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Comparator

In electronics, a comparator is a device that compares two voltages or currents and outputs a digital signal indicating which is larger. It has two analog input terminals $$V_+$$ and $$V_-$$ and one binary digital output $$V_\text{o}$$. The output is ideally
 * $$V_\text{o} =

\begin{cases} 1, & \text{if }V_+ > V_-, \\ 0, & \text{if }V_+ < V_-. \end{cases}$$ A comparator consists of a specialized high-gain differential amplifier. They are commonly used in devices that measure and digitize analog signals, such as analog-to-digital converters (ADCs), as well as relaxation oscillators.

Differential voltage
The differential voltages must stay within the limits specified by the manufacturer. Early integrated comparators, like the LM111 family, and certain high-speed comparators like the LM119 family, require differential voltage ranges substantially lower than the power-supply voltages (±15 V vs. 36 V). Rail-to-rail comparators allow any differential voltages within the power-supply range. When powered from a bipolar (dual rail) supply,
 * $$V_{S-} \le V_+, V_- \le V_{S+},$$

or when powered from an unipolar TTL/CMOS power supply,

 * $$0 \le V_+, V_- \le V_\text{cc}$$.

Specific rail-to-rail comparators with p–n–p input transistors, like the LM139 family, allow the input potential to drop 0.3 volts below the negative supply rail, but do not allow it to rise above the positive rail. Specific ultra-fast comparators, like the LMH7322, allow the input signal to swing below the negative rail and above the positive rail, although by a narrow margin of only 0.2 V. Differential input voltage (the voltage between two inputs) of a modern rail-to-rail comparator is usually limited only by the full swing of power supply.

Op-amp voltage comparator

An operational amplifier (op-amp) has a well balanced difference input and a very high gain. This parallels the characteristics of comparators and can be substituted in applications with low-performance requirements.

A comparator circuit compares two voltages and outputs either a 1 (the voltage at the plus side) or a 0 (the voltage at the negative side) to indicate which is larger. Comparators are often used, for example, to check whether an input has reached some predetermined value. In most cases a comparator is implemented using a dedicated comparator IC, but op-amps may be used as an alternative. Comparator diagrams and op-amp diagrams use the same symbols.

A simple comparator circuit made using an op-amp without feedback simply heavily amplifies the voltage difference between Vin and VREF and outputs the result as Vout. If Vin is greater than VREF, then voltage at Vout will rise to its positive saturation level; that is, to the voltage at the positive side. If Vin is lower than VREF, then Vout will fall to its negative saturation level, equal to the voltage at the negative side.

In practice, this circuit can be improved by incorporating a hysteresis voltage range to reduce its sensitivity to noise.

Because of the difference in characteristics of an operational amplifier and comparator, using an operational amplifier as a comparator presents several disadvantages as compared to using a dedicated comparator.

 * 1) Op-amps are designed to operate in the linear mode with negative feedback. Hence, an op-amp typically has a lengthy recovery time from saturation. Almost all op-amps have an internal compensation capacitor which imposes slew rate limitations for high frequency signals. Consequently, an op-amp makes a sloppy comparator with propagation delays that can be as long as tens of microseconds.
 * 2) Since op-amps do not have any internal hysteresis, an external hysteresis network is always necessary for slow moving input signals.
 * 3) The quiescent current specification of an op-amp is valid only when the feedback is active. Some op-amps show an increased quiescent current when the inputs are not equal.
 * 4) A comparator is designed to produce well-limited output voltages that easily interface with digital logic. Compatibility with digital logic must be verified while using an op-amp as a comparator.
 * 5) Some multiple-section op-amps may exhibit extreme channel-channel interaction when used as comparators.
 * 6) Many op-amps have back to back diodes between their inputs. Op-amp inputs usually follow each other so this is fine. But comparator inputs are not usually the same. The diodes can cause unexpected current through inputs.

Design
A comparator consists of a high gain differential amplifier whose output is compatible with the logic gates used in the digital circuit. The gain is high enough that a very small difference between the input voltages will saturate the output, the output voltage will be in either the low logic voltage band or the high logic voltage band of the gate input. Analogue op amps have been used as comparators, however a dedicated comparator chip will generally be faster than a general-purpose operational amplifier used as a comparator, and may also contain additional features such as an accurate, internal reference voltage, adjustable hysteresis, and a clock gated input.

A dedicated voltage comparator chip such as LM339 is designed to interface with a digital logic interface (to a TTL or a CMOS). The output is a binary state often used to interface real world signals to digital circuitry (see analog-to-digital converter). If there is a fixed voltage source from, for example, a DC adjustable device in the signal path, a comparator is just the equivalent of a cascade of amplifiers. When the voltages are nearly equal, the output voltage will not fall into one of the logic levels, thus analog signals will enter the digital domain with unpredictable results. To make this range as small as possible, the amplifier cascade is high gain. The circuit consists of mainly bipolar transistors. For very high frequencies, the input impedance of the stages is low. This reduces the saturation of the slow, large p–n junction bipolar transistors that would otherwise lead to long recovery times. Fast small Schottky diodes, like those found in binary logic designs, improve the performance significantly though the performance still lags that of circuits with amplifiers using analog signals. Slew rate has no meaning for these devices. For applications in flash ADCs the distributed signal across eight ports matches the voltage and current gain after each amplifier, and resistors then behave as level-shifters.

Open collector output
Some comparators (e.g. LM339) use open collector output to help interface to different logic families. When the inverting input is at a higher voltage than the non inverting input, the output of the comparator connects to the negative power supply. When the non inverting input is higher than the inverting input, the output is high impedance, so the output voltage in this state can be set by an external pull-up resistor to a different voltage supply.

Key specifications
While it is easy to understand the basic task of a comparator, that is, comparing two voltages or currents, several parameters must be considered while selecting a suitable comparator:

Speed and power
While in general comparators are "fast," their circuits are not immune to the classic speed-power tradeoff. High speed comparators use transistors with larger aspect ratios and hence also consume more power. Depending on the application, select either a comparator with high speed or one that saves power. For example, nano-powered comparators in space-saving chip-scale packages (UCSP), DFN or SC70 packages such as MAX9027, LTC1540, LPV7215, MAX9060, and MCP6541, are ideal for ultra-low-power, portable applications. Likewise if a comparator is needed to implement a relaxation oscillator circuit to create a high speed clock signal then comparators having few nano seconds of propagation delay may be suitable. ADCMP572 (CML output), LMH7220 (LVDS Output), MAX999 (CMOS output / TTL output), LT1719 (CMOS output / TTL output), MAX9010 (TTL output), and MAX9601 (PECL output), are examples of some good high speed comparators.

Hysteresis
A comparator normally changes its output state when the voltage between its inputs crosses through approximately zero volts. Small voltage fluctuations due to noise, always present on the inputs, can cause undesirable rapid changes between the two output states when the input voltage difference is near zero volts. To prevent this output oscillation, a small hysteresis of a few millivolts is integrated into many modern comparators. For example, the LTC6702, MAX9021, and MAX9031, have internal hysteresis desensitizing them from input noise. In place of one switching point, hysteresis introduces two: one for rising voltages, and one for falling voltages. The difference between the higher-level trip value (VTRIP+) and the lower-level trip value (VTRIP-) equals the hysteresis voltage (VHYST).

If the comparator does not have internal hysteresis or if the input noise is greater than the internal hysteresis then an external hysteresis network can be built using positive feedback from the output to the non-inverting input of the comparator. The resulting Schmitt trigger circuit gives additional noise immunity and a cleaner output signal. Some comparators such as LMP7300, LTC1540, MAX931, MAX971, and ADCMP341, also provide the hysteresis control through a separate hysteresis pin. These comparators make it possible to add a programmable hysteresis without feedback or complicated equations. Using a dedicated hysteresis pin is also convenient if the source impedance is high since the inputs are isolated from the hysteresis network. When hysteresis is added then a comparator cannot resolve signals within the hysteresis band.

Output type

Because comparators have only two output states, their outputs are either near zero or near the supply voltage. Bipolar rail-to-rail comparators have a common-emitter output that produces a small voltage drop between the output and each rail. That drop is equal to the collector-to-emitter voltage of a saturated transistor. When output currents are light, output voltages of CMOS rail-to-rail comparators, which rely on a saturated MOSFET, range closer to the rail voltages than their bipolar counterparts.

On the basis of outputs, comparators can also be classified as open-drain or push–pull. Comparators with an open-drain output stage use an external pull-up resistor to a positive supply that defines the logic high level. Open-drain comparators are more suitable for mixed-voltage system design. Since the output has high impedance for logic high level, open-drain comparators can also be used to connect multiple comparators to a single bus. Push–pull output does not need a pull-up resistor and can also source current, unlike an open-drain output.

Internal reference
The most frequent application for comparators is the comparison between a voltage and a stable reference. TL431 is widely used for this purpose. Most comparator manufacturers also offer comparators in which a reference voltage is integrated on to the chip. Combining the reference and comparator in one chip not only saves space, but also draws less supply current than a comparator with an external reference. ICs with wide range of references are available such as MAX9062 (200 mV reference), LT6700 (400 mV reference), ADCMP350 (600 mV reference), MAX9025 (1.236 V reference), MAX9040 (2.048 V reference), TLV3012 (1.24 V reference), and TSM109 (2.5 V reference).

Continuous versus clocked
A continuous comparator will output either a "1" or a "0" any time a high or low signal is applied to its input and will change quickly when the inputs are updated. However, many applications only require comparator outputs at certain instances, such as in A/D converters and memory. By only strobing a comparator at certain intervals, higher accuracy and lower power can be achieved with a clocked (or dynamic) comparator structure, also called a latched comparator. Often latched comparators employ strong positive feedback for a "regeneration phase" when a clock is high, and have a "reset phase" when the clock is low. This is in contrast to a continuous comparator, which can only employ weak positive feedback since there is no reset period.

Null detectors
A null detector identifies when a given value is zero. Comparators are ideal for null detection comparison measurements, since they are equivalent to a very high gain amplifier with well-balanced inputs and controlled output limits. The null detector circuit compares two input voltages: an unknown voltage and a reference voltage, usually referred to as vu and vr. The reference voltage is usually on the non-inverting input (+), while the unknown voltage is usually on the inverting input (−). (A circuit diagram would display the inputs according to their sign with respect to the output when a particular input is greater than the other.) Unless the inputs are nearly equal (see below), the output is either positive or negative, for example ±12 V. In the case of a null detector the aim is to detect when the input voltages are nearly equal, which gives the value of the unknown voltage since the reference voltage is known.

When using a comparator as a null detector, accuracy is limited; an output of zero is given whenever the magnitude of the voltage difference multiplied by the gain of the amplifier is within the voltage limits. For example, if the gain is 106, and the voltage limits are ±6 V, then an output of zero will be given if the voltage difference is less than 6 μV. One could refer to this as a fundamental uncertainty in the measurement.

Zero-crossing detectors
For this type of detector, a comparator detects each time an ac pulse changes polarity. The output of the comparator changes state each time the pulse changes its polarity, that is the output is HI (high) for a positive pulse and LO (low) for a negative pulse squares the input signal.

Relaxation oscillator
A comparator can be used to build a relaxation oscillator. It uses both positive and negative feedback. The positive feedback is a Schmitt trigger configuration. Alone, the trigger is a bistable multivibrator. However, the slow negative feedback added to the trigger by the RC circuit causes the circuit to oscillate automatically. That is, the addition of the RC circuit turns the hysteretic bistable multivibrator into an astable multivibrator.

Level shifter
This circuit requires only a single comparator with an open-drain output as in the LM393, TLV3011, or MAX9028. The circuit provides great flexibility in choosing the voltages to be translated by using a suitable pull up voltage. It also allows the translation of bipolar ±5 V logic to unipolar 3 V logic by using a comparator like the MAX972.

Analog-to-digital converters
When a comparator performs the function of telling if an input voltage is above or below a given threshold, it is essentially performing a 1-bit quantization. This function is used in nearly all analog to digital converters (such as flash, pipeline, successive approximation, delta-sigma modulation, folding, interpolating, dual-slope and others) in combination with other devices to achieve a multi-bit quantization.

Window detectors
Comparators can also be used as window detectors. In a window detector, a comparator is used to compare two voltages and determine whether a given input voltage is under voltage or over voltage.

Absolute-value detectors
Comparators can be used to create absolute-value detectors. In an absolute-value detector, two comparators and a digital logic gate are used to compare the absolute values of two voltages.

title: Comparator


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Compatibility

Compatibility may refer to:

Computing

 * Backward compatibility, in which newer systems can understand data generated by older ones
 * Compatibility card, an expansion card for hardware emulation of another device
 * Compatibility layer, components that allow for non-native support of components
 * Compatibility mode, software mechanism in which a software emulates an older version of software
 * Computer compatibility, of a line of machines
 * IBM PC compatible, computers that are generally similar to the original IBM PC, XT, or AT
 * Forward compatibility, in which older systems can understand data generated by newer ones
 * Hardware compatibility, between different pieces of computer hardware
 * License compatibility, of software licenses
 * Pin compatibility, in devices that have the same functions assigned to the same particular pins
 * Software compatibility, between different pieces of software
 * Software incompatibility

Science and mathematics

 * Biocompatibility, a description of materials' ability to remain performant in biological tissues
 * Blood compatibility, determines what type of donor blood to use in blood transfusions
 * Compatibility, a property of splits in phylogenetic trees
 * Compatibility (chemical), how stable a substance is when mixed with another substance
 * Compatibility (geochemistry), how readily a particular trace element substitutes for a major element within a mineral
 * Compatibility (mechanics), the study of compatible deformations in continuum mechanics
 * Electromagnetic compatibility, which studies the unintentional generation, propagation, and reception of electromagnetic energy
 * Histocompatibility, the degree of similarity of cell surface proteins between individuals
 * Genetic compatibility, how well the genes of parents function together in their offspring
 * Consistency, logical compatibility between two or more propositions
 * Compatible relation, a binary relation that commutes with each operation of an algebraic structure

Other uses

 * Astrological compatibility, a branch of astrology that studies relationships by comparing natal horoscopes
 * Compatibilism, a philosophical position
 * Interpersonal compatibility, the long-term interaction between two or more individuals in terms of the ease and comfort of communication

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

title: Compatibility


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Compatible sideband transmission

A Compatible sideband transmission, also known as amplitude modulation equivalent (AME) or Single sideband reduced-carrier (SSB-RC), is a type of single sideband RF modulation in which the carrier is deliberately reinserted at a lower level after its normal suppression to permit reception by conventional AM receivers. The general convention is to filter the lower-sideband, and communicate using only the upper-sideband and a partial carrier.

The benefits of compatible-sideband over conventional AM are increased spectral efficiency due to a reduction in bandwidth of 50% as well as a decrease in wasted power. By using compatible sideband instead of AM, less RF power is required at the transmitter to transmit the same quality of signal the same distance. This results in compatible sideband being almost 100% power-efficient, where regular AM is comparably only 16% power-efficient (84% of RF power wasted).

This modulation is currently mostly used in high frequency military communications.

title: Compatible sideband transmission


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Complementary network service

In telecommunication, a complementary network service (CNS) is a means for an enhanced-service provider customer to connect to a network and to the enhanced service provider.

Complementary network services usually consist of the customer local service, such as a business or residence, and several associated service features, such as a call-forwarding service.

title: Complementary network service


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Component

Component may refer to:

Generic systems

 * System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis
 * Lumped element model, a model of spatially distributed systems

Electrical

 * Component video, a type of analog video information that is transmitted or stored as two or more separate signals
 * Electronic component, a constituent of an electronic circuit
 * Symmetrical components, in electrical engineering, analysis of unbalanced three-phase power systems

Mathematics

 * Color model, a way of describing how colors can be represented, typically as multiple values or color components
 * Component (group theory), a quasi-simple subnormal sub-group
 * Connected component (graph theory), a maximal connected subgraph
 * Connected component (topology), a maximal connected subspace of a topological space
 * Vector component, result of the decomposition of a vector into various directions

Science

 * Component (thermodynamics), a chemically independent constituent of a phase of a system

Software

 * Component (UML), definition of component in the Unified Modeling Language
 * Component-based software engineering, a field within software engineering dealing with reusable software elements
 * Software component, a reusable software element

Other uses

 * Component (VTA), a light-rail station in San Jose, California
 * Part of the grammatical structure of a sentence, a concept relating to the catena
 * Component ingredient, in a culinary dish

title: Component


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Compromise

To compromise is to make a deal between different parties where each party gives up part of their demand. In arguments, compromise means finding agreement through communication, through a mutual acceptance of terms—often involving variations from an original goal or desires. Defining and finding the best possible compromise is an important problem in fields like game theory and the voting system.

Research indicates that suboptimal compromises are often the result of negotiators failing to realize when they have interests that are completely compatible with those of the other party, leading them to settle for suboptimal agreements. Mutually better outcomes can often be found by careful investigation of both parties' interests, especially if done early in negotiations.

The compromise solution of a multicriteria decision making or multi-criteria decision analysis problem that is the closest to the ideal could be determined by the VIKOR method, which provides a maximum utility of the majority, and a minimum individual regret of the opponent.

Politics

In international politics, compromises often discussed include infamous deals with dictators, such as Neville Chamberlain's appeasement of Adolf Hitler. Margalit calls these "rotten compromises." In the United States and other democratic countries, many politicians of recent times permanently campaign to gain reelection. Thus, United States Ambassador to Germany Amy Gutmann and political scientist Dennis F. Thompson have observed that compromise is more difficult. The problem of political compromise in general is an important subject in political ethics.

Politicians being willing to compromise can reduce partisanship and hostility. Politics is sometimes called the "art of compromise". Polling by the American Survey Center indicates that Americans take a favorable view of political compromise.

Human relationships
In human relationships, "compromise" can make no party happy because the parties involved feel that they either gave away too much or that they received too little. Compromise may be referred to as capitulation, a "surrender" of objectives, principles, or material. Extremism is often considered as an antonym to compromise, which, depending on context, may be associated with concepts of balance and tolerance.

title: Compromise


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Computer conferencing

Computer conferencing may refer to:

 * Teleconference supported by one or more computers
 * Web conferencing
 * Data conferencing
 * Distributed computer applications:
 * Instant messaging
 * Online chat

title: Computer conferencing


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Concentrator

In telecommunication, the term concentrator has the following meanings:

 * In data transmission, a functional unit that permits a common path to handle more data sources than there are channels currently available within the path. A concentrator usually provides communication capability between many low-speed, usually asynchronous channels and one or more high-speed, usually synchronous channels. Usually different speeds, codes, and protocols can be accommodated on the low-speed side. The low-speed channels usually operate in contention and require buffering.
 * A device that connects a number of links with only one destination, the main function of this device is to make a kind of load balancing between two or more servers connected together, data distribution is done according to the server processing rate.
 * A patch panel or other component in the cable plant where cable runs converge.
 * ISP used concentrators to enable modem dialing; this kind of concentrator is sometimes called a modem concentrator or a remote access concentrator. The term "access concentrator" is also used to describe similar provider edge equipment used in computer networks that doesn't rely on modems anymore, e.g. FTTH.

Theory
In the evolution of modern telecommunications systems there was a requirement to connect large numbers of low-speed access devices with large telephone company 'central office' switches over common paths. During the first generations of digital networks, analog signals were digitized on line cards attached to the telephone exchange switches. In an effort to reduce local loop costs, it was decided to push this conversion closer to the customer premises by deploying small conversion devices in customer neighborhoods. These devices would combine multiple digital signals on a single link to a larger telephone switch, which would provide service to the customer. These devices were initially called remote concentrators or simply remotes.

In fibre-optic distribution systems which offer triple-play services (voice, television, internet) the digitization has arrived at the customer premises and signals are digitized at the source and combined using customer edge routers. This traffic enters the distribution network at an Optical Network Termination and is carried to the central office using Wavelength division multiplexing and Passive optical networking.

title: Concentrator


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Concentricity error

The concentricity error of an optical fiber is the distance between the center of the two concentric circles that specify the cladding diameter and the center of the two concentric circles that specify the core diameter.

The concentricity error is used in conjunction with tolerance fields to specify or characterize optical fiber core and cladding geometry.

title: Concentricity error


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Conditioning equipment

In telecommunication, the term conditioning equipment has the following meanings:

 * 1)  At junctions of circuits, equipment used to obtain desired circuit characteristics, such as matched transmission levels, matched impedances, and equalization between facilities.
 * 2)  Corrective networks used to improve data transmission, such as equalization of the insertion loss-vs.-frequency characteristic and the envelope delay distortion over a desired frequency range.

title: Conditioning equipment


text: Conducted interference

title: Conducted interference


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Configuration management

Configuration management (CM) is a systems engineering process for establishing and maintaining consistency of a product's performance, functional, and physical attributes with its requirements, design, and operational information throughout its life. The CM process is widely used by military engineering organizations to manage changes throughout the system lifecycle of complex systems, such as weapon systems, military vehicles, and information systems. Outside the military, the CM process is also used with IT service management as defined by ITIL, and with other domain models in the civil engineering and other industrial engineering segments such as roads, bridges, canals, dams, and buildings.

Introduction
CM applied over the life cycle of a system provides visibility and control of its performance, functional, and physical attributes. CM verifies that a system performs as intended, and is identified and documented in sufficient detail to support its projected life cycle. The CM process facilitates orderly management of system information and system changes for such beneficial purposes as to revise capability; improve performance, reliability, or maintainability; extend life; reduce cost; reduce risk and liability; or correct defects. The relatively minimal cost of implementing CM is returned manyfold in cost avoidance. The lack of CM, or its ineffectual implementation, can be very expensive and sometimes can have such catastrophic consequences such as failure of equipment or loss of life.

CM emphasizes the functional relation between parts, subsystems, and systems for effectively controlling system change. It helps to verify that proposed changes are systematically considered to minimize adverse effects. Changes to the system are proposed, evaluated, and implemented using a standardized, systematic approach that ensures consistency, and proposed changes are evaluated in terms of their anticipated impact on the entire system. CM verifies that changes are carried out as prescribed and that documentation of items and systems reflects their true configuration. A complete CM program includes provisions for the storing, tracking, and updating of all system information on a component, subsystem, and system basis.

A structured CM program ensures that documentation (e.g., requirements, design, test, and acceptance documentation) for items is accurate and consistent with the actual physical design of the item. In many cases, without CM, the documentation exists but is not consistent with the item itself. For this reason, engineers, contractors, and management are frequently forced to develop documentation reflecting the actual status of the item before they can proceed with a change. This reverse engineering process is wasteful in terms of human and other resources and can be minimized or eliminated using CM.

History
Configuration Management originated in the United States Department of Defense in the 1950s as a technical management discipline for hardware material items—and it is now a standard practice in virtually every industry. The CM process became its own technical discipline sometime in the late 1960s when the DoD developed a series of military standards called the "480 series" (i.e., MIL-STD-480, MIL-STD-481 and MIL-STD-483) that were subsequently issued in the 1970s. In 1991, the "480 series" was consolidated into a single standard known as the MIL–STD–973 that was then replaced by MIL–HDBK–61 pursuant to a general DoD goal that reduced the number of military standards in favor of industry technical standards supported by standards developing organizations (SDO). This marked the beginning of what has now evolved into the most widely distributed and accepted standard on CM, ANSI–EIA–649–1998. Now widely adopted by numerous organizations and agencies, the CM discipline's concepts include systems engineering (SE), Integrated Logistics Support (ILS), Capability Maturity Model Integration (CMMI), ISO 9000, Prince2 project management method, COBIT, ITIL, product lifecycle management, and Application Lifecycle Management. Many of these functions and models have redefined CM from its traditional holistic approach to technical management. Some treat CM as being similar to a librarian activity, and break out change control or change management as a separate or stand alone discipline.

Overview
CM is the practice of handling changes systematically so that a system maintains its integrity over time. CM implements the policies, procedures, techniques, and tools that manage, evaluate proposed changes, track the status of changes, and maintain an inventory of system and support documents as the system changes. CM programs and plans provide technical and administrative direction to the development and implementation of the procedures, functions, services, tools, processes, and resources required to successfully develop and support a complex system. During system development, CM allows program management to track requirements throughout the life-cycle through acceptance and operations and maintenance. As changes inevitably occur in the requirements and design, they must be approved and documented, creating an accurate record of the system status. Ideally the CM process is applied throughout the system lifecycle. Most professionals mix up or get confused with Asset management (AM, see also ISO/IEC 19770), where it inventories the assets on hand. The key difference between CM and AM is that the former does not manage the financial accounting aspect but on service that the system supports or in other words, that the later (AM) is trying to realize value from an IT asset.

The CM process for both hardware- and software-configuration items comprises five distinct disciplines as established in the MIL–HDBK–61A and in ANSI/EIA-649. Members of an organization interested in applying a standard change-management process will employ these disciplines as policies and procedures for establishing baselines, manage and control change, and monitor and assess the effectiveness and correctness of progress. The IEEE 12207 process IEEE 12207.2 also has these activities and adds "Release management and delivery".

The five disciplines are:

 * 1) CM Planning and Management: a formal document and plan to guide the CM program that includes items such as:
 * 2) * Personnel
 * 3) * Responsibilities and resources
 * 4) * Training requirements
 * 5) * Administrative meeting guidelines, including a definition of procedures and tools
 * 6) * Baselining processes
 * 7) * Configuration control and configuration-status accounting
 * 8) * Naming conventions
 * 9) * Audits and reviews
 * 10) * Subcontractor/vendor CM requirements
 * 11) Configuration Identification (CI): consists of setting and maintaining baselines, which define the system or subsystem architecture, components, and any developments at any point in time. It is the basis by which changes to any part of a system are identified, documented, and later tracked through design, development, testing, and final delivery. CI incrementally establishes and maintains the definitive current basis for Configuration Status Accounting (CSA) of a system and its configuration items (CIs) throughout their lifecycle (development, production, deployment, and operational support) until disposal.
 * 12) Configuration Control: includes the evaluation of all change-requests and change-proposals, and their subsequent approval or disapproval. It covers the process of controlling modifications to the system's design, hardware, firmware, software, and documentation.
 * 13) Configuration Status Accounting: includes the process of recording and reporting configuration item descriptions (e.g., hardware, software, firmware, etc.) and all departures from the baseline during design and production. In the event of suspected problems, the verification of baseline configuration and approved modifications can be quickly determined.
 * 14) Configuration Verification and Audit: an independent review of hardware and software for the purpose of assessing compliance with established performance requirements, commercial and appropriate military standards, and functional, allocated, and product baselines. Configuration audits verify that the system and subsystem configuration documentation complies with the functional and physical performance characteristics before acceptance into an architectural baseline.

Software
The software configuration management (SCM) process is looked upon by practitioners as the best solution to handling changes in software projects. It identifies the functional and physical attributes of software at various points in time, and performs systematic control of changes to the identified attributes for the purpose of maintaining software integrity and traceability throughout the software development life cycle.

The SCM process further defines the need to trace changes, and the ability to verify that the final delivered software has all of the planned enhancements that are supposed to be included in the release. It identifies four procedures that must be defined for each software project to ensure that a sound SCM process is implemented. They are:
 * 1) Configuration identification
 * 2) Configuration control
 * 3) Configuration status accounting
 * 4) Configuration audits

These terms and definitions change from standard to standard, but are essentially the same.
 * Configuration identification is the process of identifying the attributes that define every aspect of a configuration item. A configuration item is a product (hardware and/or software) that has an end-user purpose. These attributes are recorded in configuration documentation and baselined. Baselining an attribute forces formal configuration change control processes to be effected in the event that these attributes are changed.
 * Configuration change control is a set of processes and approval stages required to change a configuration item's attributes and to re-baseline them.
 * Configuration status accounting is the ability to record and report on the configuration baselines associated with each configuration item at any moment of time.
 * Configuration audits are broken into functional and physical configuration audits. They occur either at delivery or at the moment of effecting the change. A functional configuration audit ensures that functional and performance attributes of a configuration item are achieved, while a physical configuration audit ensures that a configuration item is installed in accordance with the requirements of its detailed design documentation.

Configuration management database
ITIL specifies the use of a Configuration management system (CMS) or Configuration management database (CMDB) as a means of achieving industry best practices for Configuration Management. CMDBs are used to track Configuration Items (CIs) and the dependencies between them, where CIs represent the things in an enterprise that are worth tracking and managing, such as but not limited to computers, software, software licenses, racks, network devices, storage, and even the components within such items.

The benefits of a CMS/CMDB includes being able to perform functions like root cause analysis, impact analysis, change management, and current state assessment for future state strategy development. Example systems, commonly identifies themselves as IT Service Management (ITSM) systems, include FreshService, ServiceNow and Samanage.

Information assurance
For information assurance, CM can be defined as the management of security features and assurances through control of changes made to hardware, software, firmware, documentation, test, test fixtures, and test documentation throughout the life cycle of an information system. CM for information assurance, sometimes referred to as Secure Configuration Management, relies upon performance, functional, and physical attributes of IT platforms and products and their environments to determine the appropriate security features and assurances that are used to measure a system configuration state. For example, configuration requirements may be different for a network firewall that functions as part of an organization's Internet boundary versus one that functions as an internal local network firewall.

Maintenance systems
Configuration management is used to maintain an understanding of the status of complex assets with a view to maintaining the highest level of serviceability for the lowest cost. Specifically, it aims to ensure that operations are not disrupted due to the asset (or parts of the asset) overrunning limits of planned lifespan or below quality levels.

In the military, this type of activity is often classed as "mission readiness", and seeks to define which assets are available and for which type of mission; a classic example is whether aircraft on board an aircraft carrier are equipped with bombs for ground support or missiles for defense.

Operating system configuration management
Configuration management can be used to maintain OS configuration files. Many of these systems utilize Infrastructure as Code to define and maintain configuration.

The Promise theory of configuration maintenance was developed by Mark Burgess,  with a practical implementation on present day computer systems in the software CFEngine able to perform real time repair as well as preventive maintenance.

Preventive maintenance
Understanding the "as is" state of an asset and its major components is an essential element in preventive maintenance as used in maintenance, repair, and overhaul and enterprise asset management systems.

Complex assets such as aircraft, ships, industrial machinery etc. depend on many different components being serviceable. This serviceability is often defined in terms of the amount of usage the component has had since it was new, since fitted, since repaired, the amount of use it has had over its life and several other limiting factors. Understanding how near the end of their life each of these components is has been a major undertaking involving labor-intensive record keeping until recent developments in software.

Predictive maintenance
Many types of component use electronic sensors to capture data which provides live condition monitoring. This data is analyzed on board or at a remote location by computer to evaluate its current serviceability and increasingly its likely future state using algorithms which predict potential future failures based on previous examples of failure through field experience and modeling. This is the basis for "predictive maintenance".

Availability of accurate and timely data is essential in order for CM to provide operational value and a lack of this can often be a limiting factor. Capturing and disseminating the operating data to the various support organizations is becoming an industry in itself.

The consumers of this data have grown more numerous and complex with the growth of programs offered by original equipment manufacturers (OEMs). These are designed to offer operators guaranteed availability and make the picture more complex with the operator managing the asset but the OEM taking on the liability to ensure its serviceability.

Standards
A number of standards support or include configuration management, including:
 * ANSI/EIA-649-1998 National Consensus Standard for Configuration Management
 * EIA-649-A 2004 National Consensus Standard for Configuration Management
 * SAE EIA-649-C 2019 Global Consensus Configuration Management Standard
 * ISO 10007 Quality management systems – Guidelines for configuration management
 * Federal Standard 1037C
 * GEIA Standard 836–2002 Configuration Management Data Exchange and Interoperability
 * IEEE 829 Standard for Software Test Documentation
 * MIL-STD-973 Configuration Management (cancelled on 20 September 2000)
 * NATO STANAG 4427 Configuration Management in Systems Life Cycle Management including
 * NATO ACMP 2000 Policy on Configuration Management
 * NATO ACMP 2009 Guidance on Configuration Management
 * NATO ACMP 2100 Configuration Management Contractual Requirements
 * CMMI CMMI for Development, Version 1.2 Configuration Management
 * CMII-100E CMII Standard for Enterprise Configuration Management
 * Extended List of Configuration Management & Related Standards
 * ITIL Service Asset and Configuration Management
 * ISO 20000:1 2011& 2018 Service Management System.
 * ECSS-M-ST-40C Rev.1 Configuration and information management
 * ECSS-M-ST-40C Rev.1 Configuration and information management

Guidelines

 * IEEE 828-2012 Standard for Configuration Management in Systems and Software Engineering, published date:2012-03-16
 * ISO 10007:2017 Quality management – Guidelines for configuration management
 * NATO ACMP-2009 – Guidance on configuration management
 * ANSI/EIA-632-1998 Processes for Engineering a System
 * ANSI/EIA-649-1998 National Consensus Standard for Configuration Management
 * GEIA-HB-649 – Implementation Guide for Configuration Management
 * EIA-836 Consensus Standard for Configuration Management Data Exchange and Interoperability
 * MIL-HDBK-61B Configuration Management Guidance, 7 April 2020
 * MIL-STD-3046 Configuration Management, 6 March 2013 and canceled on June 1, 2015
 * Defense Acquisition Guidebook, elements of CM at 4.3.7 SE Processes, attributes of CM at 5.1.7 Lifecycle support
 * Systems Engineering Fundamentals, Chapter 10 Configuration Management
 * Configuration Management Plan United States Dept. of Defense Acquisition document

Construction
More recently configuration management has been applied to large construction projects which can often be very complex and have a huge number of details and changes that need to be documented. Construction agencies such as the Federal Highway Administration have used configuration management for their infrastructure projects. There are construction-based configuration management tools that aim to document change orders and RFIs in order to ensure a project stays on schedule and on budget. These programs can also store information to aid in the maintenance and modification of the infrastructure when it is completed. One such application, CCSNet, was tested in a case study funded by the Federal Transportation Administration (FTA) in which the efficacy of configuration management was measured through comparing the approximately 80% complete construction of the Los Angeles County Metropolitan Transit Agency (LACMTA) first and second segments of the Red Line, a $5.3 billion rail construction project. This study yielded results indicating a benefit to using configuration management on projects of this nature.

title: Configuration management


text:
Congestion

Congestion may refer to:

Medicine

 * Excessive fluid in tissues, vessels, or both, including:
 * Edema, abnormal accumulation of fluid in the interstitium, manifesting as swelling
 * Peripheral edema, edema in peripheral body parts such as limbs and feet
 * Pulmonary edema, edema in the lungs that impairs breathing
 * Congestive heart failure, heart failure resulting in congestion in one or more organs
 * Nasal congestion, the blockage of nasal passages due to swollen membranes
 * Prostatic congestion, a medical condition that happens when the prostate becomes swollen by excess fluid

Other uses

 * Network congestion, reduced quality of service when a network is carrying more data than it can handle
 * Traffic congestion, vehicles clogging streets and highways
 * Congestion pricing, a system of surcharging users of public goods that are subject to congestion through excess demand
 * Congestion (film), a 1918 silent film

title: Congestion


text:
Connectionless communication

Connectionless communication, often referred to as CL-mode communication, is a data transmission method used in packet switching networks, using data packets that are frequently called datagrams, in which each data packet is individually addressed and routed based on information carried in each packet, rather than in the setup information of a prearranged, fixed data channel as in connection-oriented communication. Connectionless protocols are usually described as stateless protocols, the Internet Protocol (IP) and User Datagram Protocol (UDP) are examples.

Attributes
Under connectionless communication between two network endpoints, a message can be sent from one endpoint to another without prior arrangement. The device at one end of the communication transmits data addressed to the other, without first ensuring that the recipient is available and ready to receive the data. Some protocols allow for error correction by requesting retransmission.

Connectionless protocols are stateless protocols because the endpoints have no protocol-defined way to remember where they are in a "conversation" of message exchanges. It has lower overhead than connection-oriented communication because, in connection-oriented communication, the communicating peers must first establish a logical or physical data channel or connection in a dialog preceding the exchange of user data. It allows for multicast and broadcast operations in which the same data are transmitted to several recipients in a single transmission.

In connectionless transmissions the service provider usually cannot guarantee that there will be no loss, error insertion, misdelivery, duplication, or out-of-sequence delivery of the packet. However, the effect of errors may be reduced by implementing error correction within an application protocol.

In connectionless mode, there is less opportunity for optimization when sending several data units between the same two peers. By establishing a connection at the beginning of such a data exchange the components (routers, bridges) along the network path would be able to pre-compute (and hence cache) routing-related information, avoiding re-computation for every packet. In connection-oriented communication, network components can also reserve capacity for the transfer of the subsequent data units of a video download, for example.

Architecture and implementations
Distinction between connectionless and connection-oriented transmission may take place at several layers of the OSI Reference Model:
 * Transport layer: TCP is a connection-oriented transport protocol. UDP is connectionless.
 * Network layer.
 * Data link layer: The IEEE 802.2 protocol at the Logical Link Control sublayer of the data link layer may provide both connectionless and connection-oriented services. In fact, some network protocols (such as SNA's Path Control in its early stages) require a connection-oriented data link layer. Others (like IP) do not. (After the appearance of APPN, SNA could operate on a connectionless data link service as well.)

Notable connectionless protocols are: Internet Protocol (IP), User Datagram Protocol (UDP), Internet Control Message Protocol (ICMP), Internetwork Packet Exchange (IPX), Transparent Inter-process Communication, NetBIOS, and Fast and Secure Protocol (FASP).

title: Connectionless communication


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Connections per circuit hour

In telecommunication, the term connections per circuit hour (CCH) has the following meanings:
 * A unit of teletraffic measurement expressed as the number of connections established at a switching point per hour.
 * A unit of traffic measurement used to express the rate at which circuits are established at a switch.

The magnitude of the CCH is an instantaneous value subject to change as a function of time (i.e. from moment to moment), and is subject to study including load curve and busy hour as other measures of traffic are.

title: Connections per circuit hour


text:
Connectivity exchange

Connectivity exchange (CONEX): In an adaptive or manually operated high-frequency (HF) radio network, the automatic or manual exchange of information concerning routes to stations that are not directly reachable by the exchange originator.

The purpose of the exchange is to identify indirect paths and/or possible relay stations to those stations that are not directly reachable.

title: Connectivity exchange


text:
Contention

Contention or contentious  may refer to:

 * Contention (telecommunications), a media access method to share a broadcast medium
 * Contention, Oregon, U.S., later known as Twickenham
 * Contention City, Arizona, U.S., a ghost town
 * Bus contention, an undesirable state in computer design
 * FV4401 Contentious, a prototype British tank destroyer
 * Resource contention, in computer science, a conflict over access to a shared resource

title: Contention


text:
Continuous operation

In telecommunication, continuous operation is an operation in which certain components, such as nodes, facilities, circuits, or equipment, are in an operational state at all times. Continuous operation usually requires that there be fully redundant configuration, or at least a sufficient X out of Y degree of redundancy for compatible equipment, where X is the number of spare components and Y is the number of operational components.

title: Continuous operation


text:
Contrast

Contrast may refer to:

Science

 * Contrast (vision), the contradiction in form, colour and light between parts of an image
 * Contrast (statistics), a combination of averages whose coefficients add up to zero, or the difference between two means
 * Behavioral contrast, a phenomenon studied in psychology (behavior analysis)
 * Contrast agent, used to distinguish structures or fluids within a body, often shortened to just "contrast"

Technology

 * Contrast ratio, a measure of a display system
 * Display contrast, of electronic visual displays

Language

 * Contrast (linguistics), expressing distinctions between words
 * Contrast (literary), describing the difference(s) between two or more entities

Music

 * The Contrast (band), an English pop band formed in 1999
 * Contrast (music), the difference between parts or different instrumental sounds
 * High Contrast, Welsh music producer and artist born in 1979

Albums

 * Contrast (Tages album), 1967
 * Contrast (Conor Maynard album), 2012
 * Contrast (Matt Fax album), 2017
 * Contrast (EP), by the Features, 2006
 * Contrast, by Emi Maria, 2010
 * Contrast, by Klinik, 1992
 * Contrast, by Signal Aout 42, 1990
 * Contrast, an EP by Toru Kitajima, 2014

Other media

 * Contrast (video game), a 2013 puzzle-based platform game
 * The Contrast (play), a 1787 play by Royall Tyler
 * The Contrast (novel), an 1832 novel by Lord Normanby

title: Contrast


text:
Control communications

In telecommunication, control communications is the branch of technology devoted to the design, development, and application of communications facilities used specifically for control purposes, such as for controlling (a) industrial processes, (b) movement of resources, (c) electric power generation, distribution, and utilization, (d) communications networks, and (e) transportation systems.

title: Control communications


text:
Controlled area

In telecommunication, a controlled area is an area in which uncontrolled movement will not result in compromise of classified information, that is designed to provide administrative control and safety, or that serves as a buffer for controlling access to limited-access areas. It can also refer to an area to which security controls have been applied to protect an information-processing system's equipment and wirelines, equivalent to that required for the information transmitted through the system.

title: Controlled area


text:
Control operation

In telecommunication, a control operation (control function) is an operation that affects the recording, processing, transmission, or interpretation of data.

Examples of control operations include a font change, or a rewind; and transmitting an end-of-transmission (EOT) control character.

title: Control operation


text:
Convolutional code

In telecommunication, a convolutional code is a type of error-correcting code that generates parity symbols via the sliding application of a boolean polynomial function to a data stream. The sliding application represents the 'convolution' of the encoder over the data, which gives rise to the term 'convolutional coding'. The sliding nature of the convolutional codes facilitates trellis decoding using a time-invariant trellis. Time invariant trellis decoding allows convolutional codes to be maximum-likelihood soft-decision decoded with reasonable complexity.

The ability to perform economical maximum likelihood soft decision decoding is one of the major benefits of convolutional codes. This is in contrast to classic block codes, which are generally represented by a time-variant trellis and therefore are typically hard-decision decoded. Convolutional codes are often characterized by the base code rate and the depth (or memory) of the encoder $$[n,k,K]$$. The base code rate is typically given as $$n/k$$, where $n$ is the raw input data rate and $k$ is the data rate of output channel encoded stream. $n$ is less than $k$ because channel coding inserts redundancy in the input bits. The memory is often called the "constraint length" $K$, where the output is a function of the current input as well as the previous $$K-1$$ inputs. The depth may also be given as the number of memory elements $v$ in the polynomial or the maximum possible number of states of the encoder (typically: $$2^v$$).

Convolutional codes are often described as continuous. However, it may also be said that convolutional codes have arbitrary block length, rather than being continuous, since most real-world convolutional encoding is performed on blocks of data. Convolutionally encoded block codes typically employ termination. The arbitrary block length of convolutional codes can also be contrasted to classic block codes, which generally have fixed block lengths that are determined by algebraic properties.

The code rate of a convolutional code is commonly modified via symbol puncturing. For example, a convolutional code with a 'mother' code rate $$n/k=1/2$$ may be punctured to a higher rate of, for example, $$7/8$$ simply by not transmitting a portion of code symbols. The performance of a punctured convolutional code generally scales well with the amount of parity transmitted. The ability to perform economical soft decision decoding on convolutional codes, as well as the block length and code rate flexibility of convolutional codes, makes them very popular for digital communications.

History
Convolutional codes were introduced in 1955 by Peter Elias. It was thought that convolutional codes could be decoded with arbitrary quality at the expense of computation and delay. In 1967, Andrew Viterbi determined that convolutional codes could be maximum-likelihood decoded with reasonable complexity using time invariant trellis based decoders — the Viterbi algorithm. Other trellis-based decoder algorithms were later developed, including the BCJR decoding algorithm.

Recursive systematic convolutional codes were invented by Claude Berrou around 1991. These codes proved especially useful for iterative processing including the processing of concatenated codes such as turbo codes.

Using the "convolutional" terminology, a classic convolutional code might be considered a Finite impulse response (FIR) filter, while a recursive convolutional code might be considered an Infinite impulse response (IIR) filter.

Where convolutional codes are used

Convolutional codes are used extensively to achieve reliable data transfer in numerous applications, such as digital video, radio, mobile communications (e.g., in GSM, GPRS, EDGE and 3G networks (until 3GPP Release 7) ) and satellite communications. These codes are often implemented in concatenation with a hard-decision code, particularly Reed–Solomon. Prior to turbo codes such constructions were the most efficient, coming closest to the Shannon limit.

Convolutional encoding
To convolutionally encode data, start with k memory registers, each holding one input bit. Unless otherwise specified, all memory registers start with a value of 0. The encoder has n modulo-2 adders (a modulo 2 adder can be implemented with a single Boolean XOR gate, where the logic is: $0+0 = 0$, $0+1 = 1$, $1+0 = 1$, $1+1 = 0$), and n generator polynomials — one for each adder (see figure below). An input bit m1 is fed into the leftmost register. Using the generator polynomials and the existing values in the remaining registers, the encoder outputs n symbols. These symbols may be transmitted or punctured depending on the desired code rate. Now bit shift all register values to the right (m1 moves to m0, m0 moves to m−1) and wait for the next input bit. If there are no remaining input bits, the encoder continues shifting until all registers have returned to the zero state (flush bit termination). The figure below is a rate 1⁄3 ($m/n$) encoder with constraint length (k) of 3. Generator polynomials are $G_{1} = (1,1,1),$ $G_{2} = (0,1,1)$, and $G_{3} = (1,0,1)$. Therefore, output bits are calculated (modulo 2) as follows:
 * n1 = m1 + m0 + m−1
 * n2 = m0 + m−1
 * n3 = m1 + m−1.

Convolutional codes can be systematic and non-systematic:

 * systematic repeats the structure of the message before encoding
 * non-systematic changes the initial structure

Non-systematic convolutional codes are more popular due to better noise immunity. It relates to the free distance of the convolutional code.

Recursive and non-recursive codes
The encoder on the picture above is a non-recursive encoder. Here's an example of a recursive one and as such it admits a feedback structure:

The example encoder is systematic because the input data is also used in the output symbols (Output 2). Codes with output symbols that do not include the input data are called non-systematic.

Recursive codes are typically systematic and, conversely, non-recursive codes are typically non-systematic. It isn't a strict requirement, but a common practice.

The example encoder in Img. 2. is an 8-state encoder because the 3 registers will create 8 possible encoder states (23). A corresponding decoder trellis will typically use 8 states as well.

Recursive systematic convolutional (RSC) codes have become more popular due to their use in Turbo Codes. Recursive systematic codes are also referred to as pseudo-systematic codes.

Other RSC codes and example applications include:

Useful for LDPC code implementation and as inner constituent code for serial concatenated convolutional codes (SCCC's).

Useful for SCCC's and multidimensional turbo codes.

Useful as constituent code in low error rate turbo codes for applications such as satellite links. Also suitable as SCCC outer code.

Impulse response, transfer function, and constraint length
A convolutional encoder is called so because it performs a convolution of the input stream with the encoder's impulse responses:

 * $$y_i^j=\sum_{k=0}^{\infty} h^j_k x_{i-k} = (x * h^j)[i],$$

where $x$ is an input sequence, $y^{j}$ is a sequence from output $j$, $h^{j}$ is an impulse response for output $j$ and $${*}$$ denotes convolution.

A convolutional encoder is a discrete linear time-invariant system. Every output of an encoder can be described by its own transfer function, which is closely related to the generator polynomial. An impulse response is connected with a transfer function through Z-transform.

Transfer functions for the first (non-recursive) encoder are:
 * $$H_1(z)=1+z^{-1}+z^{-2},\,$$
 * $$H_2(z)=z^{-1}+z^{-2},\,$$
 * $$H_3(z)=1+z^{-2}.\,$$

Transfer functions for the second (recursive) encoder are:
 * $$H_1(z)=\frac{1+z^{-1}+z^{-3}}{1-z^{-2}-z^{-3}},\,$$
 * $$H_2(z)=1.\,$$

Define $m$ by
 * $$ m = \max_i \operatorname{polydeg} (H_i(1/z)) \,$$

where, for any rational function $$f(z) = P(z)/Q(z) \,$$,
 * $$ \operatorname{polydeg}(f) = \max (\deg(P), \deg(Q)) \,$$.

Then $m$ is the maximum of the polynomial degrees of the

$$ H_i(1/z) \,$$, and the constraint length is defined as $$ K = m + 1 \,$$. For instance, in the first example the constraint length is 3, and in the second the constraint length is 4.

Trellis diagram
A convolutional encoder is a finite state machine. An encoder with n binary cells will have 2n states.

Imagine that the encoder (shown on Img.1, above) has '1' in the left memory cell (m0), and '0' in the right one (m−1). (m1 is not really a memory cell because it represents a current value). We will designate such a state as "10". According to an input bit the encoder at the next turn can convert either to the "01" state or the "11" state. One can see that not all transitions are possible for (e.g., a decoder can't convert from "10" state to "00" or even stay in "10" state).

All possible transitions can be shown as below:

An actual encoded sequence can be represented as a path on this graph. One valid path is shown in red as an example.

This diagram gives us an idea about decoding: if a received sequence doesn't fit this graph, then it was received with errors, and we must choose the nearest correct (fitting the graph) sequence. The real decoding algorithms exploit this idea.

Free distance and error distribution

The free distance (d) is the minimal Hamming distance between different encoded sequences. The correcting capability (t) of a convolutional code is the number of errors that can be corrected by the code. It can be calculated as

 * $$t=\left \lfloor \frac{d-1}{2} \right \rfloor.$$

Since a convolutional code doesn't use blocks, processing instead a continuous bitstream, the value of t applies to a quantity of errors located relatively near to each other. That is, multiple groups of t errors can usually be fixed when they are relatively far apart.

Free distance can be interpreted as the minimal length of an erroneous "burst" at the output of a convolutional decoder. The fact that errors appear as "bursts" should be accounted for when designing a concatenated code with an inner convolutional code. The popular solution for this problem is to interleave data before convolutional encoding, so that the outer block (usually Reed–Solomon) code can correct most of the errors.

Decoding convolutional codes

Several algorithms exist for decoding convolutional codes. For relatively small values of k, the Viterbi algorithm is universally used as it provides maximum likelihood performance and is highly parallelizable. Viterbi decoders are thus easy to implement in VLSI hardware and in software on CPUs with SIMD instruction sets.

Longer constraint length codes are more practically decoded with any of several sequential decoding algorithms, of which the Fano algorithm is the best known. Unlike Viterbi decoding, sequential decoding is not maximum likelihood but its complexity increases only slightly with constraint length, allowing the use of strong, long-constraint-length codes. Such codes were used in the Pioneer program of the early 1970s to Jupiter and Saturn, but gave way to shorter, Viterbi-decoded codes, usually concatenated with large Reed–Solomon error correction codes that steepen the overall bit-error-rate curve and produce extremely low residual undetected error rates.

Both Viterbi and sequential decoding algorithms return hard decisions: the bits that form the most likely codeword. An approximate confidence measure can be added to each bit by use of the Soft output Viterbi algorithm. Maximum a posteriori (MAP) soft decisions for each bit can be obtained by use of the BCJR algorithm.

Popular convolutional codes

In fact, predefined convolutional codes structures obtained during scientific researches are used in the industry. This relates to the possibility to select catastrophic convolutional codes (causes larger number of errors).

An especially popular Viterbi-decoded convolutional code, used at least since the Voyager program, has a constraint length $K$ of 7 and a rate r of 1/2.

Mars Pathfinder, Mars Exploration Rover and the Cassini probe to Saturn use a $K$ of 15 and a rate of 1/6; this code performs about 2 dB better than the simpler $$K=7$$ code at a cost of 256× in decoding complexity (compared to Voyager mission codes).

The convolutional code with a constraint length of 2 and a rate of 1/2 is used in GSM as an error correction technique.

Punctured convolutional codes

Convolutional code with any code rate can be designed based on polynomial selection; however, in practice, a puncturing procedure is often used to achieve the required code rate. Puncturing is a technique used to make a m/n rate code from a "basic" low-rate (e.g., 1/n) code. It is achieved by deleting of some bits in the encoder output. Bits are deleted according to a puncturing matrix. The following puncturing matrices are the most frequently used:

For example, if we want to make a code with rate 2/3 using the appropriate matrix from the above table, we should take a basic encoder output and transmit every first bit from the first branch and every bit from the second one. The specific order of transmission is defined by the respective communication standard.

Punctured convolutional codes are widely used in the satellite communications, for example, in Intelsat systems and Digital Video Broadcasting.

Punctured convolutional codes are also called "perforated".

Turbo codes: replacing convolutional codes

Simple Viterbi-decoded convolutional codes are now giving way to turbo codes, a new class of iterated short convolutional codes that closely approach the theoretical limits imposed by Shannon's theorem with much less decoding complexity than the Viterbi algorithm on the long convolutional codes that would be required for the same performance. Concatenation with an outer algebraic code (e.g., Reed–Solomon) addresses the issue of error floors inherent to turbo code designs.

Publications

 * Francis, Michael. "Viterbi Decoder Block Decoding-Trellis Termination and Tail Biting." Xilinx XAPP551 v2. 0, DD (2005): 1-21.
 * Chen, Qingchun, Wai Ho Mow, and Pingzhi Fan. "Some new results on recursive convolutional codes and their applications." Information Theory Workshop, 2006. ITW'06 Chengdu. IEEE. IEEE, 2006.
 * Fiebig, U-C., and Patrick Robertson. "Soft-decision and erasure decoding in fast frequency-hopping systems with convolutional, turbo, and Reed-Solomon codes." IEEE Transactions on Communications 47.11 (1999): 1646-1654.
 * Bhaskar, Vidhyacharan, and Laurie L. Joiner. "Performance of punctured convolutional codes in asynchronous CDMA communications under perfect phase-tracking conditions." Computers & Electrical Engineering 30.8 (2004): 573-592.
 * Modestino, J., and Shou Mui. "Convolutional code performance in the Rician fading channel." IEEE Transactions on Communications 24.6 (1976): 592-606.
 * Chen, Yuh-Long, and Che-Ho Wei. "Performance evaluation of convolutional codes with MPSK on Rician fading channels." IEE Proceedings F-Communications, Radar and Signal Processing. Vol. 134. No. 2. IET, 1987.

title: Convolutional code


text:
Copy

Copy may refer to:

 * Copying or the product of copying (including the plural "copies"); the duplication of information or an artifact
 * Cut, copy and paste, a method of reproducing text or other data in computing
 * File copying
 * Photocopying, a process which makes paper copies of documents and other visual images
 * Fax, a telecommunications technology used to transfer facsimile copies of documents, especially over the telephone network
 * Facsimile, a copy or reproduction that is as true to the original source as possible
 * Replica, a copy closely resembling the original concerning its shape and appearance
 * Term of art in U.S. copyright law meaning a material object in which a work of authorship has been embodied, such as a book
 * Copy (command), a shell command on DOS and Windows systems
 * Copy (publishing), written content in publications, in contrast to photographs or other elements of layout.
 * The output of journalists and authors, ready for copy editing and typesetting
 * The output of copywriters, who are employed to write material which encourages consumers to buy goods or services.
 * Camera ready copy, term used in the commercial printing industry meaning that a document is, from a technical standpoint, ready to "go to press", or be printed.
 * Copy (album), the debut album of the electronica artist Mitsuki Aira
 * Copy (musician), the Portland-based electronic music artist
 * "Copy", a procedure word or response indicating a satisfactory receipt of the last radio transmission
 * COPY, a COBOL keyword

title: Copy


text:
Cord circuit

In telecommunication, a cord circuit is a switchboard circuit in which a plug-terminated cord is used to establish connections manually between user lines or between trunks and user lines. A number of cord circuits are furnished as part of the switchboard position equipment. The cords may be referred to as front cord and rear cord or trunk cord and station cord. In modern cordless switchboards, the cord-circuit function is switch operated and may be programmable.

In early and middle 20th century telephone exchanges this task was done by a supervisory relay set known variously as junctor circuit or district junctor. Later designs made it a function of the trunk circuit or absorbed it into software.

title: Cord circuit


text:
Core

Core or cores may refer to:

Science and technology

 * Core (anatomy), everything except the appendages
 * Core (manufacturing), used in casting and molding
 * Core (optical fiber), the signal-carrying portion of an optical fiber
 * Core, the central part of a fruit
 * Hydrophobic core, the interior zone of a protein
 * Nuclear reactor core, a portion containing the fuel components
 * Pit (nuclear weapon) or core, the fissile material in a nuclear weapon
 * Semiconductor intellectual property core (IP core), is a unit of design in ASIC/FPGA electronics and IC manufacturing
 * Atomic core, an atom with no valence electrons
 * Lithic core, in archaeology, a stone artifact left over from toolmaking

Geology and astrophysics

 * Core sample, in Earth science, a sample obtained by coring
 * Ice core
 * Core, the central part of a galaxy; see Mass deficit
 * Core (anticline), the central part of an anticline or syncline
 * Planetary core, the center of a planet
 * Earth's inner core
 * Earth's outer core
 * Stellar core, the region of a star where nuclear fusion takes place
 * Solar core,

Computing

 * Core Animation, a data visualization API used in macOS
 * Core dump, the recorded state of a running program
 * Intel Core, a family of single-core and multi-core 32-bit and 64-bit CPUs released by Intel
 * Magnetic core, in electricity and electronics, ferromagnetic material around which wires are wound
 * Magnetic-core memory, the primary memory technology used before semiconductor memory
 * Central processing unit (CPU), called a core
 * Multi-core processor, a microprocessor with multiple CPUs on one integrated circuit chip
 * Server Core, a minimalist Microsoft Windows Server installation option

Mathematics

 * Core (game theory), the collection of stable allocations that no coalition can improve upon
 * Core (graph theory), the homomorphically minimal subgraph of a graph
 * Core (group theory), an object in group theory
 * Core of a triangulated category
 * Core, an essential domain of a closed operator; see Unbounded operator
 * Core, a radial kernel of a subset of a vector space; see Algebraic interior

Arts, entertainment and media

 * Core (novel), a 1993 science fiction novel by Paul Preuss
 * Core (radio station), a defunct digital radio station in the United Kingdom
 * 90.3 The Core RLC-WVPH, a radio station in Piscataway, New Jersey, US
 * C.O.R.E. (video game), a 2009 NDS game
 * Core (video game), a video game with integrated game creation system
 * "CORE", an area in the Underground in the video game Undertale
 * "The Core", an episode of The Transformers cartoon

Film and television

 * Cores (film), a 2012 film
 * The Core, a 2003 science fiction film
 * The Core, the 2006–2007 name for the programming block on Five currently known as Shake!

Music

 * Core (band), an American stoner rock band
 * Core (Stone Temple Pilots album), 1992
 * Core (Persefone album), 2006
 * "Core", a song by Susumu Hirasawa from Paranoia Agent Original Soundtrack
 * "The Core", a song from Eric Clapton's 1977 album Slowhand
 * "CORE", a track from the soundtrack of the 2015 video game Undertale by Toby Fox

Organizations

 * Core International, a defunct American computer and technology corporation
 * Core Design, a videogame developer best known for the Tomb Raider series
 * Coordenadoria de Recursos Especiais, Brazilian state police SWAT team
 * Digestive Disorders Foundation, working name Core
 * Center for Operations Research and Econometrics at the Université catholique de Louvain in Belgium
 * Central Organisation for Railway Electrification, an organization in India
 * China Open Resources for Education, an OpenCourseWare organization in China
 * Community Organized Relief Effort, a non-profit organization providing humanitarian relief
 * Congress of Racial Equality, United States civil rights organization
 * CORE (research service), a UK-based aggregator of open access content
 * C.O.R.E., a computer animation studio
 * CORE System Trust, see CORE-OM

United States

 * Core, San Diego, a neighborhood in California
 * Core, West Virginia
 * Core Banks, North Carolina
 * Core Sound, North Carolina

Other places

 * Corés, a parish in Spain
 * The Core Shopping Centre (Calgary), Alberta, Canada
 * The Core, a shopping centre in Leeds, England, on the site of Schofields

People

 * Earl Lemley Core (1902–1984), West Virginia botanist
 * Ericson Core, American director and cinematographer

Other uses

 * Core (architecture)
 * Co-ordinated On-line Record of Electors, central database in the United Kingdom
 * Coree or Cores, a Native American tribe
 * Korah, a biblical figure
 * Leadership core, concept in Chinese politics
 * Persephone, a Greek goddess also known as Kore or Cora (Greek κόρη = daughter)
 * Core countries, in dependency theory, an industrialized country on which peripheral countries depend
 * Core curriculum, in education, an essential part of the curriculum
 * CORE (Clinical Outcomes in Routine Use) System, see CORE-OM

title: Core


text:
Corner reflector

A corner reflector is a retroreflector consisting of three mutually perpendicular, intersecting flat surfaces, which reflects waves directly towards the source, but translated. The three intersecting surfaces often have square shapes. Radar corner reflectors made of metal are used to reflect radio waves from radar sets. Optical corner reflectors, called corner cubes or cube corners, made of three-sided glass prisms, are used in surveying and laser ranging.

Principle

The incoming ray is reflected three times, once by each surface, which results in a reversal of direction. To see this, the three corresponding normal vectors of the corner's perpendicular sides can be considered to form a basis (a rectangular coordinate system) (x, y, z) in which to represent the direction of an arbitrary incoming ray,. When the ray reflects from the first side, say x, the ray's x component, a, is reversed to −a while the y and z components are unchanged, resulting in a direction of. Similarly, when reflected from side y and finally from side z, the b and c components are reversed. Therefore, the ray direction goes from to  to  to, and it leaves the corner reflector with all three components of direction exactly reversed. The distance travelled, relative to a plane normal to the direction of the rays, is also equal for any ray entering the reflector, regardless of the location where it first reflects.

In radar
Radar corner reflectors are designed to reflect the microwave radio waves emitted by radar sets back toward the radar antenna. This causes them to show a strong "return" on radar screens. A simple corner reflector consists of three conducting sheet metal or screen surfaces at 90° angles to each other, attached to one another at the edges, forming a "corner". These reflect radio waves coming from in front of them back parallel to the incoming beam. To create a corner reflector that will reflect radar waves coming from any direction, 8 corner reflectors are placed back-to-back in an octahedron (diamond) shape. The reflecting surfaces must be larger than several wavelengths of the radio waves to function.

In maritime navigation they are placed on bridge abutments, buoys, ships and, especially, lifeboats, to ensure that these show up strongly on ship radar screens. Corner reflectors are placed on the vessel's masts at a height of at least 4.6 m above sea level (giving them an approximate minimum horizon distance of 8 km). Marine radar uses X-band microwaves with wavelengths of 2.5 - 3.75 cm, so small reflectors less than 30 cm across are used. In aircraft navigation, corner reflectors are installed on rural runways, to make them show up on aircraft radar.

In optics

In optics, corner reflectors typically consist of three mirrors or reflective prism faces which return an incident light beam in the opposite direction. In surveying, retroreflector prisms are commonly used as targets for long-range electronic distance measurement using a total station.

Five arrays of optical corner reflectors have been placed on the Moon for use by Lunar Laser Ranging experiments observing a laser's time-of-flight to measure the Moon's orbit more precisely than was possible before. The three largest were placed by NASA as part of the Apollo program, and the Soviet Union built two smaller ones into the Lunokhod rovers.

Automobile and bicycle tail lights are molded with arrays of small corner reflectors, with different sections oriented for viewing from different angles. Reflective paint for visibility at night usually contains retroreflective spherical beads. Thin plastic with microscopic corner reflector structures can be used as tape, on signs, or sewn or molded onto clothing.

Other examples
Corner reflectors can also occur accidentally. Tower blocks with balconies are often accidental corner reflectors for sound and return a distinctive echo to an observer making a sharp noise, such as a hand clap, nearby. Similarly, in radar interpretation, an object that has multiple reflections from smooth surfaces produces a radar return of greater magnitude than might be expected from the physical size of the object. This effect was put to use on the ADM-20 Quail, a small decoy missile which had the same radar cross section as a B-52.

title: Corner reflector


text:
Cosmic noise

Cosmic noise, also known as galactic radio noise, is a physical phenomenon derived from outside of the Earth's atmosphere. It is not actually sound, and it can be detected through a radio receiver, which is an electronic device that receives radio waves and converts the information given by them to an audible form. Its characteristics are comparable to those of thermal noise. Cosmic noise occurs at frequencies above about 15 MHz when highly directional antennas are pointed toward the Sun or other regions of the sky, such as the center of the Milky Way Galaxy. Celestial objects like quasars, which are super dense objects far from Earth, emit electromagnetic waves in their full spectrum, including radio waves. The fall of a meteorite can also be heard through a radio receiver; the falling object burns from friction with the Earth's atmosphere, ionizing surrounding gases and producing radio waves. Cosmic microwave background radiation (CMBR) from outer space is also a form of cosmic noise. CMBR is thought to be a relic of the Big Bang, and pervades the space almost homogeneously over the entire celestial sphere. The bandwidth of the CMBR is wide, though the peak is in the microwave range.

History
Karl Jansky, an American physicist and radio engineer, first discovered radio waves from the Milky Way in August, 1931. At Bell Telephone Laboratories in 1932, Jansky built an antenna designed to receive radio waves at a frequency of 20.5 MHz, which is a wavelength of approximately 14.6 meters.

After recording signals with this antenna for several months, Jansky categorized them into three types: nearby thunderstorms, distant thunderstorms, and a faint steady hiss of an unknown origin. He discovered the location of maximum intensity rose and fell once a day, which led him to believe he was detecting radiation from the Sun.

A few months went by following this signal thought to be from the Sun, and Jansky found that the brightest point moved away from the Sun and concluded the cycle repeated every 23 hours and 56 minutes. After this discovery, Jansky concluded the radiation was coming from the Milky Way and was strongest in the direction of the center of the galaxy.

Jansky's work helped to distinguish between the radio sky and the optical sky. The optical sky is what is seen by the human eye, whereas the radio sky consists of daytime meteors, solar bursts, quasars, and gravitational waves.

Later in 1963, American physicist and radio astronomer Arno Allan Penzias (born April 26, 1933) discovered cosmic microwave background radiation. Penzias's discovery of cosmic microwave background radiation helped establish the Big Bang theory of cosmology. Penzias and his partner, Robert Woodrow Wilson worked together on ultra-sensitive cryogenic microwave receivers, originally intended for radio astronomy observations. In 1964, upon creating their most sensitive antenna/receiver system, the Holmdel Horn Antenna, the two discovered a radio noise they could not explain. After further investigation, Penzias contacted Robert Dicke, who suggested it could be the background radiation predicted by cosmological theories, a radio remnant of the Big Bang. Penzias and Wilson won the Nobel Prize in Physics in 1978.

NASA's work
The Absolute Radiometer for Cosmology, Astrophysics, and Diffuse Emission (ARCADE) is a device designed to observe the transition out of the "cosmic dark ages" as the first stars ignite in nuclear fusion and the universe begins to resemble its current form.

ARCADE consists of 7 precision radiometers carried to an altitude of over 35 km (21 miles) by a scientific research balloon. The device measures the tiny heating of the early universe by the first generation of stars and galaxies to form after the Big Bang.

Sources of cosmic noise
Cosmic noise refers to the background radio frequency radiation from galactic sources, which have constant intensity during geomagnetically quiet periods.

Sun flares
Cosmic noise can be traced from solar flares, which are sudden explosive releases of stored magnetic energy in the atmosphere of the Sun, causing sudden brightening of the photosphere. Solar flares can last from a few minutes to several hours.

During solar flare events, particles and electromagnetic emissions can affect Earth's atmosphere by fluctuating the level of ionization in the Earth's ionosphere. Increased ionization results in absorption of the cosmic radio noise as it passes through the ionosphere.

Solar wind
Solar wind is a flux of particles, protons and electrons together with nuclei of heavier elements in smaller numbers, that are accelerated by the high temperatures of the solar corona to velocities large enough to allow them to escape from the Sun's gravitational field.

Solar wind causes sudden bursts of cosmic noise absorption in the Earth's ionosphere. These bursts can only be detected only if the magnitude of the geomagnetic field perturbation caused by the solar wind shock is large enough.

title: Cosmic noise


text:
Costas loop

A Costas loop is a phase-locked loop (PLL) based circuit which is used for carrier frequency recovery from suppressed-carrier modulation signals (e.g. double-sideband suppressed carrier signals) and phase modulation signals (e.g. BPSK, QPSK). It was invented by John P. Costas at General Electric in the 1950s. Its invention was described as having had "a profound effect on modern digital communications". The primary application of Costas loops is in wireless receivers. Its advantage over other PLL-based detectors is that at small deviations the Costas loop error voltage is $$\sin(2(\theta_i-\theta_f))$$ as compared to $$\sin(\theta_i-\theta_f)$$. This translates to double the sensitivity and also makes the Costas loop uniquely suited for tracking Doppler-shifted carriers, especially in OFDM and GPS receivers.

Classical implementation
In the classical implementation of a Costas loop, a local voltage-controlled oscillator (VCO) provides quadrature outputs, one to each of two phase detectors, e.g., product detectors. The same phase of the input signal is also applied to both phase detectors, and the output of each phase detector is passed through a low-pass filter. The outputs of these low-pass filters are inputs to another phase detector, the output of which passes through a noise-reduction filter before being used to control the voltage-controlled oscillator. The overall loop response is controlled by the two individual low-pass filters that precede the third phase detector, while the third low-pass filter serves a trivial role in terms of gain and phase margin.

The above figure of a Costas loop is drawn under the "locked" state, where the VCO frequency and the incoming carrier frequency have become the same due to the Costas loop process. The figure does not represent the "unlocked" state.

In the time domain
In the simplest case $$m^2(t) = 1$$. Therefore, $$m^2(t) = 1$$ does not affect the input of the noise-reduction filter. The carrier and voltage-controlled oscillator (VCO) signals are periodic oscillations $$f_{ref,vco}(\theta_{ref,vco}(t))$$ with high-frequencies $$\dot\theta_{ref,vco}(t)$$. The block $$\bigotimes$$ is an analog multiplier.

A linear filter can be described mathematically by a system of linear differential equations:
 * $$\begin{array}{ll}

\dot x = Ax + b u_{d}(t),& u_{LF} = c^*x. \end{array} $$ where $$A$$ is a constant matrix, $$x(t)$$ is a state vector of the filter, $$b$$ and $$c$$ are constant vectors.

The model of a VCO is usually assumed to be linear:

\begin{array}{ll} \dot\theta_{vco}(t) = \omega^{free}_{vco} + K_{vco} u_{LF}(t),& t \in [0,T], \end{array} $$ where $$\omega^{free}_{vco}$$ is the free-running frequency of the VCO and $$K_{vco}$$ is the VCO gain factor. Similarly, it is possible to consider various nonlinear models of VCO.

Suppose that the frequency of the master generator is constant $$ \dot\theta_{ref}(t) \equiv \omega_{ref}. $$ Equation of VCO and equation of filter yield

\begin{array}{ll} \dot{x} = Ax + bf_{ref}(\theta_{ref}(t))f_{vco}(\theta_{vco}(t)),& \dot\theta_{vco} = \omega^{free}_{vco} + K_{vco}c^*x. \end{array} $$

The system is non-autonomous and rather tricky for investigation.

In the phase-frequency domain
In the simplest case, when
 * $$\begin{align}

f_{ref}\big(\theta_{ref}(t)\big) = \cos\big(\omega_{ref} t\big),\ f_{vco}\big(\theta_{vco}(t)\big) &= \sin\big(\theta_{vco}(t)\big) \\ f_{ref}\big(\theta_{ref}(t)\big)^2 f_{vco}\left(\theta_{vco}(t)\right) f_{vco}\left(\theta_{vco}(t) - \frac{\pi}{2}\right) &= -\frac{1}{8}\Big(      2\sin(2\theta_{vco}(t)) +        \sin(2\theta_{vco}(t) - 2\omega_{ref} t) +        \sin(2\theta_{vco}(t) + 2\omega_{ref} t)     \Big) \end{align}$$

The standard engineering assumption is that the filter removes the upper sideband frequency from the input but leaves the lower sideband without change. Thus it is assumed that the VCO input is $$\varphi(\theta_{ref}(t) - \theta_{vco}(t)) = \frac{1}{8}\sin(2\omega_{ref} t - 2\theta_{vco}(t)).$$ This makes a Costas loop equivalent to a phase-locked loop with phase detector characteristic $$\varphi(\theta)$$ corresponding to the particular waveforms $$f_{ref}(\theta)$$ and $$f_{vco}(\theta)$$ of the input and VCO signals. It can be proved that filter outputs in the time and phase-frequency domains are almost equal.

Thus it is possible to study the simpler autonomous system of differential equations
 * $$\begin{align}

\dot{x} &= Ax + b\varphi(\Delta\theta), \\ \Delta\dot{\theta} &= \omega_{vco}^{free} - \omega_{ref} + K_{vco}c^*x, \\ \Delta\theta &= \theta_{vco} - \theta_{ref}. \end{align}$$.

The Krylov–Bogoliubov averaging method allows one to prove that solutions of non-autonomous and autonomous equations are close under some assumptions. Thus, the Costas loop block diagram in the time domain can be asymptotically changed to the block diagram on the level of phase-frequency relations.

The transition to the analysis of an autonomous dynamical model of the Costas loop (in place of the non-autonomous one) allows one to overcome the difficulties related to modeling the Costas loop in the time domain, where one has to simultaneously observe a very fast time scale of the input signals and slow time scale of signal's phase. This idea makes it possible to calculate core performance characteristics - hold-in, pull-in, and lock-in ranges.

Frequency acquisition
The classical Costas loop will work towards making the phase difference between the carrier and the VCO become a small, ideally zero, value. The small phase difference implies that frequency lock has been achieved.

QPSK Costas loop
The classical Costas loop can be adapted to QPSK modulation for higher data rates. The input QPSK signal is as follows

m_1(t)\cos\left(\omega_\text{ref} t\right) + m_2(t)\sin\left(\omega_\text{ref} t\right), m_1(t) = \pm 1, m_2(t) = \pm 1. $$

Inputs of low-pass filters LPF1 and LPF2 are
 * $$\begin{align}

\varphi_1(t) &= \cos\left(\theta_\text{vco}\right)\left(m_1(t)\cos\left(\omega_\text{ref} t\right) + m_2(t)\sin\left(\omega_\text{ref} t\right)\right), \\ \varphi_2(t) &= \sin\left(\theta_\text{vco}\right)\left(m_1(t)\cos\left(\omega_\text{ref} t\right) + m_2(t)\sin\left(\omega_\text{ref} t\right)\right). \end{align}$$

After synchronization, the outputs of LPF1 $$Q(t)$$ and LPF2 $$I(t)$$ are used to get demodulated data ($$m_1(t)$$ and $$m_2(t)$$). To adjust the frequency of the VCO to the reference frequency, signals $$Q(t)$$ and $$I(t)$$ are limited and cross-multiplied:
 * $$u_d(t) = I(t)\sgn(Q(t)) - Q(t)\sgn(I(t)).$$

Then the signal $$u_d(t)$$ is filtered by the loop filter and forms the tuning signal for the VCO $$u_\text{LF}(t)$$, similar to BPSK Costas loop. Thus, QPSK Costas can be described by a system of ordinary differential equations:
 * $$\begin{align}

\dot{x}_1 &= A_\text{LPF} x_1 + b_\text{LPF}\varphi_1(t),\\ \dot{x}_2 &= A_\text{LPF} x_2 + b_\text{LPF}\varphi_2(t),\\ \dot{x} &= A_\text{LF} x + b_\text{LF}\big(c_\text{LPF}^* x_1\sgn(c_\text{LPF}^* x_2) - c_\text{LPF}^* x_2\sgn(c_\text{LPF}^* x_1)\big),\\ \dot{\theta}_\text{vco} &= \omega_\text{vco}^\text{free} + K_\text{VCO}\Big(c^*_\text{LF} x + h_\text{LF}\big(c_\text{LPF}^* x_1\sgn(c_\text{LPF}^* x_2) - c_\text{LPF}^* x_2\sgn(c_\text{LPF}^* x_1)\big)\Big).\\ \end{align}$$

Here $$A_\text{LPF}, b_\text{LPF}, c_\text{LPF}$$ are parameters of LPF1 and LPF2 and $$A_\text{LF}, b_\text{LF}, c_\text{LF}, h_\text{LF}$$ are parameters of the loop filter.

title: Costas loop