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Cycloid

In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.

The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve). In physics, when a charged particle at rest is put under a uniform electric and magnetic field perpendicular to one another, the particle’s trajectory draws out a cycloid.

History
The cycloid has been called "The Helen of Geometers" as, like Helen of Troy, it caused frequent quarrels among 17th-century mathematicians, while Sarah Hart sees it named as such "because the properties of this curve are so beautiful".

Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery speculated that such a simple curve must have been known to the ancients, citing similar work by Carpus of Antioch described by Iamblichus. English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa, but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost. Galileo Galilei's name was put forward at the end of the 19th century and at least one author reports credit being given to Marin Mersenne. Beginning with the work of Moritz Cantor and Siegmund Günther, scholars now assign priority to French mathematician Charles de Bovelles  based on his description of the cycloid in his Introductio in geometriam, published in 1503. In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel.

Galileo originated the term cycloid and was the first to make a serious study of the curve. According to his student Evangelista Torricelli, in 1599 Galileo attempted the quadrature of the cycloid (determining the area under the cycloid) with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them. He discovered the ratio was roughly 3:1, which is the true value, but he incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible. Around 1628, Gilles Persone de Roberval likely learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieri's Theorem. However, this work was not published until 1693 (in his Traité des Indivisibles).

Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Mersenne passed these results along to Galileo, who gave them to his students Torricelli and Viviani, who were able to produce a quadrature. This result and others were published by Torricelli in 1644, which is also the first printed work on the cycloid. This led to Roberval charging Torricelli with plagiarism, with the controversy cut short by Torricelli's early death in 1647.

In 1658, Blaise Pascal had given up mathematics for theology but, while suffering from a toothache, began considering several problems concerning the cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons. Pascal, Roberval and Senator Carcavy were the judges, and neither of the two submissions (by John Wallis and Antoine de Lalouvère) was judged to be adequate. While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis's Tractatus Duo, giving Wren priority for the first published proof.

Fifteen years later, Christiaan Huygens had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse a segment of an inverted cycloidal arch in the same amount of time, regardless of its starting point. In 1686, Gottfried Wilhelm Leibniz used analytic geometry to describe the curve with a single equation. In 1696, Johann Bernoulli posed the brachistochrone problem, the solution of which is a cycloid.

Equations
The cycloid through the origin, generated by a circle of radius $r$ rolling over the $x$-axis on the positive side ($y ≥ 0$), consists of the points $(x, y)$, with $$\begin{align} x &= r(t - \sin t) \\ y &= r(1 - \cos t), \end{align}$$ where $t$ is a real parameter corresponding to the angle through which the rolling circle has rotated. For given $t$, the circle's centre lies at $(x, y) = (rt, r)$.

The Cartesian equation is obtained by solving the $y$-equation for $t$ and substituting into the $x$-equation:$$x = r \cos^{-1} \left(1 - \frac{y}{r}\right) - \sqrt{y(2r - y)},$$or, eliminating the multiple-valued inverse cosine:"$r \cos\!\left(\frac{x+\sqrt{y(2r-y)}}{r}\right) + y = r.$|undefined"When $y$ is viewed as a function of $x$, the cycloid is differentiable everywhere except at the cusps on the $x$-axis, with the derivative tending toward $$\infty$$ or $$-\infty$$ near a cusp. The map from $t$ to $(x, y)$ is differentiable, in fact of class $C$∞, with derivative 0 at the cusps.

The slope of the tangent to the cycloid at the point $$(x,y)$$ is given by $\frac{dy}{dx} = \cot(\frac{t}{2})$.

A cycloid segment from one cusp to the next is called an arch of the cycloid, for example the points with $$0 \le t \le 2 \pi$$ and $$0 \leq x \leq 2\pi$$.

Considering the cycloid as the graph of a function $$y = f(x)$$, it satisfies the differential equation:

 * $$\left(\frac{dy}{dx}\right)^2 = \frac{2r}{y} - 1.$$

Involute
The involute of the cycloid has exactly the same shape as the cycloid it originates from. This can be visualized as the path traced by the tip of a wire initially lying on a half arch of the cycloid: as it unrolls while remaining tangent to the original cycloid, it describes a new cycloid (see also cycloidal pendulum and arc length).

Demonstration
This demonstration uses the rolling-wheel definition of cycloid, as well as the instantaneous velocity vector of a moving point, tangent to its trajectory. In the adjacent picture, $$P_1$$ and $$P_2$$ are two points belonging to two rolling circles, with the base of the first just above the top of the second. Initially, $$P_1$$ and $$P_2$$ coincide at the intersection point of the two circles. When the circles roll horizontally with the same speed, $$P_1$$ and $$P_2$$ traverse two cycloid curves. Considering the red line connecting $$P_1$$ and $$P_2$$ at a given time, one proves the line is always tangent to the lower arc at $$P_2$$ and orthogonal to the upper arc at $$P_1$$. Let $$Q$$ be the point in common between the upper and lower circles at the given time. Then:
 * $$P_1,Q,P_2$$ are colinear: indeed the equal rolling speed gives equal angles $$\widehat{P_1O_1Q}=\widehat{P_2O_2Q}$$, and thus $$\widehat{O_1 Q P_1} = \widehat{O_2QP_2}$$ . The point $$Q$$ lies on the line $$O_1O_2$$ therefore $$\widehat{P_1 Q O_1} + \widehat{P_1QO_2}=\pi$$ and analogously $$\widehat{P_2QO_2}+\widehat{P_2QO_1}=\pi$$. From the equality of $$\widehat{O_1QP_1}$$ and $$\widehat{O_2QP_2}$$ one has that also $$\widehat{P_1QO_2}=\widehat{P_2QO_1}$$. It follows $$\widehat{P_1QO_1}+\widehat{P_2QO_1}=\pi$$.
 * If $$A$$ is the meeting point between the perpendicular from $$P_1$$ to the line segment $$O_1O_2$$ and the tangent to the circle at $$P_2$$, then the triangle $$P_1AP_2$$ is isosceles, as is easily seen from the construction: $$\widehat{QP_2A}=\tfrac{1}{2}\widehat{P_2O_2Q}$$ and $$\widehat{QP_1A} = \tfrac{1}{2}\widehat{QO_1R}=$$$$\tfrac{1}{2}\widehat{QO_1P_1}$$ . For the previous noted equality between $$\widehat{P_1O_1Q}$$ and $$\widehat{QO_2P_2}$$ then $$\widehat{QP_1A}=\widehat{QP_2A}$$ and $$P_1AP_2$$ is isosceles.
 * Drawing from $$P_2$$ the orthogonal segment to $$O_1O_2$$, from $$P_1$$ the straight line tangent to the upper circle, and calling $$B$$ the meeting point, one sees that $$P_1AP_2B$$ is a rhombus using the theorems on angles between parallel lines
 * Now consider the velocity $$V_2$$ of $$P_2$$ . It can be seen as the sum of two components, the rolling velocity $$V_a$$ and the drifting velocity $$V_d$$, which are equal in modulus because the circles roll without skidding. $$V_d$$ is parallel to $$P_1A$$, while $$V_a$$ is tangent to the lower circle at $$P_2$$ and therefore is parallel to $$P_2A$$. The rhombus constituted from the components $$V_d$$ and $$V_a$$ is therefore similar (same angles) to the rhombus $$BP_1AP_2$$ because they have parallel sides. Then $$V_2$$, the total velocity of $$P_2$$, is parallel to $$P_2P_1$$ because both are diagonals of two rhombuses with parallel sides and has in common with $$P_1P_2$$ the contact point $$P_2$$. Thus the velocity vector $$V_2$$ lies on the prolongation of $$P_1P_2$$ . Because $$V_2$$ is tangent to the cycloid at $$P_2$$, it follows that also $$P_1P_2$$ coincides with the tangent to the lower cycloid at $$P_2$$.
 * Analogously, it can be easily demonstrated that $$P_1P_2$$ is orthogonal to $$V_1$$ (the other diagonal of the rhombus).
 * This proves that the tip of a wire initially stretched on a half arch of the lower cycloid and fixed to the upper circle at $$P_1$$ will follow the point along its path without changing its length because the speed of the tip is at each moment orthogonal to the wire (no stretching or compression). The wire will be at the same time tangent at $$P_2$$ to the lower arc because of the tension and the facts demonstrated above. (If it were not tangent there would be a discontinuity at $$P_2$$ and consequently unbalanced tension forces.)

Area
Using the above parameterization $  x = r(t - \sin t), \ y = r(1 - \cos t)$, the area under one arch, $$0 \leq t \leq 2\pi,$$ is given by: $$ A = \int_{x=0}^{2 \pi r} y \, dx     = \int_{t=0}^{2 \pi} r^2(1 - \cos t)^2 dt = 3\pi r^2. $$

This is three times the area of the rolling circle.

Arc length
The arc length $S$ of one arch is given by $$\begin{align} S &= \int_0^{2\pi} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt \\ &= \int_0^{2\pi} r \sqrt{2 - 2\cos t}\, dt \\ &= 2r\int_0^{2\pi} \sin \frac{t}{2}\, dt \\ &= 8r. \end{align}$$

Another geometric way to calculate the length of the cycloid is to notice that when a wire describing an involute has been completely unwrapped from half an arch, it extends itself along two diameters, a length of $4r$. This is thus equal to half the length of arch, and that of a complete arch is $8r$.

Cycloidal pendulum
If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the string is constrained to be tangent to one of its arches, and the pendulum's length L is equal to that of half the arc length of the cycloid (i.e., twice the diameter of the generating circle, L = 4r), the bob of the pendulum also traces a cycloid path. Such a pendulum is isochronous, with equal-time swings regardless of amplitude. Introducing a coordinate system centred in the position of the cusp, the equation of motion is given by: $$\begin{align} x &= r[2\theta(t) + \sin 2\theta (t)] \\ y &= r[-3-\cos2\theta (t)], \end{align}$$ where $$\theta$$ is the angle that the straight part of the string makes with the vertical axis, and is given by $$\sin\theta (t) = A \cos(\omega t),\qquad \omega^2 = \frac{g}{L}=\frac{g}{4r},$$ where $A < 1$ is the "amplitude", $$\omega$$ is the radian frequency of the pendulum and g the gravitational acceleration.

The 17th-century Dutch mathematician Christiaan Huygens discovered and proved these properties of the cycloid while searching for more accurate pendulum clock designs to be used in navigation.

Related curves
Several curves are related to the cycloid.
 * Trochoid: generalization of a cycloid in which the point tracing the curve may be inside the rolling circle (curtate) or outside (prolate).
 * Hypocycloid: variant of a cycloid in which a circle rolls on the inside of another circle instead of a line.
 * Epicycloid: variant of a cycloid in which a circle rolls on the outside of another circle instead of a line.
 * Hypotrochoid: generalization of a hypocycloid where the generating point may not be on the edge of the rolling circle.
 * Epitrochoid: generalization of an epicycloid where the generating point may not be on the edge of the rolling circle.

All these curves are roulettes with a circle rolled along another curve of uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the similitude ratio of curve to evolute is 1 + 2q.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

Other uses

The cycloidal arch was used by architect Louis Kahn in his design for the Kimbell Art Museum in Fort Worth, Texas. It was also used by Wallace K. Harrison in the design of the Hopkins Center at Dartmouth College in Hanover, New Hampshire.

Early research indicated that some transverse arching curves of the plates of golden age violins are closely modeled by curtate cycloid curves. Later work indicates that curtate cycloids do not serve as general models for these curves, which vary considerably.

title: Cycloid


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Reference noise

In telecommunication, reference noise is the magnitude of circuit noise chosen as a reference for measurement.

Many different levels with a number of different weightings are in current use, and care must be taken to ensure that the proper parameters are stated.

Specific ones include: dBa, dBa(F1A), dBa(HA1), dBa0, dBm, dBm(psoph), dBm0, dBrn, dBrnC, dBrnC0, dBrn(f1-f2), dBrn(144-line), dBx.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Reference noise


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Reference surface

In fiber optic technology, a reference surface is that surface of an optical fiber that is used to contact the transverse-alignment elements of a component such as a connector or mechanical splice. For telecommunications-grade fibers, the reference surface is the outer surface of the cladding. For plastic-clad silica (PCS) fibers, which have a strippable polymer cladding (not to be confused with the polymer overcoat of an all-silica fiber), the reference surface may be the core.

title: Reference surface


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Reflection coefficient

In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected wave to the incident wave, with each expressed as phasors. For example, it is used in optics to calculate the amount of light that is reflected from a surface with a different index of refraction, such as a glass surface, or in an electrical transmission line to calculate how much of the electromagnetic wave is reflected by an impedance discontinuity. The reflection coefficient is closely related to the transmission coefficient. The reflectance of a system is also sometimes called a reflection coefficient.

Different specialties have different applications for the term.

Transmission lines
In telecommunications and transmission line theory, the reflection coefficient is the ratio of the complex amplitude of the reflected wave to that of the incident wave. The voltage and current at any point along a transmission line can always be resolved into forward and reflected traveling waves given a specified reference impedance Z0. The reference impedance used is typically the characteristic impedance of a transmission line that's involved, but one can speak of reflection coefficient without any actual transmission line being present. In terms of the forward and reflected waves determined by the voltage and current, the reflection coefficient is defined as the complex ratio of the voltage of the reflected wave ($$V^-$$) to that of the incident wave ($$V^+$$). This is typically represented with a $$\Gamma$$ (capital gamma) and can be written as:

 * $$\Gamma = \frac{V^-}{V^+} $$

It can as well be defined using the currents associated with the reflected and forward waves, but introducing a minus sign to account for the opposite orientations of the two currents:

 * $$\Gamma = -\frac{I^-}{I^+} = \frac{V^-}{V^+} $$

The reflection coefficient may also be established using other field or circuit pairs of quantities whose product defines power resolvable into a forward and reverse wave. For instance, with electromagnetic plane waves, one uses the ratio of the electric fields of the reflected to that of the forward wave (or magnetic fields, again with a minus sign); the ratio of each wave's electric field E to its magnetic field H is again an impedance Z0 (equal to the impedance of free space in a vacuum). Similarly in acoustics one uses the acoustic pressure and velocity respectively.

In the accompanying figure, a signal source with internal impedance $$Z_S$$ possibly followed by a transmission line of characteristic impedance $$Z_S$$ is represented by its Thévenin equivalent, driving the load $$Z_L$$. For a real (resistive) source impedance $$Z_S$$, if we define $$\Gamma$$ using the reference impedance $$Z_0 = Z_S$$ then the source's maximum power is delivered to a load $$Z_L = Z_0$$, in which case $$\Gamma=0$$ implying no reflected power. More generally, the squared-magnitude of the reflection coefficient $$|\Gamma|^2$$ denotes the proportion of that power that is reflected back to the source, with the power actually delivered toward the load being $$1-|\Gamma|^2$$.

Anywhere along an intervening (lossless) transmission line of characteristic impedance $$Z_0$$, the magnitude of the reflection coefficient $$|\Gamma|$$ will remain the same (the powers of the forward and reflected waves stay the same) but with a different phase. In the case of a short circuited load ($$Z_L=0$$), one finds $$\Gamma=-1$$ at the load. This implies the reflected wave having a 180° phase shift (phase reversal) with the voltages of the two waves being opposite at that point and adding to zero (as a short circuit demands).

Relation to load impedance
The reflection coefficient is determined by the load impedance at the end of the transmission line, as well as the characteristic impedance of the line. A load impedance of $$Z_L$$ terminating a line with a characteristic impedance of $$Z_0\,$$ will have a reflection coefficient of

 * $$ \Gamma ={Z_L-Z_0  \over Z_L+Z_0} .$$

This is the coefficient at the load. The reflection coefficient can also be measured at other points on the line. The magnitude of the reflection coefficient in a lossless transmission line is constant along the line (as are the powers in the forward and reflected waves). However its phase will be shifted by an amount dependent on the electrical distance $$\phi$$ from the load. If the coefficient is measured at a point $$L$$ meters from the load, so the electrical distance from the load is $$\phi = 2\pi L/\lambda$$ radians, the coefficient $$\Gamma'$$ at that point will be

 * $$\Gamma' =\Gamma e^{-i \, 2 \phi} $$

Note that the phase of the reflection coefficient is changed by twice the phase length of the attached transmission line. That is to take into account not only the phase delay of the reflected wave, but the phase shift that had first been applied to the forward wave, with the reflection coefficient being the quotient of these. The reflection coefficient so measured, $$\Gamma'$$, corresponds to an impedance which is generally dissimilar to $$Z_L$$ present at the far side of the transmission line.

The complex reflection coefficient (in the region $$| \Gamma| \le 1$$, corresponding to passive loads) may be displayed graphically using a Smith chart. The Smith chart is a polar plot of $$\Gamma$$, therefore the magnitude of $$\Gamma$$ is given directly by the distance of a point to the center (with the edge of the Smith chart corresponding to $$|\Gamma|=1$$). Its evolution along a transmission line is likewise described by a rotation of $$2\phi$$ around the chart's center. Using the scales on a Smith chart, the resulting impedance (normalized to $$Z_0$$) can directly be read. Before the advent of modern electronic computers, the Smith chart was of particular use as a sort of analog computer for this purpose.

Standing wave ratio
The standing wave ratio (SWR) is determined solely by the magnitude of the reflection coefficient:

 * $$ SWR = {1+| \Gamma |   \over 1- | \Gamma | } .$$

Along a lossless transmission line of characteristic impedance Z0, the SWR signifies the ratio of the voltage (or current) maxima to minima (or what it would be if the transmission line were long enough to produce them). The above calculation assumes that $$\Gamma$$ has been calculated using Z0 as the reference impedance. Since it uses only the magnitude of $$\Gamma$$, the SWR intentionally ignores the specific value of the load impedance ZL responsible for it, but only the magnitude of the resulting impedance mismatch. That SWR remains the same wherever measured along a transmission line (looking towards the load) since the addition of a transmission line length to a load $$Z_L$$ only changes the phase, not magnitude of $$\Gamma$$. While having a one-to-one correspondence with reflection coefficient, SWR is the most commonly used figure of merit in describing the mismatch affecting a radio antenna or antenna system. It is most often measured at the transmitter side of a transmission line, but having, as explained, the same value as would be measured at the antenna (load) itself.

Seismology
Reflection coefficient is used in feeder testing for reliability of medium.

Optics and microwaves
In optics and electromagnetics in general, reflection coefficient can refer to either the amplitude reflection coefficient described here, or the reflectance, depending on context. Typically, the reflectance is represented by a capital R, while the amplitude reflection coefficient is represented by a lower-case r. These related concepts are covered by Fresnel equations in classical optics.

Acoustics
Acousticians use reflection coefficients to understand the effect of different materials on their acoustic environments.

title: Reflection coefficient


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Reflective array antenna

In telecommunications and radar, a reflective array antenna is a class of directive antennas in which multiple driven elements are mounted in front of a flat surface designed to reflect the radio waves in a desired direction. They are a type of array antenna. They are often used in the VHF and UHF frequency bands. VHF examples are generally large and resemble a highway billboard, so they are sometimes called billboard antennas. Other names are bedspring array and bowtie array depending on the type of elements making up the antenna. The curtain array is a larger version used by shortwave radio broadcasting stations.

Reflective array antennas usually have a number of identical driven elements, fed in phase, in front of a flat, electrically large reflecting surface to produce a unidirectional beam of radio waves, increasing antenna gain and reducing  radiation in unwanted directions. The larger the number of elements used, the higher the gain; the narrower the beam is and the smaller the sidelobes are. The individual elements are most commonly half wave dipoles, although they sometimes contain parasitic elements as well as driven elements. The reflector may be a metal sheet or more commonly a wire screen. A metal screen reflects radio waves as well as a solid metal sheet as long as the holes in the screen are smaller than about one-tenth of a wavelength, so screens are often used to reduce weight and wind loads on the antenna. They usually consist of a grill of parallel wires or rods, oriented parallel to the axis of the dipole elements.

The driven elements are fed by a network of transmission lines, which divide the power from the RF source equally between the elements. This often has the circuit geometry of a tree structure.

Radio signals
When a radio signal passes a conductor, it induces an electrical current in it. Since the radio signal fills space, and the conductor has a finite size, the induced currents add up or cancel out as they move along the conductor. A basic goal of antenna design is to make the currents add up to a maximum at the point where the energy is tapped off. To do this, the antenna elements are sized in relation to the wavelength of the radio signal, with the aim of setting up standing waves of current that are maximized at the feed point.

This means that an antenna designed to receive a particular wavelength has a natural size. To improve reception, one cannot simply make the antenna larger; this will improve the amount of signal intercepted by the antenna, which is largely a function of area, but will lower the efficiency of the reception (at a given wavelength). Thus, in order to improve reception, antenna designers often use multiple elements, combining them together so their signals add up. These are known as antenna arrays.

Array phasing
In order for the signals to add together, they need to arrive in-phase. Consider two dipole antennas placed in a line end-to-end, or collinear. If the resulting array is pointed directly at the source signal, both dipoles will see the same instantaneous signal, and thus their reception will be in-phase. However, if one were to rotate the antenna so it was at an angle to the signal, the extra path from the signal to the more distant dipole means it receives the signal slightly out of phase. When the two signals are then added up, they no longer strictly reinforce each other, and the output drops. This makes the array more sensitive horizontally, while stacking the dipoles in parallel narrows the pattern vertically. This allows the designer to tailor the reception pattern, and thus the gain, by moving the elements about.

If the antenna is properly aligned with the signal, at any given instant in time, all of the elements in an array will receive the same signal and be in-phase. However, the output from each element has to be gathered up at a single feed point, and as the signals travel across the antenna to that point, their phase is changing. In a two-element array this is not a problem because the feed point can be placed between them; any phase shift taking place in the transmission lines is equal for both elements. However, if one extends this to a four-element array, this approach no longer works, as the signal from the outer pair has to travel further and will thus be at a different phase than the inner pair when it reaches the center. To ensure that they all arrive with the same phase, it is common to see additional transmission wire inserted in the signal path, or for the transmission line to be crossed over to reverse the phase if the difference is greater than $1/undefined$ a wavelength.

Reflectors
The gain can be further improved through the addition of a reflector. Generally any conductor in a flat sheet will act in a mirror-like fashion for radio signals, but this also holds true for non-continuous surfaces as long as the gaps between the conductors are less than about of the target wavelength. This means that wire mesh or even parallel wires or metal bars can be used, which is especially useful both for reducing the total amount of material as well as reducing wind loads.

Due to the change in signal propagation direction on reflection, the signal undergoes a reversal of phase. In order for the reflector to add to the output signal, it has to reach the elements in-phase. Generally this would require the reflector to be placed at $1/undefined$ of a wavelength behind the elements, and this can be seen in many common reflector arrays like television antennas. However, there are a number of factors that can change this distance, and actual reflector positioning varies.

Reflectors also have the advantage of reducing the signal received from the back of the antenna. Signals received from the rear and re-broadcast from the reflector have not undergone a change of phase, and do not add to the signal from the front. This greatly improves the front-to-back ratio of the antenna, making it more directional. This can be useful when a more directional signal is desired, or unwanted signals are present. There are cases when this is not desirable, and although reflectors are commonly seen in array antennas, they are not universal. For instance, while UHF television antennas often use an array of bowtie antennas with a reflector, a bowtie array without a reflector is a relatively common design in the microwave region.

Gain limits
As more elements are added to an array, the beamwidth of the antenna's main lobe decreases, leading to an increase in gain. In theory there is no limit to this process. However, as the number of elements increases, the complexity of the required feed network that keeps the signals in-phase increases. Ultimately, the rising inherent losses in the feed network become greater than the additional gain achieved with more elements, limiting the maximum gain that can be achieved.

The gain of practical array antennas is limited to about 25–30 dB. Two half wave elements spaced a half wave apart and a quarter wave from a reflecting screen have been used as a standard gain antenna with about 9.8 dBi at its design frequency. Common 4-bay television antennas have gains around 10 to 12 dB, and 8-bay designs might increase this to 12 to 16 dB. The 32-element SCR-270 had a gain around 19.8 dB. Some very large reflective arrays have been constructed, notably the Soviet Duga radars which are hundreds of meters across and contain hundreds of elements. Active array antennas, in which groups of elements are driven by separate RF amplifiers, can have much higher gain, but are prohibitively expensive.

Since the 1980s, versions for use at microwave frequencies have been made with patch antenna elements mounted in front of a metal surface.

Radiation pattern and beam steering
When driven in phase, the radiation pattern of the reflective array is a single main lobe perpendicular to the plane of the antenna, plus several sidelobes at equal angles to either side. The more elements used, the narrower the main lobe and the less power is radiated in the sidelobes.

The main lobe of the antenna can be steered electronically within a limited angle by phase shifting the drive signals applied to the individual elements. Each antenna element is fed through a phase shifter which can be controlled digitally, delaying each signal by a successive amount. This causes the wavefronts created by the superposition of the individual elements to be at an angle to the plane of the antenna. Antennas that use this technique are called phased arrays and often used in modern radar systems.

Another option for steering the beam is mounting the entire antenna structure on a pivoting platform and rotating it mechanically.

title: Reflective array antenna


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Reflectance

The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic structure of the material to the electromagnetic field of light, and is in general a function of the frequency, or wavelength, of the light, its polarization, and the angle of incidence. The dependence of reflectance on the wavelength is called a reflectance spectrum or spectral reflectance curve.

Hemispherical reflectance
The hemispherical reflectance of a surface, denoted $R$, is defined as $$R = \frac{\Phi_\mathrm{e}^\mathrm{r}}{\Phi_\mathrm{e}^\mathrm{i}},$$ where $Φ_{e}^{r}$ is the radiant flux reflected by that surface and $Φ_{e}^{i}$ is the radiant flux received by that surface.

Spectral hemispherical reflectance
The spectral hemispherical reflectance in frequency and spectral hemispherical reflectance in wavelength of a surface, denoted $R_{ν}$ and $R_{λ}$ respectively, are defined as $$R_\nu = \frac{\Phi_{\mathrm{e},\nu}^\mathrm{r}}{\Phi_{\mathrm{e},\nu}^\mathrm{i}},$$ $$R_\lambda = \frac{\Phi_{\mathrm{e},\lambda}^\mathrm{r}}{\Phi_{\mathrm{e},\lambda}^\mathrm{i}},$$ where
 * $Φ_{e,ν}^{r}$ is the spectral radiant flux in frequency reflected by that surface;
 * $Φ_{e,ν}^{i}$ is the spectral radiant flux in frequency received by that surface;
 * $Φ_{e,λ}^{r}$ is the spectral radiant flux in wavelength reflected by that surface;
 * $Φ_{e,λ}^{i}$ is the spectral radiant flux in wavelength received by that surface.

Directional reflectance
The directional reflectance of a surface, denoted RΩ, is defined as $$R_\Omega = \frac{L_{\mathrm{e},\Omega}^\mathrm{r}}{L_{\mathrm{e},\Omega}^\mathrm{i}},$$ where
 * $L_{e,Ω}^{r}$ is the radiance reflected by that surface;
 * $L_{e,Ω}^{i}$ is the radiance received by that surface.

This depends on both the reflected direction and the incoming direction. In other words, it has a value for every combination of incoming and outgoing directions. It is related to the bidirectional reflectance distribution function and its upper limit is 1. Another measure of reflectance, depending only on the outgoing direction, is I/F, where I is the radiance reflected in a given direction and F is the incoming radiance averaged over all directions, in other words, the total flux of radiation hitting the surface per unit area, divided by π. This can be greater than 1 for a glossy surface illuminated by a source such as the sun, with the reflectance measured in the direction of maximum radiance (see also Seeliger effect).

Spectral directional reflectance
The spectral directional reflectance in frequency and spectral directional reflectance in wavelength of a surface, denoted $R_{Ω,ν}$ and $R_{Ω,λ}$ respectively, are defined as $$R_{\Omega,\nu} = \frac{L_{\mathrm{e},\Omega,\nu}^\mathrm{r}}{L_{\mathrm{e},\Omega,\nu}^\mathrm{i}},$$ $$R_{\Omega,\lambda} = \frac{L_{\mathrm{e},\Omega,\lambda}^\mathrm{r}}{L_{\mathrm{e},\Omega,\lambda}^\mathrm{i}},$$ where
 * $L_{e,Ω,ν}^{r}$ is the spectral radiance in frequency reflected by that surface;
 * $L_{e,Ω,ν}^{i}$ is the spectral radiance received by that surface;
 * $L_{e,Ω,λ}^{r}$ is the spectral radiance in wavelength reflected by that surface;
 * $L_{e,Ω,λ}^{i}$ is the spectral radiance in wavelength received by that surface.

Again, one can also define a value of $I/F$ (see above) for a given wavelength.

Reflectivity

For homogeneous and semi-infinite (see halfspace) materials, reflectivity is the same as reflectance. Reflectivity is the square of the magnitude of the Fresnel reflection coefficient, which is the ratio of the reflected to incident electric field; as such the reflection coefficient can be expressed as a complex number as determined by the Fresnel equations for a single layer, whereas the reflectance is always a positive real number.

For layered and finite media, according to the CIE, reflectivity is distinguished from reflectance by the fact that reflectivity is a value that applies to thick reflecting objects. When reflection occurs from thin layers of material, internal reflection effects can cause the reflectance to vary with surface thickness. Reflectivity is the limit value of reflectance as the sample becomes thick; it is the intrinsic reflectance of the surface, hence irrespective of other parameters such as the reflectance of the rear surface. Another way to interpret this is that the reflectance is the fraction of electromagnetic power reflected from a specific sample, while reflectivity is a property of the material itself, which would be measured on a perfect machine if the material filled half of all space.

Surface type
Given that reflectance is a directional property, most surfaces can be divided into those that give specular reflection and those that give diffuse reflection.

For specular surfaces, such as glass or polished metal, reflectance is nearly zero at all angles except at the appropriate reflected angle; that is the same angle with respect to the surface normal in the plane of incidence, but on the opposing side. When the radiation is incident normal to the surface, it is reflected back into the same direction.

For diffuse surfaces, such as matte white paint, reflectance is uniform; radiation is reflected in all angles equally or near-equally. Such surfaces are said to be Lambertian.

Most practical objects exhibit a combination of diffuse and specular reflective properties.

Water reflectance

Reflection occurs when light moves from a medium with one index of refraction into a second medium with a different index of refraction.

Specular reflection from a body of water is calculated by the Fresnel equations. Fresnel reflection is directional and therefore does not contribute significantly to albedo which primarily diffuses reflection.

A real water surface may be wavy. Reflectance, which assumes a flat surface as given by the Fresnel equations, can be adjusted to account for waviness.

Grating efficiency
The generalization of reflectance to a diffraction grating, which disperses light by wavelength, is called diffraction efficiency.

title: Reflectance


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Refractive index contrast

Refractive index contrast, in an optical waveguide, such as an optical fiber, is a measure of the relative difference in refractive index of the core and cladding. The refractive index contrast, Δ, is often given by $$\Delta={n_1^2-n_2^2 \over 2 n_1^2}$$, where n1 is the maximum refractive index in the core (or simply the core index for a step-index profile) and n2 is the refractive index of the cladding. The criterion n2 < n1 must be satisfied in order to sustain a guided mode by total internal reflection. Alternative formulations include $$\Delta=\sqrt{n_1^2-n_2^2}$$ and $$\Delta = {n_1-n_2 \over n_1}$$. Normal optical fibers, constructed of different glasses, have very low refractive index contrast (Δ<<1) and hence are weakly-guiding. The weak guiding will cause a greater portion of the cross-sectional Electric field profile to reside within the cladding (as evanescent tails of the guided mode) as compared to strongly-guided waveguides. Integrated optics can make use of higher core index to obtain Δ>1 allowing light to be efficiently guided around corners on the micro-scale, where popular high-Δ material platform is silicon-on-insulator. High-Δ allows sub-wavelength core dimensions and so greater control over the size of the evanescent tails. The most efficient low-loss optical fibers require low Δ to minimise losses to light scattered outwards.

title: Refractive index contrast


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Regeneration

Regeneration may refer to:

Science and technology

 * Regeneration (biology), the ability to recreate lost or damaged cells, tissues, organs and limbs
 * Regeneration (ecology), the ability of ecosystems to regenerate biomass, using photosynthesis
 * Regeneration in humans, the ability of humans to recreate, or induce the regeneration of, lost tissue
 * Regenerative (design), a process for resilient and sustainable development
 * Regenerative agriculture, a sub-category of organic agriculture

History and politics

 * Regeneration (Colombia), La Regeneración, a 19th-century period and political movement in Colombia
 * Regeneration (Portugal), a 19th-century period in the history of Portugal
 * The ReGeneration, a cultural generation concerned with environmentalism
 * Viðreisn (Regeneration), a political party in Iceland founded in 2016

Music

 * Regeneration (Stanley Cowell album) (1976)
 * Regeneration (Roy Orbison album) (1977)
 * Regeneration (The Divine Comedy album) (2001)
 * Regeneration (Superchick album) (2003)

Film and television

 * Regeneration (1915 film), a film starring John McCann
 * Regeneration (1923 film), a Norman Studios production
 * Regeneration (1997 film) or Behind the Lines, a film adapted from the Pat Barker novel
 * ReGeneration (2010 film), a documentary film
 * "Re-generation" (The Outer Limits), a 1997 episode of The Outer Limits
 * "Regeneration" (Star Trek: Enterprise), an episode of Star Trek: Enterprise
 * "Regeneration", an episode of Transformers: Prime
 * Regeneration (Doctor Who), a biological ability exhibited by the Time Lords in the fictional context of Doctor Who

Literature

 * Regeneration (Haggard book), a 1910 non-fiction work by H. Rider Haggard
 * Regeneration (novel), a novel by Pat Barker
 * Regeneration, a book by Thomas Hunt Morgan

Other uses

 * Regeneration (sculpture), a 1975 sculpture by Alan Collins
 * Regeneration (theology), the doctrine of being born again
 * Evil Dead: Regeneration, a 2005 video game for the PlayStation 2 and Xbox

title: Regeneration


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Relative transmission level

In telecommunication, relative transmission level is the ratio of the signal power, at a given point in a transmission system, to a reference signal power.

The ratio is usually determined by applying a standard test tone at zero transmission level point (or applying adjusted test tone power at any other point) and measuring the gain or loss to the location of interest. A distinction should be made between the standard test tone power and the expected median power of the actual signal required as the basis for the design of transmission systems.

title: Relative transmission level


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Release time (telecommunication)

In telecommunication, release time is the time interval for a circuit to respond when an enabling signal is discontinued, for example:
 * 1) in an automatic gain control or echo suppressor circuit, the time interval between the end of the enabling signal and the instant at which suppression ceases,
 * 2) for a relay, the time between when the coil is de-energized, and the instant that contact closure ceases or is established.

title: Release time (telecommunication)


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Reliability

Reliability, reliable, or unreliable may refer to:

Computing

 * Data reliability (disambiguation), a property of some disk arrays in computer storage
 * Reliability (computer networking), a category used to describe protocols
 * Reliability (semiconductor), outline of semiconductor device reliability drivers

Other uses in science, technology, and mathematics

 * Reliability (statistics), the overall consistency of a measure
 * Reliability engineering, concerned with the ability of a system or component to perform its required functions under stated conditions for a specified time
 * High reliability is informally reported in "nines"
 * Human reliability in engineered systems
 * Reliability theory, as a theoretical concept, to explain biological aging and species longevity

Other uses

 * Reliabilism, in philosophy and epistemology
 * Unreliable narrator, whose credibility has been seriously compromised

title: Reliability


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Remote access

Remote access may refer to:

 * Connection to a data-processing system from a remote location, for example, through a remote access service or virtual private network
 * Remote desktop software, software allowing applications to run remotely on a server while displaying graphical output locally
 * Terminal emulation, when used to interface with a remote system. May use standard tools like:
 * – software used to interact over a network with a computer system
 * – secure shell: often used with remote applications
 * Activation of features of a business telephone system from outside the business's premises
 * RemoteAccess, a DOS-based bulletin-board system
 * Remote Database Access, a protocol standard for database access
 * Remote Access (film), a 2004 Russian drama film

title: Remote access


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Remote call forwarding

In telecommunications, remote call forwarding is a service feature that allows incoming calls to be forwarded to a remote call forwarding number, such as a cell phone or another office location, and is designated by the call receiver.

Customers may have a remote-forwarding telephone number in a central switching office without having any other local telephone service in that office.

One common purpose for this service is to enable customers to retain their telephone number when they move to a location serviced by a different telephone exchange. The service is useful for business customers with widely advertised numbers which appear on headed paper, vehicles and various marketing literature. When customers ring, their calls are forwarded to the new location.

Remote call forwarding is also a means for a suburban business to obtain a city-centre local number (with its full large-city coverage area) for inbound calls; while cheaper than a foreign exchange line, this can reduce long-distance telephony costs in markets where local calls are flat-rated but trunk calls are expensive.

One alternative to RCF is Caller Redirect whereby callers simply hear an intercept message notifying them that the number has changed. Another alternative is to port the existing number to a voice over IP carrier, which is not tied to a single physical location as the subscriber may be anywhere on broadband Internet. However, not all phone numbers can be ported.

Remote Access to Call Forwarding
Remote Call Forwarding (RCF) requires neither a physical telephone set nor physical input by customer to get calls forwarded.

In this respect, it differs from the (similarly named) Remote Access to Call Forwarding in that the number is attached to a physical line where it rings normally until a call is made to a remote number to enable redirection.

To activate Remote Access to Call Forwarding, a subscriber calls a provider-supplied Remote Access Directory Number, enters the telephone number of the line to be redirected along with a personal identification number (PIN), a vertical service code (such as 72# or *73) and the number to which the calls are to be forwarded.

Remote Access to Call Forwarding allows incoming calls to be diverted and answered elsewhere if a subscriber cannot use their telephone normally (for instance, the number is assigned to a lost or stolen wireless handset or to a landline in need of repair service). In some cases, a business which subscribes to standard call forwarding (*72) may be able to request temporary redirection of inbound calls when calling a telco repair service number (such as 6-1-1) to report a line outage.

title: Remote call forwarding


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Remote Operations Service Element protocol

The Remote Operations Service Element (ROSE) is the OSI service interface, specified in ITU-T Recommendation X.219, ISO/IEC International Standard 9072-1, that (a) provides remote operation capabilities, (b) allows interaction between entities in a distributed application, and (c) upon receiving a remote operations service request, allows the receiving entity to attempt the operation and report the results of the attempt to the requesting entity.

OSI application protocols such as X.400 and X.500 use the services provided by ROSE. The ROSE protocol itself is defined using the notation of ASN.1.


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Repeater

In telecommunications, a repeater is an electronic device that receives a signal and retransmits it. Repeaters are used to extend transmissions so that the signal can cover longer distances or be received on the other side of an obstruction. Some types of repeaters broadcast an identical signal, but alter its method of transmission, for example, on another frequency or baud rate.

There are several different types of repeaters; a telephone repeater is an amplifier in a telephone line, an optical repeater is an optoelectronic circuit that amplifies the light beam in an optical fiber cable; and a radio repeater is a radio receiver and transmitter that retransmits a radio signal.

A broadcast relay station is a repeater used in broadcast radio and television.

Overview
When an information-bearing signal passes through a communication channel, it is progressively degraded due to loss of power. For example, when a telephone call passes through a wire telephone line, some of the power in the electric current which represents the audio signal is dissipated as heat in the resistance of the copper wire. The longer the wire, the more power is lost, and the smaller the amplitude of the signal at the far end. So with a long enough wire the call will not be audible at the other end. Similarly, the greater the distance between a radio station and a receiver, the weaker the radio signal, and the poorer the reception. A repeater is an electronic device in a communication channel that increases the power of a signal and retransmits it, allowing it to travel further. Since it amplifies the signal, it requires a source of electric power.

The term "repeater" originated with telegraphy in the 19th century, and referred to an electromechanical device (a relay) used to regenerate telegraph signals.

Use of the term has continued in telephony and data communications.

In computer networking, because repeaters work with the actual physical signal, and do not attempt to interpret the data being transmitted, they operate on the physical layer, the first layer of the OSI model; a multiport Ethernet repeater is usually called a hub.

Telephone repeater
This is used to increase the range of telephone signals in a telephone line.

 * Land line repeater

They are most frequently used in trunklines that carry long distance calls. In an analog telephone line consisting of a pair of wires, it consists of an amplifier circuit made of transistors which use power from a DC current source to increase the power of the alternating current audio signal on the line. Since the telephone is a duplex (bidirectional) communication system, the wire pair carries two audio signals, one going in each direction. So telephone repeaters have to be bilateral, amplifying the signal in both directions without causing feedback, which complicates their design considerably. Telephone repeaters were the first type of repeater and were some of the first applications of amplification. The development of telephone repeaters between 1900 and 1915 made long-distance phone service possible. Now, most telecommunications cables are fiber-optic cables which use optical repeaters (below).

Before the invention of electronic amplifiers, mechanically coupled carbon microphones were used as amplifiers in telephone repeaters. After the turn of the 20th century it was found that negative resistance mercury lamps could amplify, and they were used. The invention of audion tube repeaters around 1916 made transcontinental telephony practical. In the 1930s vacuum tube repeaters using hybrid coils became commonplace, allowing the use of thinner wires. In the 1950s negative impedance gain devices were more popular, and a transistorized version called the E6 repeater was the final major type used in the Bell System before the low cost of digital transmission made all voiceband repeaters obsolete. Frequency frogging repeaters were commonplace in frequency-division multiplexing systems from the middle to late 20th century.

 * Submarine cable repeater

This is a type of telephone repeater used in underwater submarine telecommunications cables.

Optical communications repeater
This is used to increase the range of signals in a fiber-optic cable. Digital information travels through a fiber-optic cable in the form of short pulses of light. The light is made up of particles called photons, which can be absorbed or scattered in the fiber. An optical communications repeater usually consists of a phototransistor which converts the light pulses to an electrical signal, an amplifier to increase the power of the signal, an electronic filter which reshapes the pulses, and a laser which converts the electrical signal to light again and sends it out the other fiber. However, optical amplifiers are being developed for repeaters to amplify the light itself without the need of converting it to an electric signal first.

Radio repeater

This is used to extend the range of coverage of a radio signal. The history of radio relay repeaters began in 1898 from the publication by Johann Mattausch in Austrian Journal Zeitschrift für Electrotechnik (v. 16, 35 - 36). But his proposal "Translator" was primitive and not suitable for use. The first relay system with radio repeaters, which really functioned, was that invented in 1899 by Emile Guarini-Foresio.

A radio repeater usually consists of a radio receiver connected to a radio transmitter. The received signal is amplified and retransmitted, often on another frequency, to provide coverage beyond the obstruction. Usage of a duplexer can allow the repeater to use one antenna for both receive and transmit at the same time.

 * Broadcast relay station, rebroadcastor or translator: This is a repeater used to extend the coverage of a radio or television broadcasting station. It consists of a secondary radio or television transmitter.  The signal from the main transmitter often comes over leased telephone lines or by microwave relay.

 * Microwave relay: This is a specialized point-to-point telecommunications link, consisting of a microwave receiver that receives information over a beam of microwaves from another relay station in line-of-sight distance, and a microwave transmitter which passes the information on to the next station over another beam of microwaves. Networks of microwave relay stations transmit telephone calls, television programs, and computer data from one city to another over continent-wide areas.

 * Passive repeater: This is a microwave relay that simply consists of a flat metal surface to reflect the microwave beam in another direction. It is used to get microwave relay signals over hills and mountains when it is not necessary to amplify the signal.

 * Cellular repeater: This is a radio repeater for boosting cell phone reception in a limited area. The device functions like a small cellular base station, with a directional antenna to receive the signal from the nearest cell tower, an amplifier, and a local antenna to rebroadcast the signal to nearby cell phones.  It is often used in downtown office buildings.

 * Digipeater: A repeater node in a packet radio network. It performs a store and forward function, passing on packets of information from one node to another.

 * Amateur radio repeater: Used by amateur radio operators to enable two-way communication across an area which would otherwise be difficult by point-to-point on VHF and UHF. These repeaters are set up and maintained by individual operators or clubs, and are generally available for any licensed amateur to use. A hill or mountaintop location is a preferable location to construct a repeater, as it will maximize the usability across a large area.

Radio repeaters improve communication coverage in systems using frequencies that typically have line-of-sight propagation. Without a repeater, these systems are limited in range by the curvature of the Earth and the blocking effect of terrain or high buildings. A repeater on a hilltop or tall building can allow stations that are out of each other's line-of-sight range to communicate reliably.

Radio repeaters may also allow translation from one set of radio frequencies to another, for example to allow two different public service agencies to interoperate (say, police and fire services of a city, or neighboring police departments). They may provide links to the public switched telephone network as well, or satellite network (BGAN, INMARSAT, MSAT) as an alternative path from source to the destination.

Typically a repeater station listens on one frequency, A, and transmits on a second, B. All mobile stations listen for signals on channel B and transmit on channel A.  The difference between the two frequencies may be relatively small compared to the frequency of operation, say 1%. Often the repeater station will use the same antenna for transmission and reception; highly selective filters called "duplexers" separate the faint incoming received signal from the billions of times more powerful outbound transmitted signal. Sometimes separate transmitting and receiving locations are used, connected by a wire line or a radio link. While the repeater station is designed for simultaneous reception and transmission, mobile units need not be equipped with the bulky and costly duplexers, as they only transmit or receive at any time.

Mobile units in a repeater system may be provided with a "talkaround" channel that allows direct mobile-to-mobile operation on a single channel. This may be used if out of reach of the repeater system, or for communications not requiring the attention of all mobiles. The "talkaround" channel may be the repeater output frequency; the repeater will not retransmit any signals on its output frequency.

An engineered radio communication system designer will analyze the coverage area desired and select repeater locations, elevations, antennas, operating frequencies and power levels to permit a predictable level of reliable communication over the designed coverage area.

Data handling
Repeaters can be divided into two types depending on the type of data they handle:

Analog repeater
This type is used in channels that transmit data in the form of an analog signal in which the voltage or current is proportional to the amplitude of the signal, as in an audio signal. They are also used in trunklines that transmit multiple signals using frequency division multiplexing (FDM). Analog repeaters are composed of a linear amplifier, and may include electronic filters to compensate for frequency and phase distortion in the line.

Digital repeater
The digital repeater is used in channels that transmit data by binary digital signals, in which the data is in the form of pulses with only two possible values, representing the binary digits 1 and 0. A digital repeater amplifies the signal, and it also may retime, resynchronize, and reshape the pulses. A repeater that performs the retiming or resynchronizing functions may be called a regenerator.

title: Repeater


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Repeating coil

In telecommunications, a repeating coil is a voice-frequency transformer characterized by a closed magnetic core, a pair of identical balanced primary (line) windings, a pair of identical but not necessarily balanced secondary (drop) windings, and low transmission loss at voice frequencies. It permits transfer of voice currents from one winding to another by magnetic induction, matches line and drop impedances, and prevents direct conduction between the line and the drop.

It is a special application of an isolation transformer, and is often used to prevent ground loops or earth loops, which cause humming or buzzing in audio circuits. It also prevents low direct current voltages from passing.

title: Repeating coil


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Reradiation

In telecommunication, the term reradiation has the following meanings:

 * 1) Electromagnetic radiation, at the same or different wavelengths, i.e., frequencies, of energy received from an incident wave.
 * 2) Undesirable radiation of signals locally generated in a radio receiver. This type of radiation might cause interference or reveal the location of the device.

Near-field effects of an AM antenna may extend out 2 mi or more. Cellular and microwave towers within this radius can reflect the Medium Wave AM signal out at a phase which cancels the main Medium Wave AM signal. This process results in an interfering signal called reradiation.

title: Reradiation


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Resolution

Resolution(s) may refer to:

Common meanings

 * Resolution (debate), the statement which is debated in policy debate
 * Resolution (law), a written motion adopted by a deliberative body
 * New Year's resolution, a commitment that an individual makes at New Year's Day
 * Dispute resolution, the settlement of a disagreement

Mathematics and logic

 * Resolution (algebra), an exact sequence in homological algebra
 * Resolution (logic), a rule of inference used for automated theorem proving
 * Standard resolution, the bar construction of resolutions in homological algebra
 * Resolution of singularities in algebraic geometry

Measurements

 * Resolution (audio), a measure of digital audio quality
 * Resolution (electron density), the quality of an X-ray crystallography or cryo-electron microscopy data set
 * Angular resolution, the capability of an optical or other sensor to discern small objects
 * Depositional resolution, the age difference of fossils contained in one stratum
 * Display resolution, the number of distinct pixels in each dimension that can be displayed on a display device
 * Graphic display resolutions, a list of particular display resolutions
 * Distance resolution, the minimum distance that can be accurately measured
 * Image resolution, a measure of the amount of detail in an image
 * Optical resolution, the capability of an optical system to distinguish, find, or record details
 * Printing resolution, the number of individual dots a printer can produce within a unit of distance (e.g., dots per inch)
 * Sensor resolution, the smallest change a sensor can detect in the quantity that it is measuring
 * Spectral resolution, the capability of an optical system to distinguish different frequencies
 * Resolution of a spectrometer, the ability to distinguish two close-lying energies (or wavelengths, or frequencies, or masses)
 * Resolution (chromatography), the separation of peaks in a chromatogram
 * Resolution (mass spectrometry), the ability to distinguish peaks in a mass spectrum
 * Spatial resolution, the pixel width on the ground
 * Temporal resolution, the sampling frequency of a digital audio device
 * In number storage, the resolution is the reciprocal of the unit in the last place

Other uses in science and technology

 * Resolution (beam engine), an early steam engine at Coalbrookdale
 * Chiral resolution, a process in stereochemistry for the separation of racemic compounds into their enantiomers
 * URL resolution, a process in interpreting URLs

Film and television

 * "Resolutions" (Star Trek: Voyager), a 1996 second-season episode of Star Trek: Voyager
 * Resolution (film), a 2012 horror movie
 * "Resolution" (Doctor Who), a special episode of Doctor Who
 * Resolution, an episode of the HBO TV Show Boardwalk Empire.

Literature and poetry

 * Resolution (meter), the replacement of one longum with two brevia
 * "Resolution", a poem by Bertolt Brecht
 * Resolutions (short story), a 1911 short story by Franz Kafka
 * Resolution (Parker novel), a 2008 novel by Robert B. Parker, a Western, sequel to Appaloosa
 * Resolution (Wilson novel), a 2016 work of historical fiction by A. N. Wilson

Music

 * Resolution (music), the move of a note or chord from dissonance to consonance

Albums

 * Resolution (38 Special album), 1997
 * Resolution (BoDeans album), 2004
 * Resolution (Hamiet Bluiett album), 1977
 * Resolution (Hidden in Plain View album), 2007
 * Resolution (Lamb of God album), 2012
 * Resolution (Andy Timmons album), 2006
 * Resolution (Andy Pratt album), 1976
 * Resolutions (album), a 2011 album by Dave Hause

EPs

 * Resolution (EP), Matt Corby EP, 2013

Songs

 * "Resolution" (Matt Corby song), 2013
 * "Resolution" (Nick Lachey song), a 2006 song by Nick Lachey from What's Left of Me
 * "Resolution", a John Coltrane (1926–1967) song from A Love Supreme
 * "Resolution", a Mahavishnu Orchestra song from the album Birds of Fire
 * "Resolution", by Romantic Mode, the second opening theme for the anime After War Gundam X
 * "Resolution", by Haruka Tomatsu, the first opening theme for the anime Sword Art Online: Alicization War of Underworld
 * "The Resolution", a 2008 Jack's Mannequin song from The Glass Passenger

Business and organizations

 * Corporate resolution, a legal document defining which individuals are authorized to act on behalf of a corporation
 * Resolution (talent agency), a talent agency in Los Angeles, New York, and Nashville
 * Resolution Copper, a copper mining project near Superior, Arizona
 * Resolution Foundation, an independent British think tank established in 2005
 * Resolution Limited, a Guernsey-based investor in insurance companies
 * Resolution plc, a defunct manager of in-force UK life funds acquired by Pearl Group in 2008

Places

 * Resolution, United States Virgin Islands
 * Resolution Island (Nunavut), in the Arctic region of Canada
 * Resolution Island, New Zealand, the largest (uninhabited) island in Fiordland, in the southwest of New Zealand
 * Resolution Island, a fictional island in the novel Brown on Resolution
 * Fort Resolution, a hamlet in the South Slave Region of the Northwest Territories, Canada

Vessels

 * Resolution, a Douglas DC-6 aircraft, BCPA Flight 304, which crashed near San Francisco in 1953
 * Resolution (1793 ship), an American trading vessel whose crew were massacred in 1794
 * HMNZS Resolution, Royal New Zealand Navy surveillance ship
 * HMS Resolution, any of several British Royal Navy ships
 * TIV Resolution, the world's first turbine installation vessel
 * Resolution-class submarine, of the British Royal Navy
 * Resolution (1802 ship), a brig built in Spain and launched in 1802

title: Resolution