Talk:Decibel/Archive 6

This is an archive of past discussions about Decibel. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

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Archive 4

Field Quantity

The article mentions "Field Quantity" but there is no reference to what a Field Quantity is. A reference that defines "Field Quantity" should be provided.121.217.138.207 (talk) 01:49, 28 May 2013 (UTC)

See this page at the NIST site. Spinning Spark 08:33, 28 May 2013 (UTC)

Recent changes to lede

The present lede begins "The decibel (dB) is a logarithmic unit that expresses the ratio between the levels of two physical quantities (usually ones measured in units of power or intensity)". A level is a logarithm of a ratio of two like quantities, say log(X/X0). The ratio of two levels is therefore a quantity of the form log(X/X0)/log(Y/Y0). What does that ratio have to do with the decibel? Dondervogel 2 (talk) 14:30, 19 July 2013 (UTC)

Usually X0 is the same as Y0 :) ---Ehrenkater (talk) 16:30, 19 July 2013 (UTC)

Obviously nothing with that new definition of level that you added. Level may not be the most appropriate word here, so maybe you can help fix it. And in the article you made that references only a non-free standard, you could quote the definition so we'll at least know what it says. Dicklyon (talk) 15:03, 19 July 2013 (UTC)

The ISO 80000 definition is "the logarithm of the ratio of a quantity, Q, and a reference value of that quantity, Q0", exactly as stated in the article. I will try to find a more accessible source. Dondervogel 2 (talk) 15:06, 19 July 2013 (UTC)

I have added a reference to Carey 2006 (J Oceanic Eng 2006)in the Level article. An earlier Carey report that is widely accessible can be found here [1]. The content is similar. Dondervogel 2 (talk) 10:02, 20 July 2013 (UTC)

(The problem is easily fixed by SpinningSpark reverting his edit. Dondervogel 2 (talk) 15:09, 19 July 2013 (UTC))

The heart of my change is that the decibel is primarily a ratio, not an absolute level and I am not intending to revert that. This is in accordance with madern usage and the historical origin of the decibel. I agree that level is not the best word and have changed it to value, together with some other tweaks. The first two sentences now read

"The decibel (dB) is a logarithmic unit that expresses the ratio between two values of a physical quantiy (usually measured in units of power or intensity). It is commonly used as a measure of gain or attenuation. The decibel can also be used as a unit of measure of one value relative to a specified or implied reference value."

I take issue with the claim I misused the word level. I used it with its normal English meaning. I cannot find the meaning claimed be Dondervogel in any English dictionary, including the OED (or more accessible dictionaries). We have this problem a lot with standards. They frequently define terminology that they wish everyone would use but often no one does. Gbooks does not instantly show any examples of its use in this sense. In fact, it shows numerous hits for "logarithmic level", which, according to the ISO 8000 definition, would suffer from the same "loglog error" described by Dondervogel. I request that examples of usage in reliable sources (usage, as opposed to mere definitions) are provided. If no one is using this terminology in the real world, we should not be doing so in this article either. Spinning Spark 16:19, 19 July 2013 (UTC)

The trouble is that both "decibel" and "level" have some rather slippery definitions and uses. It is not uncommon to use "level" as Spinningspark has, e.g. in this book, as a non-logarithm way of indicating a reference value. The ISO standard notwithstanding, there is not just one widely accepted definition and usage. Tbird, the article has evolved over years to try to find a happy consensus compromise among viewpoints of many editors and their favorite sources; your recent pushy changes are upsetting things a bit. Let's try to work more cooperatively to find an ideal clarification here. Going back to the July 11 lead sentence might be a good step: "The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level." Though I agree with Spinningspark that the decibel as a ratio is a pervasive use even when it doesn't correspond to a level (e.g. decibels of gain). Dicklyon (talk) 16:35, 19 July 2013 (UTC)

The advantage of the ISO definition is that it is simple, unambiguous and precise. It is also precisely the meaning of the word in terms like "sound pressure level", "power level", "voltage level", "sound exposure level" and all other levels usually expressed in decibels that I can think of. All of these are examples of the use of the word "level" as defined by ISO. In the context of this article, what could be a more appropriate than that? Dondervogel 2 (talk) 17:09, 19 July 2013 (UTC)

(after ec x2) "Pushy changes?" I don't think language like that is going to be constructive. Reverting to "the decibel is a logarithmic unit of measurement..." would not only be incorrect but it also disagrees with the body of the article in the history and definition sections. The WP:LEAD is supposed to be a summary of the article, not say something different.

I note that ISO 8000 is on the subject of "data quality" and so does not cover all the fields in which decibels are used. I also note that ISO 8000 has a vocabulary volume. Is this to explain vocabulary used in the standard or recommneded vocabulary for general use? Spinning Spark 17:11, 19 July 2013 (UTC)

Perhaps not the best language, but I was referring to Tbird (Dondervogel), not your changes. You are right that the old lead was also not great. Dicklyon (talk) 17:20, 19 July 2013 (UTC)

Oh, I see. Sorry for the misunderstanding. Spinning Spark 17:27, 19 July 2013 (UTC)

@Dondervogel. Certainly "levels" are frequently expressed in dB, but that does not mean that the use of the term level implies expression in dB. A level can just as well be expressed in watts or volts. Do we have any incontrovertible examples of a reliable source using level expressly for a value in dB but using some other word when the value is expressed in non-logarithmic units? A simple definition is only an advantage if it happens to also be correct. Spinning Spark 17:45, 19 July 2013 (UTC)

@anyone you cares to listen. I think we are talking at cross purposes here. Like you, I am only trying to improve the article by making it simpler and clearer.

The ISO 80000 series is entirely about the definition of quantities and units in science and technology. It has no other purpose. In other words it defines the language in which ISO speaks. It does so in a way that it is consistent with IEC and the SI. It is the language of science and technology agreed by international consensus. Wikipedia can choose to depart from this international consensus if it wishes, but it does at the risk of introducing unnecessary confusion.
I did not intend to imply the term level implies use of the decibel. Only that the decibel is always a unit of level (or level difference).

I hope this helps clear up any misunderstanding. Dondervogel 2 (talk) 09:43, 20 July 2013 (UTC)

I would be fine with the article saying this is the definition in IEC/ISO standards (but possibly not in the lede). I am not fine with the article implying this is the standard actually in use. I see no evidence that the engineering and scientific community have actually adopted it. So it's back to requesting sources showing that engineers and/or scientists use this terminology. I can find numerous sources talking about a voltage level in volts, for instance, but sources using level exclusively for a logarithmic measure and value for a linear measure seem to be hard to come by. Spinning Spark 11:16, 20 July 2013 (UTC)

Now I am truly confused. It seems to that every time a sound engineer says "sound pressure level" he is using that definition whether he is aware of it or not. What other kind of level could it possibly mean? Dondervogel 2 (talk) 22:32, 20 July 2013 (UTC)

That may be the usage of sound engineers in the very narrow sense of SPL, but sound engineering is not the only field that decibels are used in. The lede definition should not be limited to the usage of one discipline. Even sound engineers use the word level outside of this sense, eg [2] Spinning Spark 12:40, 21 July 2013 (UTC)

I am not suggesting that there is one and only one definition of the word level. My point is that when the word is used in the context of the decibel (sound pressure level, voltage level, power level etc), it is always in the ISO 80000 sense. I am also saying that the decibel is always a unit of level (or level difference) in the same sense, and that making that simple assertion can help readers understand what a decibel is. Dondervogel 2 (talk) 13:06, 21 July 2013 (UTC)

I could equally maintain that when the word is used in the context of the volt it is never in the ISO 8000 sense. I fail to see how using this terminology helps the reader with an understanding of the decibel. It merely serves to try and reinforce a particular usage of terminology, and is unrelated to the decibel per se. Spinning Spark 14:10, 21 July 2013 (UTC)

The subject of the article is the decibel, not the volt. Using the ISO meaning of level helps because when a level is expressed in decibels, it is always in that sense. Dondervogel 2 (talk) 15:35, 21 July 2013 (UTC)

It only helps if that is the widely understood meaning. It is counter to clarity if the reader is forced to look up another article to fully understand the sentence. Spinning Spark 16:08, 21 July 2013 (UTC)

– (ec) – The ISO is trying to narrow the meaning of "level", but if you look at actual usage in books discussing decibels, it is not unusual to find a "refernce level" expressed as a power or voltage, not logarithmically. See books. If we simply adopt the ISO definition, it makes it hard to discuss the topic in terms that we find in sources on the topic. Yes, "level" in the context of dB is often per the ISO definition; but it's also often not. In my experience, the word has a sort of informal interpretation, meaning something like "steady-state intensity", indicating how strong a signal is without saying anything else about the signal. I don't mind that the ISO is trying to standardize it, but we can't wish away the typical usage. Maybe some day their definition will be more widely adhered to, but for now it's not even known, pretty much. Dicklyon (talk) 16:13, 21 July 2013 (UTC)

Talking of the lead section, I am unconvinced that "the ratio between ..." is correct English. 86.171.174.107 (talk) 22:56, 20 July 2013 (UTC)

It's less common than "the ratio of x to y", but not unusual or wrong. See [3]. Dicklyon (talk) 22:59, 20 July 2013 (UTC)

I disagree with the IP editor. I consider the word "ratio" to be critically important—it must be presented in the first sentence. The two lead sentences described above are quite satisfactory: "The decibel (dB) is a logarithmic unit that expresses the ratio between two values of a physical quantity (usually measured in units of power or intensity). It is commonly used as a measure of gain or attenuation. The decibel can also be used as a unit of measure of one value relative to a specified or implied reference value." Binksternet (talk) 16:46, 21 July 2013 (UTC)

I agree. But I don't think that's what the IP was objecting to. I'm thinking he perhaps wanted to say something more like "the ratio of one value of physical quantity to another", instead of the "the ratio between two values of a physical quantity". Or I could be wrong about what he wanted. I think it's OK either way. Dicklyon (talk) 16:50, 21 July 2013 (UTC)

The word "ratio" is correct and necessary. The problem is with the phrasing "the ratio between two values". You can talk of the ratio of one value to another, but not of the ratio "between" values. Of course, Google search will find numerous instances, as it does of everything, but it is not good English in my opinion. 86.160.208.172 (talk) 19:08, 21 July 2013 (UTC)

Your opinion is acknowledged. But the data show this as a 10% usage compared to "the ratio of X to Y". That's not so low that you can get away with simply calling it an error and pretend it doesn't happen. Dicklyon (talk) 05:35, 19 August 2013 (UTC)

Can Wikipedia ever get this right?

I asked this question rhetorically some time ago, and the answer is clearly "no". Empirical evidence is abundant in nearly all the discussion pages, especially on scientific topics, and the cause is inherent in the entire mechanism and system of rules, providing an ongoing vindiction of Larry Sanger's views. Anyway, let's return to the topic at hand.

The first sentence in the lede, "The decibel (dB) is a logarithmic unit that expresses the ratio between two values of a physical quantity", is nonsensical, since a unit is a fixed quantity the ratio to be expressed is variable. A quick patch would be replacing "that expresses" by "for expressing", but more than grammatical tinkering is needed to get a decent definition.

To start with, one must answer the question: assuming the decibel is defined as a unit (a matter of viewpoint), for what kind of quantity is it a unit? Examples:

Q: For what quantity is the meter a unit? A: Length. (Alternatively, distance or "position difference")

Q: For what quantity is the volt a unit? A: Electric potential difference.

Now, for what quantity is the (original or standard) decibel a unit? A: (power) level difference. Repeated opposition in these pages to the word level notwithstanding, this is a historical fact; see A.B.Clark, "Telephone Transmission Over Long Cable Circuits", BSTJ Jan. 1923, p. 80, and R.V.L.Hartley, "The Transmission Unit", Electrical Communication, Vol. III, No. 1, p.37. Hartleys use of the term "level" (p. 37) is preserved in the current ISO 80000 usage, with two differences

A subtle technical difference: Hartley (p. 34) (like Martin, also in 1924) defines the numerical value to be log 10^0.1, where the base of log is indefinite (essential!), whereas the standard takes the base to be e² (since the Neper was chosen to be the coherent unit), making the numerical value about 0.1151293.

A convention difference: ISO 80000 usage incorporates a reference value in the definition (which is rather clumsy).

The point is that the importance of level for defining the decibel (in the original and ISO 80000 sense) equals the importance of length for defining the meter: it is the name of the quantity being measured! Boute (talk) 09:18, 18 August 2013 (UTC)

I really don't understand what this has to do with Larry Sanger, but in any case, the article does go on to say that "One of these quantities is often a reference value, and in this case the dB can be used to express the absolute level of the physical quantity." Radiodef (talk) 18:53, 18 August 2013 (UTC)

My opening remark (which is about views, not persons!) refers to the problem that this article is endlessly going in circles about a rather simple concept. I agree with you that the word "level" appears, but after the fact (defining dB), and without defining the concept itself. A good way is defining level difference before decibel, just as in electronics potential difference is defined before volt. Example: "The level difference L between two signal power values P and P' is the logarithm of the ratio P/P' (or P'/P depending on the viewpoint, e.g., gain or loss). The decibel is a unit for level difference defined by L = (10 log₁₀ P/P') dB". Note how the subtle technicalities are circumvented by omission, so they can be discussed afterwards for interested readers. Boute (talk) 04:58, 19 August 2013 (UTC)

Afterthought (inspired by the orginal 1924 formulation and the ISO standard): "The decibel (dB) is a unit for level difference, defined such that 1 dB is the level difference representing the power ratio 10^1/10. Hence (10 log₁₀ P/P') dB is the level difference between P and P'.". Nevertheless: actual current practice reflects so many generalizations that the original definition (dB as a unit for power level difference) has de facto become an atavism, to the extent that only a unifying formulation (dB as a scaling function) can adequately cover present-day usage. Boute (talk) 06:31, 19 August 2013 (UTC)

Disadvantages

According to the article, the decibel has these disadvantages:

The decibel creates confusion
The logarithmic form obscures reasoning
Decibels are more related to the era of slide rules than that of modern digital processing
They are cumbersome and difficult to interpret

I do not have access to the source which is cited. To me, all of these except possibly the third seem dubious. Does this list reflect a general view or just one person's opinion? 81.159.105.254 (talk) 02:21, 12 July 2013 (UTC)

I don't know how widespread the expressed views are, but will try to find out. In the meantime I have reworded the section in a more neutral manner. Dondervogel 2 (talk) 12:47, 12 July 2013 (UTC)

Any system that is not understood properly can create confusion, obscure reasoning and be difficult to interpret. When it is understood it is none of those things. As for slide rules it is hard to see the point - decibels are added and one cannot perform addition on a slide rule. I don't doubt that this is the opinion of the source, but unless sources are found saying that this is a widespread attitude then I think WP:UNDUE can be invoked here. The very widespread use of decibels speaks for the very opposite attitude. The real, and widely recognised, issue with decibels, as discussed at great length on this talk page, is that they are not recognised by, or compatible with, the SI system of units. Spinning Spark 13:13, 12 July 2013 (UTC)

I have added two more articles arguing that use of the dB results in confusion, all published in the Journal of the Acoustical Society of America. Does anyone know of an article making the opposite claim, that decibels reduce confusion? Dondervogel 2 (talk) 14:07, 12 July 2013 (UTC)

Messner has produced a couple of entertaining papers extolling the virtues of decibels. Not sure quite what could usefully be cited here though. Spinning Spark 14:29, 12 July 2013 (UTC)

I have reviewed the abstract of the EE article, and noted its date. 1954 is a long time ago, and hardly a good reference to discern the attitude of an entire industry 60 years later: its relevance to "the digital age" is questionable. The three other articles referenced in the footnotes for this section are tightly grouped around 1999-2000, and are all specific to SI units. If anyone can show that the dB is an SI unit, these articles might be appropriate. I think the crux of the matter here is that dB are dimensionless, as any fraction (logarithmic or not) with the same units in both numerator and denominator are dimensionless. In this regard, the complaint against dB as useful for Dimensional Analysis (DA) is valid, but short-sighted: anyone who cared about using DA on values which are expressed in dB will start by finding out what the actual units are, and use _them_, which elevates the complaint from valid to misleading.

If anyone would like to make the referenced articles available for review (I can't afford to add JASA and IEEE to my list of memberships just to read a couple of articles) I'll be glad to see how well they support the arguments.

Which brings us to the question of whether there are articles making the opposite claim (that dB are not confusing). I find this to be a very interesting request, on the same order of whether there are articles which say that diopters are not confusing to optometrists, nm are not confusing to optical physicists, etc. There are plenty of explanatory articles and such which almost invariably start with a phrase intended to justify (often with some embarrassment) the reason for the article as "dB is one of the most commonly used units in Engineering, yet also one of the most confusing, especially when it comes to manipulating S-parameters." (vide http://www.edn.com/design/test-and-measurement/4423318/How-to-Think-in-dB), which generally show that the author has a deeper misunderstanding of units and the actual _meaning_ of dB than those he is writing to. Few ever bother to mention that, once you learn to use them and remember that they are a ratio of coefficients, not a unit, they are easy-to-use, easy-to-remember and accurately express what they are meant to express.

I believe that you could cast, as disadvantages, the requirements that you understand logarithms and ratios, that ratios between two values with the same unit are dimensionless (which may require an understanding of DA, although I doubt it) and have a reason to use them. I also believe that this would be without purpose. I bring to your attention that both the Wikipedia articles on Dimensional Analysis and Logarithm do not feel it necessary to have a "disadvantages" section.

Simply stated, dB are as useful in the fields where they are applied as any rule-of-thumb, and where someone might perceive a disadvantage, it comes from trying to misuse them. Tiorbinist (talk) 13:36, 28 March 2014 (UTC)

Advantages and disadvantages

Like many topics on WP, this article also has a list of advantages and disadvantages. These are listed randomly without examination or even any kind of qualification in which application or circumstance they are advantages or disadvantage. Just making a list without that make no sense at all. Clearly, the decibel was a very useful and easy to use unit when it was defined and used in its intended application range starting in the 1920s for telephony and for many decades. Even today, with calculators in abundance, it is still useful and easy to use, if one bothers to learn about logarithms, which seems to have fallen out of favor. Consequently, the unit is perceived as difficult. In some applications, the simplicity is surely not obvious anymore, indeed, because of the jungle of suffix versions, but the disadvantages need to be explained with context and references and not just listed without. An example is the statement (disadvantage) that the unit is not additive. Huh? It is the additivity that makes it so simple to use, that has been a major benefit throughout. Now it's not possible? Looking up reference 20 reveals that at least the abstract mentions non of the cited disadvantages, only that the unit has been misapplied where it shouldn't be. I suggest editors start cleaning this up. Kbrose (talk) 23:16, 9 May 2014 (UTC)

Does "editors" not include you? The source for "decibel units are not additive" is a deadlink but this source (amongst others) seems to put it in those terms as well. Spinning Spark 08:20, 10 May 2014 (UTC)

Horton's article is a plea to retain the use of the decibel for its traditional use as a unit of the logarithm of a power ratio, proposing a different unit (the logit) for quantities that do not satisfy this criterion. Its closing sentence reads "It is hoped that [use of the logit] may be helpful in restoring to our technical terminology some of the precision and reliability which appears to be lacking". His plea went unheeded, resulting in the mess we have today, with the decibel meaning something different to each different user, or group of users. Dondervogel 2 (talk) 09:36, 10 May 2014 (UTC)

But it's not really a mess, is it? As a physicist, I have no trouble at all in talking with my engineering colleagues about dB_this and dB_that. I don't always know the post/sub-scripts they apply after the dB, but I can always ask. For the most part though, where dBs are useful, the reference level is not so important - you just have to know that there is one.

I find this page particularly sad. Decibels are widely used, and the concept of a log scale is quite straightforward when well explained. But this page makes their use seem difficult and archaic. It is a missed opportunity, with too many editors trying to make a point rather than a clear article. If you disagree, please read the 'disadvantages' section before commenting. GyroMagician (talk) 23:02, 3 June 2014 (UTC)

If your point is that the article can be better written, I agree. If your point is that the concept of a level expressed in decibels is a useful one, if accompanied by an adequate explanation of what it is a level of, and (where appropriate) a reference value, I also agree. But if your point is that the decibel is a simple tool and no one finds it confusing I disagree. One only has to read the long list of suffixes, and bizarre accompanying explanations to become utterly baffled. A pertinent (but by no means the only) example is the confident affirmation that dB/K (or dB K^-1) means something other than decibels per kelvin. Next I shall no doubt learn that dB/m does not mean decibels per metre, and I shall not be surprised, but long before then I will have stopped trying to understand. Dondervogel 2 (talk) 17:07, 4 June 2014 (UTC)

Units: Capitalization and SPL

The article needs to explain why the B of dB is capitalized, but the unit's name (as you have written it, bel) is not. The B is capitalized because the unit's named after a person, A.G. Bell, but why isn't the unit named Bel (or even Bell) instead of bel? There's an inconsistency here.

Also, there's a unit audiophiles frequently use: dB SPL. This should at least be mentioned in the article. JKeck (talk) 19:57, 18 May 2013 (UTC)

Most units named after people use upper case symbols but lower case unit names. The article would need to explain if that was different here, but it isn't. Also, dB(SPL) is mentioned. — HHHIPPO 20:15, 18 May 2013 (UTC)

Why does the Lede call the deciBel (dB) a 'unit'? As Dondervogel points out below, 'The ISO 80000 definition is "the logarithm of the ratio of a quantity, Q, and a reference value of that quantity, Q0"'. According to Wikipedia's own definition of "Units of Measurement"^[1], "A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of measurement." ^[2]?The Bel is a ratio. Since a ratio of two different units (excluding SI multiplier-prefixes) is comparing apples to oranges, quite literally, the Bel, and thereby the deciBel is unitless. There are versions of the dB which do give a comparative value referenced to units: dBV, dBm, etc. The end of this article has a long (and necessarily incomplete list) of them. In each case, the appended letter(s) point to the reference value, and might be termed units, but the dB part remain only a ratio. This may be the biggest deficit (and greatest benefit) of the dB: calling it a unit only confuses the issue.

NIST does a good job of obscuring this by invoking Lf and Lp, without explaining that Lp (level-of-a-power-quantity) is one 'magnitude' higher than Lf (level-of-a-field-quantity). This comes from field quantities being voltage, current, etc, while power is expressed in such quantities as watts. (I limit the lists for space.) Since P = I x E, the relationship between the units is one of x^1 to x^2; i.e., through a 2-ohm resistor, two amperes of current creates a voltage drop of two volts and the power (work done, heat produced) is 2 Amp * 2 Volt = 4 Watts, a squaring relation. Since exponents are added in logarithmic expressions, dB of power is 2 times the equivalent dB of voltages or currents: this means that dB require consideration of one less aspect of the measured values than arithmetic ratios (i.e., the dimension cancels out) while requiring consideration of one more aspect (whether linear or magnitude.)

Mathematics may have a good term for the magnitude (exponentialness doesn't seem a good choice), but it does have terms for units and numbers, the latter being 'coefficient'. dB (and Nepers) are ratios of coefficients without consideration of units, beyond ensuring that they are the same. Perhaps something can be mined from Linear Algebra, which also separates coefficients from variables and computes solutions of systems using just the coefficients.

Unfortunately, Wikipedia's rules deprecate "new research", so this concept is hard to cite in a wiki article. None-the-less, once you divorce yourself from the idea that dB and Np are "units" like the SI units, and are merely ratios of the coefficients, and that you must take into account whether you are working with linear or power units, confusion usually disappears. Tiorbinist (talk) 16:33, 28 March 2014 (UTC)

References

In the International System of Quantities, the decibel, the bel and the neper, are all units of level. I see no good reason not to define them as such here. Dondervogel 2 (talk) 11:14, 21 August 2014 (UTC)

Advantages and disadvantages - continued (margin reset)

For reasons stated earlier, I rarely look at these pages any more. Since the disadvantages listed in the article are the current topic, here is a brief analysis.

The decibel creates confusion. Wrong: only sloppiness (in definition, use, and reasoning) is the cause of confusion.
The logarithmic form obscures reasoning. Right: $x\,{\text{dB}}=10^{x/10}$ reflects "modern" practice and elucidates reasoning.
Decibels are more related to the era of slide rules than that of modern digital processing. Wrong: as Ralston noted in Let's abolish pencil-and-paper arithmetic, head calculation is more important than ever in the computer era, and it is facilitated by the decibel.
Decibels are cumbersome and difficult to interpret. Wrong: see the earlier "reasoning" issue.
Decibels are not necessarily additive. Wrong: the standard decibel is additive. Unfortunately, this is precisely why dimensional analysis is applicable only after taking the logarithm (a situation that is relevant only for cables).

Soon (within minutes or weeks), I'll add some further clarification. As it happens, I just prepared a paper, available on request (my e-mail address is easy to find). Here are some "appetizers":

By the standards, levels are defined independently of the decibel by $L_{P}=(\ln(P/P_{\text{ref}}))/2$ . This corresponds to the neper (trivially defined by 1 Np = 1), and the decibel is just a scaling factor, equal to $(\ln 10)/20$ to obtain $L_{P}=(\ln(P/P_{\text{ref}}))/2=(10\lg(P/P_{\text{ref}}))\,{\text{dB}}$ .
With the standard decibel, ${\text{dB}}/{\text{m}}$ is perfectly all right and means decibel per meter, which is by no means the same as ${\text{dB}}\,({\text{m}}^{-1})$ (even if m^-1 were a "legal" reference quantity by the standards).
With the decibel that reflects actual practice, one can rigorously write $k=-228.6\,{\text{dBW/Hz/K}}$ for Boltzmann's constant. The standards would write this as $L_{k}=-228.6\,{\text{dB}}\,({\text{W/Hz/K}})$ (if they considered W/Hz/K a "legal" reference quantity).

Regards to all decibel fans. Boute (talk) 12:12, 18 August 2014 (UTC) (extra item added Boute (talk) 08:37, 19 August 2014 (UTC))

On the additivity of the decibel

The claim about non-additivity of the decibel in this source is typical for the sloppy formulations causing confusion. What the remark means (clear, if one reads the complete paragraph) is that levels are not additive. This is by itself not surprising, since log (a + a') does not equal log a + log a', but to compound the issue the example tacitly also assumes incoherent sources (in the passage about adding 3 dB; for coherent sources it would be 6 dB). All this has nothing to do with the decibel itself. The standard decibel itself is just multiplication by (ln 10)/20, which makes it trivially additive by distributivity of multiplication over addition: (x + x') dB = x dB + x' dB, and also homogeneous by associativity of multiplication: (xx') dB = x(x' dB). Boute (talk) 08:09, 21 August 2014 (UTC)

I tend to agree that levels are a problem. Decibels and nepers add just fine when they represent composed ratios, but in that usage the ratios are either not both levels, or they are levels with different reference powers. So defining decibels in terms of levels in not a net win, in my opinion. Dicklyon (talk) 20:41, 23 August 2014 (UTC)

This raises a few questions. (a) Why are levels a problem? (b) By "adding decibels and nepers", do you mean something like 10 dB + 1 Np = 2.151293 (which is correct by the standards)? (c) Can you be more precise in describing "that usage"?

I agree that defining decibels in terms of levels is not a net win. It violates the principle of "separation of concerns", a prerequisite for clarity. Here the concerns are: (i) what is the decibel (originally and by the standards: not the same!), (ii) what is its purpose (idem: originally, standards) and, to be realistic, also (iii) how is it actually used in practice. Boute (talk) 07:59, 31 August 2014 (UTC)

No, that's not the kind of additivity I mean. I've never seen anyone do anything like that, and I'm surprised that standards guys would settle on such an interpretation. In widespread normal practice, decibel addition represents cascade composition of gains. Multiplication, too. Five miles of cable with 1 dB loss per mile is 1+1+1+1+1 or 5*1 dB of loss. To do this with mixed units you'd have to convert to compatible units first. Since addition represents multiplication of the represented ratios, ordinary rules of units are inapplicable. Logs are just that way. This relationship is known as a homomorphism. Sure, it's different from the behavior of all other units. If you wanted to force it into your interpretation you'd have to write (1 db)*(1 db)*(1 db)*(1 db)*(1 db) = (1 dB)^5 = 5 dB. Nobody does that. Dicklyon (talk) 16:00, 31 August 2014 (UTC)

Still: (a) Why are levels a problem? Ad (b): Perhaps you never saw "10 dB + 1 Np = 2.151293", but (despite their sloppiness and ill-advised decisions) the standards are not open to interpretation! The ISO 80000-3 (edition 2006-03-01), page 13, item 3-22.b, clearly states: 1 Np = 1 and 1 dB = 0.1151293 Np (approximately, of course), hence (by grade school arithmetic) 10 dB + 1 Np = 2.151293. Also, by the standards, you'd have to write 1 dB + 1 dB + 1 dB + 1 dB + 1 dB = 5 dB, not the multiplication you suggest, which would be inconsistent with anyone's "interpretation" of the standards! The standard dB is a trivial homomorphism (it's also a trivial linear map, and thus behaves like an ordinary unit).

On the other hand, be very careful in saying "nobody does that". More often than not, in textbooks one finds equalities like G = P_out/P_in = 30 dB. This "direct representation" view of the decibel violates the standards, but has exactly the property you suggest: if G = 30 dB and G' = 10 dB then G*G' = (30 dB)*(10 dB) = 40 dB. This is a nontrivial, arguably more useful homomorphism, but nonstandard, and hence is better not discussed at all before other things are cleared up. In particular, (c) Can you be more precise in describing "that usage"? Boute (talk) 05:25, 1 September 2014 (UTC)

I think we can mostly agree that the standard has failed to capture practice or to make these logarithmic units understandable. Dicklyon (talk) 06:40, 1 September 2014 (UTC)

Removing confusion

None of the definitions for dB are bizarre or hard to understand in their own context: it is mixing up different definitions that causes confusion. This is not a matter of interpretation: the definitions are truly different, and there is enough variety to satisfy anyone's personal preferences. The following table will probably clear up things.

Comparison between definitions
Kind (source)	Definitions			Meaning of (10 lg(X/X')) dB	Pragmatics
Kind (source)	neper (Np)	decibel (dB)	level difference	Meaning of (10 lg(X/X')) dB	Pragmatics
Naïve use (literature)	(Np rare)	just "dB"	vague	a way to represent X/X'	ad hoc
Original (Hartley, Martin)	(see Note 0)	x dB = x log 10^1/10	L(P/P') = log(P/P')	(10 lg(P/P')) dB = L(P/P')	power quantities only
Standards (BIPM, ISO, IEC, ...)	x Np = x	x dB = x log_e² 10^1/10 = x (ln 10)/20	L(P/P') = log_e² (P/P')	(10 lg(P/P')) dB = L(P/P')	for power quantities
Standards (BIPM, ISO, IEC, ...)	x Np = x	x dB = x log_e² 10^1/10 = x (ln 10)/20	L'(F/F') = log_e (F/F')	(20 lg(F/F')) dB = L'(F/F')	for field quantities
Actual practice (literature)	(Np rare)	x dB = 10^x/10 (see Note 1)	(obviated)	(10 lg(X/X')) dB = X/X'	any quantities
Legend -->	log = logarithm with unspecified base; log_b = logarithm with base b; lg = log₁₀; ln = log_e
Legend -->	X, X': any quantities (same dimension); P, P': power quantities, F, F': field (now root-power) quantities

Note 0: the original 1924 Hartley/Martin papers do not mention "neper", which was proposed in Europe later in the same year. Hartley mentions the equivalent earlier unit $\scriptstyle \beta l$ . Martin's 1929 short note mentions "neper". The definition (in compatible style) is: x Np = x log e². Note that (1 B)/(1 Np) = (log 10)/(log e²) = (ln 10)/2, approximately 1.1513.

Note 1: this definition is inferred from expressions like "G = P/P' = 30 dB", which are ubiquitous in the literature. The preceding definitions (the original one and those from the standards) are directly taken from their sources.

Note 2: observe that the definition of "decibel" is independent of the definition of "level" (separation of concerns). For the standard variant, it is just a multiplication constant. However, it is clear that the original/standard purpose of the decibel cannot be explained properly without the concept of level difference.

Note 3: in practice, the infamous "20 log₁₀" rule is the unique major source of errors and confusion in the literature. Ideally, it should be deprecated. A compromise due to Carey essentially amounts to writing L(F²/F'²) = 10 lg(F²/F'²) dB when field quantities are involved, thus enforcing explicitness. Boute (talk) 14:31, 2 September 2014 (UTC)

Note 4 (only to be discussed if the comparison between the definitions is fully clear): throughout the literature (many references on request), usage of the decibel is quite sloppy, and authors seem continuously torn between, on one hand, writing equalities of the form G = P/P' = x dB or P = 20 dBm and, on the other hand, the additivity of the standard decibel, writing nonsense such as P = 30 dBm - 10 dB + 20 dB = 40 dBm, which they do within the same context. Clearly, this can't be done properly with the original or standard definition of dB. It is now also clear how the definition x dB = 10^x/10, inferred from G = P/P' = x dB and P = 20 dBm, allows writing rigorously P = (30 dBmW)*(-10 dB)*(20 dB) = 40 dBmW. In typical link budget tables, where figures like 30 dBm, -10 dB etc. are listed from top to bottom, the addition/multiplication sign is usually omitted, and therefore such tables become fully rigorous without rewriting them by just assuming a different underlying definition! Although this is evident, detailed examples are given in a paper (available on request). Boute (talk) 05:26, 3 September 2014 (UTC)

It is not clear to me what is being proposed here. Anything that helps clarify the decibel is valuable, but the proposed table looks complicated to me, and is unlikely to be understood without accompanying text. Would you like to clarify the main points, and propose some suitable text? Dondervogel 2 (talk) 09:24, 4 September 2014 (UTC)

This was not meant as a proposal; it is just a summary of different definitions circulating in the literature, motivated by the fact that most participants in these talks seem to have their own specific, often different definition in mind. Also, each line by itself can be understood by high school math without accompanying text; the accompanying notes are just comments on their origin, possible flaws or advantages.

However, now you mention it, a useful proposal for these talk pages is the following: whenever presenting an argument about the decibel, make clear which definition is considered: naïve ("no questions asked") and vague (abbreviated "V-dB"), original (abbreviated "O-dB"), standard (abbreviated "S-dB"), directly expressing numbers and ratios (abbreviated "D-dB"), or still something else (in that case, giving an unambiguous definition). Boute (talk) 12:27, 4 September 2014 (UTC)

To support the claim that the table can be understood with a high school math background only, here follows a detailed derivation of (20 lg(F/F')) dB = L'(F/F') (the most "complicated" case in the table) from the definitions only. Normally one would perform most steps mentally "by inspection".

(20 lg(F/F')) dB	= 20 lg(F/F') (ln 10)/20	because x dB = x (ln 10)/20 by definition
	= lg(F/F') ln 10	by basic arithmetic
	= log_e (F/F')	because (log_b y)(log_a b) = log_a y
	= L'(F/F')	because L'(F/F') = log_e (F/F') by definition

The log rules quoted assume for the argument y>0 and, for the bases, a>0, b>0 and not equal to 1. Boute (talk) 07:12, 5 September 2014 (UTC)

I guess I have too much math background to understand it. Symbol soup. Dicklyon (talk) 20:28, 6 September 2014 (UTC)

I'm not finding the table very illuminating, but starting to grok it. I'd point out that where it says "any quantities", it should be any quantities that are proportional to signal power or noise power or something like that that's being computed. Things like time, bandwidth, squared distance, antenna area, are sometimes of this sort. Dicklyon (talk) 23:23, 7 September 2014 (UTC)

Conversions between units of level

The table inserted by an anonymous user did not make sense in that form, but I would support inclusion of a similar table showing conversions between the bel, decibel and the neper. Dondervogel 2 (talk) 08:11, 23 August 2014 (UTC)

Yes, conversion from level or power to octaves is particularly nonsensical. Might be a useful concept for a VCO I suppose, but even there it would not be correct—the relationship is not that simple. SpinningSpark 10:05, 23 August 2014 (UTC)

I haven't checked the arithmetic, but I have in mind something like this

Conversion between units of level
unit	in dB	in B	in Np
1 dB	1	0.1	0.11513
1 B	10	1	1.1513
1 neper	8.68589	0.868589	1

Dondervogel 2 (talk) 11:23, 23 August 2014 (UTC)

Conversion between dB and nepers is useful, but why on earth would you want to include the bel? Bels are simply not used. Remember that a significant number of readers coming to this page will not be familiar with decibels. We should not give the impression that bels are ever used as a unit. GyroMagician (talk) 17:29, 23 August 2014 (UTC)

So that would reduce it to 1 dB in nepers and 1 neper in dB. Hardly necessary to have a table for that. Whether or not the bel is actually in use, I think it is rather trivial to include it anyway. SpinningSpark 17:56, 23 August 2014 (UTC)

In the ISQ, the bel is defined in terms of the neper, and the decibel is then defined as one tenth of a bel. I think the bel should be included, if only for that reason. Dondervogel 2 (talk) 19:40, 23 August 2014 (UTC)

That's an interesting but unusual approach they take, starting by defining "level" as the natural log of a field ratio. I wouldn't object to including the bel in a table, since people often get the factor of 10 in the wrong direction and this might help. I'd also put columns for power ratio and field ratio. Like this: Dicklyon (talk) 20:35, 23 August 2014 (UTC)

Conversion between units of level
unit	in decibels	in bels	in nepers	Power ratio	Field ratio
1 dB	1 dB	0.1 B	0.11513 Np	1.25893	1.12202
1 B	10 dB	1 B	1.1513 Np	10	3.16228
1 Np	8.68589 dB	0.868589 B	1 Np	7.38906	2.71828

What's the betting the factor of 10 keeps getting amended in the table as well? SpinningSpark 22:47, 23 August 2014 (UTC)

I would bet not. Dicklyon (talk) 00:13, 24 August 2014 (UTC)

I don't understand the wording (in the recent amendment) that says "The same standard defines 1 Np as equal to 1 ". --David Biddulph (talk) 21:28, 31 August 2014 (UTC)

That's not really an amendment, just a sentence I moved from the previous section. It means that the standard defines nepers as nondimensional. I added some text to try to explain the implications of that, which I agree is a bit bizarre and hard to understand. The point is that nepers are defined as nondimensional representation the natural log of field quantity ratio. So 1 neper is just 1, 2 neper is 2. Dicklyon (talk) 22:00, 31 August 2014 (UTC)

I find the columns "power ratio" and "field ratio" confusing. All the other columns can be read as an equality (eg 1 dB = 0.1 B), but the last two columns cannot be read in this way, and detract from the simplicity of the other conversions. I would prefer it if the ratios appeared in a separate table. Dondervogel 2 (talk) 11:22, 1 September 2014 (UTC)

Please do show us what that would look like as separate tables. Dicklyon (talk) 14:28, 1 September 2014 (UTC)

I don't know what the power ratio and field ratio columns would look like because 1 dB is not a power ratio or a field ratio, but the logarithm of such. For that reason I prefer my original proposal. A compromise might be to replace the numerical value of the power ratios with 10*lg(power ratio) and similarly for the field ratios. Dondervogel 2 (talk) 12:22, 2 September 2014 (UTC)

If 1 dB is the logarithm of a power ratio or a field ratio, why is it confusing to point out what ratio it is the logarithm of? It's not like we're saying that 2 dB would represent twice that ratio. Dicklyon (talk) 14:43, 2 September 2014 (UTC)

Agreed. If 1 Np does represent a field ratio of 2.71828, it makes sense for the table to show that, by contrast with the statement (which I still don't understand) that 1 Np is equal to 1. --David Biddulph (talk) 15:18, 2 September 2014 (UTC)

@Dick Lyon: The problem with the table as it stands is that it gives the incorrect impression that 1 dB = 1.25893. Instead it should say 1 dB = 10 lg(1.25893) dB, if that is the correct ratio. 'tis all

@David Biddulph: There is nothing to understand about 1 Np = 1. It is just a choice that someone made to define it that way. I guess it is more correct to say 1 Np := 1.

Dondervogel 2 (talk) 23:24, 3 September 2014 (UTC)

OK, but all that 1 dB = 10 lg(1.25893) dB says (up to approximation) is that 1 = 10 lg(1.25893), which is not very informative. To properly describe the relation between the original/standard decibel and power ratio, the notion of level difference is essential, because that is precisely the quantity that is originally (or by the standards) expressed in decibel (see the table below). Thus, 1 dB = L(10^1/10) and 10^1/10

\approx

1.25893 properly conveys the intended information. In words: "one decibel is the level difference corresponding to a ratio of 10^1/10, approximately 1.25893". Boute (talk) 06:28, 4 September 2014 (UTC)

@Boute: The issue here is whether we should write "1 dB = 1.25893" (the present content of the table) or "1 dB = 10 log₁₀(1.25893) dB". Which do you think is better? Dondervogel 2 (talk) 09:04, 4 September 2014 (UTC)

Clearly "1 dB = 1.25893" is wrong since 1 dB = (ln 10)/20 (approximately 0.11513), whereas (the approximation) 1 dB = 10 log₁₀1.25893 dB trivially holds since 1 = 10 log₁₀1.25893. Hence, if correctness is the only issue, 1 dB = 10 log₁₀1.25893 dB is better, but it is not more informative than f(1) = f(10 log₁₀10^1/10), which holds for any function f whose domain contains 1. Boute (talk) 11:54, 4 September 2014 (UTC)

if "1 dB = 1.25893" is "clearly wrong", why do we insist on keeping a table that says just that? Dondervogel 2 (talk) 20:34, 8 September 2014 (UTC)

It is true that 1dB is equivalent to a power ratio of 1.25893, and that's what the table says. --David Biddulph (talk) 20:52, 8 September 2014 (UTC)

Yes, those two things are indeed equivalent, in a very special sense that is not explained by the table. By contrast the first 3 columns are equalities and not vague "equivalences". The final 2 columns are at best misleading because they can only be understood by someone who knows in advance what is meant. What is the point in that? Dondervogel 2 (talk) 21:57, 8 September 2014 (UTC)

I added some text to indicate that the log differences represent ratios, and these are the ratios that a difference of unit represents. It's explained just above there, but having it by the table helps, I agree. It's certainly not trying to say "1 dB = 1.25893", but rather "1 dB represents the ratio 1.25893", which is true and useful. And "1 Np = 1" is true but irrelevant and useless. Dicklyon (talk) 02:14, 9 September 2014 (UTC)

Define "field quantity"?

No kidding the standards have favored "root-power" -- it's hard to pin down what exactly is meant by a "field quantity". It most certainly is not as in field (physics). Sure, voltage and current are good examples of field quantities, but the closest I get to a general definition is a phasor or a complex number, i.e., a quantity having magnitude and phase. But then now I learned sound pressure is a field quantity, and I don't think that has a phase. Fgnievinski (talk) 04:08, 2 October 2014 (UTC)

The ISQ is unclear on "field" vs "root power". On the one hand, ISO 80000-1:2009 deprecates "field quantity" (preferring "root power quantity"), while on the other ISO 80000-3:2006 defines "level of a field quantity" and defines the decibel in terms of that level. In the context of ISO 80000-3, sound pressure is considered a field quantity. In what sense would it not have a phase? Dondervogel 2 (talk) 06:42, 2 October 2014 (UTC)