text:
Dielectric

In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor, because they have no loosely bound, or free, electrons that may drift through the material, but instead they shift, only slightly, from their average equilibrium positions, causing dielectric polarisation. Because of dielectric polarisation, positive charges are displaced in the direction of the field and negative charges shift in the direction opposite to the field. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarised, but also reorient so that their symmetry axes align to the field.

The study of dielectric properties concerns storage and dissipation of electric and magnetic energy in materials. Dielectrics are important for explaining various phenomena in electronics, optics, solid-state physics and cell biophysics.

Terminology
Although the term insulator implies low electrical conduction, dielectric typically means materials with a high polarisability. The latter is expressed by a number called the relative permittivity. Insulator is generally used to indicate electrical obstruction while dielectric is used to indicate the energy storing capacity of the material (by means of polarisation). A common example of a dielectric is the electrically insulating material between the metallic plates of a capacitor. The polarisation of the dielectric by the applied electric field increases the capacitor's surface charge for the given electric field strength.

The term dielectric was coined by William Whewell (from dia + electric) in response to a request from Michael Faraday.

A perfect dielectric is a material with zero electrical conductivity (cf. perfect conductor infinite electrical conductivity), thus exhibiting only a displacement current; therefore it stores and returns electrical energy as if it were an ideal capacitor.

Electric susceptibility
The electric susceptibility $$\chi_e$$ of a dielectric material is a measure of how easily it polarises in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.

It is defined as the constant of proportionality (which may be a tensor) relating an electric field $$\mathbf{E}$$ to the induced dielectric polarisation density $$\mathbf{P}$$ such that

$$\mathbf{P} = \varepsilon_0 \chi_e \mathbf{E},$$

where $$\varepsilon_0$$ is the electric permittivity of free space.

The susceptibility of a medium is related to its relative permittivity $$\varepsilon_r$$ by

$$\chi_e\ = \varepsilon_r - 1.$$

So in the case of a classical vacuum,

$$\chi_e\ = 0.$$

The electric displacement $$\mathbf{D}$$ is related to the polarisation density $$\mathbf{P}$$ by

$$\mathbf{D} \ = \ \varepsilon_0 \mathbf{E} + \mathbf{P} \ = \ \varepsilon_0 \left(1 + \chi_e\right) \mathbf{E} \ = \ \varepsilon_0 \varepsilon_r \mathbf{E}. $$

Dispersion and causality
In general, a material cannot polarise instantaneously in response to an applied field. The more general formulation as a function of time is

$$\mathbf{P}(t) = \varepsilon_0 \int_{-\infty}^t \chi_e\left(t - t'\right) \mathbf{E}(t')\, dt'.$$

That is, the polarisation is a convolution of the electric field at previous times with time-dependent susceptibility given by $$\chi_e (\Delta t)$$. The upper limit of this integral can be extended to infinity as well if one defines $$\chi_e (\Delta t) = 0$$ for $$\Delta t < 0$$. An instantaneous response corresponds to Dirac delta function susceptibility $$\chi_e (\Delta t) = \chi_e \delta (\Delta t)$$.

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product, $$\mathbf{P}(\omega) = \varepsilon_0 \chi_e(\omega) \mathbf{E}(\omega).$$

The susceptibility (or equivalently the permittivity) is frequency dependent. The change of susceptibility with respect to frequency characterises the dispersion properties of the material.

Moreover, the fact that the polarisation can only depend on the electric field at previous times (i.e., $$\chi_e (\Delta t) = 0$$ for $$\Delta t < 0$$), a consequence of causality, imposes Kramers–Kronig constraints on the real and imaginary parts of the susceptibility $$\chi_e (\omega)$$.

Basic atomic model

In the classical approach to the dielectric, the material is made up of atoms. Each atom consists of a cloud of negative charge (electrons) bound to and surrounding a positive point charge at its center. In the presence of an electric field, the charge cloud is distorted, as shown in the top right of the figure.

This can be reduced to a simple dipole using the superposition principle. A dipole is characterised by its dipole moment, a vector quantity shown in the figure as the blue arrow labeled M. It is the relationship between the electric field and the dipole moment that gives rise to the behaviour of the dielectric. (Note that the dipole moment points in the same direction as the electric field in the figure. This is not always the case, and is a major simplification, but is true for many materials.)

When the electric field is removed, the atom returns to its original state. The time required to do so is called relaxation time; an exponential decay.

This is the essence of the model in physics. The behaviour of the dielectric now depends on the situation. The more complicated the situation, the richer the model must be to accurately describe the behaviour. Important questions are:
 * Is the electric field constant, or does it vary with time? At what rate?
 * Does the response depend on the direction of the applied field (isotropy of the material)?
 * Is the response the same everywhere (homogeneity of the material)?
 * Do any boundaries or interfaces have to be taken into account?
 * Is the response linear with respect to the field, or are there nonlinearities?

The relationship between the electric field E and the dipole moment M gives rise to the behaviour of the dielectric, which, for a given material, can be characterised by the function F defined by the equation: $$\mathbf{M} = \mathbf{F}(\mathbf{E}).$$

When both the type of electric field and the type of material have been defined, one then chooses the simplest function F that correctly predicts the phenomena of interest. Examples of phenomena that can be so modelled include:

 * Refractive index
 * Group velocity dispersion
 * Birefringence
 * Self-focusing
 * Harmonic generation

Dipolar polarisation
Dipolar polarisation is a polarisation that is either inherent to polar molecules (orientation polarisation), or can be induced in any molecule in which the asymmetric distortion of the nuclei is possible (distortion polarisation). Orientation polarisation results from a permanent dipole, e.g., that arising from the 104.45° angle between the asymmetric bonds between oxygen and hydrogen atoms in the water molecule, which retains polarisation in the absence of an external electric field. The assembly of these dipoles forms a macroscopic polarisation.

When an external electric field is applied, the distance between charges within each permanent dipole, which is related to chemical bonding, remains constant in orientation polarisation; however, the direction of polarisation itself rotates. This rotation occurs on a timescale that depends on the torque and surrounding local viscosity of the molecules. Because the rotation is not instantaneous, dipolar polarisations lose the response to electric fields at the highest frequencies. A molecule rotates about 1 radian per picosecond in a fluid, thus this loss occurs at about 1011 Hz (in the microwave region). The delay of the response to the change of the electric field causes friction and heat.

When an external electric field is applied at infrared frequencies or less, the molecules are bent and stretched by the field and the molecular dipole moment changes. The molecular vibration frequency is roughly the inverse of the time it takes for the molecules to bend, and this distortion polarisation disappears above the infrared.

Ionic polarisation
Ionic polarisation is polarisation caused by relative displacements between positive and negative ions in ionic crystals (for example, NaCl).

If a crystal or molecule consists of atoms of more than one kind, the distribution of charges around an atom in the crystal or molecule leans to positive or negative. As a result, when lattice vibrations or molecular vibrations induce relative displacements of the atoms, the centers of positive and negative charges are also displaced. The locations of these centers are affected by the symmetry of the displacements. When the centers do not correspond, polarisation arises in molecules or crystals. This polarisation is called ionic polarisation.

Ionic polarisation causes the ferroelectric effect as well as dipolar polarisation. The ferroelectric transition, which is caused by the lining up of the orientations of permanent dipoles along a particular direction, is called an order-disorder phase transition. The transition caused by ionic polarisations in crystals is called a displacive phase transition.

In biological cells
Ionic polarisation enables the production of energy-rich compounds in cells (the proton pump in mitochondria) and, at the plasma membrane, the establishment of the resting potential, energetically unfavourable transport of ions, and cell-to-cell communication (the Na+/K+-ATPase).

All cells in animal body tissues are electrically polarised – in other words, they maintain a voltage difference across the cell's plasma membrane, known as the membrane potential. This electrical polarisation results from a complex interplay between ion transporters and ion channels.

In neurons, the types of ion channels in the membrane usually vary across different parts of the cell, giving the dendrites, axon, and cell body different electrical properties. As a result, some parts of the membrane of a neuron may be excitable (capable of generating action potentials), whereas others are not.

Dielectric dispersion
In physics, dielectric dispersion is the dependence of the permittivity of a dielectric material on the frequency of an applied electric field. Because there is a lag between changes in polarisation and changes in the electric field, the permittivity of the dielectric is a complex function of the frequency of the electric field. Dielectric dispersion is very important for the applications of dielectric materials and the analysis of polarisation systems.

This is one instance of a general phenomenon known as material dispersion: a frequency-dependent response of a medium for wave propagation.

When the frequency becomes higher:
 * 1) The dipolar polarisation can no longer follow the oscillations of the electric field in the microwave region around 1010 Hz,
 * 2) The ionic polarisation and molecular distortion polarisation can no longer track the electric field past the infrared or far-infrared region around 1013 Hz,
 * 3) The electronic polarisation loses its response in the ultraviolet region around 1015 Hz.

In the frequency region above ultraviolet, permittivity approaches the constant ε0 in every substance, where ε0 is the permittivity of the free space. Because permittivity indicates the strength of the relation between an electric field and polarisation, if a polarisation process loses its response, permittivity decreases.

Dielectric relaxation
Dielectric relaxation is the momentary delay (or lag) in the dielectric constant of a material. This is usually caused by the delay in molecular polarisation with respect to a changing electric field in a dielectric medium (e.g., inside capacitors or between two large conducting surfaces). Dielectric relaxation in changing electric fields could be considered analogous to hysteresis in changing magnetic fields (e.g., in inductor or transformer cores). Relaxation in general is a delay or lag in the response of a linear system, and therefore dielectric relaxation is measured relative to the expected linear steady state (equilibrium) dielectric values. The time lag between electrical field and polarisation implies an irreversible degradation of Gibbs free energy.

In physics, dielectric relaxation refers to the relaxation response of a dielectric medium to an external, oscillating electric field. This relaxation is often described in terms of permittivity as a function of frequency, which can, for ideal systems, be described by the Debye equation. On the other hand, the distortion related to ionic and electronic polarisation shows behaviour of the resonance or oscillator type. The character of the distortion process depends on the structure, composition, and surroundings of the sample.

Debye relaxation
Debye relaxation is the dielectric relaxation response of an ideal, noninteracting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity ε of a medium as a function of the field's angular frequency ω:

$$\hat{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{\Delta\varepsilon}{1 + i\omega\tau},$$

where ε∞ is the permittivity at the high frequency limit, where εs is the static, low frequency permittivity, and τ is the characteristic relaxation time of the medium. Separating into the real part $$\varepsilon'$$ and the imaginary part $$\varepsilon''$$ of the complex dielectric permittivity yields:

$$\begin{align} \varepsilon' &= \varepsilon_\infty + \frac{\varepsilon_s - \varepsilon_\infty}{1 + \omega^2\tau^2} \\[3pt] \varepsilon'' &= \frac{(\varepsilon_s - \varepsilon_\infty)\omega\tau}{1+\omega^2\tau^2} \end{align}$$

Note that the above equation for $$\hat{\varepsilon}(\omega)$$is sometimes written with $$1 - i\omega\tau$$ in the denominator due to an ongoing sign convention ambiguity whereby many sources represent the time dependence of the complex electric field with $$\exp(-i\omega t)$$ whereas others use $$\exp(+i\omega t)$$. In the former convention, the functions $$\varepsilon'$$ and $$\varepsilon$$ representing real and imaginary parts are given by $$\hat{\varepsilon}(\omega)=\varepsilon'+ i \varepsilon$$ whereas in the latter convention $$\hat{\varepsilon}(\omega)=\varepsilon'- i \varepsilon''$$. The above equation uses the latter convention.

The dielectric loss is also represented by the loss tangent:

$$\tan(\delta) = \frac{\varepsilon''}{\varepsilon'} = \frac{\left(\varepsilon_s - \varepsilon_\infty\right)\omega\tau}{\varepsilon_s + \varepsilon_\infty \omega^2 \tau^2}$$

This relaxation model was introduced by and named after the physicist Peter Debye (1913). It is characteristic for dynamic polarisation with only one relaxation time.

Variants of the Debye equation

 * Cole–Cole equation: This equation is used when the dielectric loss peak shows symmetric broadening.
 * Cole–Davidson equation: This equation is used when the dielectric loss peak shows asymmetric broadening.
 * Havriliak–Negami relaxation: This equation considers both symmetric and asymmetric broadening.
 * Kohlrausch–Williams–Watts function: Fourier transform of stretched exponential function.
 * Curie–von Schweidler law: This shows the response of dielectrics to an applied DC field to behave according to a power law, which can be expressed as an integral over weighted exponential functions.
 * Djordjevic–Sarkar approximation: This is used when the dielectric loss is approximately constant for a wide range of frequencies.

Paraelectricity
Paraelectricity is the nominal behaviour of dielectrics when the dielectric permittivity tensor is proportional to the unit matrix, i.e., an applied electric field causes polarisation and/or alignment of dipoles only parallel to the applied electric field. Contrary to the analogy with a paramagnetic material, no permanent electric dipole needs to exist in a paraelectric material. Removal of the fields results in the dipolar polarisation returning to zero. The mechanisms that causes paraelectric behaviour are distortion of individual ions (displacement of the electron cloud from the nucleus) and polarisation of molecules or combinations of ions or defects.

Paraelectricity can occur in crystal phases where electric dipoles are unaligned and thus have the potential to align in an external electric field and weaken it.

Most dielectric materials are paraelectrics. A specific example of a paraelectric material of high dielectric constant is strontium titanate.

The LiNbO3 crystal is ferroelectric below 1430 K, and above this temperature it transforms into a disordered paraelectric phase. Similarly, other perovskites also exhibit paraelectricity at high temperatures.

Paraelectricity has been explored as a possible refrigeration mechanism; polarising a paraelectric by applying an electric field under adiabatic process conditions raises the temperature, while removing the field lowers the temperature. A heat pump that operates by polarising the paraelectric, allowing it to return to ambient temperature (by dissipating the extra heat), bringing it into contact with the object to be cooled, and finally depolarising it, would result in refrigeration.

Tunability
Tunable dielectrics are insulators whose ability to store electrical charge changes when a voltage is applied.

Generally, strontium titanate is used for devices operating at low temperatures, while barium strontium titanate  substitutes for room temperature devices. Other potential materials include microwave dielectrics and carbon nanotube (CNT) composites.

In 2013, multi-sheet layers of strontium titanate interleaved with single layers of strontium oxide produced a dielectric capable of operating at up to 125 GHz. The material was created via molecular beam epitaxy. The two have mismatched crystal spacing that produces strain within the strontium titanate layer that makes it less stable and tunable.

Systems such as have a paraelectric–ferroelectric transition just below ambient temperature, providing high tunability. Films suffer significant losses arising from defects.

Capacitors

Commercially manufactured capacitors typically use a solid dielectric material with high permittivity as the intervening medium between the stored positive and negative charges. This material is often referred to in technical contexts as the capacitor dielectric.

The most obvious advantage to using such a dielectric material is that it prevents the conducting plates, on which the charges are stored, from coming into direct electrical contact. More significantly, however, a high permittivity allows a greater stored charge at a given voltage. This can be seen by treating the case of a linear dielectric with permittivity ε and thickness d between two conducting plates with uniform charge density σε. In this case the charge density is given by

$$\sigma_{\varepsilon}=\varepsilon\frac{V}{d}$$

and the capacitance per unit area by

$$c=\frac{\sigma_{\varepsilon}}{V}=\frac{\varepsilon}{d}$$

From this, it can easily be seen that a larger ε leads to greater charge stored and thus greater capacitance.

Dielectric materials used for capacitors are also chosen such that they are resistant to ionisation. This allows the capacitor to operate at higher voltages before the insulating dielectric ionises and begins to allow undesirable current.

Dielectric resonator
A dielectric resonator oscillator (DRO) is an electronic component that exhibits resonance of the polarisation response for a narrow range of frequencies, generally in the microwave band. It consists of a "puck" of ceramic that has a large dielectric constant and a low dissipation factor. Such resonators are often used to provide a frequency reference in an oscillator circuit. An unshielded dielectric resonator can be used as a dielectric resonator antenna (DRA).

BST thin films
From 2002 to 2004, the United States Army Research Laboratory (ARL) conducted research on thin film technology. Barium strontium titanate (BST), a ferroelectric thin film, was studied for the fabrication of radio frequency and microwave components, such as voltage-controlled oscillators, tunable filters and phase shifters.

The research was part of an effort to provide the Army with highly-tunable, microwave-compatible materials for broadband electric-field tunable devices, which operate consistently in extreme temperatures. This work improved tunability of bulk barium strontium titanate, which is a thin film enabler for electronics components.

In a 2004 research paper, U.S. ARL researchers explored how small concentrations of acceptor dopants can dramatically modify the properties of ferroelectric materials such as BST.

Researchers "doped" BST thin films with magnesium, analyzing the "structure, microstructure, surface morphology and film/substrate compositional quality" of the result. The Mg doped BST films showed "improved dielectric properties, low leakage current, and good tunability", meriting potential for use in microwave tunable devices.

Some practical dielectrics
Dielectric materials can be solids, liquids, or gases. (A high vacuum can also be a useful, nearly lossless dielectric even though its relative dielectric constant is only unity.)

Solid dielectrics are perhaps the most commonly used dielectrics in electrical engineering, and many solids are very good insulators. Some examples include porcelain, glass, and most plastics. Air, nitrogen and sulfur hexafluoride are the three most commonly used gaseous dielectrics.
 * Industrial coatings such as Parylene provide a dielectric barrier between the substrate and its environment.
 * Mineral oil is used extensively inside electrical transformers as a fluid dielectric and to assist in cooling. Dielectric fluids with higher dielectric constants, such as electrical grade castor oil, are often used in high voltage capacitors to help prevent corona discharge and increase capacitance.
 * Because dielectrics resist the flow of electricity, the surface of a dielectric may retain stranded excess electrical charges. This may occur accidentally when the dielectric is rubbed (the triboelectric effect). This can be useful, as in a Van de Graaff generator or electrophorus, or it can be potentially destructive as in the case of electrostatic discharge.
 * Specially processed dielectrics, called electrets (which should not be confused with ferroelectrics), may retain excess internal charge or "frozen in" polarisation. Electrets have a semi-permanent electric field, and are the electrostatic equivalent to magnets. Electrets have numerous practical applications in the home and industry.
 * Some dielectrics can generate a potential difference when subjected to mechanical stress, or (equivalently) change physical shape if an external voltage is applied across the material. This property is called piezoelectricity. Piezoelectric materials are another class of very useful dielectrics.
 * Some ionic crystals and polymer dielectrics exhibit a spontaneous dipole moment, which can be reversed by an externally applied electric field. This behaviour is called the ferroelectric effect. These materials are analogous to the way ferromagnetic materials behave within an externally applied magnetic field. Ferroelectric materials often have very high dielectric constants, making them quite useful for capacitors.

title: Dielectric


text:
Dielectric strength

In physics, the term dielectric strength has the following meanings:

 * for a pure electrically insulating material, the maximum electric field that the material can withstand under ideal conditions without undergoing electrical breakdown and becoming electrically conductive (i.e. without failure of its insulating properties).
 * For a specific piece of dielectric material and location of electrodes, the minimum applied electric field (i.e. the applied voltage divided by electrode separation distance) that results in breakdown. This is the concept of breakdown voltage.

The theoretical dielectric strength of a material is an intrinsic property of the bulk material, and is independent of the configuration of the material or the electrodes with which the field is applied. This "intrinsic dielectric strength" corresponds to what would be measured using pure materials under ideal laboratory conditions. At breakdown, the electric field frees bound electrons. If the applied electric field is sufficiently high, free electrons from background radiation may be accelerated to velocities that can liberate additional electrons by collisions with neutral atoms or molecules, in a process known as avalanche breakdown. Breakdown occurs quite abruptly (typically in nanoseconds), resulting in the formation of an electrically conductive path and a disruptive discharge through the material. In a solid material, a breakdown event severely degrades, or even destroys, its insulating capability.

Electrical breakdown
Electric current is a flow of electrically charged particles in a material caused by an electric field. The mobile charged particles responsible for electric current are called charge carriers. In different substances different particles serve as charge carriers: in metals and other solids some of the outer electrons of each atom (conduction electrons) are able to move about the material; in electrolytes and plasma it is ions, electrically charged atoms or molecules, and electrons. A substance that has a high concentration of charge carriers available for conduction will conduct a large current with the given electric field created by a given voltage applied across it, and thus has a low electrical resistivity; this is called an electrical conductor. A material that has few charge carriers will conduct very little current with a given electric field and has a high resistivity; this is called an electrical insulator.

However, when a large enough electric field is applied to any insulating substance, at a certain field strength the concentration of charge carriers in the material suddenly increases by many orders of magnitude, so its resistance drops and it becomes a conductor. This is called electrical breakdown. The physical mechanism causing breakdown differs in different substances. In a solid, it usually occurs when the electric field becomes strong enough to pull outer valence electrons away from their atoms, so they become mobile. The field strength at which break down occurs is an intrinsic property of the material called its dielectric strength.

In practical electric circuits electrical breakdown is often an unwanted occurrence, a failure of insulating material causing a short circuit, resulting in a catastrophic failure of the equipment. The sudden drop in resistance causes a high current to flow through the material, and the sudden extreme Joule heating may cause the material or other parts of the circuit to melt or vaporize explosively. However, breakdown itself is reversible. If the current supplied by the external circuit is sufficiently limited, no damage is done to the material, and reducing the applied voltage causes a transition back to the material's insulating state.

Factors affecting apparent dielectric strength

 * It may vary with sample thickness. (see "defects" below)
 * It may vary with operating temperature.
 * It may vary with frequency.
 * For gases (e.g. nitrogen, sulfur hexafluoride) it normally decreases with increased humidity as ions in water can provide conductive channels.
 * For gases it increases with pressure according to Paschen's law
 * For air, dielectric strength increases slightly as the absolute humidity increases but decreases with an increase in relative humidity

Break down field strength
The field strength at which break down occurs depends on the respective geometries of the dielectric (insulator) and the electrodes with which the electric field is applied, as well as the rate of increase of the applied electric field. Because dielectric materials usually contain minute defects, the practical dielectric strength will be a significantly less than the intrinsic dielectric strength of an ideal, defect-free, material. Dielectric films tend to exhibit greater dielectric strength than thicker samples of the same material. For instance, the dielectric strength of silicon dioxide films of thickness around 1 μm is about 0.5GV/m. However very thin layers (below, say, 100 nm) become partially conductive because of electron tunneling. Multiple layers of thin dielectric films are used where maximum practical dielectric strength is required, such as high voltage capacitors and pulse transformers. Since the dielectric strength of gases varies depending on the shape and configuration of the electrodes, it is usually measured as a fraction of the dielectric strength of nitrogen gas.

Dielectric strength (in MV/m, or 10$6$⋅volt/meter) of various common materials:

Units
In SI, the unit of dielectric strength is volts per meter (V/m). It is also common to see related units such as volts per centimeter (V/cm), megavolts per meter (MV/m), and so on.

In United States customary units, dielectric strength is often specified in volts per mil (a mil is 1/1000 inch). The conversion is:
 * $$\begin{align}

1 \text{ V/mil} &= 3.94\times 10^{4} \text{ V/m} \\ 1 \text{ V/m} &= 2.54\times 10^{-5} \text{ V/mil} \end{align}$$

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

title: Dielectric strength


text:
Differential Manchester encoding

Differential Manchester encoding (DM) is a line code in digital frequency modulation in which data and clock signals are combined to form a single two-level self-synchronizing data stream. Each data bit is encoded by a presence or absence of signal level transition in the middle of the bit period, followed by the mandatory level transition at the beginning. The code is insensitive to an inversion of polarity. In various specific applications, this method is also called by various other names, including biphase mark code (CC), F2F (frequency/double frequency), Aiken biphase, and conditioned diphase.

Definition
Differential Manchester encoding is a differential encoding technology, using the presence or absence of transitions to indicate logical value. An improvement to Manchester coding which is a special case of binary phase-shift keying, it is not necessary to know the initial polarity of the transmitted message signal, because the information is not represented by the absolute voltage levels but by their transitions.

There are two clock ticks per bit period (marked with full and dotted lines in the figure). At every second clock tick, marked with a dotted line, there is a potential level transition conditional on the data. At the other ticks, the line state changes unconditionally to ease clock recovery.

One version of the code makes a transition for 0 and no transition for 1; the other makes a transition for 1 and no transition for 0.

Differential Manchester encoding has the following advantages:
 * A transition is guaranteed at least once every bit, for robust clock recovery.
 * In a noisy environment, detecting transitions is less error-prone than comparing signal levels against a threshold.
 * Unlike with Manchester encoding, only the presence of a transition is important, not the polarity. Differential coding schemes will work exactly the same if the signal is inverted (e.g. wires swapped). Other line codes with this property include NRZI, bipolar encoding, coded mark inversion, and MLT-3 encoding.
 * If the high and low signal levels have the same magnitude with opposite polarity, the average voltage around each unconditional transition is zero. Zero DC bias reduces the necessary transmitting power, minimizes the amount of electromagnetic noise produced by the transmission line, and eases the use of isolating transformers.

These positive features are achieved at the expense of doubling the clock frequency of the encoded data stream.

Differential Manchester encoding is specified in the IEEE 802.5 standard for Token Ring local area networks, and is used for many other applications, including magnetic and optical storage. As Biphase Mark Code (BMC), it is used in AES3, S/PDIF, SMPTE time code, USB PD, xDSL and DALI. Many magnetic stripe cards also use BMC encoding, often called F2F (frequency/double frequency) or Aiken Biphase, according to the ISO/IEC 7811 standard. Differential Manchester encoding is also the original modulation method used for single-density floppy disks, followed by double-density modified frequency modulation (MFM).

title: Differential Manchester encoding


text:
Diffraction grating

In optics, a diffraction grating is an optical grating with a periodic structure that diffracts light, or another type of electromagnetic radiation, into several beams traveling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structural coloration. The directions or diffraction angles of these beams depend on the wave (light) incident angle to the diffraction grating, the spacing or periodic distance between adjacent diffracting elements (e.g., parallel slits for a transmission grating) on the grating, and the wavelength of the incident light. The grating acts as a dispersive element. Because of this, diffraction gratings are commonly used in monochromators and spectrometers, but other applications are also possible such as optical encoders for high-precision motion control and wavefront measurement.

For typical applications, a reflective grating has ridges or rulings on its surface while a transmissive grating has transmissive or hollow slits on its surface. Such a grating modulates the amplitude of an incident wave to create a diffraction pattern. Some gratings modulate the phases of incident waves rather than the amplitude, and these types of gratings can be produced frequently by using holography.

James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating (in a natural form) to be discovered, about a year after Isaac Newton's prism experiments. The first human-made diffraction grating was made around 1785 by Philadelphia inventor David Rittenhouse, who strung hairs between two finely threaded screws. This was similar to notable German physicist Joseph von Fraunhofer's wire diffraction grating in 1821. The principles of diffraction were discovered by Thomas Young and Augustin-Jean Fresnel. Using these principles, Fraunhofer was the first to use a diffraction grating to obtain line spectra and the first to measure the wavelengths of spectral lines with a diffraction grating.

In the 1860s, state-of-the-art diffraction gratings with small groove period (d) were manufactured by Friedrich Adolph Nobert (1806–1881) in Greifswald; then the two Americans Lewis Morris Rutherfurd (1816–1892) and William B. Rogers (1804–1882) took over the lead. By the end of the 19th century, the concave gratings of Henry Augustus Rowland (1848–1901) were the best available.

A diffraction grating can create "rainbow" colors when it is illuminated by a wide-spectrum (e.g., continuous) light source. Rainbow-like colors from closely spaced narrow tracks on optical data storage disks such as CDs or DVDs are an example of light diffraction caused by diffraction gratings. A usual diffraction grating has parallel lines (It is true for 1-dimensional gratings, but 2 or 3-dimensional gratings are also possible and they have their applications such as wavefront measurement), while a CD has a spiral of finely spaced data tracks. Diffraction colors also appear when one looks at a bright point source through a translucent fine-pitch umbrella fabric covering. Decorative patterned plastic films based on reflective grating patches are inexpensive and commonplace. A similar color separation seen from thin layers of oil (or gasoline, etc.) on water, known as iridescence, is not caused by diffraction from a grating but rather by thin film interference from the closely stacked transmissive layers.

Theory of operation
For a diffraction grating, the relationship between the grating spacing (i.e., the distance between adjacent grating grooves or slits), the angle of the wave (light) incidence to the grating, and the diffracted wave from the grating is known as the grating equation. Like many other optical formulas, the grating equation can be derived by using the Huygens–Fresnel principle, stating that each point on a wavefront of a propagating wave can be considered to act as a point wave source, and a wavefront at any subsequent point can be found by adding together the contributions from each of these individual point wave sources on the previous wavefront.

Gratings may be of the 'reflective' or 'transmissive' type, analogous to a mirror or lens, respectively. A grating has a 'zero-order mode' (where the integer order of diffraction m is set to zero), in which a ray of light behaves according to the laws of reflection (like a mirror) and refraction (like a lens), respectively.

[[File:Diffraction Grating Equation.jpg|thumb|400x400px|A diagram showing the path difference between rays of light scattered from adjacent rulings at the same local position on each ruling of a reflective diffraction grating (actually a blazed grating). The choice of + or – in the path difference formula depends on the sign convention for$$\Theta_\mathrm m$$: plus if $$\Theta_\mathrm m = \Theta_\mathrm i$$ describes the case of backscatter, minus if it describes specular reflection.

Note that the pair of the black ray path parts and the pair of the light green ray path parts have no path difference in each pair, while there is a path difference in the red ray path part pair that matters in the diffraction grating equation derivation.]]

An idealized diffraction grating is made up of a set of slits of spacing $$d$$, that must be wider than the wavelength of interest to cause diffraction. Assuming a plane wave of monochromatic light of wavelength $$\lambda$$ at normal incidence on a grating (i.e., wavefronts of the incident wave are parallel to the grating main plane), each slit in the grating acts as a quasi point wave source from which light propagates in all directions (although this is typically limited to the forward hemisphere from the point source). Of course, every point on every slit to which the incident wave reaches plays as a point wave source for the diffraction wave and all these contributions to the diffraction wave determine the detailed diffraction wave light property distribution, but diffraction angles (at the grating) at which the diffraction wave intensity is highest are determined only by these quasi point sources corresponding the slits in the grating. After the incident light (wave) interacts with the grating, the resulting diffracted light from the grating is composed of the sum of interfering wave components emanating from each slit in the grating; At any given point in space through which the diffracted light may pass, typically called observation point, the path length from each slit in the grating to the given point varies, so the phase of the wave emanating from each of the slits at that point also varies. As a result, the sum of the diffracted waves from the grating slits at the given observation point creates a peak, valley, or some degree between them in light intensity through additive and destructive interference. When the difference between the light paths from adjacent slits to the observation point is equal to an odd integer-multiple of the half of the wavelength, l$$l(\lambda/2)$$ with an odd integer $$l$$, the waves are out of phase at that point, and thus cancel each other to create the (locally) minimum light intensity. Similarly, when the path difference is a multiple of $$\lambda$$, the waves are in phase and the (locally) maximum intensity occurs. For light at the normal incidence to the grating, the intensity maxima occur at diffraction angles $$\theta_m$$, which satisfy the relationship $$d \sin\theta_m = m\lambda$$, where $$\theta_m$$ is the angle between the diffracted ray and the grating's normal vector, $$d$$ is the distance from the center of one slit to the center of the adjacent slit, and $$m$$ is an integer representing the propagation-mode of interest called the diffraction order.

When a plane light wave is normally incident on a grating of uniform period $$d$$, the diffracted light has maxima at diffraction angles $$\theta_m$$ given by a special case of the grating equation as $$\sin\theta_m = \frac{m\lambda}{d}.$$

It can be shown that if the plane wave is incident at angle $$\theta_i$$ relative to the grating normal, in the plane orthogonal to the grating periodicity, the grating equation becomes $$\sin\theta_i + \sin\theta_m = \frac{m\lambda}{d},$$ which describes in-plane diffraction as a special case of the more general scenario of conical, or off-plane, diffraction described by the generalized grating equation: $$\sin\theta_i + \sin\theta_m = \frac{m\lambda}{d\sin\gamma},$$ where $$\gamma$$ is the angle between the direction of the plane wave and the direction of the grating grooves, which is orthogonal to both the directions of grating periodicity and grating normal. Various sign conventions for $$\theta_i$$, $$\theta_m$$ and $$m$$ are used; any choice is fine as long as the choice is kept through diffraction-related calculations. When solved for diffracted angle at which the diffracted wave intensity are maximized, the equation becomes $$\theta_m = \arcsin\!\left( \sin\theta_i -\frac{m\lambda}{d\sin\gamma}\right ).$$

The diffracted light that corresponds to direct transmission for a transmissive diffraction grating or specular reflection for a reflective grating is called the zero order, and is denoted $$m=0$$. The other diffracted light intensity maxima occur at angles $$\theta_m$$ represented by non-zero integer diffraction orders $$m$$. Note that $$m$$ can be positive or negative, corresponding to diffracted orders on both sides of the zero-order diffracted beam.

Even if the grating equation is derived from a specific grating such as the grating in the right diagram (this grating is called a blazed grating), the equation can apply to any regular structure of the same spacing, because the phase relationship between light scattered from adjacent diffracting elements of the grating remains the same. The detailed diffracted light property distribution (e.g., intensity) depends on the detailed structure of the grating elements as well as on the number of elements in the grating, but it always gives maxima in the directions given by the grating equation.

Depending on how a grating modulates incident light on it to cause the diffracted light, there are the following grating types:

 * Transmission amplitude diffraction grating, which spatially and periodically modulates the intensity of an incident wave that transmits through the grating (and the diffracted wave is the consequence of this modulation).
 * Reflection amplitude diffraction gratings, which spatially and periodically modulate the intensity of an incident wave that is reflected from the grating.
 * Transmission phase diffraction grating, that spatially and periodically modulates the phase of an incident wave passing through the grating.
 * Reflection phase diffraction grating, that spatially and periodically modulates the phase of an incident wave reflected from the grating.

An optical axis diffraction grating, in which the optical axis is spatially and periodically modulated, is also considered either a reflection or transmission phase diffraction grating.

The grating equation applies to all these gratings due to the same phase relationship between the diffracted waves from adjacent diffracting elements of the gratings, even if the detailed distribution of the diffracted wave property depends on the detailed structure of each grating.

Quantum electrodynamics

Quantum electrodynamics (QED) offers another derivation of the properties of a diffraction grating in terms of photons as particles (at some level). QED can be described intuitively with the path integral formulation of quantum mechanics. As such it can model photons as potentially following all paths from a source to a final point, each path with a certain probability amplitude. These probability amplitudes can be represented as a complex number or equivalent vector—or, as Richard Feynman simply calls them in his book on QED, "arrows".

For the probability that a certain event will happen, one sums the probability amplitudes for all of the possible ways in which the event can occur, and then takes the square of the length of the result. The probability amplitude for a photon from a monochromatic source to arrive at a certain final point at a given time, in this case, can be modeled as an arrow that spins rapidly until it is evaluated when the photon reaches its final point. For example, for the probability that a photon will reflect off of a mirror and be observed at a given point a given amount of time later, one sets the photon's probability amplitude spinning as it leaves the source, follows it to the mirror, and then to its final point, even for paths that do not involve bouncing off of the mirror at equal angles. One can then evaluate the probability amplitude at the photon's final point; next, one can integrate over all of these arrows (see vector sum), and square the length of the result to obtain the probability that this photon will reflect off of the mirror in the pertinent fashion. The times these paths take are what determines the angle of the probability amplitude arrow, as they can be said to "spin" at a constant rate (which is related to the frequency of the photon).

The times of the paths near the classical reflection site of the mirror are nearly the same, so the probability amplitudes point in nearly the same direction—thus, they have a sizable sum. Examining the paths towards the edges of the mirror reveals that the times of nearby paths are quite different from each other, and thus we wind up summing vectors that cancel out quickly. So, there is a higher probability that light will follow a near-classical reflection path than a path further out. However, a diffraction grating can be made out of this mirror, by scraping away areas near the edge of the mirror that usually cancel nearby amplitudes out—but now, since the photons don't reflect from the scraped-off portions, the probability amplitudes that would all point, for instance, at forty-five degrees, can have a sizable sum. Thus, this lets light of the right frequency sum to a larger probability amplitude, and as such possess a larger probability of reaching the appropriate final point.

This particular description involves many simplifications: a point source, a "surface" that light can reflect off of (thus neglecting the interactions with electrons) and so forth. The biggest simplification is perhaps in the fact that the "spinning" of the probability amplitude arrows is actually more accurately explained as a "spinning" of the source, as the probability amplitudes of photons do not "spin" while they are in transit. We obtain the same variation in probability amplitudes by letting the time at which the photon left the source be indeterminate—and the time of the path now tells us when the photon would have left the source, and thus what the angle of its "arrow" would be. However, this model and approximation is a reasonable one to illustrate a diffraction grating conceptually. Light of a different frequency may also reflect off of the same diffraction grating, but with a different final point.

Gratings as dispersive elements
The wavelength dependence in the grating equation shows that the grating separates an incident polychromatic beam into its constituent wavelength components at different angles, i.e., it is angular dispersive. Each wavelength of input beam spectrum is sent into a different direction, producing a rainbow of colors under white light illumination. This is visually similar to the operation of a prism, although the mechanism is very different. A prism refracts waves of different wavelengths at different angles due to their different refractive indices, while a grating diffracts different wavelengths at different angles due to interference at each wavelength.

The diffracted beams corresponding to consecutive orders may overlap, depending on the spectral content of the incident beam and the grating density. The higher the spectral order, the greater the overlap into the next order.

The grating equation shows that the angles of the diffracted orders only depend on the grooves' period, and not on their shape. By controlling the cross-sectional profile of the grooves, it is possible to concentrate most of the diffracted optical energy in a particular order for a given wavelength. A triangular profile is commonly used. This technique is called blazing. The incident angle and wavelength for which the diffraction is most efficient (the ratio of the diffracted optical energy to the incident energy is the highest) are often called blazing angle and blazing wavelength. The efficiency of a grating may also depend on the polarization of the incident light. Gratings are usually designated by their groove density, the number of grooves per unit length, usually expressed in grooves per millimeter (g/mm), also equal to the inverse of the groove period. The groove period must be on the order of the wavelength of interest; the spectral range covered by a grating is dependent on groove spacing and is the same for ruled and holographic gratings with the same grating constant (meaning groove density or the groove period). The maximum wavelength that a grating can diffract is equal to twice the grating period, in which case the incident and diffracted light are at ninety degrees (90°) to the grating normal. To obtain frequency dispersion over a wider frequency one must use a prism. The optical regime, in which the use of gratings is most common, corresponds to wavelengths between 100 nm and 10 µm. In that case, the groove density can vary from a few tens of grooves per millimeter, as in echelle gratings, to a few thousands of grooves per millimeter.

When groove spacing is less than half the wavelength of light, the only present order is the m = 0 order. Gratings with such small periodicity (with respect to the incident light wavelength) are called subwavelength gratings and exhibit special optical properties. Made on an isotropic material the subwavelength gratings give rise to form birefringence, in which the material behaves as if it were birefringent.

SR (Surface Relief) gratings
SR gratings are named due to its surface structure of depressions (low relief) and elevations (high relief). Originally, high-resolution gratings were ruled by high-quality ruling engines whose construction was a large undertaking. Henry Joseph Grayson designed a machine to make diffraction gratings, succeeding with one of 120,000 lines to the inch (approx. 4,724 lines per mm) in 1899. Later, photolithographic techniques created gratings via holographic interference patterns. A holographic grating has sinusoidal grooves as the result of an optical sinusoidal interference pattern on the grating material during its fabrication, and may not be as efficient as ruled gratings, but are often preferred in monochromators because they produce less stray light. A copying technique can make high quality replicas from master gratings of either type, thereby lowering fabrication costs.

Semiconductor technology today is also used to etch holographically patterned gratings into robust materials such as fused silica. In this way, low stray-light holography is combined with the high efficiency of deep, etched transmission gratings, and can be incorporated into high-volume, low-cost semiconductor manufacturing technology.

VPH (Volume Phase Holography) gratings
Another method for manufacturing diffraction gratings uses a photosensitive gel sandwiched between two substrates. A holographic interference pattern exposes the gel, which is later developed. These gratings, called volume phase holography diffraction gratings (or VPH diffraction gratings) have no physical grooves, but instead a periodic modulation of the refractive index within the gel. This removes much of the surface scattering effects typically seen in other types of gratings. These gratings also tend to have higher efficiencies, and allow for the inclusion of complicated patterns into a single grating. A VPH diffraction grating is typically a transmission grating, through which incident light passes and is diffracted, but a VPH reflection grating can also be made by tilting the direction of a refractive index modulation with respect to the grating surface. In older versions of such gratings, environmental susceptibility was a trade-off, as the gel had to be contained at low temperature and humidity. Typically, the photosensitive substances are sealed between two substrates that make them resistant to humidity, and thermal and mechanical stresses. VPH diffraction gratings are not destroyed by accidental touches and are more scratch resistant than typical relief gratings.

Blazed gratings
A blazed grating is manufactured with grooves that have a sawtooth-shaped cross section, unlike the symmetrical grooves of other gratings. This allows the grating to achieve maximum diffraction efficiency, but in only one diffraction order which is dependent on the angle of the sawtooth grooves, known as the blaze angle. Common uses include specific wavelength selection for tunable lasers, among others.

Other gratings
A new technology for grating insertion into integrated photonic lightwave circuits is digital planar holography (DPH). DPH gratings are generated in computer and fabricated on one or several interfaces of an optical waveguide planar by using standard micro-lithography or nano-imprinting methods, compatible with mass-production. Light propagates inside the DPH gratings, confined by the refractive index gradient, which provides longer interaction path and greater flexibility in light steering.

Examples

Diffraction gratings are often used in monochromators, spectrometers, lasers, wavelength division multiplexing devices, optical pulse compressing devices, interferometers, and many other optical instruments.

Ordinary pressed CD and DVD media are every-day examples of diffraction gratings and can be used to demonstrate the effect by reflecting sunlight off them onto a white wall. This is a side effect of their manufacture, as one surface of a CD has many small pits in the plastic, arranged in a spiral; that surface has a thin layer of metal applied to make the pits more visible. The structure of a DVD is optically similar, although it may have more than one pitted surface, and all pitted surfaces are inside the disc.

Due to the sensitivity to the refractive index of the media, diffraction grating can be used as sensor of fluid properties.

In a standard pressed vinyl record when viewed from a low angle perpendicular to the grooves, a similar but less defined effect to that in a CD/DVD is seen. This is due to viewing angle (less than the critical angle of reflection of the black vinyl) and the path of the light being reflected due to this being changed by the grooves, leaving a rainbow relief pattern behind.

Diffraction gratings are also used to distribute evenly the frontlight of e-readers such as the Nook Simple Touch with GlowLight.

Gratings from electronic components

Some everyday electronic components contain fine and regular patterns, and as a result readily serve as diffraction gratings. For example, CCD sensors from discarded mobile phones and cameras can be removed from the device. With a laser pointer, diffraction can reveal the spatial structure of the CCD sensors. This can be done for LCD or LED displays of smart phones as well. Because such displays are usually protected just by transparent casing, experiments can be done without damaging the phones. If accurate measurements are not intended, a spotlight can reveal the diffraction patterns.

Natural gratings
Striated muscle is the most commonly found natural diffraction grating and, this has helped physiologists in determining the structure of such muscle. Aside from this, the chemical structure of crystals can be thought of as diffraction gratings for types of electromagnetic radiation other than visible light, this is the basis for techniques such as X-ray crystallography.

Most commonly confused with diffraction gratings are the iridescent colors of peacock feathers, mother-of-pearl, and butterfly wings. Iridescence in birds, fish and insects is often caused by thin-film interference rather than a diffraction grating. Diffraction produces the entire spectrum of colors as the viewing angle changes, whereas thin-film interference usually produces a much narrower range. The surfaces of flowers can also create a diffraction, but the cell structures in plants are usually too irregular to produce the fine slit geometry necessary for a diffraction grating. The iridescence signal of flowers is thus only appreciable very locally and hence not visible to man and flower visiting insects. However, natural gratings do occur in some invertebrate animals, like the peacock spiders, the antennae of seed shrimp, and have even been discovered in Burgess Shale fossils. Diffraction grating effects are sometimes seen in meteorology. Diffraction coronas are colorful rings surrounding a source of light, such as the sun. These are usually observed much closer to the light source than halos, and are caused by very fine particles, like water droplets, ice crystals, or smoke particles in a hazy sky. When the particles are all nearly the same size they diffract the incoming light at very specific angles. The exact angle depends on the size of the particles. Diffraction coronas are commonly observed around light sources, like candle flames or street lights, in the fog. Cloud iridescence is caused by diffraction, occurring along coronal rings when the particles in the clouds are all uniform in size.

title: Diffraction grating


text:
Digital filter

In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is typically an electronic circuit operating on continuous-time analog signals.

A digital filter system usually consists of an analog-to-digital converter (ADC) to sample the input signal, followed by a microprocessor and some peripheral components such as memory to store data and filter coefficients etc. Program Instructions (software) running on the microprocessor implement the digital filter by performing the necessary mathematical operations on the numbers received from the ADC. In some high performance applications, an FPGA or ASIC is used instead of a general purpose microprocessor, or a specialized digital signal processor (DSP) with specific paralleled architecture for expediting operations such as filtering.

Digital filters may be more expensive than an equivalent analog filter due to their increased complexity, but they make practical many designs that are impractical or impossible as analog filters. Digital filters can often be made very high order, and are often finite impulse response filters, which allows for linear phase response. When used in the context of real-time analog systems, digital filters sometimes have problematic latency (the difference in time between the input and the response) due to the associated analog-to-digital and digital-to-analog conversions and anti-aliasing filters, or due to other delays in their implementation.

Digital filters are commonplace and an essential element of everyday electronics such as radios, cellphones, and AV receivers.

Characterization
A digital filter is characterized by its transfer function, or equivalently, its difference equation. Mathematical analysis of the transfer function can describe how it will respond to any input. As such, designing a filter consists of developing specifications appropriate to the problem (for example, a second-order low pass filter with a specific cut-off frequency), and then producing a transfer function which meets the specifications.

The transfer function for a linear, time-invariant, digital filter can be expressed as a transfer function in the Z-domain; if it is causal, then it has the form:

 * $$H(z) = \frac{B(z)}{A(z)} = \frac$$

where the order of the filter is the greater of N or M. See Z-transform's LCCD equation for further discussion of this transfer function.

This is the form for a recursive filter, which typically leads to an infinite impulse response (IIR) behaviour, but if the denominator is made equal to unity, i.e. no feedback, then this becomes a finite impulse response (FIR) filter.

Analysis techniques
A variety of mathematical techniques may be employed to analyze the behavior of a given digital filter. Many of these analysis techniques may also be employed in designs, and often form the basis of a filter specification.

Typically, one characterizes filters by calculating how they will respond to a simple input such as an impulse. One can then extend this information to compute the filter's response to more complex signals.

Impulse response
The impulse response, often denoted $$h[k]$$ or $$h_k$$, is a measurement of how a filter will respond to the Kronecker delta function. For example, given a difference equation, one would set $$x_0 = 1$$ and $$x_k = 0$$ for $$k \ne 0$$ and evaluate. The impulse response is a characterization of the filter's behaviour. Digital filters are typically considered in two categories: infinite impulse response (IIR) and finite impulse response (FIR). In the case of linear time-invariant FIR filters, the impulse response is exactly equal to the sequence of filter coefficients, and thus:

 * $$\ y_n= \sum_{k=0}^{N} b_{k} x_{n-k} =\sum_{k=0}^{N} h_{k} x_{n-k}$$

IIR filters on the other hand are recursive, with the output depending on both current and previous inputs as well as previous outputs. The general form of an IIR filter is thus:
 * $$\ \sum_{m=0}^{M} a_{m}y_{n-m} = \sum_{k=0}^{N} b_{k} x_{n-k}$$

Plotting the impulse response reveals how a filter responds to a sudden, momentary disturbance. An IIR filter is always recursive. While it is possible for a recursive filter to have a finite impulse response, a non-recursive filter always has a finite impulse response. An example is the moving average (MA) filter, which can be implemented both recursively and non recursively.

Difference equation
In discrete-time systems, the digital filter is often implemented by converting the transfer function to a linear constant-coefficient difference equation (LCCD) via the Z-transform. The discrete frequency-domain transfer function is written as the ratio of two polynomials. For example:

 * $$H(z) = \frac{(z+1)^2} {(z-\frac{1}{2}) (z+\frac{3}{4})}$$

This is expanded:
 * $$H(z) = \frac{z^2+ 2z +1} {z^2 +\frac{1}{4} z - \frac{3}{8}}$$

and to make the corresponding filter causal, the numerator and denominator are divided by the highest order of $$z$$:

H(z) = \frac{1 + 2z^{-1} +z^{-2}} {1 +\frac{1}{4} z^{-1} - \frac{3}{8} z^{-2}} = \frac{Y(z)}{X(z)} $$

The coefficients of the denominator, $$a_{k}$$, are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients, $$b_{k}$$. The resultant linear difference equation is:

y[n] = -\sum_{k=1}^{M} a_{k} y[n-k] + \sum_{k=0}^{N} b_{k} x[n-k] $$

or, for the example above:

\frac{Y(z)}{X(z)} = \frac{1 + 2z^{-1} +z^{-2}} {1 +\frac{1}{4} z^{-1} - \frac{3}{8} z^{-2}} $$

rearranging terms:

\Rightarrow (1 +\frac{1}{4} z^{-1} - \frac{3}{8} z^{-2}) Y(z) = (1 + 2z^{-1} +z^{-2}) X(z) $$

then by taking the inverse z-transform:

\Rightarrow y[n] + \frac{1}{4} y[n-1] - \frac{3}{8} y[n-2] = x[n] + 2x[n-1] + x[n-2] $$

and finally, by solving for $$y[n]$$:

y[n] = - \frac{1}{4} y[n-1] + \frac{3}{8} y[n-2] + x[n] + 2x[n-1] + x[n-2] $$

This equation shows how to compute the next output sample, $$y[n]$$, in terms of the past outputs, $$y[n-p]$$, the present input, $$x[n]$$, and the past inputs, $$x[n-p]$$. Applying the filter to an input in this form is equivalent to a Direct Form I or II (see below) realization, depending on the exact order of evaluation.

In plain terms, for example, as used by a computer programmer implementing the above equation in code, it can be described as follows:

$$y$$ = the output, or filtered value

$$x$$ = the input, or incoming raw value

$$n$$ = the sample number, iteration number, or time period number

and therefore:

$$y[n]$$ = the current filtered (output) value

$$y[n-1]$$ = the last filtered (output) value

$$y[n-2]$$ = the 2nd-to-last filtered (output) value

$$x[n]$$ = the current raw input value

$$x[n-1]$$ = the last raw input value

$$x[n-2]$$ = the 2nd-to-last raw input value

Filter design
Although filters are easily understood and calculated, the practical challenges of their design and implementation are significant and are the subject of much advanced research.

There are two categories of digital filter: the recursive filter and the nonrecursive filter. These are often referred to as infinite impulse response (IIR) filters and finite impulse response (FIR) filters, respectively.

Filter realization
After a filter is designed, it must be realized by developing a signal flow diagram that describes the filter in terms of operations on sample sequences.

A given transfer function may be realized in many ways. Consider how a simple expression such as $$ax + bx + c$$ could be evaluated – one could also compute the equivalent $$x(a + b) + c$$. In the same way, all realizations may be seen as "factorizations" of the same transfer function, but different realizations will have different numerical properties. Specifically, some realizations are more efficient in terms of the number of operations or storage elements required for their implementation, and others provide advantages such as improved numerical stability and reduced round-off error. Some structures are better for fixed-point arithmetic and others may be better for floating-point arithmetic.

Direct form I
A straightforward approach for IIR filter realization is direct form I, where the difference equation is evaluated directly. This form is practical for small filters, but may be inefficient and impractical (numerically unstable) for complex designs. In general, this form requires 2N delay elements (for both input and output signals) for a filter of order N.

Direct form II
The alternate direct form II only needs N delay units, where N is the order of the filter – potentially half as much as direct form I. This structure is obtained by reversing the order of the numerator and denominator sections of Direct Form I, since they are in fact two linear systems, and the commutativity property applies. Then, one will notice that there are two columns of delays ($$z^{-1}$$) that tap off the center net, and these can be combined since they are redundant, yielding the implementation as shown below.

The disadvantage is that direct form II increases the possibility of arithmetic overflow for filters of high Q or resonance. It has been shown that as Q increases, the round-off noise of both direct form topologies increases without bounds. This is because, conceptually, the signal is first passed through an all-pole filter (which normally boosts gain at the resonant frequencies) before the result of that is saturated, then passed through an all-zero filter (which often attenuates much of what the all-pole half amplifies).

Cascaded second-order sections
A common strategy is to realize a higher-order (greater than 2) digital filter as a cascaded series of second-order "biquadratric" (or "biquad") sections (see digital biquad filter). The advantage of this strategy is that the coefficient range is limited. Cascading direct form II sections results in N delay elements for filters of order N. Cascading direct form I sections results in N + 2 delay elements, since the delay elements of the input of any section (except the first section) are redundant with the delay elements of the output of the preceding section.

Other forms
Other forms include:
 * Direct form I and II transpose
 * Series/cascade lower (typical second) order subsections
 * Parallel lower (typical second) order subsections
 * Continued fraction expansion
 * Lattice and ladder
 * One, two and three-multiply lattice forms
 * Three and four-multiply normalized ladder forms
 * ARMA structures
 * State-space structures:
 * optimal (in the minimum noise sense): $$(N+1)^2$$ parameters
 * block-optimal and section-optimal: $$4N-1$$ parameters
 * input balanced with Givens rotation: $$4N-1$$ parameters
 * Coupled forms: Gold Rader (normal), State Variable (Chamberlin), Kingsbury, Modified State Variable, Zölzer, Modified Zölzer
 * Wave Digital Filters (WDF)
 * Agarwal–Burrus (1AB and 2AB)
 * Harris–Brooking
 * ND-TDL
 * Multifeedback
 * Analog-inspired forms such as Sallen-key and state variable filters
 * Systolic arrays

Comparison of analog and digital filters
Digital filters are not subject to the component non-linearities that greatly complicate the design of analog filters. Analog filters consist of imperfect electronic components, whose values are specified to a limit tolerance (e.g. resistor values often have a tolerance of ±5%) and which may also change with temperature and drift with time. As the order of an analog filter increases, and thus its component count, the effect of variable component errors is greatly magnified. In digital filters, the coefficient values are stored in computer memory, making them far more stable and predictable.

Because the coefficients of digital filters are definite, they can be used to achieve much more complex and selective designs – specifically with digital filters, one can achieve a lower passband ripple, faster transition, and higher stopband attenuation than is practical with analog filters. Even if the design could be achieved using analog filters, the engineering cost of designing an equivalent digital filter would likely be much lower. Furthermore, one can readily modify the coefficients of a digital filter to make an adaptive filter or a user-controllable parametric filter. While these techniques are possible in an analog filter, they are again considerably more difficult.

Digital filters can be used in the design of finite impulse response filters. Equivalent analog filters are often more complicated, as these require delay elements.

Digital filters rely less on analog circuitry, potentially allowing for a better signal-to-noise ratio. A digital filter will introduce noise to a signal during analog low pass filtering, analog to digital conversion, digital to analog conversion and may introduce digital noise due to quantization. With analog filters, every component is a source of thermal noise (such as Johnson noise), so as the filter complexity grows, so does the noise.

However, digital filters do introduce a higher fundamental latency to the system. In an analog filter, latency is often negligible; strictly speaking it is the time for an electrical signal to propagate through the filter circuit. In digital systems, latency is introduced by delay elements in the digital signal path, and by analog-to-digital and digital-to-analog converters that enable the system to process analog signals.

In very simple cases, it is more cost effective to use an analog filter. Introducing a digital filter requires considerable overhead circuitry, as previously discussed, including two low pass analog filters.

Another argument for analog filters is low power consumption. Analog filters require substantially less power and are therefore the only solution when power requirements are tight.

When making an electrical circuit on a PCB it is generally easier to use a digital solution, because the processing units are highly optimized over the years. Making the same circuit with analog components would take up a lot more space when using discrete components. Two alternatives are FPAAs and ASICs, but they are expensive for low quantities.

Types of digital filters
There are various ways to characterize filters; for example:
 * A linear filter is a linear transformation of input samples; other filters are nonlinear. Linear filters satisfy the superposition principle, i.e. if an input is a weighted linear combination of different signals, the output is a similarly weighted linear combination of the corresponding output signals.
 * A causal filter uses only previous samples of the input or output signals; while a non-causal filter uses future input samples. A non-causal filter can usually be changed into a causal filter by adding a delay to it.
 * A time-invariant filter has constant properties over time; other filters such as adaptive filters change in time.
 * A stable filter produces an output that converges to a constant value with time, or remains bounded within a finite interval. An unstable filter can produce an output that grows without bounds, with bounded or even zero input.
 * A finite impulse response (FIR) filter uses only the input signals, while an infinite impulse response (IIR) filter uses both the input signal and previous samples of the output signal. FIR filters are always stable, while IIR filters may be unstable.

A filter can be represented by a block diagram, which can then be used to derive a sample processing algorithm to implement the filter with hardware instructions. A filter may also be described as a difference equation, a collection of zeros and poles or an impulse response or step response.

Some digital filters are based on the fast Fourier transform, a mathematical algorithm that quickly extracts the frequency spectrum of a signal, allowing the spectrum to be manipulated (such as to create very high order band-pass filters) before converting the modified spectrum back into a time-series signal with an inverse FFT operation. These filters give O(n log n) computational costs whereas conventional digital filters tend to be O(n2).

Another form of a digital filter is that of a state-space model. A well used state-space filter is the Kalman filter published by Rudolf Kálmán in 1960.

Traditional linear filters are usually based on attenuation. Alternatively nonlinear filters can be designed, including energy transfer filters, which allow the user to move energy in a designed way so that unwanted noise or effects can be moved to new frequency bands either lower or higher in frequency, spread over a range of frequencies, split, or focused. Energy transfer filters complement traditional filter designs and introduce many more degrees of freedom in filter design. Digital energy transfer filters are relatively easy to design and to implement and exploit nonlinear dynamics.

title: Digital filter


text:
Digital milliwatt

In digital telephony, the digital milliwatt is a standard test signal that serves as a reference for analog signal levels in the telecommunications network. When decoding the digital milliwatt, a PCM decoder produces a sinusoidal signal with a frequency of $1 kHz$ with one milliwatt in power ($0 dBm$, a reference for dBm0).

The digital milliwatt signal is encoded by eight 8-bit words corresponding to one pulse-code modulated cycle of the signal, sampled 8000 times per second. It is typically stored in read-only memory (ROM) in the telecommunication equipment.

The digital milliwatt signal is often generated in instruments in place of separate test equipment. It has the advantage of being tied in frequency and amplitude to the relatively stable digital clock signal and power (voltage) supply, respectively, that are used by the digital channel bank.

title: Digital milliwatt


text:
Digital multiplex hierarchy

In telecommunications, a digital multiplex hierarchy is a hierarchy consisting of an ordered repetition of tandem digital multiplexers that produce signals of successively higher data rates at each level of the hierarchy.

Digital multiplexing hierarchies may be implemented in many different configurations depending on; (a) the number of channels desired, (b) the signaling system to be used, and (c) the bit rate allowed by the communications media.

Some currently available digital multiplexers have been designated as Dl-, DS-, or M-series, all of which operate at T-carrier rates.

In the design of digital multiplex hierarchies, care must be exercised to ensure interoperability of the multiplexers used in the hierarchy.

Digroup
Digroup is an abbreviation for digital group. In telephony, a basic group in the digital multiplex hierarchy.

In the North American and Japanese T-carrier digital hierarchies, each digroup supports 12 PCM voice channels or their equivalent in other services. The DS1 line rate (2 digroups plus overhead bits) is 1.544 Mbit/s, supporting 24 voice channels or their equivalent in other services.

In the E-carrier European hierarchy, each digroup supports 15 PCM channels or their equivalent in other services. The DS1 line rate (2 digroups plus overhead bits) is 2.048 Mbit/s, supporting 30 voice channels or their equivalent in other services.

title: Digital multiplex hierarchy


text:
Digital Signal 0

Digital Signal 0 (DS0) is a basic digital signaling rate of 64 kilobits per second (kbit/s), corresponding to the capacity of one analog voice-frequency-equivalent communication channel. The DS0 rate, and its equivalents E0 in the E-carrier system and T0 in the T-carrier system, form the basis for the digital multiplex transmission hierarchy in telecommunications systems used in North America, Europe, Japan, and the rest of the world, for both the early plesiochronous systems such as T-carrier and for modern synchronous systems such as SDH/SONET.

The DS0 rate was introduced to carry a single digitized voice call. For a typical phone call, the audio sound is digitized at an 8 kHz sample rate, or 8000 samples per second, using 8-bit pulse-code modulation for each of the samples. This results in a data rate of 64 kbit/s.

Because of its fundamental role in carrying a single phone call, the DS0 rate forms the basis for the digital multiplex transmission hierarchy in telecommunications systems used in North America. To limit the number of wires required between two involved in exchanging voice calls, a system was built in which multiple DS0s are multiplexed together on higher capacity circuits. In this system, twenty-four (24) DS0s are multiplexed into a DS1 signal. Twenty-eight (28) DS1s are multiplexed into a DS3. When carried over copper wire, this is the well-known T-carrier system, with T1 and T3 corresponding to DS1 and DS3, respectively.

Besides its use for voice communications, the DS0 rate may support twenty 2.4 kbit/s channels, ten 4.8 kbit/s channels, five 9.67 kbit/s channels, one 56 kbit/s channel, or one 64 kbit/s clear channel.

E0 (standardized as ITU G.703) is the European equivalent of the North American DS0 for carrying a single voice call. However, there are some subtle differences in implementation. Voice signals are encoded for carriage over E0 according to ITU G.711. Note that when a T-carrier system is used as in North America, robbed bit signaling can mean that a DS0 channel carried over that system is not an error-free bit-stream. The out-of-band signaling used in the European E-carrier system avoids this.

title: Digital Signal 0