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Practical Significance

Hi, I came to this article to get an idea of the approximate loudness of various decibel values e.g. perception threshold, whispering, shouting, a jet engine, etc. I realize this article is aiming for scientific validity, but doesn't this merit inclusion somewhere? Pinochet (3) (talk) 02:54, 31 December 2010 (UTC)

Look at the "main" link in the Acoustics section. Dicklyon (talk) 04:55, 31 December 2010 (UTC)

i had same problem in locating the right article, so i added to the hatnote: This article is about the ratio of measures. For sound or acoustic level, see Sound pressure. For other uses, see Decibel (disambiguation). Petersam (talk) 21:07, 12 June 2011 (UTC)

"also 20log(optical power) which is why it's relevant"

Where in the cited text is the decibel used for 20*lg(optical power)? Dondervogel 2 (talk) 14:32, 6 February 2011 (UTC)

More to the point, the cited text does not support the claim that CCD volts are (or can be) proportional to optical power, which is what lies behind the alleged confusion. Most texts I've looked at give a relationhsip between stored charge and incident light flux but do not venture to say what this means in terms of readout voltage. SpinningSpark 16:10, 6 February 2011 (UTC)

It's not the ideal source, since the linear conversion of intensity to voltage is not often mentioned at the same time of the use of that voltage in the dB calculation, but you'll find it a few pages on, on p.130, where it says that a CCD is linear. In general, CMOS and CCD and other detectors get a very linear conversion of optical power, or photon count, to charge, and then a fairly linear charge-to-voltage conversion. Attempts to use a more compressive conversion for better DR have not been very successful, so most sensors are still essentially linear, and when they're not, their DR in dB is calculated after linearizing. Dicklyon (talk) 02:44, 10 February 2011 (UTC)

Here's a good one. See section. 3.4.2.1 defining dynamic range as 20*log(electron count ratio), thereby sidestepping the assumption of linear conversion via voltage. See also p.83, 3.4.2.6 where it mentions that photon-to-electron conversion is inherently linear. And realize that optical power is proportional to photons. Then it all follows. Maybe this is a better source, but it still requires looking at several pages. Dicklyon (talk) 02:54, 10 February 2011 (UTC)

Thank you for the explanation. My immediate reaction is that this use seems controversial and that the article can be improved by stating so explicitly. I need more time to understand the implications better. Dondervogel 2 (talk) 14:52, 10 February 2011 (UTC)

It has always seemed a bit crazy to me, but I don't know any sources that treat it as controversial. I just tried to explain it as it is. Dicklyon (talk) 15:27, 10 February 2011 (UTC)

Definition and power ratios again

If the definition of decibel includes "the ratio of two power quantities", then why does an RF power meter output a measure in dBW (dB-Watts) or dBm (dB-milliWatts)? What is the ratio? If it's a ratio, then wouldn't the Watts units cancel out and leave the output unitless (and it wouldn't matter if it's 'milli' or not)??? .... This is a loaded question because I already have the answer, most of us were taught incorrectly, and even respected dictionaries and other references carry on this misconception that a decibel is a ratio. It is not. The decibel is (in it's many forms as evidenced by all the entries in this article) simply a mapping from a logarithmic to a linear scale that may or may not be unitless and may or may not include ratios at any step of the process, nothing more, nothing less. Even the 'logarithmic scale' article in Wikipedia incorrectly defines 'logarithmic scale' in terms of ratios... what a sad state..... As an electrical engineering student I was taught that dB is the log of the ratio of outputs over inputs (voltages or powers), but that definition is not strictly true, it is only the common use in my field. Many other engineers I have met have the same, similarly learned, misconception. This skewed view of this concept hides it true elegance and usefulness as a purely mathematical tool that enables all KINDS of great things. The logarithm itself was discovered/developed by John Napier in the early 17th century and was used in the first slide-rule-like tool that he created. This tool was heralded as a great advancement and enabled rapid advancements in engineering, navigation, and even astronomy. I read a quote somewhere that said that Kepler himself used the new tool to reduce years of calculations to months. NASA engineers in the Apollo program can be seen in old videos using slide-rules! Napier's invention literally took us to the moon! The typical use of the decibel in adding/subtracting dB numbers instead of having to multiply and divide the original numbers is simply an abbreviated slide-rule calculation. Getting back to the real definition, independent of "ratios", will simplify the concept for those trying to learn and apply it and hopefully slowly remove all the confusion out there as people are re-educated. So, other than having time to post here, I don't have the time to re-write the article (and others: "neper", "logarithmic scale") accordingly... who does? 147.81.129.17 (talk) 01:39, 10 February 2011 (UTC)

To be more specific, ONE particular definition of a decibel IS tied to power, and since power in electrical engineering is expressed in units of Watts, the definition of "dBW" is simply y dBW = 10log(x Watts), similarly, y dBm(W) = 10log(x milli-Watts). This is just a mapping from a logarithmic Watts scale to a linear dBW scale. "Power" was chosen as part of the definition, so a similar mapping from a log Volts scale to a linear dBV scale must include the factor of 2 because of the square law relationship between Volts and Power. THEN, it easily follows that dB (unitless) must have a unitless argument of the log function. Since this particular definition is tied to power, it doesn't make sense to apply the mapping to purely mathematical, unitless quantities (that would be a different definition), the argument of the log function must be, or be related to, a power quantity, so the only way to end up with a unitless result is to have a ratio of powers or the unitless result of a ratio of powers (gain). Or, obviously, a ratio of Volts will qualify, if the factor of 2 is included.

Trying to define dBW as a ratio to a "specified" or "fixed" reference makes no sense. The reference level for the linear dB[unit] scale is obvious: it's 0 dB[unit] (what's the reference point on your ruler? it's zero inches obviously). Because 10^0 = 1, that means that 0 dB is equivalent to 1[unit] in the original unit's linear scale (10log(1[unit]) = 0 dB[unit]). There's no "implied" or "specified" reference level. The reference is zero in the linear dB scale just as zero is the reference in any other linear scale. Change the unit's linear scale to logarithmic and zero disappears in negative infinity (can't have negative numbers!), but 1[units] still means 1[units] in this scale. Map that log scale to a new linear scale (dB[units]), and 1[units] got mapped to 0 dB[units]. 10[units] on the log scale got mapped to 1 dB[units] on the new linear scale.

So, yes, at least ONE definition of decibel REQUIRES power quantities which includes units, and therefore, a unitless dB result must incorporate a ratio of powers so the units will cancel. But it does NOT follow that all forms of dB<units> ALSO require ratios, in fact, just the opposite, the presence of units ALLOWS for the ABSCENCE of ratios. Closely studying the definitions of dB<units> that require ratios will reveal the circular arguments that confuse so many people trying to understand what is going on.

How about an example that MAY help: Link margin calculations of RF sytems that I have looked at include many unitless dB gain parameters for the various series links in the path, but always have two parameters that are expressed in dBW, one being the transmitter (the start of the signal obviously, expressed in units of power in a dB scale so that all the subsequent downstream gains expressed in pure dB can just be added/subtracted to it). The second dBW parameter is (the name of which I don't remember, it's been a while ;-) is always (and HAS to be) negative,... why? Because the final result of the link margin has to be unitless and the way to cancel out the transmitter dBW quantity is to SUBTRACT another dBW qty. Going further, remember the identity log(y/x)=log(y)-log(x), or similarly 10log((y Watts)/(x Watts)) = 10log(y Watts) - 10log(x Watts). Since the Watts obviously cancel in the left side of the eqn, they will cancel in the right side as well:

c dB = 10log(y/x) = 10log((y Watts)/(x Watts)) = 10log(y Watts) - 10log(x Watts) = a dBW - b dBW = (a-b) dB = c dB

207.211.59.252 (talk) 01:00, 11 February 2011 (UTC)

(Aside: it might be a good idea to register) You are correct on two main issues: (a) reference to power in the definition of decibel is misleading (restrictive); (b) the decibel is simply a mapping from a logarithmic to a linear scale. However, your calculations are incorrect since the logarithm is defined only for dimensionless quantities, so 'log(y Watts)' is meaningless. (comment first posted 13 Feb. 2011 - full text includes continuation below dated Feb. 13)Boute (talk) 09:48, 3 March 2011 (UTC)

THANK YOU!!! First, I appreciate all the responses both above and below!!, but I disagree with the statement 'log(y Watts)' is meaningless. I think it's too obvious: 'log(y Watts)' = log(y) Watts. So 10log(y Watts) = z dB Watts. The "Watts" can't just disappear. This still works in the inverse: 10^(z dB Watts) = 10^(z dB) Watts = y Watts. The definition of dBW requiring a ratio of units is an attempt to MAKE the units disappear to avoid having to take the log function of them, but then they're simply reintroduced again after the function evaluation, and a kludged definition is cited as the rationale. OR...and maybe this is better: you could just as easily define "dBW" = 10log(Watts), then the inverse exponentiation works as well: 10^(dBW) = Watts. Look at the link margin example above, either definition fits in well with the cancelling of units on the right side of the equation which follows from the cancelling of units on the left side using one of the log identities... It just seems too obvious that defining dBW FIRST (sticking with the convention of defining in terms of power) makes a LOT more sense without ratios, works mathematically, and the definition of a unitless dB then easily just follows from ratios of like units as arguments of the log function. .... Admittedly, I'm not an expert on all the available sources (I'm going to try reading some of the reference links provided below, hopefully they're free! ;-), but it seems too obvious as I've outlined above. .... Now, for some of the dB-unit definitions in the article that REQUIRE ratios involving non-unity denominator "reference" values... well, I think I would simply change the definition to only use the non-unity value in the denominator without the unit. This would perfectly fit in with the intention of using those reference values in the first place and still make the definition consistent with other dB-unit definitions and unitless dB expressions would have the same meaning in those fields as well... I'm really going to have to try to read those references you listed, I just don't see how and why we're all wrapped around the axle on this, it seems too simple ;-)...207.211.59.252 (talk) 03:36, 3 March 2011 (UTC)

Generally, if you have an expression with units, you can convert it to a different but equivalent expression with different units. This is not possible with log(y Watts), and log(y) Watts makes not sense at all. How would you convert both of those into mW consistently? Dicklyon (talk) 06:00, 3 March 2011 (UTC)

(Aside: be careful when posting in the middle of a piece of text; it makes unclear who signed it) I agree that dB can be made very simple, but (paraphrasing Einstein) not simpler than possible. The equality log(y Watts) = log(y) Watts does not hold; pulling the Watts out of the parentheses is incorrect. The argument of functions like sin, cos, lg, ln, exp must be dimensionless. Intuition: consider exp x = 1 + x + x^2/2! + x^3/3! + ..., so if x were not dimensionless, you would be adding terms with different dimension! Boute (talk) 09:48, 3 March 2011 (UTC)

Here is the problem with explaining the decibel satisfactorily: any definition of the decibel based on its historical roots (instead of mathematical "good practice") will remain troublesome. A cleanup yields the following.

(a) Define B (bel) as a function such that x B = $10^{x}$ for any (real number) x. Define d (deci) for any x by x d = x/10. Define dB (decibel) by x dB = (x d)B, hence x dB = $10^{x/10}$. Example: 20 dB = 100. Similarly x B' = $10^{x/2}$ and correspondingly dB' (example: 20 dB' = 10) for purposes shown next. (comment first posted 13 Feb. 2011 - full text includes continuation below dated Feb. 13) Boute (talk) 09:48, 3 March 2011 (UTC)

I think you meant to have x/20 in that last expression for B'.207.211.59.252 (talk) 03:36, 3 March 2011 (UTC)

No, that would be for dB', thus: x dB' = $10^{x/20}$.Boute (talk) 09:48, 3 March 2011 (UTC)

(b) Define dBW by x dBW = (x dB)W (example: 20 dBW = 100 W) and dBV by x dBV = (x dB')V (example: 20 dBV = 10 V). Similarly, x dB$\mu$W = ((x dB)$\mu$)W and so on.

This is essentially the same as one of the ideas I just wrote a little further above, but I went back just a little further to address "log(units)"...207.211.59.252 (talk) 03:36, 3 March 2011 (UTC)

Done. Insofar as a definition (which is always a convention) can be called "correct", the criterion is conceptual clarity plus convenience and safety in calculation. You can verify that the definitions (a) and (b) also work out nicely and correctly for your examples. Boute (talk) 10:25, 13 February 2011 (UTC)

It's normal to be somewhat baffled by the confusing and informal uses of decibels. But dBW is just a logarithmic expression of the ratio of a power to reference level of 1 Watt; for dBmW, the ratio is with respect to 1 mW. If you want to take log10(power in Watts), that's fine, we'll all understand that "in Watts" can be interpreted as a ratio if we care that the argument of a log really needs to be dimensionless. The argument and the resulting dB number are indeed "unitless"; the dBW notation is not a unit, just a label that tells you what the number represents. Trying to make a modern cleanup of decibels is not something to be undertaken on wikipedia. Bring us sources if you think there's a better way out there than in currently represented in the article. Dicklyon (talk) 17:03, 13 February 2011 (UTC)

As you say, traditional use of the decibel is confusing and informal. Precisely for this reason it is poor engineering practice. Your correct observation that (in the traditional view) the dBW notation "is" just a label points to the heart of the problem: as a label, dB or dBW is more of a comment in an expression than a calculationally useful part of the expression.

I disagree that dBW is not a "calculationally useful part of the expression" look at my example above from the real world of link margin analysis, you HAVE to have that second negative dBW input in order to end up with a unitless dB margin for the whole link. OTHERWISE, you would end up with a result in dBWATTS.207.211.59.252 (talk) 03:36, 3 March 2011 (UTC)

The relevant sources on the traditional view are (a) standardization documents such as International System of Units and BIPM The International System of Units (SI), 8th edition; (b) for elaboration, NIST Guide for the Use of the International System of Units, page 29; (c) for extensive discussion, the papers "Definition of the units radian, neper, bel and decibel" (by Mills, Taylor and Thor in Metrologia, vol. 38, pp. 353-361, 2001) and "On logarithmic ratio quantities and their units" (by Mills and Morfey in Metrologia, vol. 42, pp. 1-7, 2005).

In this view, (deci)bel is introduced in two steps as follows. First, given a (power-like) quantity $X$ and a reference value $X_{0}$, the level $L_{X}$ is defined by $L_{X}=\lg(X/X_{0})$. Second, the statement "$L_{X}=m{\text{ B}}=10m{\text{ dB}}$" is interpreted to mean that $\lg(X/X_{0})=m$. Observation: mathematically, this means that $m{\text{ B}}=10m{\text{ dB}}=m$. Consequences: (i) B and dB "do nothing interesting" (just linear scaling after the logarithm has already been "taken"); (ii) One cannot say "the ratio $X/X_{0}$ is (for instance) 20 dB", but only "the ratio $X/X_{0}$ corresponds to 20 dB". Similarly, assuming $X$ is a power quantity and letting $X_{0}$ = 1 W, the statement "$L_{X}=10m{\text{ dBW}}$" is interpreted to mean that $\lg(X/X_{0})=m$. So one cannot say "the output power is (for instance) 20 dBW, only "the output power corresponds to 20 dBmW". Practically and calculationally this is highly inconvenient.

This confuses me. First, I think it's clear that this equality: mB = 10mdB = m , is false. It would be true in this form: mB=10mdB, but just dropping the 'B' (or it's 10dB equivalent) without replacing it with something equivalent is not allowed. I don't see how 'B' is "linear scaling", it REPRESENTS the log function, the 'd' is obviously a scalar.... For the rest of the complaints about falsely equating ratios directly to dB expressions through the use of the word "is", well, that (and it's equivalent X/Y "=" Z dB) is a rookie calculus student mistake that is supposed to be beaten out of you by your teacher. ....

Yes, it is strictly incorrect to say "10W is 10dBW" (because 10W \= 10dBW), the relationship between them is a logarithmic transform. But, I don't find it calculationally inconvenient in written math form because I would never write it that way (and I will beat that mistake out of my students ;-), and conversationally in engineering discussions, I think the use of the word "is" is NOT strictly held to mean ONLY '=', in that context, the word is is an adequate (and understood) substitute for corresponds to.207.211.59.252 (talk) 03:36, 3 March 2011 (UTC)

Now comes the clincher: informal engineering practice gravitates towards the more terse "is", which is sloppy in the strict traditional view, but is fully justified in the alternative formalization based on "$x{\text{ dB}}=(x/10){\text{ B}}=10^{x/10}$. Here B and dB do something interesting, are calculationally useful (allow working via equalities, not "correspondences") and more intuitive. This formalization fits in with Occam's razor principle, and thereby eliminates a lot of unnecessary grief.

once again, you've lost me. 'B' represents the log function and 'dB' is a scaled representation, of course they both do something. The correct use of the "=" sign in this expression ("$x{\text{ dB}}=(x/10){\text{ B}}=10^{x/10}$) is NOT the same as trying to use it in "10W = 10dBW", because the first, correct, use does not neglect a mathematical operation, but the second one does. So, I'm not really sure this argument buys anything. I'll keep working on understanding it though ;-)... I need to read some or all (if I can!) of the sources you've referenced! Thanks!207.211.59.252 (talk) 03:36, 3 March 2011 (UTC)

The alternative formulation needs no further "trying" on Wikipedia but is fully developed and reported in the refereed scientific literature. A source is "The decibel done right: a matter of engineering the math", IEEE Antennas and Propagation Magazine, vol. 51, issue 6, pp. 177-184 (Dec. 2009).

Hopefully the brief outlines I have given for both views will be helpful to the discussion. Boute (talk) 16:09, 22 February 2011 (UTC)

The interventions by an anonymous colleague are appreciated (yet, please register!), but I gave up on trying to patch the discontinuities caused by posting in the middle of text that has been signed. Does anyone know the Wikipedia procedure for putting everything in clean order without falsifying history (which is not an acceptable solution)?

As regards "I think it's clear that this equality: m B = 10m dB = m , is false": that equality happens to be what the traditional interpretation in the standardization documents says! It makes B and dB trivial (certainly not logarithmic) and saying that it is "not allowed" to take the letter B away (justified by m B = m) cannot be enforced mathematically, only by a convention that says nothing more than "you may not take away this comment" in a program.

I refrain from further writing until the ordering of the postings can be cleaned up. People interested in the referenced papers (including those explaining the standards) can obtain them from their authors or by sending me an e-mail (address not written here but like everyone else's easy to find). Boute (talk) 10:23, 3 March 2011 (UTC)

You guys are totally missing the point... draw yourself a linear axis and label it "Watts". Now change the scale of the axis to logarithmic. The units are still "Watts". Now, transform the scale of that (now logarithmic) axis to another linear scale by performing 10log(.) on every tick value. What are your units now? Did the "Watts" just disappear? NO. You DID NOT perform a log function on the "Watts" unit itself. You only performed a log function on the VALUES of the ticks. The new axis label is "dBW" or "dB-Watts". So, maybe it was incorrect to write 10log(x Watts). Maybe it would be more correct to write 10log(x) Watts. 'x', can or can NOT be a ratio, by itself it's unitless, but by definition, it HAS to be associated with power, therefore, 'x' must be a ratio of powers that ends up unitless. SO, in order to get y dB = 10log(x), you had to START with y dB = 10log((a1 Watts)/(a2 Watts)). Nothing wrong with having the Watts inside the log function there obviously.... log identity: 10log((a1 Watts)/(a2 Watts)) = 10log(a1 Watts)-10log(a2 Watts) <--That HAS TO BE TRUE BY THE LOG IDENTITY!....Now what? 10log(a1 Watts) = b1 dB-W obviously. Of course you can't take the log of the "Watts". Introducing ANOTHER Watts unit in the denominator (even as qty '1' in an attempt to maintain equality) VIOLATES the equality. So what happens to the "Watts"??... Consider 'sin' as suggested above somewhere: the arguments of sin, cos, etc. aren't unitless, the argument of sin(2*pi*f*t) has units of radians. What happens to the 'rad' when evaluating the trig function? Well, it looks like it is defined out of the result. Is this the right answer for 'log'? I say NO, when taking the measurement of the output power of a transmitter and writing that measure down somewhere (in dBW or dBmW or the shortened form dBm representation because THAT'S WHAT THE METER OUTPUTS), you NEED to have the W (or mW) to clarify what was JUST measured! . .... Look, I'm not into all the esoteric mathematical and philosophical underpinnings of this. I'm just saying there's NOTHING wrong with discussing/suggesting alternative definitions alongside the archeologically archived definitions that HAVE been used. The REQUIREMENT that dB-W be derived by INTRODUCING a 1W denominator in the argument of the log function is very unintuitive to most people. If this requirement is driven by a fundamentally sound math/philosophical argument, it would be nice to see that explained ALONG WITH, a discussion of how much more intuitive (although maybe admittedly WRONG by current (or ancient) math/philosophy) it is to just consider 10log(x Watts) = y dB-W without having to introduce seemingly unmotivated ratios and how this FITS IN PERFECTLY with the rest of the established definitions (even requiring the unitless 'dB' to originate from a RATIO of powers).... Want to use a different "reference" other than '1 <unit>'? I call that "normalization", you just normalize your log argument input by any chosen number (without the unit), this has the desired effect of transforming scales and then drops neatly into all the arguments above.... 'log' itself is not a ratio, it is a transformation. Ratios are not required for 'log' functions.... Now, I haven't studied "complex logarithms", or other probable variations that I don't even know about, so maybe the only worthwhile input from all this is just the suggestion to consider a more intuitive rationale but remember that it is technically incorrect. 207.211.59.252 (talk) 20:15, 3 March 2011 (UTC)

Also, I'm sorry to offend sensibilities by putting comments where they seem more intuitively placed. But, really... saying that x dB = x, is clearly not mathematically correct. That suggests that this is then true: x dB-W = x W, which is DEFINITELY not true. This is too obvious, the 'B' or 'dB' is a representation that means something. It may not be strictly a "unit" in and of itself like "feet", but it represents an operation/transformation that cannot be ignored. 207.211.59.252 (talk) 20:15, 3 March 2011 (UTC)

First, putting comments in an orderly way is not a matter of sensibilities but of avoiding creating a mess. Looking back at the preceding paragraphs where you put new text in the middle of text signed by others, you will notice that text with which you disagree is now signed by you! To people who read the comments, the structure of the discussion is totally lost.

As regards the mathematics, be careful to avoid incorrect conclusions. (a) 10log((a1 Watts)/(a2 Watts)) is meaningful, because the Watts cancel inside the argument of the log. Writing 10log((a1 Watts)/(a2 Watts)) = 10log(a1 Watts)-10log(a2 Watts) is nonsensical, because 10log(a1 Watts) and 10log(a2 Watts) are nonsensical. In general, avoid the mistake f(g(x)) = g(f(x)), an equality that holds only very rarely. (b) The arguments of sin, cos etc. are dimensionless because radians are dimensionless; in fact, mathematically x radians = x (as also explained in the standardization documents), but watts are not dimensionless. (c) Writing x dB = x (watch out: decibel) is of course incorrect, but x B = x (for bel) is the basic interpretation of the traditional standard (rescaled only when the neper is also introduced, but let's stick to the essentials). Still, concluding x dB-W = x W (or, in view of the preceding sentence, x B-W = x W) to derive a contradiction is fallacious reasoning, because you are not allowed to split up dBW in this manner: the standard defines dBW as an "atomic" symbol just indicating that 1 W is used in the denominator of the power ratio, in your example: 10 lg((a W)/(1 W)). Summarizing: if you want to criticize the standard, better use correct arguments.

A final observation: talking about literature, Boileau remarked that well-conceived ideas can be expressed clearly. Paraphrased for mathematics: if a certain intuitive understanding leads to sloppy or misused notation, better start thinking again. As said before, I can send you copies of the publications mentioned earlier. Also, registering will not put you in mortal danger. Boute (talk) 10:42, 4 March 2011 (UTC)

On the matter of interrupting anothers post, the relevant guideline is WP:TALK. Interruptions are allowed, if properly indented, but are best avoided if possible. I have to say that the length of your posts does tend to encourage interruptions. If you want, you can copy the final signature to the broken end of the post. Personally, I tend to avoid these difficulties by quoting a snippet of the point I wish to reply to in my own post. SpinningSpark 13:28, 4 March 2011 (UTC)

Thanks for your comment. The rules indeed allow brief interruptions, but (a) The rules also state "When introducing an interruptive break, please add {{subst:interrupted|USER NAME OR IP}} before the interruption." and (b) "Allowed" is not synonymous with "wise": it might be better to keep the structure as clear as possible for other people who read the posts. If you find the length of my posts excessive, you may wish to read the papers by Mills et al. (the people who drafted the standards) referenced earlier, which illustrate the subtleties of the issue in two full papers. I tried presenting both the traditional and the alternative view in about a dozen sentences. Any suggestion to make this even shorter while keeping it understandable are welcome in view of a final writeup. Boute (talk) 18:07, 4 March 2011 (UTC)

Taking stock

The discussion around the decibel seems doomed to be a never-ending story. Indeed, ever since I became familiar with the decibel about half a century ago, I have always seen people wrestling with the concept. It is time for taking stock.

A. Observations

A.1 In practice, the decibel does not appear to present any difficulty. As probably envisioned by its original "designers", it can be explained extremely simply: given a power ratio, take its base-10 logarithm, multiply by 10, and write dB after the result. Done. If the denominator of the ratio was 1 W by convention, write dBW instead of dB after the aforementioned result. Done. The reverse process is evident. Billions of pages of specs and numerical calculations have been written and used, causing no well-known disasters. So where is the problem?

A.2 The problem is that general practice leaves the decibel without a proper definition (what does the symbol dB "written after the result" mean?), and as a result the mathematics around it remains substandard (what are the algebraic rules?). Different textbooks explain the decibel differently. Numerical calculations proceed by tacitly patching up the abuses of notation (on the basis that "everyone knows what the notation means"). Symbolic calculation cannot be done properly and is simply avoided. From the viewpoint of "good practice" in engineering and also educationally, this is disgraceful.

A.3 Standardization documents (listed earlier) provide formal definitions of bel, decibel and neper, characterizing them as "units" (some also mention that using the term "unit" here is debatable). The numerical values of these (dimensionless) "units" depend on the chosen defining equations (involving a natural or decimal logarithm), resulting in B = 1 and/or Np = 1 and/or correspondences of the form 1 Np = (20/ln 10) dB depending on the chosen equations. Also depending on the chosen equations, the symbol Np or B (whichever equals 1) may be omitted mathematically, but is "commonly added" as it "reminds the reader of the nature of the quantity" --- in other words: as nothing more than a comment. The resulting subtleties are discussed in the two cited papers by Mills et al., where it is also explained why the choices mentioned have no impact on practical use as observed in A.1 above.

A.4 The current Wikipedia page gives a definition not found in any standard, and does not even mention the reference used. Apparently its writers recognized that various consequences from A.3 (e.g., B = 1) are unpalatable to readers desiring a good intuitive grasp, and therefore they have provided an explanation that reflects their own understanding. However, such a compromise ends up by providing neither clean mathematical expression, nor intuitive clarity. As a result, the essential difficulties are not eliminated, and justified questions (sometimes even venting some anger) keep appearing on the discussion pages.

B. Defining decibel For easy reference, here is a brief writeup of earlier explanations, highlighting also the similarities and differences between formulations.

B.1 Common parts of the formulations For any positive real $y$ (typically, but not necessarily, a power ratio) and real $x$,

a) $x{\text{ dB}}=(x/10){\text{ B}}$. Warning: at this stage, one should not yet assume $(x/10){\text{ B}}=(x{\text{ B}})/10$.

b) ${\text{L}}_{y}=x{\text{ B}}$ means that $x=\lg {\text{ }}y$ or, equivalently, that $y=10^{x}$.

Via (b), any chosen definition for ${\text{L}}$ defines ${\text{B}}$ and hence, via (a), also ${\text{dB}}$.

B.2 Two different formulations

Traditional formulation (reflected in the standards): ${\text{L}}_{y}=\lg {\text{ }}y$, hence, by (b), $x{\text{ B}}=x$ and, by (a), $x{\text{ dB}}=x/10$. Standard terminology is as follows.

If $P$ is a power quantity and $R$ a reference power, then ${\text{L}}_{(P/R)}$ is called the power level of $P$ relative to $R$.

If $P$ and $P'$ are powers, then ${\text{L}}_{(P/P')}$ is called the power level difference between $P$ and $P'$ because ${\text{L}}_{(P/P')}={\text{L}}_{(P/R)}-{\text{L}}_{(P'/R)}$, clearly irrespective of $R$.

One can say that a power level or power level difference of 20 equals $20{\text{ dB}}$, but for a power ratio of 100 one can only say that it corresponds to $20{\text{ dB}}$. The commonly used definition of ${\text{dBW}}$ is via the power level relative to 1 W.

Alternative formulation: ${\text{L}}_{y}=y$, hence, by (b), $x{\text{ B}}=10^{x}$ and, by (a), $x{\text{ dB}}=10^{x/10}$.

Here one can say for a power ratio of 100 that it equals $20{\text{ dB}}$. Defining ${\text{ dBW}}$ by $x{\text{ dBW}}=(x{\text{ dB}}){\text{ W}}$, one can write $20{\text{ dBW}}=100{\text{ W}}$.

Since circuit analysis and differential equations involve physical quantities directly rather than via their logarithms, this alternative formulation allows direct embedding of decibel expressions in calculations via equalities (not mere correspondences).

Hopefully this writeup answers many questions and provides a first start for addressing item A.4 above. Boute (talk) 14:25, 5 March 2011 (UTC)

The only thing we really need to do is to make improvments to the article that are supported by reliable sources. If our definition is not found in a source, it should be replaced by one that is. What source do you recommend? If your improved treatment is published some place, we can incorporate that and cite it. Dicklyon (talk) 19:09, 5 March 2011 (UTC)

What Boute says makes sense. My view is that the article should include a definition and use based on international standards (like IEC 60027), but there is also a need to explain common use, as such standards are only rarely followed. Dondervogel 2 (talk) 17:20, 7 March 2011 (UTC)

Boute's alternative formulation (which I suspect is what he really wants to get into the article) is based neither on international standards nor common usage and would completely outlaw the common practice of the 20 log representation of field quantities even when unrelated to power. SpinningSpark 17:36, 7 March 2011 (UTC)

The most important point Boute makes is that the definition quoted in the article is not supported by authoritative sources. If that assertion is correct, shouldn't the criticism be addressed? Dondervogel 2 (talk) 18:11, 7 March 2011 (UTC)

Yes, that's what my suggestion was about. It may not be what he's angling for, however. Feel free to work on it. Dicklyon (talk) 20:06, 7 March 2011 (UTC)

Since Dicklyon's remark I have been busy comparing the various standardization documents, and found the best definition in the BIPM documents. It needs a minor tweak (change of variables) to also reflect the more general common engineering practice. Due to other work, I could not provide the writeup yet. Spinningspark may rest assured: there is no question of replacing the standard definition with my alternative. Still, I must say that all his technical remarks are incorrect: in fact, the alternative is precisely a formalization of common usage, and also supports the "20 log" representation without any problem. Anyway, if no other work intervenes, I have the writeup accompanied with various ISO references ready by tonight. Boute (talk) 08:38, 8 March 2011 (UTC)

For what it's worth, ANSI S1.1 defines the decibel as "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." Dondervogel 2 (talk) 18:02, 10 March 2011 (UTC)

So they must define "level" or "unit of level" some place, too? Dicklyon (talk) 19:45, 10 March 2011 (UTC)

Indeed, other standards define decibel in a way that does not involve other terms that have to be defined in turn. I consulted several hundred of pages from standardization organizations (BIPM, NIST, ISO, IEC, ITU-T), and am awaiting one final document before I complete the writeup (now nearly finished). Sorry for the delay. Boute (talk) 01:10, 11 March 2011 (UTC)

More from ANSI S1.1 "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." Dondervogel 2 (talk) 15:49, 11 March 2011 (UTC)

So, way too narrow to be useful, being tied to acoustics, and a form that nobody has ever heard of elsewhere. Dicklyon (talk) 16:14, 11 March 2011 (UTC)

Yes, it's unfortunate that only the application to acoustics is mentioned, but the definition itself is not a narrow one. I would be surprised if it differs much from that of other standards bodies. Let's see what Boute comes up with and compare then. Dondervogel 2 (talk) 16:59, 11 March 2011 (UTC)

The def for "level" appears to be "in acoustics" only. I think they made it up, as it's an odd notion. We do talk about "level" in acoustics, by I've never heard it defined to be a logarithm, or seen it used to define decibel. This is just odd. Dicklyon (talk) 01:41, 12 March 2011 (UTC)

In both cases the application is acoustics because S1.1 is the ANSI standard for "Acoustical Terminology". In both cases though, the definitions are completely general and seem to me equally applicable to (say) radio or electrical engineering. Dondervogel 2 (talk) 17:54, 15 March 2011 (UTC)

The writeup is as good as finished. It evolved (degenerated?) into a tutorial, which may be too long. I have some ideas for reducing the length drastically, but some prior input from the various contributors following this talk page may be helpful. Does anyone know a more suitable place than this talk page to post initial versions? Probably I can post what I have tomorrow morning (local time). Boute (talk) 19:10, 12 March 2011 (UTC)

Yes, in your own userspace. Create a page with a name such as User:Boute/Decibel draft or some such and then link back to it here. SpinningSpark 21:10, 12 March 2011 (UTC)

Thanks for laying out a red carpet to create User:Boute/Decibel draft. The first half of the draft is now available on that page. Urgent family business (parents in law celebrating 60 years of marriage) requires temporarily postponing the rest. It is the most interesting but also challenging part, about root-power quantities, neper and engineering usages. Boute (talk) 15:55, 13 March 2011 (UTC)

The draft is now complete on User:Boute/Decibel draft. Boute (talk) 15:07, 14 March 2011 (UTC)

Is this a proposal for a whole article replacement, or a new section, or what? From the start, I think I object; if you want to assert that "The literature on this topic is highly nonuniform", it would be good to have a source that says so; otherwise, just skip the wordy intro. Dicklyon (talk) 18:26, 14 March 2011 (UTC)

The draft is too long for direct incorporation. It is meant as a coherent source of material with detailed references, and can be shortened, complemented and rearranged. Your objection against pointing out the nonuniformity of the literature is surprising, as a random literature or web search clearly reveals the sad state of affairs. If one is getting soaked, does one need a source that says it rains? Boute (talk) 21:25, 14 March 2011 (UTC)

I know, I've done that myself; but I think it's WP:SYN, so we should avoid it if possible. I almost always object when I see things like "people are often confused by" or "this term is often misused", as these are just an editor's interpretation, edging toward prescription, unless there's a secondary source that says so. Stating that the literature is mixed up is similar; not that I disagree, but that I don't think we can justify saying so. Dicklyon (talk) 21:36, 14 March 2011 (UTC)

I see wat you mean with WP:SYN. Still, judging by their stated rationale and the examples given, these recommendations seem tailored to history, sociology etc. where an editor's interpretation can easily suggest unjustified conclusions. For the sciences, readers can in some cases reliably verify the conclusions themselves on the basis of their logic, especially if proper examples are given (documented by references). Anyway, just to be safe, I will implement the WP:SYN recommendations rather strictly, and change the wordings where necessary in the next version (version 1). The subsection about practices and literature with "negligent" conventions will certainly look different. Boute (talk) 15:58, 15 March 2011 (UTC)

But Boute is right, it is a mess. There are plenty of articles we can cite, so we should not be afraid to say so. I have in mind several papers by Horton from the 1950s, but there are probably more recent publications as well. Dondervogel 2 (talk) 17:45, 15 March 2011 (UTC)

I am most interested in these references, so we can quote them. Boute (talk) 08:57, 16 March 2011 (UTC)

We all seem to agree on the facts, but WP:SYN suggests that value judgements, however evidently justified, are better avoided unless explicitly found in reference material, and even then clearly identified as the opinion of those sources. On Wikipedia, we can also try to work by example, providing the best possible work despite the Wikipedia restrictions (especilly WP:NOR and WP:SYN, which are reasonable for sociological and political issues, but overly stifling for conclusions and simple results reached and verifiable by pure logic). In fact, the decibel is only the tip of the iceberg, considering the vast multitude of negligent and even inconsistent conventions and notations in engineering and in mathematics, of which a very small fraction are mentioned in a paper by Lee and Varaiya. Paradoxically, in an attitude that a recent incoming e-mail described as a form of machismo, mathematicians even more than engineers tend to impose junk notation and conventions on their students, thereby (citing Lee and Varaiya) "undermining their confidence in mathematics". This is a disservice to future generations, especially since it takes surprisingly little effort to use defect-free conventions from the start (although the prior cleaning up often presents a serious intellectual challenge that requires rethinking familiar material). Boute (talk) 08:51, 16 March 2011 (UTC)

Your Mills article has a bad link; I'd like to see what it says. And the recent attempt by the BIPM to rationalize the definition of B and dB through a footnote is probably not enough reason to adopt their approach as the basis for the article; it will be very foreign to everyone who has learned this stuff in the last 8 decades or so. I'd be more comfortable seeing it incorporated as just a section on "Modern rationalization of definition" or something like that. Dicklyon (talk) 18:37, 14 March 2011 (UTC)

The bad link should now work. The BIPM definition also appears in many other sources (including the Mills papers); it can be safely skipped by readers who dislike its style, omitted or postponed, since it is not the basis for the article and is immediately followed by a definition (with multiple references) in a style that makes it look more familiar while still avoiding abuse of notation. If you have a reference to an earlier definition of B and dB (such as the one from which you consider the BIPM formulation to be a rationalization) it might perhaps be an even better source, provided it avoids abuse of notation to a comparable degree. Boute (talk) 21:25, 14 March 2011 (UTC)

Oh, and congrats on your in-laws' 60th anniversary; my parents had that recently, too. And my wife's parents a few years back, as you can see at Larned B. Asprey. Dicklyon (talk) 18:40, 14 March 2011 (UTC)

Thank you. Boute (talk) 21:25, 14 March 2011 (UTC)

A gradual approach

A more gradual approach to the decibel is now posted as Draft Version 1 on User:Boute/Decibel draft. Version 0 was too long to start with and, as Dicklyon observed, taking the recent rigorous definitions found in the standards as the point of departure for a unified treatment led to excessive abstraction for a Wikipedia entry. Version 1 remains informal for the most part (except perhaps at the end, still to be written), concentrates on practical issues and proceeds mostly by examples.

Three entirely new sections concern (a) understanding the reason for using decibels at all, based on calculation examples, (b) the historical origins of the definition, (c) the use in acoustics, including a very small case study illustrating typical pitfalls. Many reference documents considered difficult to find are made downloadable by clicking in the reference list.

For more general interest, when trying to find the original 1924 and 1929 decibel papers by Martin by Googling BSTJ, I got the complete BSTJ collection on the Alcatel-Lucent BSTJ site. A real treasure trove! Boute (talk) 14:48, 22 March 2011 (UTC)

That BSTJ site was terribly slow when it first came online a few months back; it's good to see that it's working well now. It's great that they're making it all freely available. Dicklyon (talk) 01:52, 23 March 2011 (UTC)

It's good that you found good sources for the confusion around decibels; still, I would prefer not to see that driving the organization of the article. A more normal or typical presentation first would be OK, followed by a section on why it's confusing, and approaches to fix that. Dicklyon (talk) 01:52, 23 March 2011 (UTC)

The organization of the article is not driven by the fact that sources about the confusion are available, but rather by the desire to give readers an informal confusion-free understanding right from the start, as opposed to repeating the usual confusion in the introduction. The informal introduction reflects the typical presentation and use found in dozens of engineering textbooks (more references will be added), except that inadequate definitions are skipped (to be discussed in the section on engineering practice and textbooks). If you have an even more normal or typical presentation in mind, some references would be welcome. I think that the organization you propose would be more appropriate for a scientific paper intended for peers: first describe the problem, analyze it, and then describe fixes. Non-specialist readers may find it easier to start with something that needs no fixing, which happens to be the informal understanding underlying engineering practice. Therefore the cautioning for pitfalls is also done gradually. Boute (talk) 11:13, 23 March 2011 (UTC).

Your "informal approach" starts by talking about dB as representing a "number". This is not the traditional approach, and completely misses the point about the difference between the 10log and 20log formulas. The decibel is really about power ratios, intensity ratios, energy ratios, or the equivalent field quantities. Trying to adapt the definition to numbers is bound to lead to more errors (like the traditional error in how the decibel is used with image sensors). I haven't read beyond that yet. Dicklyon (talk) 20:35, 23 March 2011 (UTC)

Unlike 80 years ago, nowadays the decibel is also used for temperatures, resistances, frequencies and so on. First, these cannot be considered as power quantities: the decibel has outgrown this narrow use, and ratios are pure numbers, so the definition is not "adapted to" numbers, it always has been about numbers (sometimes camouflaged as power ratios)! Second, for clear exposition, the principle of "separation of concerns" is crucial: separating the concept (numbers, exponents, logarithms) from appliction areas (power, temperature etc.) keeps ideas clear and helps avoiding errors (certainly does not lead to them). The use of 20 log formulas for root-power ("field") quantities is exceptional, especially in view of the current wider use of decibels. It was poorly engineered from the start (for instance, Mills et al. had a better idea using a distinct symbol dB' for exposition and symbolic calculation, and independently I did exactly the same). As a result its use is anomalous and a source of errors, even in books devoting a full chapter to "decibel math". Therefore the 20 log issue should be presented as the exception it really is, with due warnings and caveats, and only after the reader has had the opportunity for obtaining a "feel" for the decibel's essential (exponential/logarithmic) properties. In fact, nearly all textbooks using the "traditional approach" also avoid talking about the 20 log issue too soon! All this explains why I think a gradual approach makes the decibel easier to understand for the reader. It is good that you bring these issues up, I will address them in the "engineering practice" subsection with examples from current practice (including use of dB for formulas with a mix of quantities like power, distance, frequency). Boute (talk) 04:16, 24 March 2011 (UTC)

I haven't seen such uses: temperatures, resistances, frequencies, really? I suppose that in each of those cases one can come up with a power or energy that's proportional (kT, kTR, hf); is that what they're doing? Where can I see examples? Dicklyon (talk) 07:10, 24 March 2011 (UTC)

A quick review of a bunch of books on GBS doesn't show any that present decibel as anything but a power or intensity ratio. Dicklyon (talk) 07:15, 24 March 2011 (UTC)

I think that many consider the use of the dB for anything other than power (or energy) controversial. If they are nevertheless used in that way (eg for temperature ratios), it makes to say so - backed up with references - but the controversy created should also be explained IMO. Dondervogel 2 (talk) 10:45, 24 March 2011 (UTC)

On User:Boute/Decibel draft, version 1 examples are added. Just ask if you want more. In satellite communication and receiver design, decibel is used routinely for temperatures, distances and frequencies, even for heterogeneous quantities like G/T (gain / temperature) expressed in dB/K. Even for the kT example, k and T are often logarithmically separated (assumed made dimensionless via matching units). It is very strange that, regarding a topic where confusion reigns and abuse of notation prevails (IMO more than anything else the cause of the confusion), uses that are common in certain fields (for formulas more complex than $P=V^{2}/R$) and can be supported by a rigorous mathematical formulation in flawless notation would suddenly cause squeamishness! If there are sources considering such uses controversial, I am interested and will certainly add them. Boute (talk) 12:15, 24 March 2011 (UTC)

All of those terms are factors in the SNR; either signal power gain or noise power; the fact that they take a shortcut and apply dB directly to T is therefore just that, a shortcut for the effect on power; not the application of dB to arbitrary numbers. Dicklyon (talk) 19:12, 24 March 2011 (UTC)

The main source I have in mind is one you already have (Carey 2006), which reads (see p62)

• even though [the Department of Commerce and the Department of the Navy] have access to talented scientists, they did not use the current standards; rather, they incorrectly redefined the decibel and sonar equation terms [28]: "Decibel (dB): Decibel is a dimensionless ratio term that can be applied to any two values; temperature, rainfall, the number of jellybeans in a jar, or sound. Decibels are expressed as 10 times the logarithm of the ratio of a value (V) to its reference value (Vref), or N decibels. The decibel originated in electrical engineering measurements of transmission line losses, but it is also physiologically significant in that the response of biological ears to sound is logarithmic. Decibels should always be accompanied by a reference value that defines the ratio being expressed, unless the reference is clearly stated at the start of the paper.".

In the subsequent text on the same page, Carey makes it clear that he considers the use of the decibel to express ratios of "temperature, rainfall, the number of jellybeans in a jar" to be incorrect, because they are not ratios of power. I will look out for other examples. Dondervogel 2 (talk) 16:53, 24 March 2011 (UTC)

This paper contains more examples, quoting from several references. Dondervogel 2 (talk) 16:57, 24 March 2011 (UTC)

Examples of what? Just decibel applied to ratios of power-like quantities? Dicklyon (talk) 20:22, 24 March 2011 (UTC)

Examples of statements (by Urick, Horton, Camp, Kuperman) that make it clear the authors of these statements consider use of the decibel for anything other than a power (or intensity) ratio to be incorrect. Dondervogel 2 (talk) 08:28, 25 March 2011 (UTC)

From Carey I only have the 1995 paper I referenced. Can you provide more on (Carey 2006), full title or link? Anyway, correctness here is not a matter of traffic regulations but of mathematical consistency. The fact that Marconi originally invented radio for telegraphy does not make telephony or TV incorrect. Dicklyon's mentioning of kTR suggests also considering $V^{2}/R$, reflecting dependencies on different exponents for R (+1 and -1). Furthermore, the people who use dB for resistances (for S-parameters, transconductance amplifiers etc.) consider I = V/R as the basic equation and use the 20 log rule (with a minus sign). The conclusion is that consistent and useful application of decibels should not be prejudiced by incorporating certain exponents in the definition, but that all exponents should kept outside the definition of dB and made explicit in the formulas used. Hence there is more reason for saying that the 20 log rule is incorrect (because it is prejudicial) than objecting against mathematically consistent wider use of decibels. More bluntly, causing a mess is incorrect. Boute (talk) 18:09, 24 March 2011 (UTC)

The mess that we live with is that historically, decibel is for power ratios, so it's clear when the 20log rule is used, and when the 10log rule is used; plus the new mess that some body has defined the decibel to be for any ratio, and always the 10log rule. Calling the latter "controversial" is an understatement in my opinion. I've still seen no evidence that there's any adoption of that radical departure from the historical usage. Is there any secondary source that has looked at that issue, beside the criticism quoted above? Anything positive suggesting any impact of the new def? Dicklyon (talk) 19:08, 24 March 2011 (UTC)

Here is Carey 2006. Dicklyon (talk) 19:26, 24 March 2011 (UTC)

Thanks for the Carey reference. In User:Boute/Decibel draft, Version 1, I cited two relevant sources, one being already 25 years old, and reproduced a full example from that source. These and other sources show that in fields such as satellite communication and receiver design it is common to use the decibel for any numbers and to adopt 10 log as the basic rule, with multiplication factors 2, -1 etc. automatically arising from the exponents in the formulas translated into dB. This is not a "new mess", but a very safe and straightforward approach used by many people in a manner that, especially in formulas more complex than $P=V^{2}/R$, has proven to be much less error-prone than introducing exceptions like a 20 log rule (which makes some sense only if no other resistances exist than 600 ohms). You can easily find a dozen more sources. Temperatures and frequencies are just two examples, distances also appear. The G/T (gain to noise temperature) expressed in dB/K appears as an antenna characteristic in many places in the literature. If you look into this matter starting with those references, you will see that I am not some body making things up but am reporting current practices. Most controversy comes from the acoustics sector; their criticism of the chaos is fully justified but the cause of the problems is incorrectly identified. Attaching greater weight to old history than to proven usages is throwing the baby out with the bathwater. Boute (talk) 21:48, 24 March 2011 (UTC)

I refuted that interpretation above already. In the satellite example, all of the factors are factors in the signal and noise power, and the use of dB there is a computational shortcut that is OK, but not part of any standard or endorsed way of using decibel; they get away with it because they are all factors in powers, and because they've separately set up an equation where all the units work out to nondimensional before taking the logs of all the factors. The decibel of a temperature makes no sense outside of such an equation. If there are fields within which this is routinely done, we can talk about that in a section, but still pretty much all texts define the decibel only in terms of power ratios, so that should be our main interpretation. Dicklyon (talk) 22:06, 24 March 2011 (UTC)

At least we now agree that what I described are actual practices in certain fields, not something I made up. This allows concentrating on the real issue: the merits. Your objections thus far are only of a historical nature, not a refutation in the scientific sense. In the satellite example, not all factors are in signal power: frequency and distance appear squared, which would make them "root-power" (in the power-centric view). Yet the definitions in the literature use the 10 log rule, and in formulas a factor 2 automatically arises from a square (if any) in an error-free manner. So this is not a matter of "getting away" but a systematic approach that is always general and safe. The technique for combining quantities into a dimensionless expression and then cancelling the units to obtain pure numbers is not a "set up", but the very essence of using a coherent set of units. Moreover, the definition of dBK follows the same pattern as the standards do for powers: for representing a power P in dB (1 W) (or dBW as a nonstandard engineering shorthand) one uses by definition 10 lg (P/(1 W)), for representing an absolute temperature T in dB (1 K) (dBK in engineering) one uses by definition 10 lg (T/(1 K)). Similarly for dBHz. Such definitions are self-contained and hence make perfect sense independently of the equations in which they are used. In brief, the engineering practices just described are systematic and application-independent, thus making the decibel easier to explain to novices than the intertwining of the decibel concept with one specific historical application (power ratios). As regards "pretty much all texts": ironically, the many texts that do not attempt defining the decibel but introduce it casually by example currently provide the best service to their readers! So arguably a separation of concerns optimally serves the interests of the Wikipedia readers who are hoping for some clarity. Boute (talk) 09:13, 25 March 2011 (UTC)

Sorry about the confusion over Carey. I noticed his name in the references list and just assumed it was the same paper. My fault. The quote is from W M Carey, Sound sources and levels in the ocean, IEEE J Oceanic Engineering 31, January 2006, pp 61-75. Dondervogel 2 (talk) 08:33, 25 March 2011 (UTC)

Thanks for the Carey reference. This detailed paper is very revealing. Especially the formulas in the conclusion are significant, much more than the words in the introduction, which at first are difficult to interpret without access to Horton's paper. The recommendations can be seen as a special instance (in acoustics) of the general engineering practice described earlier. Indeed, rather than using a 20 log rule for root-mean-square pressures with reference to 1 μPa, Carey recommends the 10 log rule for mean-square pressures with reference to $1{\text{ μPa}}^{2}$. Translated back to electronics, this would amount to using ${\text{dB }}(1{\text{ V}}^{2})$ defined via the rule $10{\text{ }}\lg(V^{2}/1{\text{ V}}^{2})$, recognizing that not voltages but squares thereof are being represented. This is precisely what happens in the formula manipulation in the earlier antenna examples.

The quote from Horton, "The term decibel has been used for quantities for which it is not the assigned designation. Confusion and error have resulted." contains 2 parts: a purely regulatory remark, and an observation. Using the decibel in a uniform way even for other than the designated purpose cannot cause confusion; to the contrary: it repeats a mathematically consistent pattern. A reasonable conjecture is that the confusion resulted from nonuniform use, needless to say: the 20 log rule (I am awaiting a copy of Horton's paper to see what he really means).

Also, Carey's criticism about the definition quoted from the report by [the Department of Commerce and the Department of the Navy] primarily concerns confusing the decibel with a relative level, which "is simply the order of magnitude" (not clear what this phrase means: "level" is not mentioned in the quote, nor defined by Carey as order of magnitude). It does not explicitly object to the "temperature, rainfall, the number of jellybeans in a jar" generalization. Although Carey mentions power ratios in defining decibels, his recommendations clearly suggest how to use decibels safely and unambiguously for other quantities. In fact, in one of Carey's equations, the reference level is $1{\text{ }}{\text{W}}/{\text{m}}^{2}{\text{Hz}}$: by allowing the presence of various other dimensions (length squared, frequency), the power dimension has effectively lost the restrictive normative status that some people consider essential for historical reasons. Finally, using decibels as mere representations for numbers (as in the antenna examples) can more than anything else contribute to reducing confusion in the long run: quoting decibels without context would be considered as uninformative as saying "42" without specifying to what quantity or ratio the number refers.

So, from a scientific viewpoint (i.e., independently of regulatory issues), one can see a convergence across discipline boundaries.

[Due to some deadlines, it may take some time before I can respond to further comments.] Boute (talk) 12:57, 25 March 2011 (UTC)

The draft looks GREAT! You guys are really doing great work on this... unbelievably thorough and exhaustive... but it's more than just mindless thoroughness, you're really helping to make "sense" of the subject and I'm sure this product will be considered not only helpful, but the next best thing to a new standard! The mountain of work you've done on this already is clearly evident and even awe-inspiring! Pages like this are so much more than just an encyclopedia entry and are what make Wikipedia great! 207.211.59.252 (talk) 22:48, 24 May 2011 (UTC)

Needs introductory section for non-technical reader

A reader who does not already know what "decibel" means needs basic foundational information via examples to introduce the basic concept, e.g., examples of how loud something is if it emits X decibels.

Readers need this basic framework to get a grasp on the rest of the article. The article should start with sound, because it is by far the most accessible introductory point and can easily be illustrated with examples that someone who does not know what a "decibel" is can readily grasp. This backdrop can then be used as a base as the article goes on to mention and develop other areas where decibel measurements are used, for readers who want to go deeper. Even when writing for sophisticated readers, beginning with the concrete before proceeding to the abstract is better exposition.

Sound examples also provide salient information for most consumers, who use decibels readings or estimates primarily to make decisions:

"On the highest setting, are these 50 db earphones going to damage my hearing?"

"When ordering discount on the web, should I spring for the 75 db jackhammer, or save with the 90 db tool?"

"Is this strato-amp with the 111 setting going to really kick up a reaction at the rock concert?

They also need a reference to articles on biological effects. "How loud are rock concerts? Will one permanently damage my hearing? What if I'm on stage right in front of the strato amp feed 4x4 speaker set to 111?"

Readers new to "decibel" need news they can use, first to understand the rest of this article, or just what they needed to get on with their lives.

My quick search failed to find a basic introduction like this in related WP articles. But what a "decibel" means should be clearly explained to someone who has no clue in this article, not by reference to another article. Other articles should refer to this one for that basic, non-technical introduction foundational for understanding the math examples; not the other way around.

The information I have bolded in the following F&OS article illustrates the kind of basic fundamental information needed to provide for a much better introduction for the general reader, i.e., the foundational material necessary to understand the rest of our wonky WP article on decibels.

"This article is free for republishing

Source: http://www.articlealley.com/article_1720271_17.html"

"Measuring Hearing Loss" http://www.articlealley.com/article_1720271_17.html Date Published: 27th August 2010 Author: specialtytas

Measuring hearing loss is frequently confusing and, oftentimes, people get decibels and the concept of a percentage mixed up. Decibel is an open-ended scale of loudness that begins with zero, which is defined as the faintest sound that can be heard by a human ear. From there, regular conversation is usually around 60 decibels, most power tools fall somewhere between 100 and 120 decibels, and a gunshot might measure at 150 decibels or more. For every 10 dB (decibels) the volume increases, the perceived loudness doubles and there is a tenfold increase in sound intensity.

If someone, for example, had hearing loss that made it impossible for him or her to hear sound fainter than 30 dB, you could say this person has a 30-decibel loss. Likewise, if they have a 50-decibel loss, they cannot hear sound that is less than 50 decibels. As you might guess, the greater the decibel loss, the less a person can hear. If someone has 40-decibel loss, this person probably will not be able to hear you whisper and may have some difficulty with even regular volume speaking.

Although 0 dB is the lowest volume a human ear can hear, it should not be confused with the lowest volume a human ear usually hears. According to most classifications, even people who cannot hear below 15 or 20 dB are considered to have "normal" hearing. In addition, since there is no highest number of decibels, you cannot simply come up with a percentage for hearing loss. The decibel scale is logarithmic, meaning that an increase of 10 dB is a 10-fold increase and not simply a set percentage. For example, 50 dB is 3000 times louder than 30 dB, not a 66 % increase as one might assume.

The implications of this in measuring hearing loss are that you cannot say that someone with 20 dB of hearing loss has a 20% hearing loss. On the other hand, you can use percentages to describe one's ability to discriminate between sounds. If you have had or will ever have your hearing tested, the audiologist will ask you to put on headphones. The will then play a recording of a list of words and then ask you to say them back. The audiologist then compares what you think you heard to what was actually spoken. The ratio of correct words to the number of words spoken can be seen as the percentage of your discrimination. This can be an important measurement, but it is not a way of gauging hearing loss.

Therefore, the most accurate way of quantifying hearing loss is to say someone has a certain decibel of hearing loss, which can be equated to general guidelines describing the loss. For example, a 20-40 dB hearing loss might be considered as "mild" hearing loss by an audiologist. Above this level, but below 60 or 70 decibels might be considered as "moderate" hearing loss. Levels above this can be considered "serious" or "severe" hearing loss, with 90 to 100 dB or more of hearing loss being "profound" or "extreme" hearing loss. Again, these are just guidelines, but they help in explaining the level of hearing loss in a much more helpful way than inaccurate and meaningless percentages.

Another way of putting hearing loss in perspective is to consider what the effects of this kind of loss are on an individual as well as what they might do to remedy the situation. Those who can hear sounds at around 20 dB or less have what could be considered "normal" hearing. Mild hearing loss usually means that, although the loss is not very extensive, one might have trouble following a conversation or distinguishing other sounds when in a loud environment. Moderate hearing loss is more severe than this, but usually less than 70 dB of hearing loss. Those with this level of loss will have trouble listening to a conversation without a hearing aid of some sort. More severe hearing loss means that an individual is very dependent on a hearing aid or implant in order to hear. Extreme hearing loss is what most would consider borderline deaf, at best. Those with this type of loss depend on sign language, written text, captioned media, and lip reading. Some may benefit from an implant.

Another example of a straightforward simple direct introductory lead-in to the subject is at http://www.howstuffworks.com/question124.htm I wasn't easily able to see if howstuffworks is F&OS.

This article is free for republishing Source: http://www.articlealley.com/article_1720271_17.html" — Preceding unsigned comment added by Ocdnctx (talk • contribs) 16:37, 24 April 2011 (UTC)

What makes you think that decibels are only, or even mainly, about loudness? SpinningSpark 20:20, 24 April 2011 (UTC)

Plus that article is essentially anonymous, not WP:RS, and has nonsense in it like "50 dB is 3000 times louder than 30 dB"! In reality, 50 dB is about 4 times louder than 30 dB, or 100 times more intense. Dicklyon (talk) 05:26, 25 April 2011 (UTC)

Does this make sense?

The flow at the very start of the "Definition" section seems confused, but in looking to improve it I encountered this:

"The bel is the logarithm of the ratio of two power quantities of 10:1, and for two field quantities in the ratio ${\sqrt {10}}:1$."

This sentence is grammatically garbled, but I'm not sure it's right technically either. Although a (power) ratio of 10:1 equates to 1 B or 10 dB, I don't think it makes sense to define the bel as "the logarithm of the ratio of two power quantities of 10:1", or any grammatically corrected version thereof. I'm tempted simply to delete this sentence since the ensuing explanation of the calculation seems much clearer, and there has already been a summary definition in the opening paragraph of the article. Anyone have any feelings on this? 86.179.114.69 (talk) 22:17, 6 June 2011 (UTC)

I agree it's nonsense. You might want to try to rewrite it to be sensible. Dicklyon (talk) 22:22, 6 June 2011 (UTC)

formula for decibel

the defination of decibel is tenth of log ratio, therefore, the formula should be something like dB=1/10*log(P1/P0), I think the current version misses the tenth factor same goes for Bel, which is 10 times of decibel, right formula should be Bel=log(P1/P2) Maybe I miss-understand something here? — Preceding unsigned comment added by Yuanquan74 (talk • contribs) 23:52, 22 June 2011 (UTC)

Since log(P1/P0) gives the number of bels, the number of decibels is 10 times greater. Jc3s5h (talk) 01:42, 23 June 2011 (UTC)

Advantages of an image over a table

10 log10 x x 100 10 000 000 000 90 1 000 000 000 80 100 000 000 70 10 000 000 60 1 000 000 50 100 000 40 10 000 30 1 000 20 100 10 10 0 1 -10 0.1 -20 0.01 -30 0.001 -40 0.000 1 -50 0.000 01 -60 0.000 001 -70 0.000 000 1 -80 0.000 000 01 -90 0.000 000 001 -100 0.000 000 000 1 An example scale showing x and 10 log10 x. It is easier to grasp and compare 2 or 3 digit numbers than to compare up to 10 digits.

An example scale showing x and 10 log x. It is easier to grasp and compare 2 or 3 digit numbers than to compare up to 10 digits.

The image was more compact, and looked better. It also was somewhat more tamper-resistant than the table. It's not like these numbers are ever going to change, so we don't need them in a text format. --Wtshymanski (talk) 13:34, 24 August 2011 (UTC)

I'm unsure how it looks on your browser, but the table is far more compact on mine. Preference of looks is pretty subjective but at very least, the final "0" of 100 dB and "1" of -100 dB are truncated in the image due to the librsvg bug. I also prefer the digit-grouping (hence, right alignment), italicised "x" and non-bold text in the table. In the spirit of Wikipedia, I think being tamper-resistant is a drawback; we want contributors to improve articles, and tamper-resistance makes it much harder to do so. The numerical values might not change, but the presentation and formatting may. cmɢʟee☺τaʟκ 17:57, 24 August 2011 (UTC)

Bugs are ever with us but it looks ok on the browsers I use. It's also *shorter* - less verical space, so the text seems to wrap around the image better. The amplitudes column is just Wikiclutter and takes away from the point of the table, in my opinion. The version currently in the article doesn't even have the - and 100 on the same line, owing to a bad column width; I'm sure some Wiki table expert can fix that too, but the image already did this correctly. --Wtshymanski (talk) 18:10, 24 August 2011 (UTC)

I also don't like the way the (implied) decimal point jumps around at 0, spoiling the visual effect. Looking at the image just makes me happy. Oh, and if you take out' width="1" ' in the table header, it just gets worse; you either have a table taht takes up half the screen width, or else a table that can't put -100 properly in a column. --Wtshymanski (talk)

dB power ratio amplitude ratio 100   10 000 000 000 100 000 90 1 000 000 000 31 620 80 100 000 000 10 000 70 10 000 000 3 162 60 1 000 000 1 000 50 100 000 316 .2 40 10 000 100 30 1 000 31 .62 20 100 10 10 10 3 .162 0 1 1 -10 0 .1 0 .316 2 -20 0 .01 0 .1 -30 0 .001 0 .031 62 -40 0 .000 1 0 .01 -50 0 .000 01 0 .003 162 -60 0 .000 001 0 .001 -70 0 .000 000 1 0 .000 316 2 -80 0 .000 000 01 0 .000 1 -90 0 .000 000 001 0 .000 031 62   -100 0 .000 000 000 1 0 .000 01 An example scale showing power ratios x and amplitude ratios √x and dB equivalents 10 log10 x. It is easier to grasp and compare 2- or 3-digit numbers than to compare up to 10 digits.

An example scale showing x and 10 log x. It is easier to grasp and compare 2 or 3 digit numbers than to compare up to 10 digits.

I agree about the decimal point jumping, so have fixed it. The table is shorter than the image in Firefox, but I see what you mean on Internet Explorer, so I've also fixed the split "-100", reduced the padding and removed gridlines. Incidentally, User:Dicklyon added the amplitudes column, not me. cmɢʟee☺τaʟκ 12:00, 25 August 2011 (UTC)

Much better, though now it's illustrating a different topic than the original figure. It's good that you've made it accesible even to that tiny minority of Web users who use Internet Explorer. --Wtshymanski (talk) 14:24, 25 August 2011 (UTC)

Constants for 1 dB not mentioned?

Shouldn't the two constants:

• 10 ^ 0.1 = 1.258925411794167
• sqrt(10) ^ 0.1 = 1.122018454301963

be mentioned? They are the ratio of power (or amplitude) of two fields/waves that differ by one decibel, no? --Noleander (talk) 23:54, 15 November 2011 (UTC)

Sure they should, as long as you cite a source. Dicklyon (talk) 23:59, 15 November 2011 (UTC)

Okay, I found some sources that define 1 dB as those constants. Feel free to improve the wording to make it clearer. --Noleander (talk) 00:34, 16 November 2011 (UTC)