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Definitions of the units radian, neper, bel and decibel
Metrologia
Abstract
The definition of coherent derived units in the International System of Units (SI) is reviewed, and the important role of the equations defining physical quantities is emphasized in obtaining coherent derived units. In the case of the dimensionless quantity plane angle, the choice between alternative definitions is considered, leading to a corresponding choice between alternative definitions of the coherent derived unit - the radian, degree or revolution. In this case the General Conference on Weights and Measures (CGPM) has chosen to adopt the definition that leads to the radian as the coherent derived unit in the SI. In the case of the quantity logarithmic decay (or gain), also sometimes called decrement, and sometimes called level, a similar choice of defining equation exists, leading to a corresponding choice for the coherent derived unit - the neper or the bel. In this case the CGPM has not yet made a choice. We argue that for the quantity logarithmic decay the most logical choice of defining equation is linked to that of the radian, and is that which leads to the neper as the corresponding coherent derived unit. This should not prevent us from using the bel and decibel as units of logarithmic decay. However, it is an important part of the SI to establish in a formal sense the equations defining physical quantities, and the corresponding coherent derived units.
... The obvious remedy is a precise definition. Such a definition can be found in the standards [9], but here we present one that is far simpler and universally applicable. It even supports dimensional analysis, which no other variant does. ...
... 324 m Colloquially, one often says "a quantity Q" to abbreviate "the value Q of a quantity." A quantity Q can be written as , x u where x is a number, called the numerical value of the quantity, and u is a unit or a combination of units, as in l = l m l m as the quantity equation and hence, u u u = l m without any numerical factors [9]. ...
... In the terminology of the standards, the neper, being equal to 1, is the coherent unit and, hence, the (deci)bel is not. A detailed discussion can be found in [9]. In practical use, the decibel is dominant. ...
The decibel is widely used in signal propagation. Yet, as commonly presented, it does not behave like normal units in formulas, calculations, or dimensional analysis, which hampers understanding and practical use. All issues are resolved by defining the decibel as just an elementary function in calculus, called the
direct
decibel, which reflects actual practice and is the simplest for a first introduction. Since students will certainly meet other variants, it is crucial they understand the wider picture and all definitional choices. Recognizing that units and scales are functions yields a unifying framework that explains the common conventions and eliminates unwanted effects. The standard decibel and the concept of level are explained rigorously and issues with the (in)famous “20 lg” rule are resolved. The use of the direct decibel in calculations is illustrated in examples from engineering in general, for propagation in cables and free space, noise, and satellite link budgets.
... " 2. An additional argument for declaring a plane angle a dimensionless quantity is also considered to be the power series expansion of trigonometric functions (sinθ, cosθ) based on θ (see, for example, [7,8]). 3. A statement in CIPM: Recommendation 1 [3] concerning the fact that the possibility of "interpreting the radian and the steradian as the base SI units ... violates the internal consistency of the SI system based on only seven base units." ...
... 1 Expressional (6) and (7) are provided in all mathematical reference books and encyclopedias (see, for example, [11, p. 15]). 2 In the physical literature, there is a defi nition of dimensional and dimensionless values [12]. Quantities whose numerical values depend on the selection of measurement units are called dimensional quantities. ...
... Otherwise, all terms on the right side will have different dimensions, and it is impossible to add quantities having different dimensions. The fact of θ being dimensionless in Eq. (12) was used by some authors [7,8] as a serious argument in favor of dimensionless nature of a plane angle. ...
The situation was analyzed that has been established in the International System of Units (SI) as a result of adoption of the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities. It was shown that such decision was based on a misunderstanding of the mathematical formula, relating the arc length of a circle with its radius and corresponding central angle, as well as series expansions of trigonometric functions. As follows from the conducted analysis, the plane angle value does not depend on any of the SI quantities and should be treated as a base quantity, while its unit (radian) should be added to the population of base SI quantities. Therefore, the value of a solid angle becomes a derived quantity from a plane angle, and its unit (steradian) represents a coherent derived unit equal to a square radian.
... " 2. An additional argument for declaring a plane angle a dimensionless quantity is also considered to be the power series expansion of trigonometric functions (sinθ, cosθ) based on θ (see, for example, [7,8]). 3. A statement in CIPM: Recommendation 1 [3] concerning the fact that the possibility of "interpreting the radian and the steradian as the base SI units ... violates the internal consistency of the SI system based on only seven base units." ...
... 1 Expressional (6) and (7) are provided in all mathematical reference books and encyclopedias (see, for example, [11, p. 15]). 2 In the physical literature, there is a defi nition of dimensional and dimensionless values [12]. Quantities whose numerical values depend on the selection of measurement units are called dimensional quantities. ...
... Otherwise, all terms on the right side will have different dimensions, and it is impossible to add quantities having different dimensions. The fact of θ being dimensionless in Eq. (12) was used by some authors [7,8] as a serious argument in favor of dimensionless nature of a plane angle. ...
The situation that has developed in the International System of Units (SI) as a result of adopting the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities, is analyzed. It is shown that the basis for such a solution was a misunderstanding of the mathematical formula relating the arc length of a circle with its radius and corresponding central angle, as well as of the expansions of trigonometric functions in series. From the analysis presented in the article, it follows that a plane angle does not depend on any of the SI quantities and should be assigned to the base quantities, and its unit, the radian, should be added to the base SI units. A solid angle, in this case, turns out to be a derived quantity of a plane angle. Its unit, the steradian, is a coherent derived unit equal to the square radian.
... In the papers by other authors [8][9][10][11][12], it was proposed to consider the plane angle as dimensional and refer it to the base quantities, and its unit, the radian, to the base SI units. In [13][14][15][16][17][18], the difficulties of the agreement of the non-dimensional status of angles and the existing equations of mathematics and physics are discussed. Quincey and his colleagues [19][20][21] analyzed various versions of the treatment of angles and concluded that so far it is not possible to eliminate all the contradictions associated with the current status of the angles in the SI. ...
... There is another point that may arise confusion the issue of dimensional character of angles-trigonometric functions. For example, in [13,19] a series expansion of a trigonometric function according to the powers of its argument is considered as strong evidence of the dimensionless character of the plane angle. Such confusion arises because the argument of the trigonometric functions is usually considered only angles [17]. ...
... The corresponding function θ(ϕ) will be the constant θ, independent of ϕ. The integral in (13) gives the following value of the solid angle ...
The article analyzes the arguments that have become the basis for the 1980 CIPM recommendations declaring plane and solid angles as dimensionless derived quantities. This decision was the result of an incorrect interpretation of mathematical relationships that connect the ratio of two lengths with the plane angle, and the ratio of area to square of length with the solid angle. The analysis of these relationships, presented in the article, showed that they determine neither the dimensions of the angles nor their units, but only the numerical values of the angles expressed in radians and steradians. It is shown that the series expansions of trigonometric functions sometimes used to prove the dimensionless character of the plane angle is also incorrect because in this case the trigonometric functions of two different types, independent of each other, are offen confused. It is established that the plane angle is an independent quantity and therefore should be assigned to the base quantities and its unit, the radian, should be added to the base SI units. It is shown that the solid angle is the derived quantity of a plane angle. Its unit, the steradian, is a coherent derived unit equal to the square radian.
... for brevity and adopt the convention that successive units are multiplied. The units radian, degree, and quaternion are equivalent to unity with dimensionless units meter-per-meter [17], but developers know and use them. The unknown unit δ is useful in expressing and tracking uncertainty in units. ...
... during addition, they are 'coherent units of measure' [17], meaning that they cannot be added to a dissimilar unit, even though they are dimensionless. ...
... Others like tmp point out.point.x have semantic meaning but are misleading in the artifact context (Q 10 ) which is about force. We found 'Good identifiers' in 13/20 questions(3,4, 5,6,8, 9,11,12,14,16,17,18,19, in Appendix D). ...
Robot software risks the hazard of dimensional inconsistencies. These inconsistencies occur when a program incorrectly manipulates values representing real-world quantities. Incorrect manipulation has real-world consequences that range in severity from benign to catastrophic. Previous approaches detect dimensional inconsistencies in programs but require extra developer effort and technical complications. The extra effort involves developers creating type annotations for every variable representing a real-world quantity that has physical units, and the technical complications include toolchain burdens like specialized compilers or type libraries. To overcome the limitations of previous approaches, this thesis presents novel methods to detect dimensional inconsistencies without developer annotations. We start by empirically assessing the difficulty developers have in making type annotations. In a human study of 83 subjects, we find that developers are only 51% accurate and require more than 2 minutes per annotation. We further find that type suggestions have a significant impact on annotation accuracy. We find that when showing developers annotation suggestions, three suggestions are better than a single suggestion because they are as helpful when correct and less harmful when incorrect. Since developers struggle to make type annotations accurately, we present a novel method to infer physical unit types without developer annotations. This is novel because it is the first method to detect dimensional inconsistencies in ROS C++ without developer annotations, and this is important because robot software and ROS are increasingly used in real-world applications. Our method leverages a property of robotic middleware architecture that reuses standardized data structures, and we implement our method in an open-source tool, Phriky. We evaluate our method empirically on a corpus of 5.9 M lines of code and find that it detects real inconsistencies with an 87% TP rate. However, our method only assigns physical unit types to 25% of variables, leaving much of the annotation space unaddressed. To overcome these limitations, we extend our method to utilize uncertain evidence in identifiers using probabilistic reasoning. We implement our new probabilistic method in a tool Phys and find that it assigns units to 75% of variables while retaining a TP rate of 82%. We present the first open dataset of dimensional inconsistencies in open-source robotics code, to our knowledge. Lastly, we identify extensions to our work and next steps for software tool developers to build more powerful robot software development tools. Advisers: Sebastian Elbaum and Carrick Detweiler
... e operator ' * ' means multiplication, unit is identity, ut −1 is a unit's inverse, and δ represents the unknown unit. We also include radian, deree, and quaternion because they are familiar to developers, even though they are equivalent to unity with dimensionless units meter-per-meter[22]. e seven base units can be combined to represent other physical quantities and these combinations are called derived units. ...
... We use a visitor paaern in each statement's AST to apply units and evaluate expressions with unit resolution rules. During implementation, we realized that radian and quaternion require special handling: during multiplication, radian and quaternion act as unit since their units are meters-per-meter; during addition, they are 'coherent units of measure'[22], meaning that they cannot be added to a dissimilar unit, even though they are dimensionless. An example inconsistency message for the code inFigure 4reads: Addition of inconsistent units on line 1094 with high confidence. ...
Systems interacting with the physical world operate on quantities measured with physical units. When unit operations in a program are inconsistent with the physical units' rules, those systems may suffer. Existing approaches to support unit consistency in programs can impose an unacceptable burden on developers. In this paper, we present a lightweight static analysis approach focused on physical unit inconsistency detection that requires no end-user program annotation, modification, or migration. It does so by capitalizing on existing shared libraries that handle standardized physical units, common in the cyber-physical domain, to link class attributes of shared libraries to physical units. Then, leveraging rules from dimensional analysis, the approach propagates and infers units in programs that use these shared libraries, and detects inconsistent unit usage. We implement and evaluate the approach in a tool, analyzing 213 open-source systems containing +900,000 LOC, finding inconsistencies in 11% of them, with an 87% true positive rate for a class of inconsistencies detected with high confidence. An initial survey of robot system developers finds that the unit inconsistencies detected by our tool are 'problematic', and we investigate how and when these inconsistencies occur.
... SI8 states (p 127): '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', although they are not listed in table 6 'Non-SI units accepted for use with the SI'. It has been suggested that the neper should be adopted as a coherent derived SI unit [34,35]. Another argument [36] is that these dimensionless ratios are of a different logarithmic nature (natural and decadic) and do not require units. ...
... The radian is a derived unit in the SI, defined from the identity s = r · θ (in which r is the radius of a circle and s is the length of the arc subtended by the angle θ ). From this definition, the radian has the unit m/m, and is said to be a dimensionless derived unit [34]. This definition has several undesirable consequences for rotational quantities, for example the SI unit for a rate of rotation is a 'per second', without any reference to the angle through which rotation takes place, or its unit [50]. ...
The International System of Units (SI) was declared as a practical and evolving system in 1960 and is now 50 years old. A large amount of theoretical and experimental work has been conducted to change the standards for the base units from artefacts to physical constants, to improve their stability and reproducibility. Less attention, however, has been paid to improving the SI definitions, utility and usability, which suffer from contradictions, ambiguities and inconsistencies. While humans can often resolve these issues contextually, computers cannot. As an ever-increasing volume and proportion of data about physical quantities is collected, exchanged, processed and rendered by computers, this paper argues that the SI definitions, symbols and syntax should be made more rigorous, so they can be represented wholly and unambiguously in ontologies, programs, data and text, and so the SI notation can be rendered faithfully in print and on screen.
... In 2001, Mills, Taylor, and Thor looked at the definitions of the units radian, neper, bel, and decibel [12]. They made the following general comments on units for dimensionless quantities: 'It is clear that as the value of a dimensionless quantity is always simply a number, the coherent unit... is always the number one, symbol 1. ...
The SI brochure’s treatment of quantities that it regards as dimensionless, with the associated unit one, requires certain physical quantities to be regarded as simply numbers. The resulting formal system erases the nature of these quantities and excludes them from important benefits that quantity calculus provides over numerical value calculations, namely, that accidental confusion of different units and different kinds of quantities is sometimes prevented. I propose a better treatment that entails removing from the SI brochure those prescriptions that conflict with common practices in the treatment of dimensionless quantities, especially the definition and use of non-SI dimensionless units that are distinguished by kind.
... The neper and bel are somewhat different and may be considered as names for the coherent SI unit one, used to express the value of logarithmic ratio quantities. (At the turn of the century there had been serious discussion about recognising the neper and bel as coherent SI units [16] but in the end no change to their status as non-SI units was made.) The sizes of some of these units were defined traditionally, long before the French Revolution and are so deeply embedded in the history and culture of the human race that they will likely always be used (e.g. the units of time and angle) although the SI Brochure is perhaps the only place these are authoritatively elaborated. ...
The International System of Units (SI) is one of the greatest scientific, technological, and political achievements of recent times. After the formal introduction of the SI, it was necessary to accept the ongoing usage of some non-SI units because of their long history of previous use, despite this losing the benefit of coherence that the SI brings. This work reviews the history of non-SI units that have been allowed for use with the SI in successive SI Brochures, observes the rules that have been required for this use especially with SI prefixes, and notes that this set of non-SI units has decreased significantly as the SI has become more deeply embedded in modern society. The current situation, where a relatively small set of non-SI units remain accepted for use with the SI, is analysed. It is observed that these non-SI units acquire quasi-SI status because of their listing in the SI Brochure. As a result, it is proposed that this set of non-SI units should be as small as possible and that clear rules for their use are necessary, most obviously with SI prefixes where the current guidance needs improvement. Historical precedent has been examined to assist in shaping these recommendations, which could be discussed and considered when the SI Brochure is next updated.
... Decibel (dB) stands for the ratio between two physical quantity values and is actually a logarithmic unit [7]. Specifically, one Bel is the ratio between two power quantities of 10:1, whereas one Decibel (dB) is one tenth of the Bel (B) or in other words: ...
... That is to say, the natural logarithm of a number is the inverse of the exp() function. The number e is used to express values of such logarithmic quantities as field level, power level, sound pressure level, and logarithmic decrement [67]. Affective issues concerning humans could be defined as logarithmic quantities as well. ...
Individual emotions play a crucial role during any learning interaction. Identifying a student's emotional state and providing personalized feedback, based on integrated pedagogical models, has been considered to be one of the main limits of traditional tools of e-learning. This paper presents an empirical study that illustrates how learner mood may be predicted during online self-assessment tests. Here a previous method of determining student mood has been refined based on the assumption that the influence on learner mood of questions already answered declines in relation to their distance from the current question. Moreover, this paper sets out to indicate that ldquoexponential logicrdquo may help produce more efficient models, if integrated adequately with affective modelling. The results show that these assumptions may prove useful to future research.
We show the implications of angles having their own dimension, which facilitates a consistent use of units as is done for lengths, masses, and other physical quantities. We do this by examining the properties of complete trigonometric and exponential functions that are generalizations of the corresponding functions that have dimensionless numbers for arguments. These generalizations provide functions that are independent of units in which the angles are expressed. This property also provides a consistent framework for including quantities involving angles in computer algebra programs without ambiguity that may otherwise occur. This is in contrast to the conventional practice in scientific applications involving trigonometric or exponential functions of angles where it is assumed that the argument is the numerical part of the angle when expressed in units of radians. That practice also assumes that the functions are the corresponding radian-based versions. These assumptions allow angles to be treated as if they had no dimension and no units, an approach that can lead to important difficulties such as incorrect factors of , which can be avoided by assigning an independent dimension to angles.
We examine implications of angles having their own dimension, in the same sense as lengths, masses, and so forth. The conventional practice in scientific applications involving trigonometric or exponential functions of angles is to assume that the argument is the numerical part of the angle when expressed in units of radians. It is also assumed that the functions are the corresponding radian-based versions. These (usually unstated) assumptions generally allow one to treat angles as if they had no dimension and no units, an approach that sometimes leads to serious difficulties. Here we consider arbitrary units for angles and the corresponding generalizations of the trigonometric and exponential functions. Such generalizations make the functions complete, that is, independent of any particular choice of unit for angles. They also provide a consistent framework for including angle units in computer algebra programs.
This is a description of the overall structure of an absolute ballistic gravimeter in which a test object moves freely in a vacuum in the gravitational field. This system is intended for determining the acceleration of gravity using measurements of length and time intervals in the equation of motion of the test object. These intervals are measured by a laser displacement interferometer and a system for precise measurement of time intervals, which are incorporated in the gravimeter. Uncertainties in the measured acceleration of gravity and metrological support of absolute ballistic gravimeters for length and time measurements are discussed
Ben-Gurion International Airport operates 24 hours at day for landing airplanes. The areas in vicinity of Ben-Gurion International Airport are very populated. The noise of the landing airplanes exceeds the common thresholds. This paper examines the current situation and suggests a possible solution.
The possibility of establishing a relationship among the conceptual apparatus of mathematics, physics, and metrology with a change in the requirement for the formation of the SI units of measurement is examined on the basis of new approaches to mathematical operations, including the introduction of the new concept of a qualitative (dimensional) unit.
The various features and future prospects of the International System of Units (SI) are described. The SI is based on seven selected base units, corresponding to the seven quantities such as length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Fundamental metrology faces the anomalous situation, in which the realization of the SI electrical units is difficult to perform according to their SI definitions, although new highly reproducible quantum standards are available, allowing the National Metrology Institutes the use of practical units of voltage and resistance. From time to time, mathematical considerations give rise to change proposals, such as introduction of the number 1 as an SI unit.
This is a description of the overall structure of an absolute ballistic gravimeter in which a test object moves freely in a vacuum in the gravitational field. This system is intended for determining the acceleration of gravity using measurements of length and time intervals in the equation of motion of the test object. These intervals are measured by a laser displacement interferometer and a system for precise measurement of time intervals, which are incorporated in the gravimeter. Uncertainties in the measured acceleration of gravity and metrological support of absolute ballistic gravimeters for length and time measurements are discussed.
Keywords: nanometrology, dynamic measurements, absolute gravimeter, laser displacement interferometer
In recent years the Comité International des Poids et Mesures has assigned the unit one to all dimensionless quantities and to some countable quantities. This article examines the reasons for that decision and questions their logical basis. It draws attention to some of its undesirable consequences.
All quantities of dimension one are said to have the SI coherent derived unit "one" with the symbol '1'. (Single quotation marks are used here sometimes to indicate a quote, name, term or symbol; double quotation marks flag a concept when necessary.) Conventionally, the term and symbol may not be combined with the SI prefixes (except for the special terms and symbols for one and 1: radian, rad, and steradian, sr). This restriction is understandable, but leads to correct yet impractical alternatives and ISO deprecated symbols such as ppm or in some cases redundant combinations of units, such as mg/kg. "Number of entities" is dimensionally independent of the current base quantities and should take its rightful place among them. The corresponding base unit is "one". A working definition is given. Other quantities of dimension one are derived as fraction, ratio, efficiency, relative quantity, relative increment or characteristic number and may also use the unit "one", whether considered to be base or derived. The special term 'uno' and symbol 'u' in either case are proposed, allowing combination with SI prefixes.
The 21 st Conférence Générale des Poids et Mesures (CGPM) considered in 1999 a resolution proposing that the neper rather than the bel should be adopted as the coherent derived SI unit. Discussions remain open for further considerations until the next CGPM in 2003. In this paper further arguments are presented showing the confusions generated by the use of some dimensionless units, while the changes that the SI will have to face in the future are of a quite different nature. .
The use of special units for logarithmic ratio quantities is reviewed. The neper is used with a natural logarithm (logarithm to the base e) to express the logarithm of the amplitude ratio of two pure sinusoidal signals, particularly in the context of linear systems where it is desired to represent the gain or loss in amplitude of a single-frequency signal between the input and output. The bel, and its more commonly used submultiple, the decibel, are used with a decadic logarithm (logarithm to the base 10) to measure the ratio of two power-like quantities, such as a mean square signal or a mean square sound pressure in acoustics. Thus two distinctly different quantities are involved. In this review we define the quantities first, without reference to the units, as is standard practice in any system of quantities and units. We show that two different definitions of the quantity power level, or logarithmic power ratio, are possible. We show that this leads to two different interpretations for the meaning and numerical values of the units bel and decibel. We review the question of which of these alternative definitions is actually used, or is used by implication, by workers in the field. Finally, we discuss the relative advantages of the alternative definitions.
In recent years there has been considerable interest in the application of the principles of measurement science to chemistry. This has led to the recognition of 'metrology in chemistry' as an area of relevance to analytical chemistry research. This tutorial review describes the benefits to chemistry of the implementation of the principles of measurement science and explains how they are able to improve the reliability and accuracy of chemical measurements.