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Cable modem

A cable modem is a type of network bridge that provides bi-directional data communication via radio frequency channels on a hybrid fibre-coaxial (HFC), radio frequency over glass (RFoG) and coaxial cable infrastructure. Cable modems are primarily used to deliver broadband Internet access in the form of cable Internet, taking advantage of the high bandwidth of a HFC and RFoG network. They are commonly deployed in the Americas, Asia, Australia, and Europe.

MITRE Cablenet
Internet Experiment Note (IEN) 96 (1979) describes an early RF cable modem system. From pages 2 and 3 of IEN 96: The Cable-Bus System

The MITRE/Washington Cablenet system is based on a technology developed at MITRE/Bedford. Similar cable-bus systems are in operation at a number of government sites, e.g. Walter Reed Army Hospital, and the NASA Johnson Space Center, but these are all standalone, local-only networks.

The system uses standard Community Antenna Television (CATV) coaxial cable and microprocessor based Bus Interface Units (BIUs) to connect subscriber computers and terminals to the cable. ... The cable bus consists of two parallel coaxial cables, one inbound and the other outbound. The inbound cable and outbound cable are connected at one end, the headend, and electrically terminated at their other ends. This architecture takes advantage of the well developed unidirectional CATV components. The topology is dendritic (i.e. branched like a tree). ... The BIUs contain Radio Frequency (RF) modems which modulate a carrier signal to transmit digital information using 1 MHz of the available bandwidth in the 24 MHz frequency range. The remainder of the 294 MHz bandwidth can be used to carry other communication channels, such as off-the-air TV, FM, closed circuit TV, or a voice telephone system, or, other digital channels. The data rate of our test-bed system is 307.2 kbps.

IEEE 802.3b (10BROAD36)
The IEEE 802 Committee defined 10BROAD36 in 802.3b-1985 as a 10 Mbit/s IEEE 802.3/Ethernet broadband system to run up to 3600 m over CATV coax network cabling. The word broadband as used in the original IEEE 802.3 specifications implied operation in frequency-division multiplexed (FDM) channel bands as opposed to digital baseband square-waveform modulations (also known as line coding), which begin near zero Hz and theoretically consume infinite frequency bandwidth. (In real-world systems, higher-order signal components become indistinguishable from background noise.) In the market 10BROAD36 equipment was not developed by many vendors nor deployed in many user networks as compared to equipment for IEEE 802.3/Ethernet baseband standards such as 10BASE5 (1983), 10BASE2 (1985), 10BASE-T (1990), etc.

IEEE 802.7
The IEEE 802 Committee also specified a broadband CATV digital networking standard in 1989 with 802.7-1989. However, like 10BROAD36, 802.7-1989 saw little commercial success.

Hybrid networks
Hybrid Networks developed, demonstrated and patented the first high-speed, asymmetrical cable modem system in 1990. A key Hybrid Networks insight was that in the nascent days of the Internet, data downloading constitutes the majority of the data traffic, and this can be served adequately with a highly asymmetrical data network (i.e. a large downstream data pipe and many small upstream data pipes). This allowed CATV operators to offer high speed data services immediately without first requiring an expensive system upgrade. Also key was that it saw that the upstream and downstream communications could be on the same or different communications media using different protocols working in each direction to establish a closed loop communications system. The speeds and protocols used in each direction would be very different. The earliest systems used the public switched telephone network (PSTN) for the return path since very few cable systems were bi-directional. Later systems used CATV for the upstream as well as the downstream path. Hybrid's system architecture is used for most cable modem systems today.

LANcity
LANcity was an early pioneer in cable modems, developing a proprietary system that was widely deployed in the U.S. LANcity, which was led by the Iranian-American engineer Rouzbeh Yassini, was then acquired by Bay Networks. Bay Networks was subsequently acquired by Nortel. Nortel at the time had formed a joint-venture with Antec called ARRIS Interactive. Because of contractual agreements with Antec involving this joint venture, Nortel spun the LANCity group out into the ARRIS Interactive joint-venture. ARRIS continues to make cable modems and cable modem termination system (CMTS) equipment compliant with the DOCSIS standard.

Zenith homeworks
Zenith offered a cable modem technology using its own protocol which it introduced in 1993, being one of the first cable modem providers. The Zenith Cable Modem technology was used by several cable television systems in the United States and other countries, including Cox Communications San Diego, Knology in the Southeast United States, Ameritech's Americast service (later to be sold off to Wide Open West after the SBC / Ameritech merger), Cogeco in Hamilton Ontario and Cablevision du Nord de Québec in Val-d'Or. Zenith Homeworks used BPSK (Bi-Phase Shift Keyed) modulation to achieve 500 Kbit/sec in 600 kHz, or 4 Mbit/sec in 6 MHz.

Com21
Com21 was another early pioneer in cable modems, and quite successful until proprietary systems were made obsolete by the DOCSIS standardization. The Com21 system used a ComController as central bridge in CATV network head-ends, the ComPort cable modem in various models and the NMAPS management system using HP OpenView as platform. Later they also introduced a return path multiplexer to overcome noise problems when combining return path signals from multiple areas. The proprietary protocol was based on Asynchronous Transfer Mode (ATM). The central ComController switch was a modular system offering one downstream channel (transmitter) and one management module. The remaining slots could be used for upstream receivers (2 per card), dual Ethernet 10BaseT and later also Fast-Ethernet and ATM interfaces. The ATM interface became the most popular, as it supported the increasing bandwidth demands and also supported VLANs. Com21 developed a DOCSIS modem, but the company filed for bankruptcy in 2003 and closed. The DOCSIS CMTS assets of COM21 were acquired by ARRIS.

CDLP
CDLP was a proprietary system manufactured by Motorola. CDLP customer premises equipment (CPE) was capable of both PSTN (telephone network) and radio frequency (cable) return paths. The PSTN-based service was considered 'one-way cable' and had many of the same drawbacks as satellite Internet service; as a result, it quickly gave way to "two-way cable." Cable modems that used the RF cable network for the return path were considered 'two-way cable', and were better able to compete with the bi-directional digital subscriber line (DSL) service. The standard is in little use now as new providers use, and existing providers having changed to, the DOCSIS standard. The Motorola CDLP proprietary CyberSURFR is an example of a device that was built to the CDLP standard, capable of a peak 10 Mbit/s downstream and 1.532 Mbit/s upstream. CDLP supported a maximum downstream bandwidth of 30 Mbit/s which could be reached by using several cable modems.

The Australian ISP BigPond employed this system when it started cable modem tests in 1996. For a number of years cable Internet access was only available in Sydney, Melbourne and Brisbane via CDLP. This network ran parallel to the newer DOCSIS system for several years. In 2004, the CDLP network was terminated and replaced by DOCSIS.

CDLP has been also rolled out at the French cable operator Numericable before upgrading its IP broadband network using DOCSIS.

DVB/DAVIC
Digital Video Broadcasting (DVB) and Digital Audio Visual Council (DAVIC) are European-formed organizations that developed some cable modem standards. However, these standards have not been as widely adopted as DOCSIS.

IEEE 802.14
In the mid-1990s the IEEE 802 committee formed a subcommittee (802.14) to develop a standard for cable modem systems. IEEE 802.14 developed a draft standard, which was ATM-based. However, the 802.14 working group was disbanded when North American multi system operators (MSOs) instead backed the then-fledgling DOCSIS 1.0 specification, which generally used best-effort service and was IP-based (with extension codepoints to support ATM for QoS in the future). MSOs were interested in quickly deploying service to compete for broadband Internet access customers instead of waiting on the slower, iterative, and deliberative processes of standards development committees. Albert A. Azzam was Secretary of the IEEE 802.14 Working Group, and his book, High-Speed Cable Modems, describes many of the proposals submitted to 802.14.

IETF
Although the Internet Engineering Task Force (IETF) generally does not generate complete cable modem standards, the IETF chartered Working Groups (WGs) that produced various standards related to cable modem technologies (including 802.14, DOCSIS, PacketCable, and others). In particular, the IETF WGs on IP over Cable Data Network (IPCDN) and IP over Digital Video Broadcasting (DVB) produced some standards applicable to cable modem systems, primarily in the areas of Simple Network Management Protocol (SNMP) Management Information Bases (MIBs) for cable modems and other networking equipment that operates over CATV networks.

DOCSIS
In the late 1990s, a consortium of US cable operators, known as "MCNS" formed to quickly develop an open and interoperable cable modem specification. The group essentially combined technologies from the two dominant proprietary systems at the time, taking the physical layer from the Motorola CDLP system and the MAC layer from the LANcity system. When the initial specification had been drafted, the MCNS consortium handed over control of it to CableLabs which maintained the specification, promoted it in various standards organizations (notably SCTE and ITU), developed a certification testing program for cable modem equipment, and has since drafted multiple extensions to the original specification.

While deployed DOCSIS RFI 1.0 equipment generally only supported best-effort service, the DOCSIS RFI 1.0 Interim-01 document discussed quality of servce (QoS) extensions and mechanisms using IntServ, RSVP, RTP, and Synchronous Transfer Mode (STM) telephony (as opposed to ATM). DOCSIS RFI 1.1 later added more robust and standardized QoS mechanisms to DOCSIS. DOCSIS 2.0 added support for S-CDMA PHY, while DOCSIS 3.0 added IPv6 support and channel bonding to allow a single cable modem to use concurrently more than one upstream channel and more than one downstream channel in parallel.

Virtually all cable modems operating in the field today are compliant with one of the DOCSIS versions. Because of the differences in the European PAL and US's NTSC systems two main versions of DOCSIS exist, DOCSIS and EuroDOCSIS. The main differences are found in the width of RF-channels: 6 MHz for the US and 8 MHz for Europe. A third variant of DOCSIS was developed in Japan and has seen limited deployment in that country.

Although interoperability "was the whole point of the DOCSIS project," most cable operators only approve a very restricted list of cable modems on their network,   identifying the 'allowed' modems by their brand, models, sometimes firmware version and occasionally going as far as imposing a hardware version of the modem, instead of simply allowing a supported DOCSIS version.

Multimedia over Coax Alliance
In 2004, the Multimedia over Coax Alliance (MoCA) was established to develop industry standard for the connected home, using the existing coaxial cabling. Initially developed for in-home networking with MoCA 1.0/1.1, the MoCA standards has continued to develop with MoCA 2.0/2.1 in 2010 and MoCa 2.5 in 2016.

In 2017, Multimedia over Coax Alliance introduced MoCA Access specification, based on the MoCA 2.5 standard, suitable for addressing broadband network access in-building using coaxial cabling. MoCA Access extends MoCA 2.5 in-home networking to fit operators and ISPs that are installing fiber-to-the-basement/drop point (FTTB/FTTdp) and want to use the existing coax for connection to each apartment or house."

Multimedia terminal adapter
With the development of voice over Internet Protocol (VoIP) telephony, analog telephone adapters (ATA) have been incorporated into many cable modems for providing telephone service. An embedded ATA is known as an embedded multimedia terminal adapter (E-MTA).

Many cable TV service providers also offer VoIP-based telephone service via the cable infrastructure (PacketCable). Some high-speed Internet customers may use VoIP telephony by subscribing to a third-party service, such as Vonage, MagicJack+ and NetTALK.

Network architectural functions
In network topology, a cable modem is a network bridge that conforms to IEEE 802.1D for Ethernet networking (with some modifications). The cable modem bridges Ethernet frames between a customer LAN and the coax network. Technically, it is a modem because it must modulate data to transmit it over the cable network, and it must demodulate data from the cable network to receive it.

With respect to the OSI model of network design, a cable modem is both physical layer (layer 1) device and a data link layer (layer 2) forwarder. As an IP addressable network node, cable modems support functionality at other layers.

Layer 1 is implemented in the Ethernet PHY on its LAN interface, and a DOCSIS defined cable-specific PHY on its HFC cable interface. The term cable modem refers to this cable-specific PHY. The Network Layer (Layer 3) is implemented as an IP host in that it has its own IP address used by the network operator to maintain the device. In the transport layer (layer 4) the cable modem supports UDP in association with its own IP address, and it supports filtering based on TCP and UDP port numbers to, for example, block forwarding of NetBIOS traffic out of the customer's LAN. In the Application Layer (Layer 7), the cable modem supports certain protocols that are used for management and maintenance, notably Dynamic Host Configuration Protocol (DHCP), SNMP, and TFTP.

Some cable modems may incorporate a router and a DHCP server to provide the LAN with IP network addressing. From a data forwarding and network topology perspective, this router functionality is typically kept distinct from the cable modem functionality (at least logically) even though the two may share a single enclosure and appear as one unit, sometimes called a residential gateway. So, the cable modem function will have its own IP address and MAC address as will the router.

Cable modem flap
Cable modems can have a problem known in industry jargon as "flap" or "flapping". A modem flap is when the connection by the modem to the head-end has been dropped (gone offline) and then comes back online. The time offline or rate of flap is not typically recorded, only the incidence. While this is a common occurrence and usually unnoticed, if a modem's flap is extremely high, these disconnects can cause service to be disrupted. If there are usability problems due to flap the typical cause is a defective modem or very high amounts of traffic on the service provider's network (upstream utilization too high). Types of flap include reinsertions, hits and misses, and power adjustments.

Known vulnerabilities
In January 2020, a vulnerability affecting cable modems using Broadcom chipsets was disclosed and named Cable Haunt. Security researchers say that the vulnerability affects hundreds of millions of devices. Exploits are possible because of the use of default credentials in the spectrum analyzer component of the modem (mostly used for debugging purposes) accessible through a network port which is open by default in the vulnerable models.

title: Cable modem


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Kensington, Maryland

Kensington is a U.S. town in Montgomery County, Maryland. The population was 2,122 at the 2020 census. Greater Kensington encompasses the entire 20895 ZIP code, with a population of 19,753 in 2020.

History
The area around Rock Creek where Kensington is located was primarily agricultural until 1873, when the B&O Railroad completed the Metropolitan Branch across Montgomery County. A community arose where the new railroad line intersected the old Rockville-to-Bladensburg road. This early settlement was first known as "Knowles Station".

In the early 1890s, Washington developer Brainard Warner began purchasing land parcels to build a planned Victorian community, complete with church, library, and local newspaper. Fascinated by a recent trip to London, Warner named his development "Kensington Park", the tenth and largest subdivision in the area. Upon incorporation in 1894, Warner convinced the mayor and council to name the town "Kensington".

Originally a farming community at Knowles Station, Kensington developed into a summer refuge for Washington residents wishing to escape the capital's humid summers. As years passed and its residents increasingly remained year-round, Kensington evolved into a commuter suburb. Commuters reached downtown Washington, D.C., on B&O passenger trains and, beginning in 1895, a streetcar line founded as Chevy Chase Lake & Kensington Railway and later called the Kensington Railway Company.

The large southernmost section originally mapped out by Warner remains largely unchanged since inception, and is a historically preserved zone. Indeed, the only major changes in the town's basic layout have been the bridging of the original railroad crossing in 1937, and the extension and widening of Connecticut Avenue, the town's main thoroughfare, in 1957. As well, the right-of-way of the streetcar line, which was replaced with bus service in 1935, was paved to become Kensington Parkway.

In March 1975, Kensington gained attention regionally when Sheila and Katherine Lyon walked to Wheaton Plaza, a local shopping mall, and never returned home. Their abduction and murder was solved only in 2017.

The historic core of Kensington was listed on the National Register of Historic Places as the Kensington Historic District in 1980.

In the early 21st century, the town gained national attention three times in 10 months for events within a quarter-mile radius. In December 2001, the town responded to complaints from anonymous citizens by banning Santa Claus from the annual holiday parade. Protesters arrived at the parade en masse, including dozens of Santas riding everything from motorcycles to fire trucks. Eight months later, an Amtrak train derailed near the town center when the tracks separated at an overheated joint, injuring 72. And on October 2, 2002, Lori Ann Lewis-Rivera became the fifth victim of the "Beltway snipers" while cleaning her car at a Kensington gas station.

Geography
Kensington is located in Montgomery County, northwest of Silver Spring, northeast of Bethesda, west of Wheaton and southeast of Rockville. Its latitude is 39°1′48″N, longitude 77°4′30″W.

According to the U.S. Census Bureau, the town has a total area of 0.50 sqmi, all land.

While the town proper is but one-half square mile in size, the Kensington post office (ZIP Code 20895) serves a larger area, including all of the neighboring town of Chevy Chase View as well as most of the unincorporated neighborhoods known by the Census Bureau as North Kensington and South Kensington. The 20895 ZIP Code also extends into smaller portions of North Bethesda and the Wheaton Planning District.

The look and white color of the Washington D.C. Temple located in South Kensington, coupled with its location near the Capital Beltway, have made it a local landmark. D.C.-area traffic reports often refer to the "Mormon temple" or "temple".

2010 census
As of the census of 2010, there were 2,213 people, 870 households, and 563 families residing in the town. The population density was 4610.4 PD/sqmi. There were 902 housing units at an average density of 1879.2 /sqmi. The racial makeup of the town was 82.0% White, 6.1% African American, 0.1% Native American, 5.7% Asian, 0.1% Pacific Islander, 2.8% from other races, and 3.2% from two or more races. Hispanic or Latino of any race were 6.4% of the population.

There were 870 households, of which 33.7% had children under the age of 18 living with them, 51.8% were married couples living together, 10.0% had a female householder with no husband present, 2.9% had a male householder with no wife present, and 35.3% were non-families. 27.1% of all households were made up of individuals, and 10.5% had someone living alone who was 65 years of age or older. The average household size was 2.54 and the average family size was 3.17.

The median age in the town was 42.1 years. 26.2% of residents were under the age of 18; 5% were between the ages of 18 and 24; 24% were from 25 to 44; 30% were from 45 to 64; and 14.8% were 65 years of age or older. The gender makeup of the town was 47.6% male and 52.4% female.

2000 census
As of the census of 2000, the median income for a household in the town was $76,716, and the median income for a family was $96,394. Males had a median income of $65,804 versus $41,364 for females. The per capita income for the town was $35,919. About 0.9% of families and 2.1% of the population were below the poverty line, including none of those under age 18 and 1.3% of those age 65 or over.

Commerce
Kensington is primarily a bedroom community for workers who commute to jobs in the Washington, D.C., area, but it has some commercial enterprises, including "Antique Row" on Howard Avenue, the West Howard Antique District, and Kaiser-Permanente's Kensington facility, plus art shops, restaurants, supermarkets, auto repair shops, hardware stores, and others. The town hosts a website, Explore Kensington, listing businesses, services, news and events.

The town hosts a farmers' market on Saturdays between 9 a.m. and 1 p.m. at the train station.

The town is home to the Noyes Library for Young Children, Montgomery County's oldest public library.

Events

 * April 23 or the closest Sunday: The Kensington Day of the Book Festival or World Book Day is celebrated on the Sunday closest to April 23. This afternoon street festival is celebrated with live music, author readings, open mic, activities for children and adults, storytellers, and books. Local authors, book artists, publishers, booksellers, and literary groups line Howard Ave in Historic Old Town Kensington to show, sell, and discuss their works. In conjunction is the Gala Craft Fair on Armory Avenue featuring crafts.
 * Labor Day: Kensington holds an annual Labor Day parade.
 * Late September: Held since 1994, the Kensington 8K Race includes 8-km, 2-mile, and 1-km races through the historic core of the town and along nearby Beach Drive and in the Rock Creek Hills neighborhood.
 * Saturdays: Kensington parkrun, a free, timed 5K run/walk on the Rock Creek Trail.

Law and government
A mayoral election is held in even-numbered years for a two-year term. Kensington has a four-member council, elected for two-year terms. Terms are staggered. Every year there are two council seats up for election. For contested elections, election winners are noted in bold.

The 2021 Kensington Town Election was held on June 7, 2021
 * Conor D. Crimmins, Council Member, 318 votes
 * Darin R. Bartram, Council Member, 246 votes
 * Tim Willard, Council Member, 151 votes

The 2020 Kensington Town Election was held on June 1, 2020
 * Tracey Furman, Mayor, 420 votes
 * Peter Fosselman, Mayor, 368 votes
 * Bridget Hill-Zayat, Council Member, 549 votes
 * Nate Engle, Council Member, 511 votes
 * Jon Gerson, Council Member, 356 votes

The 2019 Kensington Town Election was held on June 3, 2019
 * Darin R. Bartram, Council Member, 72 votes
 * Conor D. Crimmins, Council Member, 71 votes

The 2018 Kensington Town Election was held on June 4, 2018
 * Tracey Furman, Mayor, 147 votes
 * Bridget Hill-Zayat, Council Member, 113 votes
 * Duane L. Rollins, Council Member, 136 votes

The 2017 Kensington Town Election was held on June 5, 2017
 * Darin R. Bartram, Council Member, 210 votes
 * Conor D. Crimmins, Council Member, 233 votes
 * Tom H. Rodriguez, Council Member, 207 votes

The 2016 Kensington Town Election was held on June 6, 2016
 * Tracey Furman, Mayor, 129 votes
 * Duane Rollins, Councilman, 125 votes
 * Sean McMullen, Councilman, 108 votes

The 2015 Kensington Town Election was held on June 1, 2015
 * Tom Rodriguez, Councilman, 158 votes
 * Darin Bartram, Councilman, 157 votes

Education
The town of Kensington is served by the Montgomery County Public Schools system:
 * Kensington-Parkwood Elementary School
 * North Bethesda Middle School
 * Walter Johnson High School

Montgomery County Public Schools serving Greater Kensington include:
 * Garrett Park Elementary School
 * Kensington-Parkwood Elementary School
 * North Chevy Chase Elementary School
 * Oakland Terrace Elementary School
 * Rock View Elementary School
 * Newport Mill Middle School
 * North Bethesda Middle School
 * Tilden Middle School (formerly Charles W. Woodward)
 * Silver Creek Middle School
 * Albert Einstein High School
 * Bethesda-Chevy Chase High School
 * Walter Johnson High School

Kensington is also home to:
 * Kensington Nursery School
 * Holy Redeemer Catholic School, a Roman Catholic parochial school
 * Grace Episcopal Day School
 * Academy of the Holy Cross, an all-girls Roman Catholic high school

Transportation
Four state highways serve Kensington. The most prominent of these is Maryland Route 185 (Connecticut Avenue), which provides the most direct link between Kensington and both Interstate 495 (the Capital Beltway) and Washington, D.C. The other major state highway serving the town is Maryland Route 193, which follows University Boulevard and Greenbelt Road east from Kensington across the northern and northeastern suburbs of Washington. The other two state highways, Maryland Route 192 and Maryland Route 547, are short connectors linking Kensington to nearby communities.

The MARC Brunswick Line stops at Kensington station. It connects downtown Kensington with Washington, D.C. via Union Station, as well as with points west including Point of Rocks, Maryland, and Harpers Ferry, West Virginia.

Multiple Ride On buses serve Kensington, including lines 4, 5, 33, 34, and 37. These buses take passengers from downtown Kensington to a variety of destinations including Silver Spring, Glenmont station, Medical Center station, and Wheaton.

Places of worship

 * Cedar Lane Unitarian Universalist Church
 * Christ Episcopal Church
 * First Baptist Church, Kengar
 * Holy Redeemer Catholic Church
 * Jehovah's Witnesses Kensington Kingdom Hall
 * Kensington Baptist Church
 * Lee Memorial AME Church
 * Saint Paul's United Methodist Church
 * Temple Emmanuel
 * Warner Memorial Presbyterian Church
 * Washington D.C. Temple of the Church of Jesus Christ of Latter-day Saints
 * Kensington Ward of The Church of Jesus Christ of Latter-day Saints

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

title: Kensington, Maryland


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Gaithersburg, Maryland

Gaithersburg is a city in Montgomery County, Maryland, United States. At the time of the 2020 census, Gaithersburg had a population of 69,657, making it the ninth-most populous community in the state. Gaithersburg is located to the northwest of Washington, D.C., and is considered a suburb and a primary city within the Washington metropolitan area. Gaithersburg was incorporated as a town in 1878 and as a city in 1968.

Gaithersburg is located east and west of Interstate 270. The eastern section includes the historic area of the town. Landmarks and buildings from that time can still be seen in many places but especially in the historic central business district of Gaithersburg called "Olde Towne". The east side also includes City Hall, the Montgomery County Fairgrounds, and Bohrer Park (a well-known joint community recreation center and outdoor water park for kids and families). The west side of the city has many wealthier neighborhoods that were designed with smart growth techniques and embrace New Urbanism. These include the Kentlands community, the Lakelands community, and the Washingtonian Center (better known as Rio), a shopping/business district. Two New Urbanism communities are under construction, including Watkins Mill Town Center (Casey East and West), and the massive "Science City". The state has a bus rapid transit line, Corridor Cities Transitway or "CCT", planned for the western portion of the city starting at Shady Grove Metro Station and connecting all the high-density western Gaithersburg neighborhoods with a total of eight stops planned in the city.

The National Institute of Standards and Technology (NIST) is headquartered in Gaithersburg directly west of I-270. Other major employers in the city include IBM, Lockheed Martin Information Systems and Global Services business area headquarters, AstraZeneca, and the French multinational corporation, Sodexo. Gaithersburg is also the location of the garrison of the United States Army Reserve Legal Command.

Gaithersburg is noted for its ethnic and economic diversity; it was ranked second for ethnic diversity among the 501 largest U.S. cities, and first among smaller U.S. cities, by WalletHub in 2021. In 2023, Wallethub announced that Gaithersburg was back in the number one spot for diversity in the U.S.

History
Gaithersburg was settled in 1765 as a small agricultural settlement known as Log Town near the present day Summit Hall on Ralph Crabb's 1725 land grant "Deer Park". The northern portion of the land grant was purchased by Henry Brookes, and he built his brick home "Montpelier" there, starting first with a log cabin in 1780/3. This 1,000-acre tract became part of the landmark IBM Headquarters complex built on the then-new I-270 Interstate "Industrial", now "Technology", Corridor in the late 1960s to the 1970s. Benjamin Gaither married Henry's daughter Margaret, and Benjamin and Margaret inherited a portion of Henry's land prior to Henry's death in 1807. Gaither built his home on the land in 1802. By the 1850s the area had ceased to be called Log Town and was known to inhabitants as Gaithersburg.

19th century
The Forest Oak Post Office, named for a large tree in the town, was located in Gaither's store in 1851.

On July 10, 1864, using the route of present-day 355, over 10,000 Confederate troops camped overnight in the area, including the present Bohrer Park, after a one-day march from Frederick after the Battle of Monocacy. The next day the troops continued towards Washington in an unsuccessful attempt to take the city.

When the railroad was built through town in 1873, the new station was called Gaithersburg, an officially recognized name for the community for the first time. Also in 1873 the Baltimore and Ohio Railroad constructed a station at Gaithersburg, designed by Ephraim Francis Baldwin as part of his well-known series of Victorian stations in Maryland. Rapid growth occurred shortly thereafter, and on April 5, 1878, the town was officially incorporated as the Town of Gaithersburg.

Gaithersburg boomed during the late 19th century and churches, schools, a mill, grain elevators, stores, and hotels were built. Much of this development focused around the railroad station.

In 1899, Gaithersburg was selected as one of six global locations for the construction of an International Latitude Observatory as part of a project to measure the Earth's wobble on its polar axis. The Gaithersburg Latitude Observatory is (as of 2007) the only National Historic Landmark in the City of Gaithersburg. The observatory and five others in Japan, Italy, Russia, and the United States gathered information that is still used by scientists today, along with information from satellites, to determine polar motion; the size, shape, and physical properties of the earth; and to aid the space program through the precise navigational patterns of orbiting satellites. The Gaithersburg station operated until 1982 when computerization rendered the manual observation obsolete.

Late 20th century
In 1968, Gaithersburg was upgraded from a town to a city.

Gaithersburg remained a predominantly rural farm town until the 1970s when more construction began. As the population grew, with homes spreading throughout the area, Gaithersburg began taking on a suburban and semi-urban feel, leaving its farming roots behind. During the late 1990s and 2000s, it had become one of the most economically and ethnically diverse areas in the Washington, D.C. Metropolitan Area as well as the State of Maryland, with people from all walks of life calling Gaithersburg home. This can be seen in the local schools, with Gaithersburg High School and Watkins Mill High School having two of the most diverse student bodies in the region.

During a 1997 rainstorm, the 295-year-old forest oak tree that gave its name to the Forest Oak Post Office crashed down. The tree served as the inspiration for the city's logo, which is also featured prominently on the city's flag.

21st century
In 2007, parts of the film Body of Lies were filmed in the city, at a building on 100 Edison Park Drive. The film was released in 2008 and the building is now the Montgomery County Police Department's headquarters.

On July 16, 2010, Gaithersburg was part of the area where a 3.6 magnitude earthquake was felt, one of the strongest to occur in Maryland.

After years of decline and loss of tenants, including three of its four anchor stores in 2019, Lakeforest Mall closed on March 31, 2023, with plans to demolish it and redevelop the area.

Gaithersburg is also the location of the United States Army Reserve Legal Command.

Geography
According to the U.S. Census Bureau, the city has a total area of 10.34 sqmi, of which 10.20 sqmi is land and 0.14 sqmi is water.

2022 ACS
As of the 2022 American Community Survey, there were 68,952 people and 24,523 households in the town. The racial makeup of the town was 33% White, 13% Black, 15% Asian, and 1% from other races. Hispanic people of any race were 36% of the population.

The median household income was 95,453, and 6% of people were under the poverty line.

The average time to work was 30 minutes, 57% of people drove alone, 11% carpooled, 8% took public transit, 1% biked, 2% walked and 20% work from home.

2010 census
As of the census of 2010, there were 59,933 people, 22,000 households, and 14,548 families residing in the city. The population density was 5875.8 PD/sqmi. There were 23,337 housing units at an average density of 2287.9 /sqmi. The racial makeup of the city was 31.9% non-Hispanic White, 16.3% African American, 0.5% Native American, 16.9% Asian (6.01 Chinese, 4.77% Indian, 2.03% Korean, 1.69% Filipino, 1.02% Vietnamese, 0.62% Burmese), 0.1% Pacific Islander, 10.7% from other races, and 4.8% from two or more races. Hispanic or Latino of any race were 24.2% of the population (8.3% Salvadoran, 2% Honduran, 1.9% Mexican, 1.9% Peruvian, 1.7% Guatemalan).

There were 22,000 households, of which 37.1% had children under the age of 18 living with them, 48.3% were married couples living together, 12.7% had a female householder with no husband present, 5.1% had a male householder with no wife present, and 33.9% were non-families. 26.7% of all households were made up of individuals, and 7.5% had someone living alone who was 65 years of age or older. The average household size was 2.70 and the average family size was 3.24.

The median age in the city was 35.1 years. 24.2% of residents were under the age of 18; 7.9% were between the ages of 18 and 24; 33.8% were from 25 to 44; 24.6% were from 45 to 64; and 9.5% were 65 years of age or older. The gender makeup of the city was 48.6% male and 51.4% females.

2000 census
As of the census of 2000, there were 52,613 people, 19,621 households, and 12,577 families residing in the city. The population density was 5,216.2 PD/sqmi. There were 20,674 housing units at an average density of 2,049.7 /sqmi. The racial makeup of the city is 34.7% White, 19.5% Black or African American, 0.2% Native American, 13.9% Asian, 0.1% Pacific Islander, 3.6% from other races, and 3.2% from two or more races. 24.8% of the population were Hispanic or Latino of any race. 34.3% of Gaithersburg's population was foreign-born.

There were 19,621 households, out of which 34.8% had children under the age of 18 living with them, 48.6% were married couples living together, 11.2% had a female householder with no husband present, and 35.9% were non-families. 27.8% of all households were made up of individuals, and 7.2% had someone living alone who was 65 years of age or older. The average household size was 2.65 and the average family size was 3.14 the population was spread out, with 25.0% under the age of 18, 9.0% from 18 to 24, 37.7% from 25 to 44, 20.0% from 45 to 64, and 8.2% who were 65 years of age or older. The median age was 34 years. For every 100 females, there were 95.1 males. For every 100 females age 18 and over, there were 92.4 males.

Economy
According to the city's 2020 Comprehensive Annual Financial Report, the top employers in the city are:

Gaithersburg also receives significant income from its conference organization platform including prominent conferences such as the CHI 84 conference.

Patton Electronics was founded in Gaithersburg during 1984.

Government
Gaithersburg has an elected, five-member City Council, which serves as the legislative body of the city. The mayor, who is also elected, serves as non-voting president of the council. The day-to-day administration of the city is overseen by a career city manager.

The city's current mayor is Jud Ashman, who has held the office since 2014. On October 6, 2014, the Gaithersburg City Council selected city council member Jud Ashman to serve as mayor until the next City of Gaithersburg election in November 2015, replacing resigning mayor Sidney Katz. Ashman was re-elected in November 2015 and would be re-elected to full terms in 2017 and 2021.

Previous mayors include:

 * 1) George W. Meem 1898–1904
 * 2) Carson Ward 1904–1906
 * 3) John W. Walker 1906–1908
 * 4) E. D. Kingsley 1908–1912
 * 5) Richard H. Miles 1912–1918
 * 6) John W. Walker 1918–1924
 * 7) Walter M. Magruder 1924–1926
 * 8) William McBain 1926–1948
 * 9) Harry C. Perry, Sr. 1948–1954
 * 10) Merton F. Duvall 1954–1966
 * 11) John W. Griffith 1966–1967
 * 12) Harold C. Morris 1967–1974
 * 13) Susan E. Nicholson, May–September 1974
 * 14) Milton M. Walker 1974–1976
 * 15) B. Daniel Walder 1976–1978
 * 16) Bruce A. Goldensohn 1978–1986
 * 17) W. Edward Bohrer, Jr. 1986–1998
 * 18) Sidney A. Katz 1998 – 2014
 * 19) Jud Ashman, November 2014 – Present

The departments of the city of Gaithersburg and their directors include:
 * Office of the City Manager, Tanisha R. Briley
 * Finance and Administration, Janice Hartman
 * Planning and Code Administration, John Schlichting
 * Community, Neighborhood and Housing Services, Tom Lonergan-Seeger
 * Human Resources, Kimberly Yocklin
 * Information Technology, Ruth Lutero
 * Parks, Recreation, and Culture, Carolyn Muller
 * Chief of Police, Mark Sroka
 * Public Works, Anthony Berger

Education
The following Montgomery County Public Schools are located in Gaithersburg:

Elementary schools

 * Brown Station
 * Darnestown
 * Diamond
 * DuFief
 * Fields Road
 * Flower Hill
 * Gaithersburg
 * Goshen
 * Harriet R. Tubman
 * Jones Lane
 * Judith A. Resnik
 * Laytonsville
 * Rachel Carson
 * Rosemont
 * South Lake
 * Stedwick
 * Strawberry Knoll
 * Summit Hall
 * Thurgood Marshall
 * Washington Grove
 * Watkins Mill
 * Whetstone
 * Woodfield

Middle schools

 * Forest Oak
 * Gaithersburg
 * Lakelands Park Middle School
 * Ridgeview
 * Shady Grove

High schools

 * Gaithersburg High School
 * Quince Orchard High School
 * Watkins Mill High School

Media
Gaithersburg is primarily served by the Washington, D.C. media market.

Newspapers

 * The Town Courier newspaper is based in Kentlands and focuses on Gaithersburg's west side neighborhoods, in addition to publishing Rockville and Urbana editions.

Police
Being a city, Gaithersburg also has its own police department, which was created in 1963.

Roads and highways
The most prominent highways serving Gaithersburg are Interstate 270 and Interstate 370. I-270 is the main highway leading northwest out of metropolitan Washington, D.C., beginning at Interstate 495 (the Capital Beltway) and proceeding northwestward to Interstate 70 in Frederick. I-370 is a short spur, starting just west of I-270 in Gaithersburg and heading east to its junction with Maryland Route 200. Via MD 200, I-370 connects Gaithersburg with Interstate 95 near Laurel.

Maryland Route 355 was the precursor to I-270 and follows a parallel route. It now serves as the main commercial roadway through Gaithersburg and neighboring communities. Other state highways serving Gaithersburg include Maryland Route 117, Maryland Route 119 and Maryland Route 124. Maryland Route 28 passes just outside the Gaithersburg corporate limits.

Transit
Gaithersburg is connected to the Washington Metro via Shady Grove station, which is located just outside the city limits and is the north-western terminus of the Red Line.

The Corridor Cities Transitway is a proposed bus rapid transit line that would have 8 stops in Gaithersburg, generally in the western half of the city.

Maryland's MARC system operates commuter rail services connecting Gaithersburg to Washington, D.C., with two stations in the city, at Old Town Gaithersburg and Metropolitan Grove, and a third station — Washington Grove — just outside city limits.

Bus service in Gaithersburg consists of Metrobus routes operated by WMATA and Ride-On routes operated by Montgomery County, as well as paratransit service provided by MetroAccess.

Airport
Montgomery County Airpark is located 3 miles (5 km) northeast of the city.

Notable people

 * Sankar Adhya, member of the National Academy of Sciences
 * Utkarsh Ambudkar, actor, rapper
 * Lawson Aschenbach, NASCAR driver
 * Georges C. Benjamin, former secretary of the Maryland Department of Health and Mental Hygiene
 * Kimberly J. Brown, actress who starred in Halloweentown
 * Mark Bryan, lead guitarist of Hootie & the Blowfish
 * Isabel McNeill Carley, published music teacher, lived in Gaithersburg from 2004 until her death in 2011
 * Justin Carter (born 1987), basketball player for Maccabi Kiryat Gat of the Israeli Premier League
 * Kiran Chetry, CNN anchor
 * Chris Coghlan, Major League Baseball player
 * Jeanine Cummins, author
 * Dominique Dawes, three-time women's Olympic gymnastics team member, member of the Magnificent Seven
 * Stefon Diggs, football player for the Houston Texans
 * Trevon Diggs, football player for the Dallas Cowboys
 * Brandon Victor Dixon, American actor, singer and theatrical producer
 * Astrid Ellena, Miss Indonesia 2011
 * Hank Fraley, former football player in the NFL
 * Judah Friedlander, actor, most notably from the television show 30 Rock
 * Jake Funk, football player for the Los Angeles Rams and Super Bowl LVI champion
 * Joshua Harris, author and former Christian pastor
 * Dwayne Haskins, NFL quarterback for the Pittsburgh Steelers
 * Matt Holt, former singer of Nothingface and Kingdom of Snakes
 * Paul James, actor, most notably from the television show Greek
 * Kelela, R&B singer
 * Courtney Kupets, 2004 Olympic gymnast and three-time NCAA champion
 * Tim Kurkjian, ESPN baseball analyst, appears on SportsCenter and Baseball Tonight
 * Charles Lee, Charlotte Hornets head coach
 * Matthew Lesko, author of Free Money from the government books
 * Logic (Robert Bryson Hall II), hip hop musician, rapper, musical engineer
 * Lucas and Marcus, dancers and YouTube personalities
 * Shane McMahon, WWE wrestler and commissioner of WWE SmackDown Live
 * Jim Miklaszewski, chief Pentagon correspondent for NBC News
 * Malcolm Miller, basketball player and NBA champion for the Toronto Raptors
 * Nick Mullen, a comedian
 * John Papuchis, college football coach
 * Andrew Platt, former Maryland House of Delegates member
 * Guy Prather, football player
 * Paul Rabil, lacrosse player (midfield), four-time All-American at Johns Hopkins University
 * Eddie Stubbs, country musician, disc jockey, and Grand Ole Opry announcer
 * Jodie Turner-Smith, actress and model
 * Wale, hip hop musician and rapper
 * Jessica Watkins, NASA astronaut
 * David P. Weber, former assistant inspector general for the U.S. Securities and Exchange Commission
 * James White, basketball player who played for the San Antonio Spurs and Houston Rockets
 * Frederick Yeh, biologist and animal welfare activist
 * 6ix, record producer
 * Chop Robinson defensive end for the Miami Dolphins

In popular culture

 * Part of the 2006 film Borat was filmed in Gaithersburg in 2005.
 * Part of an episode of Da Ali G Show was filmed in Gaithersburg in 2004.
 * It is mentioned by character Fox Mulder in episodes of The X-Files and as a story location.
 * Some of The Blair Witch Project was filmed in Seneca Creek State Park

title: Gaithersburg, Maryland


text:
Delaware County

Delaware County is the name of six counties in the United States:

 * Delaware County, Indiana
 * Delaware County, Iowa
 * Delaware County, New York
 * Delaware County, Ohio
 * Delaware County, Oklahoma
 * Delaware County, Pennsylvania

title: Delaware County


text:
Hamilton County

Hamilton County is the name of ten counties in the United States of America, eight of them named for Alexander Hamilton, first United States Secretary of the Treasury:

 * Hamilton County, Florida
 * Hamilton County, Illinois
 * Hamilton County, Indiana
 * Hamilton County, Iowa (named for William W. Hamilton, president of the Iowa state senate)
 * Hamilton County, Kansas, the least populous county on the list
 * Hamilton County, Nebraska
 * Hamilton County, New York, the most sparsely populated county in the eastern half of the United States
 * Hamilton County, Ohio, the most populous county on the list
 * Hamilton County, Tennessee
 * Hamilton County, Texas (named for James Hamilton Jr., Governor of South Carolina from 1830 to 1832)

title: Hamilton County


text:
Fulton County

Fulton County is the name of eight counties in the United States of America. Most are named for Robert Fulton, inventor of the first practical steamboat:
 * Fulton County, Arkansas, named after Governor William Savin Fulton
 * Fulton County, Georgia
 * Fulton County, Illinois
 * Fulton County, Indiana
 * Fulton County, Kentucky
 * Fulton County, New York
 * Fulton County, Ohio
 * Fulton County, Pennsylvania

Other uses

 * Fulton County (novel) by James Goldman
 * Atlanta–Fulton County Stadium, former home to the Atlanta Braves (1966–96) and the Atlanta Falcons (1966–91)

title: Fulton County


text:
Howard County

Howard County is the name of seven counties in the United States of America:

 * Howard County, Arkansas
 * Howard County, Indiana
 * Howard County, Iowa
 * Howard County, Maryland
 * Howard County, Missouri
 * Howard County, Nebraska
 * Howard County, Texas

title: Howard County


text:
Marion County

Marion County is the name of seventeen counties in the United States of America, mostly named for General Francis Marion:

 * Marion County, Alabama
 * Marion County, Arkansas
 * Marion County, Florida
 * Marion County, Georgia
 * Marion County, Illinois
 * Marion County, Indiana
 * Marion County, Iowa
 * Marion County, Kansas
 * Marion County, Kentucky
 * Marion County, Mississippi
 * Marion County, Missouri
 * Marion County, Ohio
 * Marion County, Oregon
 * Marion County, South Carolina
 * Marion County, Tennessee
 * Marion County, Texas
 * Marion County, West Virginia

title: Marion County


text:
Wayne County

Wayne County may refer to:

Counties in the United States

 * Wayne County, Georgia
 * Wayne County, Illinois
 * Wayne County, Indiana
 * Wayne County, Iowa
 * Wayne County, Kentucky
 * Wayne County, Michigan
 * Wayne County, Mississippi
 * Wayne County, Missouri
 * Wayne County, Nebraska
 * Wayne County, New York
 * Wayne County, North Carolina
 * Wayne County, Ohio
 * Wayne County, Pennsylvania
 * Wayne County, Tennessee
 * Wayne County, Utah
 * Wayne County, West Virginia

Entertainment

 * Wayne County & the Electric Chairs, an American punk rock band
 * Wayne County (now Jayne County), an American musician in the above band

title: Wayne County


text:
Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms.

Functions of a continuous variable
Consider two functions $$u(x)$$ and $$v(x)$$ with Fourier transforms $$U$$ and $$V$$:

 * $$\begin{align}

U(f) &\triangleq \mathcal{F}\{u\}(f) = \int_{-\infty}^{\infty}u(x) e^{-i 2 \pi f x} \, dx, \quad f \in \mathbb{R}\\ V(f) &\triangleq \mathcal{F}\{v\}(f) = \int_{-\infty}^{\infty}v(x) e^{-i 2 \pi f x} \, dx, \quad f \in \mathbb{R} \end{align}$$

where $$\mathcal{F}$$ denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically $$2\pi$$ or $$\sqrt{2\pi}$$) will appear in the convolution theorem below. The convolution of $$u$$ and $$v$$ is defined by:

 * $$r(x) = \{u*v\}(x) \triangleq \int_{-\infty}^{\infty} u(\tau) v(x-\tau)\, d\tau = \int_{-\infty}^{\infty} u(x-\tau) v(\tau)\, d\tau.$$

In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol $$\otimes$$ is sometimes used instead.

The convolution theorem states that:

Applying the inverse Fourier transform $$\mathcal{F}^{-1},$$ produces the corollary:

$$

The theorem also generally applies to multi-dimensional functions.

Consider functions $$u,v$$ in Lp-space $$L^1(\mathbb{R}^n),$$ with Fourier transforms $$U,V$$:

\begin{align} U(f) &\triangleq \mathcal{F}\{u\}(f) = \int_{\mathbb{R}^n} u(x) e^{-i 2 \pi f \cdot x} \, dx, \quad f \in \mathbb{R}^n\\ V(f) &\triangleq \mathcal{F}\{v\}(f) = \int_{\mathbb{R}^n} v(x) e^{-i 2 \pi f \cdot x} \, dx, \end{align} $$

where $$f\cdot x$$ indicates the inner product of $$\mathbb{R}^n$$:  $$f\cdot x = \sum_{j=1}^{n} {f}_j x_j,$$   and   $$dx = \prod_{j=1}^{n} d x_j.$$

The convolution of $$u$$ and $$v$$ is defined by:

 * $$r(x) \triangleq \int_{\mathbb{R}^n} u(\tau) v(x-\tau)\, d\tau.$$

Also:

 * $$\iint |u(\tau)v(x-\tau)|\,dx\,d\tau=\int \left( |u(\tau)| \int |v(x-\tau)|\,dx \right) \,d\tau = \int |u(\tau)|\,\|v\|_1\,d\tau=\|u\|_1 \|v\|_1.$$

Hence by Fubini's theorem we have that $$r\in L^1(\mathbb{R}^n)$$ so its Fourier transform $$R$$ is defined by the integral formula:

\begin{align} R(f) \triangleq \mathcal{F}\{r\}(f) &= \int_{\mathbb{R}^n} r(x) e^{-i 2 \pi f \cdot x}\, dx\\ &= \int_{\mathbb{R}^n} \left(\int_{\mathbb{R}^n} u(\tau) v(x-\tau)\, d\tau\right)\, e^{-i 2 \pi f \cdot x}\, dx. \end{align} $$

Note that $$|u(\tau)v(x-\tau)e^{-i 2\pi f \cdot x}|=|u(\tau)v(x-\tau)|,$$ Hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration):

\begin{align} R(f) &= \int_{\mathbb{R}^n} u(\tau) \underbrace{\left(\int_{\mathbb{R}^n} v(x-\tau)\ e^{-i 2 \pi f \cdot x}\,dx\right)}_{V(f)\ e^{-i 2 \pi f \cdot \tau}}\,d\tau\\ &=\underbrace{\left(\int_{\mathbb{R}^n} u(\tau)\ e^{-i 2\pi f \cdot \tau}\,d\tau\right)}_{U(f)}\ V(f). \end{align} $$

This theorem also holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform (see Mellin inversion theorem). It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.

Periodic convolution (Fourier series coefficients)
Consider $$P$$-periodic functions $$u_{_P}$$  and   $$v_{_P},$$ which can be expressed as periodic summations:

 * $$u_{_P}(x)\ \triangleq \sum_{m=-\infty}^{\infty} u(x-mP)$$  and   $$v_{_P}(x)\ \triangleq \sum_{m=-\infty}^{\infty} v(x-mP).$$

In practice the non-zero portion of components $$u$$ and $$v$$ are often limited to duration $$P,$$ but nothing in the theorem requires that.

The Fourier series coefficients are:

 * $$\begin{align}

U[k] &\triangleq \mathcal{F}\{u_{_P}\}[k] = \frac{1}{P} \int_P u_{_P}(x) e^{-i 2\pi k x/P} \, dx, \quad k \in \mathbb{Z}; \quad \quad \scriptstyle \text{integration over any interval of length } P\\ V[k] &\triangleq \mathcal{F}\{v_{_P}\}[k] = \frac{1}{P} \int_P v_{_P}(x) e^{-i 2\pi k x/P} \, dx, \quad k \in \mathbb{Z} \end{align}$$

where $$\mathcal{F}$$ denotes the Fourier series integral.

 * The pointwise product: $$u_{_P}(x)\cdot v_{_P}(x)$$ is also $$P$$-periodic, and its Fourier series coefficients are given by the discrete convolution of the $$U$$ and $$V$$ sequences:

 * $$\mathcal{F}\{u_{_P}\cdot v_{_P}\}[k] = \{U*V\}[k].$$

 * The convolution:

 * $$\begin{align}

\{u_{_P} * v\}(x)\ &\triangleq \int_{-\infty}^{\infty} u_{_P}(x-\tau)\cdot v(\tau)\ d\tau\\ &\equiv \int_P u_{_P}(x-\tau)\cdot v_{_P}(\tau)\ d\tau; \quad \quad \scriptstyle \text{integration over any interval of length } P \end{align}$$

is also $$P$$-periodic, and is called a periodic convolution.

 * $$\begin{align}

\int_{-\infty}^\infty u_{_P}(x - \tau) \cdot v(\tau)\,d\tau &= \sum_{k=-\infty}^\infty \left[\int_{x_o+kP}^{x_o+(k+1)P} u_{_P}(x - \tau) \cdot v(\tau)\ d\tau\right] \quad x_0 \text{ is an arbitrary parameter}\\ &=\sum_{k=-\infty}^\infty \left[\int_{x_o}^{x_o+P} \underbrace{u_{_P}(x - \tau-kP)}_{u_{_P}(x - \tau), \text{ by periodicity}} \cdot v(\tau + kP)\ d\tau\right] \quad \text{substituting } \tau \rightarrow \tau+kP\\ &=\int_{x_o}^{x_o+P} u_{_P}(x - \tau) \cdot \underbrace{\left[\sum_{k=-\infty}^\infty v(\tau + kP)\right]}_{\triangleq \ v_{_P}(\tau)}\ d\tau \end{align}$$

The corresponding convolution theorem is:

 * $$\begin{align}

\mathcal{F}\{u_{_P} * v\}[k] &\triangleq \frac{1}{P} \int_P \left(\int_P u_{_P}(\tau)\cdot v_{_P}(x-\tau)\ d\tau\right) e^{-i 2\pi k x/P} \, dx\\ &= \int_P u_{_P}(\tau)\left(\frac{1}{P}\int_P v_{_P}(x-\tau)\ e^{-i 2\pi k x/P} dx\right) \, d\tau\\ &= \int_P u_{_P}(\tau)\ e^{-i 2\pi k \tau/P} \underbrace{\left(\frac{1}{P}\int_P v_{_P}(x-\tau)\ e^{-i 2\pi k (x-\tau)/P} dx\right)}_{V[k], \quad \text{due to periodicity}} \, d\tau\\ &=\underbrace{\left(\int_P\ u_{_P}(\tau)\ e^{-i 2\pi k \tau/P} d\tau\right)}_{P\cdot U[k]}\ V[k]. \end{align}$$

Functions of a discrete variable (sequences)
By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now $$\mathcal{F}$$ denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences $$u[n]$$ and $$v[n]$$ with transforms $$U$$ and $$V$$:

 * $$\begin{align}

U(f) &\triangleq \mathcal{F}\{u\}(f) = \sum_{n=-\infty}^{\infty} u[n]\cdot e^{-i 2\pi f n}\;, \quad f \in \mathbb{R}, \\ V(f) &\triangleq \mathcal{F}\{v\}(f) = \sum_{n=-\infty}^{\infty} v[n]\cdot e^{-i 2\pi f n}\;, \quad f \in \mathbb{R}. \end{align}$$

The of $$u$$ and $$v$$ is defined by:

 * $$r[n] \triangleq (u * v)[n] = \sum_{m=-\infty}^\infty u[m]\cdot v[n - m] = \sum_{m=-\infty}^\infty u[n-m]\cdot v[m].$$

The convolution theorem for discrete sequences is:

Periodic convolution
$$U(f)$$ and $$V(f),$$ as defined above, are periodic, with a period of 1. Consider $$N$$-periodic sequences $$u_{_N}$$ and $$v_{_N}$$:

 * $$u_{_N}[n]\ \triangleq \sum_{m=-\infty}^{\infty} u[n-mN]$$  and   $$v_{_N}[n]\ \triangleq \sum_{m=-\infty}^{\infty} v[n-mN], \quad n \in \mathbb{Z}.$$

These functions occur as the result of sampling $$U$$ and $$V$$ at intervals of $$1/N$$ and performing an inverse discrete Fourier transform (DFT) on $$N$$ samples (see ). The discrete convolution:

 * $$\{u_{_N} * v\}[n]\ \triangleq \sum_{m=-\infty}^{\infty} u_{_N}[m]\cdot v[n-m] \equiv \sum_{m=0}^{N-1} u_{_N}[m]\cdot v_{_N}[n-m]$$

is also $$N$$-periodic, and is called a periodic convolution. Redefining the $$\mathcal{F}$$ operator as the $$N$$-length DFT, the corresponding theorem is:

And therefore:

Under the right conditions, it is possible for this $$N$$-length sequence to contain a distortion-free segment of a $$u*v$$ convolution. But when the non-zero portion of the $$u(n)$$ or $$v(n)$$ sequence is equal or longer than $$N,$$ some distortion is inevitable. Such is the case when the $$V(k/N)$$ sequence is obtained by directly sampling the DTFT of the infinitely long impulse response.

For $$u$$ and $$v$$ sequences whose non-zero duration is less than or equal to $$N,$$ a final simplification is:

This form is often used to efficiently implement numerical convolution by computer. (see and )

As a partial reciprocal, it has been shown that any linear transform that turns convolution into pointwise product is the DFT (up to a permutation of coefficients).

A time-domain derivation proceeds as follows:

\begin{align} \scriptstyle{\rm DFT}\displaystyle \{u_{_N} * v\}[k] &\triangleq \sum_{n=0}^{N-1} \left(\sum_{m=0}^{N-1} u_{_N}[m]\cdot v_{_N}[n-m]\right) e^{-i 2\pi kn/N}\\ &= \sum_{m=0}^{N-1} u_{_N}[m] \left(\sum_{n=0}^{N-1} v_{_N}[n-m]\cdot e^{-i 2\pi kn/N}\right)\\ &= \sum_{m=0}^{N-1} u_{_N}[m]\cdot e^{-i 2\pi km/N} \underbrace{ \left(\sum_{n=0}^{N-1} v_{_N}[n-m]\cdot e^{-i 2\pi k(n-m)/N}\right)}_{\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\quad \scriptstyle \text{due to periodicity}}\\ &= \underbrace{ \left(\sum_{m=0}^{N-1} u_{_N}[m]\cdot e^{-i 2\pi km/N}\right)}_{\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]} \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right). \end{align} $$

A frequency-domain derivation follows from, which indicates that the DTFTs can be written as:

\mathcal{F}\{u_{_N} * v\}(f) = \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle \{u_{_N} * v\}[k]\right)\cdot \delta\left(f-k/N\right). \quad \scriptstyle \mathsf{(Eq.5a)} $$

\mathcal{F}\{u_{_N}\}(f) = \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot \delta\left(f-k/N\right). $$

The product with $$V(f)$$ is thereby reduced to a discrete-frequency function:

\begin{align} \mathcal{F}\{u_{_N} * v\}(f) &= G_{_N}(f) V(f) \\ &= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot V(f)\cdot \delta\left(f-k/N\right)\\ &= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot V(k/N)\cdot \delta\left(f-k/N\right)\\ &= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right) \cdot \delta\left(f-k/N\right), \quad \scriptstyle \mathsf{(Eq.5b)} \end{align} $$

where the equivalence of $$V(k/N)$$ and $$\left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right)$$ follows from. Therefore, the equivalence of (5a) and (5b) requires:

 * $$\scriptstyle{\rm DFT}

\displaystyle {\{u_{_N} * v\}[k]} = \left(\scriptstyle{\rm DFT} \displaystyle\{u_{_N}\}[k]\right)\cdot \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right).$$

We can also verify the inverse DTFT of (5b):

\begin{align} (u_{_N} * v)[n] & = \int_{0}^{1} \left(\frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\cdot \delta\left(f-k/N\right)\right)\cdot e^{i 2 \pi f n} df \\ & = \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\cdot \underbrace{\left(\int_{0}^{1} \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\right)}_{\text{0, for} \ k\ \notin\ [0,\ N)} \\ & = \frac{1}{N} \sum_{k=0}^{N-1} \bigg(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\bigg)\cdot e^{i 2 \pi \frac{n}{N} k}\\ &=\ \scriptstyle{\rm DFT}^{-1} \displaystyle \bigg( \scriptstyle{\rm DFT}\displaystyle \{u_{_N}\}\cdot \scriptstyle{\rm DFT}\displaystyle \{v_{_N}\} \bigg). \end{align} $$

Convolution theorem for inverse Fourier transform
There is also a convolution theorem for the inverse Fourier transform:

Here, "$$\cdot$$" represents the Hadamard product, and "$$*$$" represents a convolution between the two matrices.

 * $$\begin{align}

&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\ &\mathcal{F}\{u \cdot v\}= \mathcal{F}\{u\}*\mathcal{F}\{v\} \end{align}$$

so that

 * $$\begin{align}

&u*v= \mathcal{F}^{-1}\left\{\mathcal{F}\{u\}\cdot\mathcal{F}\{v\}\right\}\\ &u \cdot v= \mathcal{F}^{-1}\left\{\mathcal{F}\{u\}*\mathcal{F}\{v\}\right\} \end{align}$$

Convolution theorem for tempered distributions
The convolution theorem extends to tempered distributions. Here, $$v$$ is an arbitrary tempered distribution:

 * $$\begin{align}

&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\ &\mathcal{F}\{\alpha \cdot v\}= \mathcal{F}\{\alpha\}*\mathcal{F}\{v\}. \end{align}$$

But $$u = F\{\alpha\}$$ must be "rapidly decreasing" towards $$-\infty$$ and $$+\infty$$ in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if $$\alpha = F^{-1}\{u\}$$ is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.

In particular, every compactly supported tempered distribution, such as the Dirac delta, is "rapidly decreasing". Equivalently, bandlimited functions, such as the function that is constantly $$1$$ are smooth "slowly growing" ordinary functions. If, for example, $$v\equiv\operatorname{\text{Ш}}$$ is the Dirac comb both equations yield the Poisson summation formula and if, furthermore, $$u\equiv\delta$$ is the Dirac delta then $$\alpha \equiv 1$$ is constantly one and these equations yield the Dirac comb identity.

Additional resources
For a visual representation of the use of the convolution theorem in signal processing, see:

 * Johns Hopkins University's Java-aided simulation: http://www.jhu.edu/signals/convolve/index.html

Faltung (Mathematik) Produit de convolution

title: Convolution theorem


text:
Ramstein Air Base

Ramstein Air Base is a United States Air Force base located in Rhineland-Palatinate, a state in southwestern Germany. It serves as the headquarters for the United States Air Forces in Europe – Air Forces Africa (USAFE-AFAFRICA) and NATO Allied Air Command (AIRCOM). Ramstein is near the town of Ramstein-Miesenbach, which is situated outside the base's west gate, in the rural district of Kaiserslautern. The base supports forward elements deploying to Eastern Europe and Africa.

The construction of the air base was a project designed and undertaken by the French Army and the U.S. Army Corps of Engineers from 1949 to 1952.

Ramstein AB is part of the Kaiserslautern Military Community (KMC), where more than 54,000 American service members and over 5,400 U.S. civilian employees live and work. U.S. organizations in the KMC also employ the services of more than 6,200 German workers. Air Force units in the KMC alone employ almost 9,800 military members, who are accompanied by nearly 11,100 family members. There are more than 16,200 military personnel, U.S. civilians, and U.S. contractors assigned to Ramstein AB alone.

The east gate of Ramstein Air Base is approximately 16 km from Kaiserslautern (locally referred to by Americans as "K-Town"). Other nearby civilian communities include Landstuhl, which is about 5 km from the west gate.

Development and establishment
In 1940, construction of today's Bundesautobahn 6 was stopped when a bridge that was being built across the Rhine River near Mannheim collapsed, leaving a section of autobahn that could not be used. A part of the unused autobahn to the west of Mannheim, near Kaiserslautern, was used as an airstrip by the Luftwaffe. The airstrip was also used by the advancing U.S. Army Air Forces during the final months of World WarII. The old autobahn section is still used as the access road to the east and west gates of the base and the A6 was rebuilt south of the air base after the war.

During the initial postwar era, the USAAF repaired several former Luftwaffe airfields in Bavaria, part of the American occupation zone of Germany. With the advent of the Berlin Blockade and the chilling of relations with the Soviet Union, by 1948, it became obvious to United States Air Force planners that these bases were tactically untenable because of their proximity to the East German and Czechoslovakian borders.

With the creation of NATO in response to Cold War tensions in Europe in 1949, USAFE wanted its vulnerable fighter units in West Germany moved west of the Rhine River to provide greater air defense warning time. France agreed to provide air base sites within their zone of occupation in the Rhineland-Palatinate as part of the NATO expansion program.

Construction of the modern USAF base near Kaiserslautern began in April 1948 under the provisions of a Franco-American reciprocal agreement. Two separate, but adjoining bases were designed. A headquarters base for Twelfth Air Force, along with several NATO organizations, designated as Ramstein Air Station; and an operational fighter base, designated as Landstuhl Air Base. What is today known as Kisling Memorial Drive would separate the two facilities.

The area was a swamp that had to be built up by two meters (six feet). A train line was laid out from Einsiedlerhof-Kaiserslautern in a yoke shape around to the current base and back down to the Landstuhl spur in 1948, by agreement of the U.S. and French Occupational Forces. Trainloads of earth were moved over the line and spread over the base's current area to raise it to its current level. Once the ground was level, construction work began. Two bases were laid out. Landstuhl Air Base on the south side and Ramstein Air Station (station, no airstrip) on the north. From 1948 to the opening of the bases in 1953, it was the largest one spot construction site in Europe employing over 270,000 workers at one time.

Enough construction was completed in mid-1952 that LandstuhlAB was opened on 5 August. Its facilities included a runway, dispersal hardstands, a control tower, ramps, and other flight-related facilities and the associated flying and support units. On, Det1, 86th Fighter-Bomber Wing arrived at LandstuhlAB from Neubiberg Air Base near Munich.

On, Ramstein Air Station was opened. Ramstein was the location of headquarters, Twelfth Air Force, and supported family housing, base exchange, commissary, dependents' schools and other administrative offices for the WAFs (Women's Air Force). The barracks that were built at Ramstein AS were used to house WAFs and single women that worked as U.S. Government employees at both RamsteinAS and LandstuhlAB. On, Headquarters, Twelfth Air Force was activated on Ramstein Air Base, having moved from its joint facilities with HQ USAFE at Wiesbaden AB. What was not generally known at the time and not made public until after the end of the Cold War in 1993, was the desire to have HQTwelfth Air Force in close proximity to the Air Defense Operations Center (ADOC)Kindsbach, a.k.a. 'Kindsbach Cave'the site of NATO's underground combat operations center.

Operational history
The 86th Air Base Group was activated as the main base support unit for Landstuhl while the 7030th HQ Support Group was the main base support unit for Ramstein. On, the two bases were consolidated into the largest NATO-controlled air base in service on the continent. It was called "Ramstein–Landstuhl Air Base", but later, after the West German government continued construction of the A6 autobahn from Kaiserslautern to Saarbrücken, the autobahn cut off access at the south of the base which is where the main gate was in the city limit of Landstuhl. The main gate was moved to the west side of the base which was in the town of Ramstein. The two bases were joined and the current Kisling Memorial Drive cut off to the public which made one base. In 1961, the base was officially named "Ramstein Air Base".

One legacy of the two separate air bases is that the north side of Ramstein retained a separate APO from the south side. The north side (Ramstein AB) is APOAE09012, while the south side (LandstuhlAB) is APOAE09009. Also separate Combat Support Groups, the 7030th for the north side, and the 86th for the south side existed. These were consolidated in the 1980s, and the two Combat Support units were merged into the 377th Combat Support Wing. There is still a north and south side Fitness Centers. The current northside Community Center before housed the WAF NCO Club. As well, there were two Movie Theaters on the North side and two on the South side. Currently, only two still stand on the north side: a remodeled Nightingale Theater (known before as the Four Corners Theater) on the corner across from the Base gas Station, and the north side AAFES dry cleaners that was known as the Ramstein Rocket Theater. On the South side, there are the current Hercules Theater (Falcon Theater) next to HQ 86th Air Wing and a non-existent theater for which new barracks are currently under construction.

Near the Ramstein Air Base is the Landstuhl Regional Medical Center (LRMC), operated by the United States Army. Although part of the Kaiserslautern Military Community, LRMC has a separate history and was never a part of Ramstein or Landstuhl Air Bases, although both facilities have utilized the medical facilities at LRMC, since they were established in 1953. Currently there are plans on the drawing board from the U.S. Department of Defense to build a new Medical Center on the current U.S. Army Weilerbach Storage Installation just to the east of RamsteinAB. Construction is to be completed in and around 2024.. It will be a twelve-story facility to house all departments of LRMC and the current Ramstein AB Clinic along with Dental Clinic facilities for the whole KMC. In turn, the East Gate to RamsteinAB will be extended from its current location to just off the Autobahn 6 Einsiedlerhof exit to the base at what is known as the Elvis Gate.

From 2004 to 2006, Ramstein Air Base underwent an extensive expansion with a major construction project – including an all-new airport terminal, among other new facilities, through the so-called Rhein-Main Transition Program which was initiated in support of the total closure of Rhein-Main Air Base on and transferring all its former capacities to Ramstein Air Base (70%) and Spangdahlem Air Base (30%).

Ramstein served as temporary housing for the United States men's national soccer team during the 2006World Cup.

Name changes

 * Landstuhl Air Base,
 * Ramstein Air Base,
 * Landstuhl and Ramstein were separate bases until

 * Ramstein–Landstuhl Air Base,
 * Ramstein Air Base, present

Major USAF units assigned
Source: Fletcher, Air Force Bases, Volume II

 * 86th Air Base Group, 5 April 195214 November 1968
 * 86th Wing (various designations), 21 August 195218 November 1960; 31 January 1973present
 * (86th Air Base Group (later under various designations) assigned as subordinate unit, 21 August 1952)
 * 86th Air Division, 18 November 196014 November 1968

 * 7030th Combat Support Group, 6 April 19531 May 1960
 * 7030th Air Base Wing, 5 October 196614 June 1985

 * Twelfth Air Force, 27 April 19531 January 1958
 * 7486th Air Defense Group, 1 May 195425 September 1957
 * 322d Air Division, 22 March 195412 August 1955
 * 322d Airlift Division, 23 June 19781 April 1992

 * 7455th Tactical Intelligence Wing, 1 September 19851 July 1992
 * Seventeenth Air Force, 15 November 19597 October 1972; 1 October 200824 April 2012

 * HQ, Atlantic Air Rescue Center (various designations), 8 October 196130 June 1973
 * 26th Tactical Reconnaissance Wing, 5 October 196631 January 1973
 * HQ, USAFE, 10 March 1973present
 * 306th Strategic Wing, 15 August 197630 June 1978
 * 7th Air Division, 1 July 19781 February 1992
 * 316th Air Division, 14 June 19851 May 1991
 * NHQ119 Civil Air PatrolRamstein Cadet SquadronUnit 3395, 1984present
 * 377th Combat Support Wing, 14 June 19851 May 1991
 * 435th Air Base Wing, 15 January 200416 July 2009
 * 435th Air Ground Operations Wing, 16 July 2009–present
 * 38th Combat Support Wing, 24 May 200530 June 2007
 * 521st Air Mobility Operations Wing, 4 September 2008present

Major U.S. Army units assigned
Source: Fletcher, Air Force Bases, Volume II
 * 21st TSC / 39th Movement Control BN. (2008Present)
 * USAREUR Movement Control Team / AMC Logistic Center
 * USAREUR Overseas Replacement CenterContingency Operations / AMC Passenger Terminal

86th Wing
Reassigned from Neubiberg Air Base, West Germany in 1952 and except for a period between 1968 and 1973, the 86th Wing, under various designations, has been the main operational and host unit at Ramstein Air Base.

Throughout the 1950s, the 86th was primarily a Fighter-Bomber Wing. In 1960, it was realigned to an air defense mission and became the 86th Air Division (Defense). The 86th AD was inactivated in 1968. Returning as an F-4 Phantom II Tactical Fighter Wing in 1973, the 86th TFW performed that mission until 1994, deploying components to the Middle East during the 1990 Gulf War.

On, the Strategic Air Command 306th Strategic Wing was activated at Ramstein with a KC-135 air refueling and an RC-135 reconnaissance mission. The 306th also functioned as the focal point for all SAC operations in Europe and as liaison between SAC and USAFE. The wing moved to RAF Mildenhall, England on.

In June 1985, the 316th Air Division was activated, centralizing command authority at Ramstein. The 86 TFW became the division's flight operations arm, while the newly formed 377th Combat Support Wing, also activated in 1985, became responsible for the logistical and administrative support on base, replacing the 86th and 7030 Combat Support Wings. On, Ramstein Air Base was the site of the tragic Ramstein airshow disaster, which killed 72 spectators and three pilots, and injured hundreds.

After the Cold War, the 86th was realigned to become the 86th Airlift Wing. On, the 55th Aeromedical Airlift Squadron moved from the 435thAW at Rhein-Main Air Base Germany to Ramstein. On 1 October, the 75th and 76th Airlift Squadron arrived at Ramstein from the 60th AW at Travis Air Force Base California, and 437thAW at Charleston AFB South Carolina, respectively. A year later on, the 37thAirlift Squadron was transferred to Ramstein from Rhein-Main.

In 1999, the activation of the 86th Contingency Response Group brought the airfield and aerial port operations and provision of force protection at contingency airfields mission to the wing.

On, the 38th Combat Support Wing was activated to enhance support to USAFE geographically separated units. This wing was inactivated in 2007. The 521st Air Mobility Operations Wing was activated on. The wing is the headquarters for the existing 721st Air Mobility Operations Group at Ramstein and the 521st AMOG at Naval Station Rota, Spain. The 521st AMOW provides an enhanced level of control for the AMC route structure in Europe, which includes critical locations for getting people, cargo and patients to and from current war zones.

By 1984, the Kindsbach Cave had become too small and its cost for renovation too high. The USAFE vacated the facility and, on, control was returned to the German government and the German government returned the facility to the original owner of the land. Today, the Kindsbach Cave is private property, through tours of the cave can be arranged. The cave is overgrown by vegetation, trees, and new housing.

The new 521st Air Mobility Operations Wing stood up on. , the commander of the 521st AMOW is Colonel Adrienne Williams.

In December 2023 a new Space Force component stands up at Ramstein Air Base.

26th Tactical Reconnaissance Wing
On, French President Charles de Gaulle announced that France would withdraw from NATO's integrated military structure. The United States was informed that it must remove its military forces from France by.

As a result, the 26th Tactical Reconnaissance Wing, based at Toul-Rosieres Air Base, France, and two of its squadrons, the 38th and 32d, equipped with the RF-4C Phantom II, were relocated to Ramstein on.

Assigned squadrons of the 26th TRW at Ramstein were:
 * 38th Tactical Reconnaissance (RF-4C, Tail Code: RR)
 * 526th Fighter Interceptor/Tactical Fighter (F-102/F-4E (1970) Tail Code: RS)
 * 7th Special Operation (C-130, C-47, UH-1)

While at Ramstein, the 26th TRW acquired a number of other units with different flying missions. One function gained by the 26 TRW, almost immediately after arriving at Ramstein, was the maintenance and flying of the HQ USAFE liaison aircraft. In addition, the Wing was responsible for flying members of the HQ USAFE staff to Air Force and NATO bases throughout Europe. In addition, the 26th TRW was only designated as a flight, because of its small size. It consisted of a mixture of aircraft, including: T-29s, T-33s, T-39s, C-54s, O-2s, H-19s, and UH-1s.

In 1971 a detachment of the 630th Military Airlift Support Squadron from Rhein-Main Air Base was assigned to Ramstein and a large cargo aerial port constructed. This allowed Military Airlift Command C-141 and C-5 Galaxy aircraft to use Ramstein as a transshipment point for material, which was then moved within USAFE by C-130 tactical transports.

In the spring of 1972, the 7th Special Operations Squadron (SOS) was assigned flying C-130Es, C-47As, and UH-1Ns. Because of the special operations mission of the 7 SOS, it reported directly to HQ USAFE for operational control.

As part of operation "Creek Action", a command-wide effort to realign functions and streamline operations, HQ USAFE transferred the 26th TRW from Ramstein to Zweibrücken Air Base and the 86th Tactical Fighter Wing was reassigned from Zweibrücken to Ramstein on.

NATO command center
From its inception, Ramstein was designed as a NATO command base. In 1957, Ramstein provided support for NATO's HQ Fourth Allied Tactical Air Force, which moved to Ramstein from Trier Air Base on upon the closure of that facility. Also on that date, HQ Twelfth Air Force was transferred to Bergstrom Air Force Base, Texas, and was assigned to Tactical Air Command. It was replaced by HQ Seventeenth Air Force (USAFE) which was moved from North Africa. In turn, the 17th AF was replaced by its mother unit HQ USAFE from Lindsey Air Station, Wiesbaden, West Germany in 1973. The HQ 17th AF was moved to Sembach AB at that time and controlled all USAF Air Divisions and Wings north of the Alps, with the exception of the British Isles and Scandinavia, which were controlled by HQ 3rd AF at Mildenhall.

On, several headquarters were relocated into and out of Ramstein, when SeventeenthAF moved to Sembach Air Base to make room for the expected move of HQUSAFE to Ramstein. This entire operation, code-named "Creek Action", was carried out as part of the USAF's new worldwide policy of locating the most vital headquarters in thinly populated rural areas rather than near cities. Later, HQ USAFE was moved, due to the fact that US Intelligence found that the Soviets had plans to invade Western Europe through the Fulda Gap in West Germany. The military thought to move vital HQs on the other side of the Rhein River for protection.

As a result of this policy change, Ramstein air base became a large multi-national NATO center: in addition to the USAFE's headquarters, it also housed the new NATO headquarters of the Allied Air Forces Central Europe (AAFCE).

The AAFCE also commanded the 2nd Allied Tactical Air Force (2ATAF) and the 4th ATAF. The 4th ATAF, which had been headquartered at Ramstein for many years, included the 1st Canadian Air Group, 1st and 2nd Divisions of the West German Air Force, and units of the USAFE's 3rd and 17th Air Force.

HQ USAFE fully completed its move from Wiesbaden to Ramstein in early 1991.

With USAFE's arrival in 1973, Ramstein entered a period of expansion. The duel commander of the 316th AD / 86 TFW became host commander of Americans living in the Kaiserslautern Military Community instead of the US Army 21st Commanding General. The Wiesbaden USAF Community was then traded to the US Army Control as for an even Kaiserlautern switch. The KMC from the 1950s to the early 1990s had an average population of Americans of 110,000, outnumbering those Germans in the City of Kaiserslautern for that period.

Allied Air Forces Central Europe was established at Ramstein on. Ramstein subsequently provided support for other headquarters including the 322nd Airlift Division that arrived on, and SAC's 7th Air Division that arrived on.

In December 1980, HQ Fourth Allied Tactical Air Force was moved from Ramstein to Heidelberg and co-located with HQ Central Army Group.

In 1984, an enlisted airman (Sgt Darrel Dietlein), assigned to the 1964th Communications Group, solicited National Headquarters Civil Air Patrol to charter the first "Cadet Squadron" in West Germany, naming the unit "Ramstein Cadet Squadron" and becoming the unit's first commander as a CAP First Lieutenant. The Ramstein Cadet Squadron was formed with Captain Mark Bailey serving as the unit's first liaison officer, as well as other like minded military volunteers and roughly six cadets. To this day, the squadron enjoys vibrant member participation, as well as base support, hosting the European Encampments along with their traditional military studies, cadet programs and aerospace education efforts. The Ramstein Cadet Squadron commander is 1st Lt Cody Chenowith. The squadron is the central hub for all CAP units in Europe. Membership was 55 members. In the subsequent years, a companion cadet squadron was formed at Spangdahlem Air Base. Distance learning cadets are located at SHAPE, Belgium and Hohenfels, Germany. Additionally, the Ramstein Cadet Squadron is the second oldest and continuous operating unit on the installation and even predating the 86th Air Wing.

Today, the base is home to the Allied Air Command, which is responsible to Joint Force Command Brunssum, the only and main NATO command unit on Ramstein AB.

ADOC Kindsbach
Close to Ramstein was the site of Air Defense Operations Center (ADOC)Kindsbach, AKA 'Kindsbach Cave' – the site of Europe's underground combat operations center.

The facility was located in a former German western front command headquarters. The French took control of the underground bunker after World War II, and USAFE assumed control in 1953. After major renovations, USAFE opened the center on.

The center was a state-of-the-art, 67-room, 37000 sqft facility where USAFE could have led an air war against the Soviet Union. The center had a digital computer to work out bombing problems, cryptographic equipment for coded message traffic and its own photo lab to develop reconnaissance photos. Responsible for an air space extending deep behind the Iron Curtain, the center interacted directly with The Pentagon, NATO, Supreme Headquarters Allied Powers Europe and all USAFE bases. With its massive telephone switchboard and 80 teletype machines, the cave was plugged into everything in the outside world. The center was receiving more than 1,000 calls a day.

As a further measure of protection, the cave was fully self-contained with its own water supply, electric backup-generators, climate controls, dining facilities and sleeping accommodations for its 125-man crew. Visitor passes were rarely issued to this secret facility.

Throughout the years, leadership changed but USAFE led the operations through numbered Air Forces. The center's commander was the USAFE Advanced Echelon. The glassed-in office was on the top floor of the three-story underground command center. Directly under the office was the management for offensive air operations. And the bottom floor office was the management for defensive air operations – to include support for U.S. Army forces and West German Civil Defense. All three offices had a full view of the massive Air Operations Center map on the opposing wall.

The AOC was the largest room in the complex. Its three-story map was used to plot minute-by-minute movements of friendly and unidentified aircraft. But the center was much more than just a tracking station, because it could also react to threats. They always knew the current operational status of air weapons in theater including missiles, and could dispatch armed response "at a moment's notice".

By the early 1960s, the manual plotting system used to track aircraft at the cave and elsewhere throughout West Germany was too slow and inaccurate for the quick responses necessary. Beginning in 1962, airmen trained in the new 412L air weapons control system began to arrive in Germany and at the cave. Over the next year, the new GE semi-automatic system was installed. When complete at the cave, the current air picture over East and West Germanies, as well as parts of the eastern Soviet bloc countries, was displayed on a 40 × screen with radar information provided by various 412L sites located throughout West Germany. Senior U.S. staff monitored the dynamic display 24/7. Over the next several years, additional 412L sites throughout Germany joined the network, until the manual system had been totally replaced.

Drone war control center
In April 2015, Ramstein Air Base was reported by German and international media as an important control center in the drone war staged under the Obama administration against targets in areas like Pakistan, Yemen, Afghanistan and Somalia. The German government claimed not to have been informed about this function of the U.S. base.

In a TV and online documentary, the German Das Erste channel cited 2014 reports from Norddeutscher Rundfunk, WDR, and the Süddeutsche Zeitung that revealed Ramstein to be an important hub in the drone war against terror suspects. New data, provided by Edward Snowden affiliate Glenn Greenwald, supported these reports with classified documents from inside the U.S. administration and were also presented in the Citizenfour video documentary. The revelation of US drone activities from Ramstein lead to nationwide anti-drone protests under the banner of "Stop Ramstein Air Base".

In 2019, three Yemenis who lost relatives in a 2012 US drone strike took legal action against the German government for aiding the breaking of international law by the United States, by tolerating these operations from Ramstein. The German Higher Administrative Court in Münster ruled that the German government must take appropriate measures to control whether the US army follows international law at Ramstein Air Base. However, the possibilities of Germany to control US activities on their territory are very limited as the United States have jurisdiction over Ramstein Air Base.

Illegal arms and munition transports
In 2015, the Serbian newspaper Večernje novosti reported about Ramstein Air Base being used by the United States Armed Forces to transport arms and munitions to Syria. At the end of 2017, an anonymous U.S. official stated that the U.S. does, indeed, use Ramstein Air Base to supplement Syrian rebels with arms and munition. The German Federal Government stated that it did not grant the necessary permits for these transports, nor that it had been informed about them. The Kaiserslautern public prosecutor's office did not initiate any official investigations. She checked generally accessible sources, on which some of the press articles mentioned were based, and obtained information from the authorities involved in monitoring foreign trade. According to the review, there was "no evidence of concrete arms transports to Syria via Ramstein". However, the journalistic conclusion cannot be ruled out, according to the prosecution authority. Previous investigations of the Ramstein Air Base, such as the Abu Omar case, have proven to be unsuccessful. If the investigations into the arms and ammunition transports had been successful, this would have constituted a violation of the German Kriegswaffenkontrollgesetz (War Weapons Control Act) and the Foreign Trade and Payments Act.

Afghanistan Evacuation
In late Summer 2021, Ramstein Air Base became the transfer point for thousands of Afghan civilians fleeing Afghanistan following the fall of Kabul back into Taliban hands as U.S. and NATO forces withdrew from the country. Civilian airlines under the Civil Reserve Air Fleet, such as Delta, American, Eastern, and United were allowed to land at Ramstein and serve as flights to take Afghan families and other supporters of the U.S. and NATO forces to new lives in the United States. Those who had to stay on the base were cared for in makeshift living centers which were set up to tend to their needs until they could be processed and flown to the United States.

Russo-Ukrainian War
On 26 April 2022, Ramstein Air Base hosted a meeting of the International Advisory Group on Ukraine's Defense and Counteraction to Russia, convened by US Secretary of Defense Lloyd Austin to synchronize and coordinate Ukraine's military assistance in the war with Russia. The event was attended by heads of defense agencies of 42 countries. The meeting was attended by Minister of Defense of Ukraine Oleksii Reznikov.

The participating countries agreed on financial assistance to Ukraine and the supply of "heavy" weapons. They also allowed Ukraine to strike at Russian strategic sites with Western weapons. The next meetings are planned to be held monthly.

An RAAF E-7A Wedgetail is based at Ramstein from October 2023, for a six-month deployment to supplement NATO AEW&C.

Airfield
Ramstein Air Base consists of two runways09/27 and 08/26two large aprons, one near a hangar north of Runways 27 and 26, and one to the north of 09 and 08. The north-western apron also has a small passenger terminal with two jetways. This means the air base is capable of joint-use operations, although currently there are no scheduled airlines running flights to and from Ramstein.

Schools
There are four schools at Ramstein Air Base: Ramstein Elementary School (grades PreK–2), Ramstein Intermediate School (grades 3–5), Ramstein American Middle School (grades 6–8), and Ramstein High School (grades 9–12). All of these schools are run by DoDDS, a component of DoDEA.

Role and operations

The host unit is the 86th Airlift Wing (86AW), commanded by Brigadier General Otis C. Jones. The 86th Airlift Wing is composed of six groups, 30 squadrons and four bases in Germany, Spain, the Azores, and Belgium. Its mission is the operation and maintenance of airlift assets consisting of C-130Js, C-21s, and C-37A Gulfstream aircraft throughout Europe, Africa and the Middle East.

Also at Ramstein is the 435th Air Ground Operations Wing (formerly the 435th Air Base Wing) (435AGOW), which focuses on base-support responsibilities within the KMC. It is composed of five groups and 20 squadrons. The wing provides rapid mobility and agile combat support for military operations, and maintains expeditionary forces and infrastructure.

, the commander of the 435th AGOW is Colonel Bryan T. Callahan.

Ramstein's wings are assigned to the headquarters 3rd Air Force also based at RamsteinAB that controls most of the USAF Wings throughout Europe.

There is often a Summer Camp to Ramstein from British CCF (RAF) and ATC cadets, as well as Civil Air Patrol encampments and tours like the ones held in July 2015 and June 2016.

Based units
Flying and notable non-flying units based at Ramstein Air Base.

Units marked GSU are Geographically Separate Units, which although based at Ramstein, are subordinate to a parent unit based at another location.

United States Air Force
United States Air Forces in Europe – Air Forces Africa (USAFE-AFAFRICA)
 * Headquarters United States Air Forces in EuropeAir Forces Africa
 * Third Air Force
 * Headquarters Third Air Force
 * 86th Airlift Wing
 * 86th Operations Group
 * 37th Airlift SquadronC-130J Hercules
 * 76th Airlift SquadronC-21A, C-37A
 * 86th Aeromedical Evacuation Squadron
 * 86th Operational Support Squadron
 * 86th Civil Engineer Group
 * 786th Civil Engineer Squadron
 * 86th Logistics Readiness Group
 * 86th Maintenance Group
 * 86th Medical Group
 * 86th Mission Support Group
 * 786th Force Support Squadron
 * 406th Air Expeditionary Wing
 * 435th Air Ground Operations Wing
 * 4th Air Support Operations Group
 * 2nd Air Support Operations Squadron
 * 7th Combat Weather Squadron
 * 435th Air and Space Communications Group
 * 1st Air & Space Communications Operations Squadron
 * 1st Combat Communications Squadron
 * 1st Communications Maintenance Squadron
 * 435th Contingency Response Group
 * 435th Construction and Training Squadron
 * 435th Contingency Response Squadron
 * 435th Contingency Response Support Squadron
 * 435th Security Forces Squadron

Air Combat Command (ACC)
 * Sixteenth Air Force
 * 480th Intelligence, Surveillance and Reconnaissance Wing
 * 693rd Intelligence, Surveillance and Reconnaissance Group (GSU)
 * 24th Intelligence Squadron
 * 402nd Intelligence Squadron
 * 450th Intelligence Squadron
 * 693rd Intelligence Support Squadron

Air Mobility Command (AMC)
 * United States Air Force Expeditionary Center
 * 521st Air Mobility Operations Wing
 * 721st Air Mobility Operations Group
 * 10th Expeditionary Aeromedical Evacuation Flight
 * 313th Expeditionary Operations Support Squadron
 * 721st Aerial Port Squadron
 * 721st Aircraft Maintenance Squadron

United States Army
United States Army Europe and Africa (USAREUR-AF)
 * 21st Theater Sustainment Command
 * Ramstein Army Reception Center (RARC)

NATO
Supreme Headquarters Allied Powers Europe (SHAPE)
 * Allied Air Command (AIRCOM)

Fictional entities

 * Lois Lane was born at Ramstein Air Base despite the base opening in 1948 and Lane debuting in 1938.

In films

 * Ramstein was the location where Colonel Masters is taken after being rescued by his son in Iron Eagle (1986).
 * Ramstein was the location of the aborted landing of Air Force One when it is hijacked by a group of terrorists in Air Force One (1997). Rickenbacker Air National Guard Base stood in for Ramstein as the film was shot mostly in the state of Ohio.
 * Ramstein was the location where Ethan Hunt and his team are given a second chance to retrieve stolen plutonium in Mission: Impossible – Fallout (2018)

In games

 * Ramstein Air Base appears in the video game Tom Clancy's EndWar as a possible battlefield. In the game, NATO has since collapsed, and the base is controlled by the European Federation.
 * In Call of Duty: Modern Warfare 3, Ramstein Air Base suffers a surprise invasion by Russian Ground Forces after a gas attack.
 * In Wargame: European Escalation, one mission of the NATO campaign takes place in the base.

In literature

 * Both Ramstein Air Force Base and the Ramstein air disaster figure as plot points in Donna Leon's second Guido Brunetti novel, Death in a Strange Country (1993)
 * The base was mentioned in Walter Dean Myers' book Sunrise over Fallujah.

In music

 * Rammstein, a German metal band formed in 1994, have stated that they take their name from the Ramstein air show disaster; in turn, the asteroid 110393 Rammstein is named after the band. The band's self-titled song (on the album Herzeleid (1995)) refers to the event.

In television

 * In The West Wing episode "Memorial Day", Donna Moss is flown to Ramstein to be treated at Landstuhl Regional Medical Center nearby.
 * In The West Wing episode "Red Haven's on Fire", Air Force Veteran Leo McGarry refers to "Ramstein Air Force Base".
 * In the pilot of Homeland, Marine Sergeant Nicholas Brody is debriefed in Ramstein after being rescued from an al-Qaeda base in Iraq.
 * In the Madam Secretary episode "The Seventh Floor", journalist Colin Mitchell is flown to Ramstein after being released by the Sudanese government.
 * In Law & Order: Special Victims Unit, chief medical examiner Melinda Warner reveals in season 7's "Blast" that she served on the base in the U.S. Air Force during the Gulf War.

Red Army Faction car bombing,
The U.S. Air Forces in Europe (USAFE) headquarters became the target of a bombing attack at 07:21 on, carried out by a Red Army Faction commando called 'Sigurd Debus'. A total of 20 victims were injured, some seriously.

Ramstein air show disaster,
The Ramstein air show disaster was a mid-air collision that occurred during the Flugtag '88 air show on Sunday,, killing 70 people.

C-5 crash,
On, a C-5 Galaxy Transport plane was carrying U.S. servicemen to the Persian Gulf when the thrust reverser on one of the engines of the large jet suddenly deployed, causing it to crash shortly after takeoff, killing 13 people and injuring 4 others.

Environmental scandal
In 2014, it was revealed that poisonous extinguishing foams (PFCs) were used on Ramstein Air Base and other U.S. air bases in the region. These are now contaminating lakes, rivers and the ground water in the region. In one river, the contamination was 7700 times higher than the safety limit set by the European Union. These contaminations are linked to cancer and birth defects.

title: Ramstein Air Base