Talk:Decibel

Template:Vital article

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)

Proposal: the definition from the standards as the primary definition

For nearly all units, scales etc., Wikipedia articles use the definition from the standards as the primary definition. For the sake of neutrality, we should do the same for the decibel. For the sake of completeness, other definitions should be presented as well. For the sake of clarity, the separation should be kept explicit. As Dondervogel suggested, this can be done in text form.

Definition In the standards ISO 80000-3 and IEC 60027-3, the decibel (dB) is defined as one-tenth of a Bel, more specifically, 1 dB = 1 B/10 = (ln 10)/20, which is approximately 0.11513.

It is used for scaling level differences between power quantities or between field quantities. The level difference L(P/P') between power quantities P and P' is defined by L(P/P') = (1/2) ln (P/P'). Hence, expressed in decibel, L(P/P') = 10 lg (P/P') dB. Conversely, x dB = L(10^x/10), for instance, 1 dB = L(1.25893).

For field quantities F and F', level difference is defined by L'(F/F') = ln (F/F'), hence L'(F/F') = 20 lg (F/F') dB.

Taking for P' or F' be a reference value P_ref or F_ref allows expressing "absolute" levels.

Standard notations The IEC 60027-3 (page 19) writes L(P/P_ref) as L_P/Pref or as L_P (re P_ref). It also condones writing L_P = x dB (P_ref).

This is an initial example. Any comments? Boute (talk) 13:12, 6 September 2014 (UTC)

While the wording can be improved, I agree in principle. I suggest starting with "The decibel is a unit of level, defined as one tenth of a bel", before going on to explain how the bel is related to the neper. Dondervogel 2 (talk) 20:11, 6 September 2014 (UTC)

I don't care for it. The decibel has a long and useful history before the standards org tried to pin it down as just a unit of level. It's more widely used to represent gain and loss than level. Where our lead says "often" I wouldn't want to imply "usually" or "always" as Dondervogel does. The standard org's "level differences" is better. Dicklyon (talk) 20:18, 6 September 2014 (UTC)

And the statement In the standards ISO 80000-3 and IEC 60027-3, the decibel (dB) is defined as one-tenth of a Bel, more specifically, 1 dB = 1 B/10 = (ln 10)/20, which is approximately 0.11513, which makes the decibel just the nondimensional value 0.11513, gives no clue to what the decibel is, or is for. Dicklyon (talk) 20:24, 6 September 2014 (UTC)

@Dondervogel: Even the literature before the original definition and the "defining papers" by Martin and Hartley indicate that level difference is indeed more fundamental than level, which depends on a fixed reference value, irrelevant for gain and attenuation. The IEC 60027-3 implicitly recognizes this, but writes sloppy equations like Q_(F) = ln(F₁/F₂).

@Dick Lyon: Although the standards are notationally sloppy (but "all" textbooks using dB are even worse), at least they give a clear and unambiguous definition of the decibel. It is fully self-consistent, and the fact that it defines dB to be just the number (ln 10)/20 does not hamper understanding --- perhaps the contrary. It also helps understanding if definition is separated from use (which is also subject to evolution): if two issues can be explained separately, it is always simpler to do so. Anyway, the "normal use" is explained immediately in the very next sentence: "It is used for ...", so nobody will miss it.

Since you seem to prefer the original definition over the standards for historical reasons, here is a proposed summary. Stating the intended use from the very start is justified here, because that was the only purpose --- a single-mindedness that arguably is the source of most problems that still plague us today!

Definition In the original papers (Martin, Hartley), the decibel (dB) is defined under its early name Transmission unit (TU) as a unit for level difference between power quantities. The level difference L(P/P') between power quantities P and P' is defined by L(P/P') = log (P/P') where log is the logarithm with unspecified base. The decibel is a unit defined by dB = log 10^1/10. Hence L(P/P')/dB = (log(P/P'))/(log10^1/10) = 10 lg (P/P'). Equivalently, L(P/P') = 10 lg (P/P') dB. Conversely, x dB = L(10^x/10).

Note: log with unspecified base is formally handled just like log_b for arbitrary --- but consistently used --- base b, with b > 0 not equal to 1. Thus, log (P/P') is indeed a fundamental "quantity" and dB is truly a unit of measurement, as emphasized in Horton's 1954 paper. The neper (Np) would be defined as log e² and the bel (B) as log 10. There used to be a Wikipedia article about the indefinite logarithm, possibly by Michael Frank, but some "mathematicians" who clearly missed the point removed it.

Clearly there are enough possibilities to choose from (see the earlier table). If your own preference is not included, it would be most helpful if you stated it completely and unambiguously. Sloppy notation (if any) can always be cleaned up afterwards. Boute (talk) 04:28, 7 September 2014 (UTC)

The Martin 1929 and Harrison (NBS) 1931 papers don't say anything about level or level difference; those concepts were made up later. Not sure about Hartley, as I don't seem to have that one. Harrison has a pretty traditional definition: The decibel may be defined by the statement that two amounts of power differ by 1 decibel when they are in the ratio of 10^{0.1} and any two amounts of power differ by N decibels when they are in the ratio of 10^N(0.1). The number of transmission units [decibels] expressing the ratio of any two powers is therefore ten times the common logarithm of that ratio. I don't see any ambiguity there. Why not stick with it? Looking at your table, I'd note that Martin does include the neper, and I don't see what the distinction is between the "naive" and "original Martin/Hartley" definitions, or why you can the latter in terms of level difference when that concept wasn't there yet (or was it in Hartley?). Dicklyon (talk) 05:50, 7 September 2014 (UTC)

Let's consider these issues one at a time.

(a) It is not helpful denying existence on the basis of "not having seen". The term "power level" (as a logarithmic quantity) is used in the 1923 paper by Alva Clark "Telephone transmission over long cable circuits" (pages 79-80, with an illustration) and in Hartley's 1924 paper "The Transmission Unit" (page 37) (sent to you on request) in a manner indicating that it was already common terminology at that time.

(b) I appreciate it that you make a concrete proposal. Still, Harrison's statement has some serious flaws, enough reason for not sticking with it. First, since you mention ambiguity (I didn't!) Harrison's statement leaves "difference" between "amounts of power" and "decibel" undefined. If one formalizes his definition as "D(P,P') = N dB iff P/P' = 10^N/10", both D and dB are still undefined. In fact, the pair D(P,P') = log_b (P/P') and N dB = log_b 10^N/10 satisfies the statement for any base b (>0 and /=1). Moreover, even the pair D(P,P') = P/P' and N dB = 10^N/10 satisfies it (and arguably better reflects current practice)! Second, it perpetuates the bias towards power quantities. This is mathematically nonsensical, since a ratio of quantities of the same dimension is a pure number: (1.34102209 hp)/(1 kW) = (3.2808399 ft)/(1 m) = 1. Sound notational engineering means treating all dimensionless ratios on the same footing. Third, your earlier argument: in no way does Harrison's restriction to power quantities reflect current practice.

(c) The naïve usage just says: let's represent P/P' by the expression 10 lg(P/P') dB, without caring about what it means. The Martin/Hartley definition is quite precise.

(d) Even if the term level hadn't been there in 1924 (but it was), there is no problem in the table. In general, recasting old concepts in new terminology is rarely a problem in science; to the contrary: it helps unification, streamlining and clarification.

(e) Thanks for the "neper" remark. I found "neper" in Martin's 1929 paper, and will amend the table ASAP. Boute (talk) 10:46, 7 September 2014 (UTC)

The table has now been amended by adding Note 0. Boute (talk) 15:41, 7 September 2014 (UTC)

Clark is using "power level" more informally, though I see you're right that he does plot "Comparative transmission levels in miles of standard cable". Maybe from your collection of sources you can find when the use of "level" to mean explicitly a logarithmic quantification of power came in. I'd be interested in knowing. I think it was a retrospective rationalization of the informal concept of level as represented by decibels. Dicklyon (talk) 16:43, 7 September 2014 (UTC)

On Hartley 1924

Boute, thanks for the copy of the Hartley paper. I see you are right that he very explicitly talked about how to interpret the decibel or transmission unit as a unit for measuring the log of a ratio (he discussed both current ratios and power ratios, and the advantage of the latter, in various parts of the paper). He says:

... Here the quantity which the unit expresses is the logarithm of a current ratio. The number of units x is the logarithm of the ratio being measured divided by the unit, which is log b. Thus the nature of the unit is the logarithm of a current ratio. Its magnitude is the logarithm of that particular current ratio b which is chosen for defining it; in this case 1.115. It should be noted that (11) is true regardless of the base of the system of loga­rithms used. The numerical value of the unit will, of course, vary with the base chosen, but the number of units corresponding to the particular current ratio will not.

... we see that the TU is a unit for expressing the logarithm of the ratio of two amounts of power, and that it is numerically equal to the logarithm of a power ratio of 10^{0.1}. When common logarithms are used its value is 0.1 and the number of units corresponding to any power ratio is ten times the common logarithm of the ratio.

So, this is like what the modern standards do, except that they use base e for current ratios. What bothers me most about this approach is how arbitrary, technical, and non-useful it is, in the sense he that notes: the value of the unit doesn't matter; a decibel could have any value, depending on what log base you choose. The only possible reason to tie it to a value this way would be so that you can write the nice math to relate the decibel to the neper. But in my experience, nobody ever need to know that to work with decibel or nepers, or to convert between them. It's arbitrary and useless to try to pin down a value for the unit. As Hartley says, "The numerical value of the unit will, of course, vary with the base chosen, but the number of units corresponding to the particular current ratio will not."

The official definition the decibel (dB) is defined as one-tenth of a Bel, more specifically, 1 dB = 1 B/10 = (ln 10)/20, which is approximately 0.11513 is of this sort. It defines the decibel to have a value—a valid that doesn't matter at all to what the decibel actually means—but the only clue to what that value measures would have to be traced down via the reference to the Bel, which would redirect to the Neper, which would still leave the answer very cryptic. As a definition, it's the sort only useful to a mathematician, perhaps, or to a standards committee.

As for "Level" as a logarithmic measure of power, that's not what Hartley says. He defines level more informally, suggesting it's just a power or a power ratio, but that it's convenient to display it logaritihmically (my bold):

... Corresponding to each point along the circuit is plotted the “level” at that point relative to some point, usually the entrance to the long distance line, which is taken as a reference level. The level at any point is determined by the ratio of the power passing that point to that passing the point of reference. The purpose of such a chart is to indicate on the one hand what power the various repeater tubes will be called upon to handle, since they are limited in this respect, and, on the other hand, what is the ratio of the power of the voice currents to that of the inter­ fering currents. This ratio is important because it determines the detrimental effect, of the inter­ference when it reaches the listener. These relative levels are most conveniently plotted in logarithmic units, so it was natural to use 800 cycle miles.

It was some time later that the practice of treating level logarithmically got turned around and level got defined as log of power. I haven't quite found where that happened. In 1933, Fletcher and Munson define level in terms of dB: "The intensity level of a sound is the number of db above the reference intensity," as opposed to taking dB as a unit of level; seems like it's not quite there yet. ... Reviweing a ton of papers from the 1930s and 1940s that use both "dB" and "level", I find none that suggest that level means log intensity or anything that. They all use level more informally, the way I always did, as just a way to high higher and lower, without implying how to quantify or measure it. So when did it become log power or log intensity? I can't find it. Dicklyon (talk) 23:20, 7 September 2014 (UTC)

I fully agree that the numerical value of the decibel as a unit for logarithmic ratios does not matter in practice. This is exactly what I meant by saying that Harrison's "definition", written symbolically as "D(P,P') = N dB iff P/P' = 10^N/10" (let's call this the "generic specification"), specifies an uncountable number of D, dB pairs where D(P,P') = log_b (P/P'), and the value of b remains "invisible" in usage. However, when writing down a "primary" definition for the matching decibel, one has to make a choice, and the choice in the standards is not really worse than any other. Arguably it is the most "neutral" from the Wikipedia viewpoint. Personally, of course, I prefer the (D, dB) solution that reflects actual practice: D(P,P') = P/P' and N dB = 10^N/10, where no such choice need be made.

As regards "level" as a log power, the earliest reference I found (without looking for anything earlier) is Clark's 1923 paper, Figure 7, where the vertical axis on the left is labeled as "Comparative transmission level in miles of standard cable at 800 cycles". This "800 cycle mile" (M) was a unit for the logarithm of a power ratio, and is the unit that was replaced in 1924 by the TU (1 TU $\scriptstyle \approx$ 1.057 M), in 1929 renamed dB. The vertical axis on the right is labeled by the power ratio, which Clark calls "comparative power".

Comparative level in M	-25	-20	-15	-10	-5	0	5	10
Comparative power	0.0043	0.013	0.04	0.11	0.33	1	3.0	8.9

This unambiguously defines "comparative level" (or level difference) as a "log power ratio" quantity. Clark seems to consider this usage as self-evident; probably it was common. Anyway, regardless of its origin, "level" is more succinct than "amount of power", which is also sometimes identified with the power quantity itself. Boute (talk) 06:46, 8 September 2014 (UTC)

I don't see it as unambiguous, but I'll grant you that it appears that even as early as 1922 the use of log power ratio or log current ratio as "level" was "in the air". I found a more explicit description in Espenshied's 1922 "Application to Radio of Wire Transmission Engineering", which says This necessity of having to keep the power of the received waves above the interference level may be visualized by reference to Fig. 4. Here we have what in wire practice is called a “ transmission level” diagram. Such a diagram is useful in showing what goes on in the system from the power and interference standpoints. The vertical scale is plotted in terms of the transmission level expressed as the logarithm of the current or field intensity ratios, and the horizontal scale represents progression along the system. It's still not unambiguous that is plotted in terms of the transmission level expressed as the logarithm of the current or field intensity ratios means that level is defined as the logarithm, as opposed to just being plotted as the logarithm, but eventually I grant you that's what happened. Perhaps never very explicitly, which is why I never learned it.

Anyway, I agree that if you want to pin down the definitions of things like dB most precisely, then picking a log base and defining the dB as a unit of logarithm is a way to go. Emphasizing the value of the unit before explaining what it means is not a good way to go, though, since the value is irrelevant. And I don't agree that it would be less neutral to pick log base 10 and power ratio and bel as starting points, instead of base e, current ration, and neper. Most of the world, with the exception of those standards bodies, does the former. We could do both. Dicklyon (talk) 23:17, 8 September 2014 (UTC)

But wait, I have found evidence against level meaning a log. In the 1959 BTL Transmission Systems for Communications vol. 1 p. 2-3 there's a section called "Level" that says

... To put this in the form of a defintion:

The transmission level at any point in a transmission system is the ratio of the power of a test signal at that point to the test signal power applied at some point in the system chosen as a reference point. This ratio is expressed in decibels. In toll systems, the transmitting tool switch-board is usually taken as the zero level or reference point."

So, it says level is a ratio; but it is expressed as log; the "zero level" means the 0 dB or 1:1 ratio. This is like what I've always thought: that the dB value represents the level or ratio via a log, not that the level is a log.

Checking more books, I find more:

Cooke 1942 Mathematics for Electricians and Radiomen clearly says level is a power, not a log: Because the decibel is an expression for a power ratio, it would be meaningless to say, for example, that an amplifier has an output of so many decibels unless that output is referred to some power level. Several zero-decibel levels are in use. For example, telephone engineers commonly use 0.006 watt as the reference, or zero, level.

Everitt's 1937 2nd ed. Communication Engineering defines the neper, bel, and decibel clearly as units of power ratio, not of level. For level, he does not use log (his axis label has "power in watts" with 10^{-2} and 10^{-5}), and notes that for a long transmission line's level plot If the ordinates are logarithmic, the decay curves will be linear as shown.

Frederick Emmons Terman's various books don't mention level but have a traditional simple non-rigorous definition of decibel.

Frankly, I don't find anyone defining level as a log, and decibel as a unit of level, until the modern standards committees. Dicklyon (talk) 23:56, 8 September 2014 (UTC)

I found a handful of papers that include "level is defined as the logarithm", going back to Fletcher 1935 with "loudness level is defined as the logarithm". And similarly few in books. It seems a rarity, though it may be expressed in different words; can anyone find? My friends in the speech business see it as I do: the level is not a log, but is often expressed in dB. Dicklyon (talk) 02:36, 9 September 2014 (UTC)

Of course, the term level has been used as sloppily as the decibel during the 90 years since its definition. Still, what ambiguity do you see in Clark's usage?

I don't like the standards any more than you do (although my criticisms have quite different grounds), but at least they made "level" precise, as distinct from "ratio". This is a useful distinction. Consider the original definition of TU (dB) mentioning "difference between amounts of power". In English, difference means a - b. If "amounts of power" is interpreted as the values of the power quantities, this would mean P - P', --- not what we want. The term "difference" suggests logarithms of the represented ratios.

Anyway, let's postpone choice of words until concepts are settled. Your suggestion to use base 10 rather than base e² would amount to 1 B = 1, an option considered in Mill's "tutorial" on the standards, so you have a reference. This choice is no problem in view of the "generic specification" I mentioned earlier, allowing to choose any base.

When making some final changes to the paper I mentioned, I realized that one of my examples illustrates that the value chosen by the standards is the "most natural" one, and that its numerical value does matter: if cable loss is expressed as N dB/m, then, with dB = (ln 10)/20, this is the correct coefficient b in e^-bx (for the voltage ratio at distance x). Epistemologically, I came to the conclusion that many useful aspects of the decibel remain unexploited because "everyone" says that they are unimportant, such as precise definitions, and as a result don't become common engineering knowledge: a vicious circle.

I'm off for a few days of vacation, without Wikipedia reading. Boute (talk) 15:06, 9 September 2014 (UTC)

It matters not what we like, but the empirical facts. I can't speak for other sciences, but I know that in acoustics, the term "level", as formally defined by ANSI since the 1960s, has been the logarithm of a ratio, and the decibel has been a unit of level, so that is what the article should say. I do not understand the arguments of those who resist that. Dondervogel 2 (talk) 15:09, 14 September 2014 (UTC)

Early definitions of "level", "bel" and "decibel"

The earliest formal definition of "level" I am aware of is from ANSI S1.1-1960 Acoustical Terminology. The following definitions are all from that (American National) Standard

Level: In acoustics, the level of a quantity is the logarithm of the ratio of that quantity to a reference quantity of the same kind. The base of the logarithm, the reference quantity, and the kind of level must be specified.

Bel: The bel is a unit of level when the base of the logarithm is 10. Use of the bel is restricted to levels of quantities proportional to power.

Decibel: The decibel is one tenth of a bel. Thus, the decibel is a unit of level when the base of the logarithm is the tenth root of ten, and the quantities concerned are proportional to power.

Dondervogel 2 (talk) 09:21, 15 September 2014 (UTC)

Prior to 1960, the American National Standard was ASA Z1.24-1951, which contains the following definitions of "bel" and "decibel" (none of level):

Bel: The bel is a dimensionless unit for expressing the ratio of two values of power, the number of bels being the logarithm to the base 10 of the power ratio.

Decibel: The decibel is one-tenth of a bel. The abbreviation "db" is commonly used for the term decibel.

It seems reasonable to conclude from this that the modern definition of level was introduced in 1960, and has been with us for more than half a century. Dondervogel 2 (talk) 09:29, 15 September 2014 (UTC)

Good find. Yet, though the standarrds committee defined it thus, I don't think it was ever taught that way; certainly not in my engineering education in the 1970s. Has such a definition ever been widely adoped in texts? I have no objection to putting the standards-based definitions in the article, and attributing them as such, but we should also put the more conventional or "informal" way that people have been taught, which avoid the funny ideas of the neper being equal to 1 and such. Dicklyon (talk) 13:15, 15 September 2014 (UTC)

In that case we are in complete agreement :) Dondervogel 2 (talk) 13:22, 15 September 2014 (UTC)

Good. I also find the definition in terms of level to be awkward to apply to the usual uses for dB for gains and losses, since those don't really involve any reference quantities. They presume linearity and treat ratios of output to input, not to reference. Dicklyon (talk) 15:38, 15 September 2014 (UTC)

The way the standards deal with gains and losses is by defining them as level differences where a "level difference" is just that, the difference between two levels. Dondervogel 2 (talk) 16:39, 15 September 2014 (UTC)

I understand. But it's a fiction, since the gain or loss in dB as a level difference doesn't involve any actual levels or references. And it gets ever further from the original "transmission unit" that was a gain or loss and had nothing to do with level. Dicklyon (talk) 05:14, 21 September 2014 (UTC)

Definitions evolve with time, usually for the better. If the intensity falls from 500 W/m^2 to 5 W/m^2, the level falls from 27 dB re 1 W/m^2 to 7 dB re 1 W/m^2. The ratio of the two intensities is 100 and the difference between the two levels is 20 dB. Where is the fiction? Dondervogel 2 (talk) 06:38, 21 September 2014 (UTC)

I deal a lot in s-parameters for passive RF circuits. For example, the coupling between two antenna-like structures may be described as -20 dB. Where is the level in that measure? Assuming electromagnetic coupling is linear (it should be), the coupling is independent of the input power - which is why EM coupling is usually given as a ratio. GyroMagician (talk) 15:52, 1 October 2014 (UTC)

If EM coupling is a ratio of power (or root-power) quantities, it does not make sense to express it in units of dB. It is only when such quantities are expressed in logarithmic form (as a level) that the dB makes sense. Dondervogel 2 (talk) 12:38, 4 October 2014 (UTC)

This is where we differ in opinion. I do not consider a dB to be a unit, in the sense that a volt or a meter is. I consider the (deci)bel to be a convenient way to represent a unitless ratio in logarithmic form. I did not invent this usage - it's standard practice, at least in the physical sciences and RF/MW engineering. One example would be the one I gave above, where EM coupling (or any other s-parameter measurement) is rarely quoted in anything other than dB. I chose this example because it demonstrates a case where there really are no units - the coupling itself is a unitless ratio - and the level has no meaning. To limit the measurement to a particular level is to ignore the generality of the result. As Dickylon says, the level is a fiction. GyroMagician (talk) 01:19, 2 November 2014 (UTC)

specific proposal

How about this to replace the opening sentence of the lede?

The decibel (dB) is a unit of level that is formally defined as one tenth of a bel. Historically, the bel is defined as a dimensionless unit for expressing the ratio of two values of power, the number of bels being the logarithm to the base 10 of the power ratio. Modern standards define the bel by linking it to the neper. The decibel is a logarithmic unit used ...

Dondervogel 2 (talk) 17:05, 17 September 2014 (UTC)

Sounds good. Dicklyon (talk) 01:49, 21 September 2014 (UTC)

I agree with the anon IP that the new version of the lede is unclear, so I re-reverted back to the old version. Think of someone coming to this page who does not know what a dB is (or who has forgotten). The lede is the single most important paragraph in the article, and should be clear and concise. Is the linking of the decibel to the neper really the most important fact about it? Then why should it appear in the lede? Similarly, anyone with a basic grounding in science or engineering (i.e. anyone who knows enough to care about units and dimensional analysis) will already understand that a ratio of two values of the same physical quantity is dimensionless - so it does not need discussing in the lede. These are details that belong further down in the article. Dondervogel2 - I also don't understand why you're so determined to state the the dB is historically dimensionless - it still is. GyroMagician (talk) 15:45, 1 October 2014 (UTC)

I still object to this sentence: "The bel is named in honor of Alexander Graham Bell, but is seldom used." Firstly it seems to be implying that we'd expect a unit named after such an illustrious person to be used rather a lot (in fact there's no need to link these two unconnected facts in one sentence). Secondly, it ought to be made clear that the decibel is not covered by "seldom used". W. P. Uzer (talk) 16:05, 1 October 2014 (UTC)

The present version of the lede now completely ignores the modern definition, and only mentions the historical definition. Are we writing a Wikipedia in the 21st Century or the 20th one? Dondervogel 2 (talk) 17:12, 1 October 2014 (UTC)

Sorry, I should not have supported putting "level" in the opening sentence. I've put that info at the end of the lead instead. Perhaps that's more appropriate? Dicklyon (talk) 17:30, 1 October 2014 (UTC)

I support DickLyon's solution. Good edit. Dondervogel 2 (talk) 19:25, 1 October 2014 (UTC)

No worries, I think we're getting a better lede now. For the 'level' and 'level difference' paragraph at the end of the lede, the linked SI document doesn't say anything about levels. What it does say (p127) is actually rather sensible (emphasis mine):

Table 8 also gives the units of logarithmic ratio quantities, the neper, bel, and decibel. These are dimensionless units that are somewhat different in their nature from other dimensionless units, and some scientists consider that they should not even be called units. They are used to convey information on the nature of the logarithmic ratio quantity concerned. The neper, Np, is used to express the values of quantities whose numerical values are based on the use of the neperian (or natural) logarithm, ln = log_e. The bel and the decibel, B and dB, where 1 dB = (1/10) B, are used to express the values of logarithmic ratio quantities whose numerical values are based on the decadic logarithm, lg = log_10. The way in which these units are interpreted is described in footnotes (g) and (h) of Table 8. The numerical values of these units are rarely required. The units neper, bel, and decibel have been accepted by the CIPM for use with the International System, but are not considered as SI units.

I guess I consider myself one of those scientists. In case you're wondering, table 8 lists a selection of non-SI 'units' that SI consider acceptable to use. GyroMagician (talk) 22:34, 3 October 2014 (UTC)

The edits to the lede were intended to reflect the definition of the International System of Quantities (ISQ), of which the SI is a part. In the ISQ, the decibel is defined as a unit of level. Your opinion has no bearing on that statement, and neither does mine. Dondervogel 2 (talk) 12:34, 4 October 2014 (UTC)

My opinion is of little importance, but the cited document is. The section I quoted was the only mention I could find of the decibel in said document, and it does not support what is written in the lede. This was, and is, my point.GyroMagician (talk) 21:21, 4 October 2014 (UTC)

Point taken. I agree the cited reference did not support the claim so I replaced it with a reference to the specific ISO standard that defines the decibel in this way. Dondervogel 2 (talk) 21:31, 4 October 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)

self-contradiction

In sec. "Properties" there is this unsourced statement: "The unit is an additive function". Taken literally, it contradicts the following sourced statement, in sec. Disadvantages: "quantities in decibels are not necessarily additive". I'm about to delete the first one. Fgnievinski (talk) 05:35, 2 October 2014 (UTC)

Fixed. Fgnievinski (talk) 23:29, 31 October 2014 (UTC)

120 dB vs 20 Pa

In the article it is claimed that 120 dB SPL "may be clearer" than "a trillion times more intense than the threshold of hearing" or "20 pascals of sound pressure". Why would an obscure logarithmic quantity ever be considered clearer than a description of the physical quantity in clear physical units??? Dondervogel 2 (talk) 20:19, 12 April 2015 (UTC)

Bogus math

In the "Power quantities" section, what's the point of asserting "Lp = 0.5*ln(P/Po) = 10*Log(P/Po)" when it is true only for the degenerative case: P == Po? Also, why term "Po" as "reference power"? "P" & "Po" should be "P1" & "P2" (that is, any arbitrary power measurements). Likewise for the "Field quantities" section. --MarkFilipak (talk) 16:04, 14 May 2015 (UTC)

Looks OK to me. The equation serves to define the decibel and is valid for any value of P/Po. Dondervogel 2 (talk) 17:48, 14 May 2015 (UTC)

Looks OK to you, eh? ...valid for any value of P/Po, eh? Okay, how about the following?

Let P/Po = 2. Then

0.5*ln(P/Po) = 10*Log(P/Po) becomes

0.5*ln(2) = 10*Log(2) becomes

0.3466 = 3.0103

--MarkFilipak (talk) 21:46, 16 May 2015 (UTC)

You missed the factor of 1 dB, which is defined as 1 Np * (1/20)*ln(10). Further, 1 Np is defined as 1, from which it follows that 1 dB =~ 0.115. If you put the missing factor back in you will see that it works. Dondervogel 2 (talk) 22:03, 16 May 2015 (UTC)

Thanks Dondervogel 2, but what I missed on the 14th was that Fgnievinski would fix it on the 15th. Fgnievinski added "Np" to the first equation. --MarkFilipak (talk) 22:32, 16 May 2015 (UTC)