of products can be re-written as a sum of logs -- only applies to the numerical values of a decibel quantity, not to their units. The article currently does not state how units are to be manipulated. Worse still, if the only rule stated is misinterpreted as being valid for both value and units, then it is incorrect. Fgnievinski (talk) 04:16, 27 March 2012 (UTC)

I agree with SpinningSpark - this is simply a case of keeping track of your units, as in any other calculation, whether that is in dB's or linear units. I don't understand what you think is missing, or what needs explaining, wrt handling of units. Most of the discussion on this page seems to be creating confusion where there isn't really any. Vague statements like "units might require a more careful combination..." only reinforce the idea that dB's are somehow complicated, when they really are not. Once the user understands the basic rules for manipulating logarithms, it's all pretty straight forward. And there is certainly no need to create a whole new system for dealing with logs (OR), based on an uncited magazine article, when that system is at-odds with very wide common usage.GyroMagician (talk) 21:27, 27 March 2012 (UTC)

Exactly, "keeping track of your units"! But how exactly do you do that? For example, try explaining how to combine 10 dB (re 1 W) of transitted power with 2 dB (re 1 m^2) of path loss to obtain 8 dB (re 1 W/m^2) of power density. The numerical values you simply subtract (8 = 10 - 2). The units you divide, in the original units (i.e., remove the log, divide, then restore the log). Clearly it's not the same rule. And the article only states the first one. The article does not make it explicit how to keep track of the units. Worse still, it might be misinterpreted as saying that units are to be added just like the numerical values (W + m^-2???). Which you could do with ordinary, non-decibel, units: you just divide the numerical values, and then correctly apply the same rule for the units. That's not what one does in decibels. Keeping track of the units in decibels is not much more difficult than manipulating the numerical values, but it is a different rule -- and it needs to be stated explicitly. Disaster happens if the only rule stated is applied blindly to both numerical values and units, as one could happily do with ordinary units. Fgnievinski (talk) 22:33, 27 March 2012 (UTC)

Fgnievinski led me indirectly to an article about decibels in Mathcad, which also shows on page 3 how the current practice in communications engineering can be expressed rigorously. Up to some syntactic inconveniences (apparently Mathcad supports only postfix syntax for functions), this also supports units. The rules that follow from the definitions on page 3 of the reference are obvious. Boute (talk) 09:12, 29 March 2012 (UTC)

"forward" gain of antenna

What does the "forward" mean in "forward gain of antenna", under "dBi"? I also looked in the 'antenna gain' article, but found no answer. 193.140.194.148 (talk) 12:54, 28 March 2012 (UTC)

The forward gain is the gain in the direction of the main antenna beam. SpinningSpark 14:18, 28 March 2012 (UTC)

Claimed confusion

Please list below any claimed confusions found in the literature. Quote literally; at least one sentence, at most one paragraph, per reference. Additionally, you may also want for offer a synthesis, however understand it may be disputed. Fgnievinski (talk) 02:56, 30 March 2012 (UTC)

In the Chapman & Ellis article [2], multiple confusions are claimed, due to: multiple uses (different physical quantities expressed in same units); used frequently without stating a reference value (without which it is "useless"); different reference values used in air and water for same physical quantity; unclear what weighting applied (or not) to correct for human (or other) hearing. Their conclusions include the statement:

Some sensible acousticians have advocated abandoning the use of the decibel — which is partly to blame for our woes — in favor of the good old SI (i.e., metric) units for sound pressure, acoustic intensity, power, etc. Until that happy day dawns, let us include reference values with our decibels, so we don’t end up with fruit salad dBs.

Other references: [3] (to be detailed)

More references in User:Boute/Decibel_draft (to be detailed).

Even more: [4], [5], [6], [7].

See "Possible Irritations" in [8]

More [9] (see citing -- not cited -- articles).

Satcomm units

These need to be sourced and described: dB (re 1 Hz), dB (re 1 W/(Hz*K)), dB (re 1 m^-2). There are more. Some references exist in User:Boute/Decibel_draft. Fgnievinski (talk) 03:00, 30 March 2012 (UTC)

Some work in that direction is certainly needed. Still, sourcing expressions like dB (re 1 W/(Hz*K)) will be difficult, since the "re-"convention appears only in some standards, but is rarely (if ever) used. More importantly, they make correct handling of units impractical.

Here is a representative example, assuming the traditional level unit (LU) view.

Cable losses are typically specified in dB/m (dB per meter) or similar (per foot, per mile). This seems OK, since for a cable of length x m we find (b dB/m) x x m = (b x x) dB. However, this dB/m is by no means translatable to dB (re 1/m).

Consider now free space "losses" (path loss) for field quantities, which are also expressed in dB/m, say, p dB/m (wavelength dependent).

However, field quantities along a lossy cable vary as e^{-bd}, whereas in free space they vary as 1/d. Therefore (p dB/m) x x m = (p x x) dB obviously yields the wrong result for path losses. On the other hand, the dB/m for path losses is translatable to dB (re 1/m)! There is no simple way (if any) out of this conundrum with the LU view.

The International System of Units brochure warns on p. 127 that viewing the decibel as a unit is contentious. Viewing the decibel as a scaling function, as done tacitly by satellite communications engineers and explicitly in Mathcad (decibels on p. 3) and in The decibel done right: a matter of engineering the math constitutes a simple and clear solution. Application to the conundrum yields the following.

Writing unit distance loss for exp -d propagation as (b/m) dB , hence (d x b/m) dB loss for a cable of length d.

Consistently, writing unit distance loss for 1/d propagation as p dB/m, hence d x (p dB)/m loss for path length d. Incidentally: if we write d as D dBm (m for meter, not milliwhat?), clearly d x (p dB)/m = (D + p) dB.

Remark: here field quantities were considered to enhance similarities but also to hide yet another problem with the LU view, namely, aggravation by the 20 log convention, already criticized 60 years ago, e.g., by Horton. This (additional) problem is most easily overcome by making explicit that the 20 log convention defines a different decibel, (for instance dB' as proposed independently by various authors).

To avoid the endless discussions the decibel has caused thus far, editors who still strongly adhere to the LU view should first of all show a way out of the dB/m conundrum with their preferred conventions. Boute (talk) 09:33, 30 March 2012 (UTC)

Nobody needs to fix or defend a system that's out there. We just need to report on it, and other viewpoints, with appropriate weight based on what's in reliable secondary sources. Dicklyon (talk) 15:42, 30 March 2012 (UTC)

Wikipedia is not the Flat Earth Society: it has a duty to its readers, and the burden of proof is on those who want to burden the readers with inconsistent material. Arguments towards escaping this duty are counterproductive. Note that Mathcad (decibels on p. 3) and The decibel done right: a matter of engineering the math are also "out there". The criterion for "appropriate weight" in scientific/mathematical matters is consistency, clarity, and explanation of empirical evidence (consistent actual practice), not statistics on sources, however numerous, that have been rendered obsolete by current practice. Basic knowledge of high school mathematics about exponents and logarithms (Algebra 2 level) suffices to apply these criteria, and hence to verify reliability. This is exactly what "Routine Calculation" in WP:NOR means. Inconsistent viewpoints may be reported, but their place is in the history section, certainly not as the primary exposition. Editors who keep evading the most pertinent question: "Should Wikipedia reflect current practice that evolved towards consistency, or material whose inconsistency is clear to any student who understood high school math?" have a genuine WP:conflict of interest with Wikipedia.

I humbly confess to stupidly falling into the trap of responding seriously to every artificial interpretation of the rules that actually harms the goals of Wikipedia. Others may have time for "editor games", but I will refrain from further comments until I finally see someone else do some serious work on improving contents. Boute (talk) 18:40, 30 March 2012 (UTC)

I support Dicklyons comment. The dBm continues to be the most widely used unit of level in the field of RF and microwaves. We do not need to justify its inclusion. On the contrary, it is those who seek to remove it or relegate it to history who need to provide the justification. I completely disagree with Boute's interpretation of "routine calculation". The exact wording is;

This policy allows routine mathematical calculations, such as adding numbers, converting units, or calculating a person's age, provided there is consensus among editors that the calculation is an obvious, correct, and meaningful reflection of the sources.

Consensus among editors can hardly be claimed when numerous editors have opposed such changes. What is proposed is a new method of expressing these (decibel) units contrary to numerous sources so is more than routine in the sense meant in the policy. SpinningSpark 19:35, 31 March 2012 (UTC)

These arguments are spurious. (a) The above technical discussion was not about dBm but about dB/m (decibel per meter). (b) None of the editors have disputed the routine calculations (Algebra 2 level anyway - see Exponents and logarithms). (c) Which changes have been proposed and opposed? Boute (talk) 05:03, 2 April 2012 (UTC)

Boute, appropriate weight is a matter of judgement, but it's hard to make the case that we should provide more than zero weight to a paper with a new approach, when the only known citation to that paper (according to Google scholar) is another paper by the same author (you). Given this situation, you really don't even have standing to keep bothering us about it, since you have a clear conflict of interest in promoting your own work. Get others to accept, refer to, and write about it elsewhere first, and then wikipedia will have secondary sources about the approach, to help us weigh it. Dicklyon (talk) 20:02, 31 March 2012 (UTC)

Be careful with accusations and the many misinterpretations of Wikipedia policy that are indicative of "editor games". WP:COI clearly says: Where advancing outside interests is more important to an editor than advancing the aims of Wikipedia, that editor stands in a conflict of interest. Now (a) I am not promoting my own work, but consistency, in the interest of the readers; (b) even if I were promoting my own work, accusers would have to demonstrate that I have a personal interest in doing so and, most importantly, that it is in conflict with the aims of Wikipedia.

Ceterum censeo: I would like to see some other editors doing some real work to give a clear account of the decibel to the Wikipedia readers. Boute (talk) 05:43, 1 April 2012 (UTC)

Having noticed the date, I would insist on serious work. Boute (talk) 05:46, 1 April 2012 (UTC)

From WP:COI; When in doubt, defer to the community's opinion". Citing your own work naturally puts you in a conflict of interest. It is not necessarily disruptive, but it is a COI situation. You should stand back and let others evaluate the contribution. SpinningSpark 08:55, 1 April 2012 (UTC)

For the record: nowhere in this entire discussion have I proposed citing my own work. To the contrary, I have given at least four other references, and I could give many more, but to what end if some "others" want the clock stopped around 1950? Note also that citations/contributions in Wikipedia carry no (favorable) weight whatsoever in the scientific community, since (even by policy) they are neither based on quality criteria, nor evaluated by experts (peer review). Hence no editor has any "personal interest" in contributing to a scientific topic to which no commercial consequences are attached. Finally, you have not demonstrated any conflict with the aims of Wikipedia. So please refrain from personal innuendo so I can carry out my intention of not losing more valuable time on this talk page, and wait until I see someone else make a constructive proposal. Boute (talk) 05:03, 2 April 2012 (UTC)

On the importance of consistency in an encyclopaedia article

An admirable (arguably essential) goal for any encyclopaedia article is clarity. In order to achieve clarity it is important to make unambiguous statements. Inconsistency leads to ambiguity, so ultimately the requirement for clarity leads to one for consistency. For an animal so well endowed with inconsistent use as the decibel (the proliferation of suffixes with this or that interpretation is but one example), the question becomes one of how to convey that inconsistent use in a clear and concise way. I don’t have a simple answer to the question, but it is right that it be debated. Boute is making a valuable contribution to that debate. Dondervogel 2 (talk) 11:10, 1 April 2012 (UTC)

I agree that the SpinningSpark's accusation of COI against Boute is unfair. Fgnievinski (talk) 02:56, 4 April 2012 (UTC)

That being said, Boute is being disruptive IMHO; his repetitive and long edits are not time conscious towards other editors. If I could ask this, Boute please keep your comments short (< 1 paragraph). We know your position. You don't need to repeat it ad nauseam. If you need more space please use your user talk page and add a link to it. Fgnievinski (talk) 02:56, 4 April 2012 (UTC)

<1 paragraph: (a) technically I was not repetitive but always brought new examples and references; (b) the only repetition (even characterized as such by myself) was the question many (still) seem afraid to face; (c) I announced my intention to refrain from further edits and to wait until I saw something constructive by others, but leaving unfounded accusations unanswered is usually an admission of guilt. Boute (talk) 09:26, 4 April 2012 (UTC)

Notation variants

The case raised by Boute of dB/m is perhaps the most vexing example of the confusion in decibel notation. There are three notations currently in use in the literature:

• fractions: "x dB/m" means x decibels of loss per meter of propagation distance (wired or wireless).
• SI reference level: "x dB (re 1 mW)" means x decibels with reference to one milli-watt; this is a consistent rule which leads to unambiguous interpretation but is seldom used in practice. It could be used in conjunction with fractions, e.g., "x dB (re 1 mW) per meter", albeit it's admittedly cumbersome.
• discipline-specific suffixes: sometimes the suffix is a unit symbol ("W","K"), sometimes it's a transliteration of a unit symbol ("uV" instead of μV for micro volt), sometimes it's an acronym for the units name ("sm" for m^2, "m" for mW -- although that can be the symbol for meters), other times it's a mnemonic for the type of quantity being calculated ("i" for antenna gain w.r.t. an isotropic antenna, "λ" for anything normalized by the EM wavelength). Sometimes the suffix is connected with a dash (dB-Hz), most of the time it's not. These may be consistent internally to a community but are inconsistent across different communities. They are widely used.

The label written just after the numerical value is not to be interpreted as multiplicative units, rather as a label, which can be more or less ambiguous depending on the context. But in any case they cannot be used as in standard dimensional analysis, i.e., units in decibels cannot be handled the same way as numerical values in decibels.

Boute's/Mathcad's interpretation of dB as a non-multiplicative scaling function in postfix notation (very much like degrees Fahrenheit being a whole different temperature scale, not just different units) is consistent but requires abondoning the useful summation of numerical values in decibels in favor of a multiplication sign. We don't want to throw out the baby with the bathwater. That means either rejecting the practice of summing arguments nor rejecting Boute's quest for dimensional consistency.

The comparison to Celsius and Farenheit is very illustrative for the scaling concept. Still, for treating dB as scaling, one does not have to reject the summing of arguments at all: writing (x dB) x (y dB) = (x + y) dB puts the addition where it counts, namely with the arguments. See also below (middle ground). Boute (talk) 10:09, 4 April 2012 (UTC)

I proposed a middle ground in which the addition of the decibels is retained, interpreting it as being applicable only for the numerical values, and acknowledging that units must be handled separately by different rules (most of the time simply multiplication of the original units).

IMHO the article needs:

1. general confusion to be quoted
2. satcomm units to be listed
3. notation variants to be described
4. dimensional analysis to be explained

Previously I saw no impediment for 1 and 2; now I am seeking your consent on 3; finally 4 will take more time to reach consensus. Fgnievinski (talk) 03:30, 4 April 2012 (UTC)

This is a workable programme. Boute (talk) 10:09, 4 April 2012 (UTC)

Glad to find some convergence.Fgnievinski (talk) 08:23, 12 April 2012 (UTC)

For point 3: as Dicklyon often remarked, current practice (since the 1960's) goes far beyond the original 1924 decibel (tightly tailored to telephone cables) and exhibits essential conceptual differences. Hence a choice has to be made. Boute (talk) 10:09, 4 April 2012 (UTC)

Can we leave that choice up to the reader? Is there any notation in use in the literature other than three described above? Also: any objections to including the block above ("• fractions, • SI reference level, • discipline-specific suffixes" -- up to "They are widely used.") under Decibel#Suffixes_and_reference_levels? Fgnievinski (talk) 08:23, 12 April 2012 (UTC)

The reader's interests (in both meanings) are indeed primary. However, the professional reader does not need a Wikipedia article unless it brings something interesting, and the reader who wants clarification is not served by perpetuating confusion. Even if the article should cover all interpretations floating around (why? to what extent?), it has to start with one that best serves those interests.

No objections against the block, although the classification may overemphasize lexical details at the cost of overall syntax and semantics. Also, the SI (see the Thompson and Taylor Guide and CEI/IEC 60027-3) does not allow " x dB (re 1 mW)", but only the following: prescribed is "L_P(re 1 mW) = 7 dB" or " L_{P/1 mW} = 7 dB"; tolerated is "L_P/mW = 7 dB" (1 omitted) and "7 dB (1 mW)" (space after dB obligatory). No "re" after dB. The notation (albeit clumsy - and not just lexically!) indeed allows unambiguous interpretation, but is a dead end w.r.t. current practice. Boute (talk) 07:10, 15 April 2012 (UTC)

For point 4, there is common middle ground that is formally correct in all interpretations. This is illustrated by the following example, reflecting the earlier noise calculation (Pratt and Bostian). Expressions like (dB) stand for "expressed in dB" (this should be square brackets or subscripting, both widespread in the dB literature, but these don't fit well in Wikipedia editing).

N (dBW) = kTB (dBW)

= k (dBW/K/Hz) + T (dBK) + B (dBHz)

= -228.6 + 27 + 44.8

= -156.8

The layout looks even better when turned sideways, as typically shown in link budget tables:

k (dBW/K/Hz): -228.6

T (dBK)  : 27

B (dBHz)  : 44.8

So N (dBW)  : -156.8

This symbolic punning (same form, conceptually different interpretations) is fully rigorous. Boute (talk) 10:09, 4 April 2012 (UTC)

Are you saying you're okay with the summation signs as long as there is an accompanying explanation for how the units should be handled? Fgnievinski (talk) 08:23, 12 April 2012 (UTC)

Yes, but there is more to it. Some convenient terminology to avoid lengthy circumscriptions: given Q := P/P' , we call the 1924 view the levels view, with defining equation L = 10 lg Q dB (plus linearity), and the view best reflecting current practice the direct view, with defining equation Q = 10 lg Q dB (nothing extra). Even those professing the levels view often (in fact, nearly always) write Q(dB) or Q_dB where they should have written L(dB) or L_dB (many examples on the web). With the direct view, Q(dB) and Q_dB would be fully correct.

Consider now the equality T (dBK) = 27. Allowing this ubiquitous abuse of notation in the levels view, T (dBK) is a pure number expressing a temperature level in dB (1 K). In the direct view, T (dBK) is also a pure number, but expressing the temperature itself in dBK. Formally, x dBK = (xdB)K = 10^x/10K and T (dBK) = (T(K))(dB) = 10 lg (T(K)) dB, as for function inverses. Hence, in T (dBK) the (dBK) first strips off the unit K, giving a number, and then converts that number in a number of decibels.

Finally, for any accompanying explanation of how the units are handled in either view, however evident even for high school students, some will insist on mentioning a source. Boute (talk) 07:10, 15 April 2012 (UTC)

Use with SI: Though not an SI unit, the decibel is permitted (as is the neper) for use alongside SI units, with the proviso that its symbol dB is free from suffixes, subscripts or other adornments. A statement like this needs to find its way into any rewrite, if only because it explains the cumbersome nature, previously noted by Boute, of "SI approved" notation. A similar comment applies also to ISO. Dondervogel 2 (talk) 17:34, 15 April 2012 (UTC)

Dimensional analysis

I found these quotes:

Not all scale units will change in this [standard] manner. A pressure measured in decibels does not fit this relation.

Occasionally units are defined that do not obey an equation of the form [above], for example the decibel in acoustics...

Not quite all units are additive: there are units like pH and decibels that are logarithms of ratios...

They might serve as useful citations. Fgnievinski (talk) 03:55, 4 April 2012 (UTC)

"1 joule = 1 watt per hertz"????

In section 5.5, the term dBJ contains the explanation that "1 joule = 1 watt per hertz". Isn't 1 joule equal to 1 watt per second, not 1 watt per hertz? (I'd make the change myself if it hadn't been so long since 6.002...) -- Dan Griscom (talk) 21:21, 31 May 2012 (UTC)

A joule is a watt-second, not watt/second. SpinningSpark 23:51, 31 May 2012 (UTC)

And a watt second is the same as a watt per Hz, which is why joule can be a unit of power spectral density. It's usually just called a watt per hertz though. Dicklyon (talk) 00:11, 1 June 2012 (UTC)

I agree that there is no physical difference between a watt per hertz and a joule; they are one and the same thing. But to express a power spectral density in units of dB re 1 J is confusing. Much clearer to write as dB re 1 W/Hz. Dondervogel 2 (talk) 10:40, 3 June 2012 (UTC)

Base 10^x and y*log₁₀ multipliers are missing in the article!

Which base multiplier 10^x and log multiplier y*log₁₀ that shall be used is not mentioned in the article in a clear and consistent manner.

This issue has been mentioned earlier under the heading "Satcomm units"

"Remark: here field quantities were considered to enhance similarities but also to hide yet another problem with the LU view, namely, aggravation by the 20 log convention, already criticized 60 years ago, e.g., by Horton." by Boute (talk) 09:33, 30 March 2012 (UTC)

And he has worked towards a new article: User:Boute/Decibel draft (last edit 24 March 2011‎)

Where the confusion issue is mentioned under User:Boute/Decibel draft#Writing (and reading) expressions with decibels:

"Some caution is needed: "Knowing when to multiply by 10 or 20 in such calculations is an endless source of confusion"

Going from linear unit to dB:

$G_{\mathrm {dB} }=y\times \log _{10}{\bigg (}{\frac {linear~unit\times 10^{-x}}{1}}{\bigg )}=..~\mathrm {dB} \,$

Going from dB to linear unit:

$G=10^{x}\times 10^{\frac {..~\mathrm {dB} }{y}}=..~linear~unit\,$

Which base x and y*log₁₀ to use has to be mentioned! Electron9 (talk) 14:46, 9 June 2012 (UTC)

I think the article is clear. You use y = 10 for power ratios and y = 20 for field-quantity ratios. The "linear unit" is taken care of by what your reference level is. So for dbm, the "m" means a ratio relative to milliwatts. Watts are power, so you use 10. If further clarification is needed, lets talk here about how to do it, instead adding fragments with novel notation. Dicklyon (talk) 16:18, 9 June 2012 (UTC)

When it says "dBm" in other documentation how do you know if it refers to mV, mW, mA etc..? and is the rule y=10 for power ratios and y=20 for anything else without exception? and if information lacks on say mV vs mW then the value of y becomes undefined. Electron9 (talk) 16:40, 9 June 2012 (UTC)

dBm is referred to mW because that is the convention. One doesn't know unless one knows the convention. It isn't "y=20 for anything else", it is for field quantities. That is, the vector field quantity of field strength, anything directly proportional to it such as flux density, or scalar integrals of them over a metric such as voltage and current. There really is no need to single out μV and mW which are fairly obvious. If those ones need clarifying then every single entry needs clarifying. We certainly don't need an example or formula in each one. Possibly a table might be better if clarification is really needed. SpinningSpark 16:58, 9 June 2012 (UTC)

The convention needs to be stated explicitly. One doesn't know out of the blue, particularly if one is in a hurry. I hope the formula above 10^x*10^(dB/y) is general enough to cover all units used in the article. Electron9 (talk) 18:58, 9 June 2012 (UTC)

In what way is expalaining dBm as "dB(mW) – power relative to 1 milliwatt" not explicit? SpinningSpark 21:40, 9 June 2012 (UTC)

There are other units without statement on base. And in this case dBmW still misses log specification (ie 10log or 20log). Electron9 (talk) 01:24, 10 June 2012 (UTC)

I started to add explicit statements in a couple of sections. It's not clear to me that it's a good idea to continue, as it does get repetitive. Dicklyon (talk) 03:35, 10 June 2012 (UTC)

Yes, I agree, explicit statements are only needed where the reader may not know whcih is the correct group. dBm is not such a case since it already explicitly states its reference is 1 mW which is clearly a measure of power. More radically, we could reorganise the article into two groups: power quantities and field quantities. SpinningSpark 10:24, 10 June 2012 (UTC)

I don't think making a distinction between "power quantities" and "field quantities" would help, because the only difference between them is that one is a squared version of the other. In other words, the 20log rule applies to *any* power quantity if you first take its square root. I would prefer to see the confusion addressed by using the 10 log (power) convention consistenly throughout the article, and then pointing out that the same quantity can also be written 20 log sqrt(power), without changing the meaning one iota. Dondervogel 2 (talk) 02:22, 11 June 2012 (UTC)

That's what I used to think, too. The decibel to me was always defined in terms of power ratios. But the standards docs now make the distinction with field quantities, even making them primary in the definition, as the citations show, so it might be hard to just ignore that. Dicklyon (talk) 02:34, 11 June 2012 (UTC)

The standards I know have dropped the term "field variable" and replaced it with "root power variable" (or similar - I forget the precise words). What is the difference between "power" and "square of the square root of power"? Dondervogel 2 (talk) 02:40, 11 June 2012 (UTC)

OK, then. I shall be happy to learn that my knowledge of the standards is obsolete. Dicklyon (talk) 03:09, 11 June 2012 (UTC)

Standards are good, everyone has their own.. :-) Electron9 (talk) 23:09, 11 June 2012 (UTC)

use for things other than power?

The article says "usually power or intensity". I challenge the writer of this to find a use of dB for anything other than power. As far as I know (been doing electrical engineering for 15 years) dB is only a power ratio unit. (And this makes it distinct from the neper, which is a generic logarithm unit). Jurgen Hissen (talk) 08:25, 29 December 2012 (UTC)

Table suggestion

Going from linear to dB:

$G_{\mathrm {dB} }=y\times \log _{10}{\bigg (}{\frac {G\times 10^{-x}}{1}}{\bigg )}=..~\mathrm {dB} \,$

Going from dB to linear:

$G=10^{x}\times 10^{\frac {G_{\mathrm {dB} }}{y}}=..\,$

Unit	Base multiplier (x)	Logarithm divisor (y)
dBμV or dBuV	-6	20
dBm or dBmW	-3	10

Electron9 (talk) 23:10, 11 June 2012 (UTC)

Seems like a novel (read "bad") idea. If one were to use a table, it would list reference levels, with their units, and the 10 or 20 factor, I think, depending on whether the units are power-type quantities or not. This 10^x nonsense is just off. What is 1 nV in dBuV? No 10^6 involved there, just the ratio of nV to uV. Dicklyon (talk) 05:50, 12 June 2012 (UTC)

1nV is -60dBuV. It is the ratio of the powers represented by 1nV versus 1uV. I quite agree with Dicklyon's objection. The units must always be mapped to power (at least mentally) before converting to dB. Jurgen Hissen (talk) 08:30, 29 December 2012 (UTC)

Acoustic section confusion

In the acoustic section it mentions that the loudest sound intensity we can hear that causes permanent damage during short exposure (120 dB) is 1 trillion times the quietest sound we can hear. This is confusing, seeing as the loud sound pressure would be 1,000,000 times the quiet sound pressure (not 1 trillion), if we are measuring in dB SPL (which is almost always used when describing sound) since dB SPL = 20 * log10( A₁ / A₀ ) where A₀ = the reference level ( 20µPa ).

The sound intensity article also says that: "Note 1^ : The term "intensity" is used exclusively for the measurement of sound in watts per unit area. To describe the strength of sound in terms other than strict intensity, one can use "magnitude" "strength", "amplitude", or "level" instead. Sound intensity is not the same physical quantity as sound pressure. Hearing is directly sensitive to sound pressure which is related to sound intensity. In consumer audio electronics, the level differences are called "intensity" differences, but sound intensity is a specifically defined quantity and cannot be sensed by a simple microphone. Sound intensity is not valuable in music recording."

The 1 trillion should be changed to 1 million since sound pressure level is the more relevant measurement and what the ear senses (as well as what colloquial unlabeled "dB" is referring to).

2602:306:CFC9:E390:E4BF:81E1:83C6:3787 (talk) 06:12, 17 August 2012 (UTC)

I have no idea what that rambilng paragraph is trying to say, but intensity is a power type quantity, proportional to the square of pressure. A million-to-one pressure ratio is a trillion-to-one (that's 10¹²) intensity ratio; both are 120 dB. Dicklyon (talk) 06:27, 17 August 2012 (UTC)

Sound pressure (which the acoustic section links to) is a field quantity and is almost always measured that way, so why include the intensity (power quantity) measurement? It would be less confusing to keep everything in field quantities when referring to acoustic measurements 2602:306:CFC9:E390:E4BF:81E1:83C6:3787 (talk) 06:47, 17 August 2012 (UTC)

It's certainly not unusual to compare sound levels by intensity, too. See? Dicklyon (talk) 05:29, 24 August 2012 (UTC)

dBc

CURRENT TEXT: dBc – relative to carrier—in telecommunications, this indicates the relative levels of noise or sideband peak power, compared with the carrier power.

SUGGESTED NEW TEXT (removed the word "peak") dBc – relative to carrier—in telecommunications, this indicates the relative levels of noise or sideband power, compared with the carrier power.

217.76.85.199 (talk) 12:36, 10 October 2012 (UTC)

Volume

This article really should mention volume somewhere, since I'm fairly sure that according to most people's understanding decibels are the standard unit for measuring volume levels. In fact, even once you realise it talks about 'loudness' instead (are the two strictly synonymous?) it seems oddly hazy on how the unit relates to perceived loudness. Without digging in too deeply, I'm not seeing an answer to the basic question 'is a sound that is ten decibels louder perceived as ten times as loud, or more like 3.162 times as loud?' I'm thinking ten times, given it's ten times as powerful, but ten minutes' reading of this and related articles hasn't totally convinced me one way or the other. Is there a clear answer to this? --Oolong (talk) 00:14, 3 November 2012 (UTC)

"Volume" is not a standard concept in sound. Roughly speaking, a mid-frequency sound 10 dB higher is perceived as a bit more than twice as loud. But dB is not a measure of loudness. See loudness. Dicklyon (talk) 00:38, 3 November 2012 (UTC)

Neper paragraph

The paragraph that mentions the Neper scale does not have an explicit topic sentence. It consists of only two tangentially-related sentences. Using two sentences in a paragraph is usually considered bad form. I believe I was fixing the issue by adding the (semi-obvious) implied topic sentence but it was reverted. How can this issue be improved? Jason Quinn (talk) 03:18, 8 November 2012 (UTC)

The trouble with "Any other valid logarithmic base could be used for an alternative logarithmic scale" is that it's not clear what it means by "valid", or why anyone would consider such alternatives. It's a sort of good point, if you can source it, but as a made-up topic sentence, I don't really see that it improves the paragraph. Dicklyon (talk) 05:24, 8 November 2012 (UTC)

It is perfectly clear what is meant by "a valid base" to anybody who knows what a logarithm is. Logarithms are background material perfectly justified assumed to be known to the readers of this article. No source ought be required and the request hints of gaming the system. It's akin to asking for a source for the domain of the sine function on the harmonic motion article or for a source when somebody wants to use "57=50+7" in an article. It's just not the proper place. The statement under question is no less vague than statements like "an arbitrary base", "among all choices for the base", and "logarithms to other bases" (among many others similar statements) used abundantly on the logarithm article without issue. Saying "a valid base" is too vague to use feels purposely obtuse. For what it's worth, other rephrasings are perfectly acceptable too:

• "An arbitrary logarithmic base could be used for an alternative logarithmic scale"
• "Any acceptable base logarithmic base could be used for an alternative logarithmic scale"
• "Other bases could be used for an alternative logarithmic scale"

I addressed a very specific concern above: that the paragraph has no topic sentence and the two sentences present are not well linked. You either agree with that or you do not. If you disagree, fine. Just say so and explain. If you agree, then please tell me what the implied topic sentence is that links the ideas presented in the two sentences. I suggest you'll have trouble coming up with something different than my own sentence, so please do not label my sentence as "made-up".

I have given two grammatical reasons why the paragraph is currently bad and how the inclusion of the sentence improves it. It was a small lateral change. It's very disappointing that it was even reverted in the first place. I also question how independent your idea that the phrase is "vague" from the first reverter's. To me it seems like it provided an anchor bias to the discussion. In any case, this is the type of over-protective article "ownership" that is giving Wikipedia a bad reputation. Twice in a short period of time while I was attempting to improve this article I get the dreaded "edit conflict", which ended up derailing me from bothering to continue improving the references here. And over what? Nothing worth the time. Jason Quinn (talk) 08:15, 8 November 2012 (UTC)

Whoa, slow down with the accusations. The dreaded "edit conflict", while very annoying, is a result of two people trying to improve an article at the same time. But you know that. Dickylon and SpinningSpark are very regular contributors, but I have never found them obstructive. Challenging, often, but that only improves what we all write.

I think maybe we have a mathematician/engineer disconnect here. It is true that a logarithm can be taken to any base you fancy. As a physicist, I could guess what a "valid" base might be, but I would be left with the impression the phrase has some formal meaning that I missed - so while the phrase may be correct, I don't like it. Of the options you suggest, I prefer "Other bases could be used for an alternative logarithmic scale". However, I don't think this article is the right place to talk about all possible log scales (that's what the logarithm article is for). Decibels are very widely used for practical measurements. It is interesting to note that natural and binary logs are also sometimes used, so that is relevant here. I think that is as far as we need to go, unless there are other bases in actual use? GyroMagician (talk) 09:38, 8 November 2012 (UTC)

I find it a little unsavoury being accused of article ownership, but sadly all too common from relatively inexperienced editors when they are not getting their own way. It is entirely Jason Quinn's own fault that he got edit conflicted twice. If he had followed WP:BRD there would never have been a second reversion. Trying to get your own way by edit warring and then complaining you are being reverted is a bit rich. After being reverted by two different editors you should at least consider the possibility that you might be wrong and discuss on that basis.

On the substantive issue, "any valid logarithmic base" will not do. Logarithms can validly be taken to complex and imaginary bases, but these are completely useless for our purposes. In an article about units, we should describe the bases that are actually used as units, not some theoretical mathematical possibility. That means base 10 and base e; base 2 logarithms certainly have many uses in science and engineering, but I am not sure that they are used to measure a level in the sense that decibel is and hence should be excluded.

Grammatical issues, while of concern, take second place to the actual content of the article. The required content should not be expanded, contracted or mangled in order to comply with some perceived grammatical rule. However, in my view the topic of the paragraph is "decibels use logarithms to base 10" and the first sentence of the paragraph thus counts as a topic sentence. SpinningSpark 11:50, 8 November 2012 (UTC)

I can at least see what Jason is trying to get at here. Basically the definition of the neper is included to further define the decibel by association and exclusion, differentiating it from logarithmic units with other bases, and also because the neper has a similar application. The distinction is really only vaguely necessary, in my opinion, and I am not sure the location of the relational mention is appropriate. If the problem really needs a solution, I am more one to question that paragraph's inclusion at all. If the reader wants examples of other logarithmic units, the neper is already included at Logarithmic_unit#Example_scales and in the SI Units navigation box. The first sentence ("The definitions of the decibel and bel use base 10 logarithms.") is also redundant because the definition is already stated: "A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities. A decibel is one tenth of a bel, a seldom-used unit named in honor of Alexander Graham Bell." While the relation between the two units is logical, I find its location there extraneous and I don't really see a compelling reason to keep it unless the paragraph is altered to make its purpose more self-apparent. This article is indeed about the decibel unit, and including the definition of another, although similar, unit in the opening section is a little peculiar. Radiodef (talk) 16:31, 8 November 2012 (UTC)

It seems clear to me that the neper needs to be discussed, as a different scaling of the same kind of measurement – but probably not in the lead, which is supposed to summarize the high points of the article, not add new tangents. If a topic sentence is needed, something more factual, rather than conjectural would be good, like maybe "Power ratios are something quantified with logarithms to other bases." So add a section on neper. Dicklyon (talk) 17:49, 8 November 2012 (UTC)

Uses and Suffixes and reference levels seem like appropriate places to me. It should also probably be added to the See also, where the neper article already makes the respective inclusion. Radiodef (talk) 19:45, 8 November 2012 (UTC)

I am not inexperienced, Spinningspark, and this has little to do with "getting my way". In some sense, however, this is the discussion you wanted when you reverted. And it's the hurdle that an inexperience editor would have to jump if they wanted to keep their change. Do you think they would bother? An inexperienced editor would have already left the project.

Please do not cite essays like gospel because essays can conflict with each other. If you want to cite Wikipedia:BRD, allow me to cite the essay Wikipedia:Revert only when necessary and the related help page Help:Reverting. According to these, you should generally not be reverting (in your case "undoing") good faith edits in the first place. If you do, it asks you to explain your revert, which you did in your edit summary but in a way I deem unfair. Your misquote of "valid logarithmic base" to "valid logarithm" changes a clearer phrase to a vaguer one which makes it unfair to accuse my edit of being "vague" when you added vagueness to it. Ultimately, I believe, it had the end result of anchor biasing the discussion. I maintain that the phrasing is not vague (and yes bases like $\pi$ or even complex bases could be used) but even if it is slightly vague, I fail to see why such a small change in a non-good status article was warranted a revert in the first place, especially considering how much there is to improve here. Is the quality of this article really so high that an almost ambiguous or imperceptible changes in quality deserve reverts? If so, lets get good or featured article status ASAP. Upon the second revert of my change — also without discussion, I point out — I began a discussion here and posted clear issues prompting my change. Those points were mostly sidestepped. Accusing me of "edit warring" (after a single revert nonetheless) when I am the first to start the discussion is flat out wrong. My claims of obstructionism and article ownership are too strong and I wish I had worded it differently but I think that there's at least a hint of that here. I am curious what would have happened had I not initiated the discussion. Would either of my reverters have done the favor in kind?

Back to the paragraph: I see benefit to pointing out that other bases can be used. If you don't want to do that, a rewrite of the paragraph to "The neper, an alternative logarithmic ratio unit sometimes used, uses the natural logarithm (base e).", where the first sentence has been dropped as you suggest, or "The neper, an alternative logarithmic ratio unit sometimes used, uses the natural logarithm (base e), whereas the definitions of the decibel and bel use base 10 logarithms." solves the topic sentence problem (the topic sentence is now explicit and merely about the existence of the Neper scale.) but at the expense of having a one-sentence paragraph. I still prefer my original version. Jason Quinn (talk) 18:55, 8 November 2012 (UTC)

Please refrain from furthering the needless personal back and forth. As you've stated, only a few reverts have been made. There's no ownership going on like you asserted at the beginning and there is only a conflict if the insinuations continue to heat up. Edits have been made, edits have been reverted, discussion is taking place, policy is being followed and it would be great if we could continue along that track. If other contributors could refrain from perpetuating this irrelevant tangent, that would also be great. Radiodef (talk) 19:55, 8 November 2012 (UTC)

The neper is not mentioned in the article because it's also a logarithmic unit; it is mentioned because it has parallel applications for measuring physical quantities. That is the source of confusion that I believe GyroMagician has pointed out with his mathematician/engineer comments. I've also suggested the article fails to make the clarification. The relation is only implied ("an alternative...sometimes used"), and that is probably why you've observed that the inclusion of the definition seems grammatically awkward. Other contributers have disagreed with your revision because, while true that a logarithm can have any base, your addition is a fact about the mathematical concept of the logarithm, not the decibel unit of measurement. If that paragraph is to be kept, the phrase could be more effectively changed by appending something like "an alternative logarithmic ratio unit sometimes used for related applications". However, I think the solution here is to more clearly state the reason for the neper inclusion and move it to somewhere that's more relevant to the topic of the article. Radiodef (talk) 20:42, 8 November 2012 (UTC)

Why the logarithm with unspecified base is essential to the traditional definition of the decibel

First, a sobering observation. The reactions to the contributions of Jason Quinn (talk) are part of a familiar phenomenon. The ownership reflex is a definite and serious problem throughout Wikipedia. Often "experienced" is confused with long-standing inability to learn and improve. For Wikipedia, improvement means better serving the readers, not playing the Wikipedia game. Typical "experienced" players keep enforcing the Wikipedia guidelines according to their own counterproductive interpretation to obstruct "inexperienced" editors (and even seem to take pride in doing so). One can also verify that, in the discussions, such players never contribute any literature references or other reliable sources about the subject matter itself. The Wikipedia game is the reason why potential editors with expertise in the subject matter gradually lose interest in Wikipedia.

Anyway, the remarks by Jason Quinn (talk) about the base are crucial to the traditional definition of the decibel. The paper by Martin The Transmission Unit and Telephone Transmission Reference Systems and by Hartley ("The Transmission Unit", Electrical Communication, Vol. 111, No. 1, pp. 34-42, July 1924) clearly define the decibel as log 10^1/10 with base essentially unspecified. This is why Horton (in "The Bewildering Decibel", Electrical Engineering Vol. 73, No. 6, pp. 550-555, June 1954) emphasises that the (traditional) decibel is not a dimensionless unit, but reflects a fundamental dimension. Later interpretations that dB is a pure number (equal to 1/10 in the bel-centered definition or 0.1151293 in the neper-centered definition) fail to grasp this and are mathematically inconsistent (they make dB into a comment, which is not mathematically meaningful). Note: the Martin reference is ready for downloading, the others are more difficult to find, but can be obtained by sending me an e-mail. Boute (talk) 08:31, 21 November 2012 (UTC)

I'll ignore your first paragraph. Without reading too closely, the Martin paper appears to very clearly define the logarithm as base ten. In the first paragraph of the section "Definition of the Transmission Unit", Martin states "The number of transmission units corresponding to the ratio of any two powers P1 and P2, is then the common logarithm (logarithm to the base 10) of the ratio P1/P2, divided by 0.1." The equation in the middle of page 402 is a conversion from a logarithm of arbitrary base to base 10. The numbers are clearly chosen to work with a base 10 log - the ratio 10^0.1 is not accidental.

But this is hijacking the original discussion. The original point was that, while logarithms can be taken to all kinds of different bases, the decibel is base ten. This article is about the decibel. The neper is relevant, in that it is another unit used for similar purposes. The logarithm article is a suitable place to discuss logarithms to different bases, not here. GyroMagician (talk) 15:40, 21 November 2012 (UTC)

Both Martin's formulas (page 402) and Hartley's formulas (identical) must be read closely since there is a subtlety: the difference between log₁₀ and log without subscript is crucial. As you say, the "number of transmission units" is given by $N=10\log _{10}\textstyle {\frac {P_{1}}{P_{2}}}$ (with subscript, base 10). However, the transmission unit (or decibel) itself is the denominator in

$N={\frac {\log {\frac {P_{1}}{P_{2}}}}{\log 10^{0.1}}}$

namely, log 10^0.1, and has unspecified base (log 10^0.1 is not just 0.1). Of course, numerator and denominator must have the same (unspecified) base. There is some abuse of notation, easily corrected in various ways (e.g. by Hartley, using an explicit but arbitrary base in a later paper "A new system of logarithmic units", 1954).

Whether or not the neper should be mentioned here depends on the extent to which it is desirable in this article to reflect the ISO standard (ISO 800000-3, "quantities and units, Part 3: Space and Time, 2006-03-01). There the neper is defined as the coherent unit (1 Np = 1) and the decibel as a derived unit satisfying (20 lg r) dB = (ln r) Np (where r is a ratio of root power quantities), hence 1 dB = (ln 10)/20 Np = 0.1151293 Np = 0.1151293. As said earlier, and explained in the comments on the SI (by Mills et al. in Metrologia, by Thompson et al. on the BIPM website), this makes 1 Np and 1 dB pure numbers with the symbols Np and dB as mere comments. However, this deviates from the original unspecified base definition, and comments are not enforcable mathematically in a sound way, so I consider it rather poor design. Anyway, the definition of the decibel as a "unit" in the standard depends on the neper. Boute (talk) 13:32, 22 November 2012 (UTC)

The transmission unit is not the denominator, it's the whole fraction. Let's apply a little basic manipulation to the formula in question:

$N={\frac {\log _{b}{\frac {P_{1}}{P_{2}}}}{\log _{b}10^{0.1}}}=10{\frac {\log _{b}{\frac {P_{1}}{P_{2}}}}{\log _{b}10}}=10\log _{10}{\frac {P_{1}}{P_{2}}}$

where b is an arbitrary base. The final result looks a little more familiar. What you are saying is that any base may be used, so long as the final quoted number it is converted to a base-10 logarithm, using the base-changing identity. There's no magic or slight-of-hand here, it's simple. GyroMagician (talk) 17:59, 22 November 2012 (UTC)

Your basic manipulation is exactly Hartley's 1954 variant I described above and evidently leads to the earlier $N=10\log _{10}\textstyle {\frac {P_{1}}{P_{2}}}$ (with subscript, base 10). Of course it would be intolerable if the symbol N in all formulas I gave earlier did not always consistently denote the same number! No one said there is more to it than high school math, but confusing (or identifying) Hartley's 1954 explicit but arbitrary base formulation with the unspecified base formulation (whose rigorous formalisation is quite different) would miss the subtlety.

However, it is incorrect thinking that "the transmission unit is not the denominator, it's the whole fraction". Indeed, the whole fraction equals N, the number of transmission units (like "the number of meters"), the numerator log P₁/P₂ is the logarithmic quantity of interest, called level difference, and the denomonator log 10^0.1 is the transmission unit (or decibel). Analogy: for length, the number of length units N = L/u where L is the length and u the unit. The pun with L also suggesting level difference is intentional. Boute (talk) 19:25, 22 November 2012 (UTC)

Then I think I'm missing the subtlety. To quote Martin "The transmission unit (TU) has been chosen so that two amounts of power differ by one transmission unit when they are in the ratio of 10^0.1." Base-10 is right there. The only case where I have seen the logarithm written to a base other than 10 is the above quoted equation, which is a conversion from an arbitrary base to base ten. Without the conversion, the units would not be dB. I don't see how you could argue that this means Bels are defined to an arbitrary base. I'm not sure I see your point. GyroMagician (talk) 22:11, 22 November 2012 (UTC)

The phrase "chosen so that" indicates a desired effect and the full sentence does not define the TU. Regarding the "N-formula" of the form N = L/u (where the unit u = TU = dB), the point (Martin's, Hartley's and Horton's, not mine!) is that one wants to distinguish the TU (or decibel) as a unit from anything else. The formula can also be rewritten via the level difference L (and Martin/Hartley's correctable abuse of notation)

$L=\log \textstyle {\frac {P_{1}}{P_{2}}}=(10\log _{10}\textstyle {\frac {P_{1}}{P_{2}}})\cdot \log 10^{0.1}=(10\log _{10}\textstyle {\frac {P_{1}}{P_{2}}})\mathrm {dB} =N\mathrm {dB}$

Martin's sentence "the transmission unit is a logarithmic measure of power ratio and is numerically equal to log 10^0.1 " (and a similar sentence by Hartley in his 1924 paper) tries to express this non-rigorously by leaving the base unspecified (not showing it as a parameter). Using a specific base (like base 10) would ruin this scheme since log₁₀ 10^0.1 = 0.1, just a number. So does the ISO standard by effectively choosing the base e² (to make the neper the coherent unit), resulting in

${\text{dB}}=\log _{{\text{e}}^{2}}10^{0.1}=(\ln 10)/20=0.1151293$ --- which fails to distinguish dB from numbers!

Note that $N=10\log _{10}\textstyle {\frac {P_{1}}{P_{2}}}$ holds in all formulations, including the "functional" definition (x B = 10^x and x d = x/10, so x dB = 10^x/10). This explains why these subtleties always remain hidden in practice - unless one wants to define what the dB itself actually is! If this explanation does not fully clarify the matter, I can send you some more detailed papers (Hartley, ISO, surveys on dB). Boute (talk) 09:37, 23 November 2012 (UTC)

Addendum: the opposite of more detail is the bottom line: to calculate the number of decibels corresponding to a given power ratio, you use base 10, as in $N=10\log _{10}\textstyle {\frac {P_{1}}{P_{2}}}$. The subtleties only come in when defining the precise semantics of the symbol dB. Boute (talk) 09:47, 23 November 2012 (UTC)

Apology: a source of confusion may have been the loss of a factor 10 in some formulas by copy/paste laziness. I have now corrected this in (hopefully) all formulas where this happened. Boute (talk) 10:00, 23 November 2012 (UTC)

Simplest possible explanation of the preceding observations re. Martin, Hartley, Horton etc.

Let the level function L from positive reals to reals be specified (not fully defined!) by

L(b^x) = x L(b) for any positive real b and any real x.

One can prove that L(a b) = L(a) + L(b) for any positive real a and b. Now the traditional decibel as a unit is "defined" by (not fully, preserving the "secret code") dB = L(10^0.1). If p is a ratio (say, P₁/P₂) of power quantities, then the level difference is defined as L(p) and one can prove that L(p) = (10 log₁₀ p) dB. I will not be able to follow further discussion for a few weeks. Boute (talk) 11:49, 25 November 2012 (UTC)