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Broadway (Manhattan)

Broadway is a road in the U.S. state of New York. Broadway runs from the south at State Street at Bowling Green for 13 mi through the borough of Manhattan, over the Broadway Bridge, and 2 mi through the Bronx, exiting north from New York City to run an additional 18 mi through the Westchester County municipalities of Yonkers, Hastings-On-Hudson, Dobbs Ferry, Irvington, Tarrytown, and Sleepy Hollow, after which the road continues, but is no longer called "Broadway". The latter portion of Broadway north of the George Washington Bridge/I-95 underpass comprises a portion of U.S. Route 9.

It is the oldest north–south main thoroughfare in New York City, with much of the current street allegedly beginning as the Wickquasgeck trail before the arrival of Europeans. This then formed the basis for one of the primary thoroughfares of the Dutch New Amsterdam colony, which continued under British rule, although most of it did not bear its current name until the late 19th century. Some portions of Broadway in Manhattan are interrupted for continuous vehicle traffic, including Times Square, Herald Square, and Union Square, and instead used as pedestrian-only plazas. South of Columbus Circle, the road is one-way going southbound.

Broadway in Manhattan is known widely as the heart of the American commercial theatrical industry, and is used as a metonym for it, as well as in the names of alternative theatrical ventures such as Off-Broadway and Off-off-Broadway.

Colonial history
Broadway was originally the Wickquasgeck trail, carved into the brush of Manhattan by its Native American inhabitants. This trail originally snaked through swamps and rocks along the length of Manhattan Island.

Upon the arrival of the Dutch, the trail was widened and soon became the main road through the island from Nieuw Amsterdam at the southern tip. The Dutch explorer and entrepreneur David Pietersz. de Vries gives the first mention of it in his journal for the year 1642 ("the Wickquasgeck Road over which the Indians passed daily"). The Dutch called it the Heeren Wegh or Heeren Straat, meaning "Gentlemen's Way" or "Gentlemen's Street" – echoing the name of a similar street in Amsterdam – or "High Street" or "the Highway"; it was renamed "Broadway" after the British took over the city, because of its unusual width. Although currently the name of the street is simply "Broadway", in a 1776 map of New York City, it is labeled as "Broadway Street".

18th century
In the 18th century, Broadway ended at the town commons north of Wall Street. The part of Broadway in what is now Lower Manhattan was initially known as Great George Street. Traffic continued up the East Side of the island via Eastern Post Road and the West Side via Bloomingdale Road, which opened in 1703, continued up to 117th Street and contributed to the development of the modern Upper West Side into an upscale area with mansions.

In her 1832 book Domestic Manners of the Americans, Fanny Trollope wrote of her impressions of New York City in general and of Broadway in particular:

"This noble street may vie with any I ever saw, for its length and breadth, its handsome shops, neat awnings, excellent trottoir, and well-dressed pedestrians. It has not the crowded glitter of Bond Street equipages, nor the gorgeous fronted palaces of Regent Street; but it is magnificent in its extent, and ornamented by several handsome buildings, some of them surrounded by grass and trees."

19th century
In 1868, Bloomingdale Road between 59th Street (at the Grand Circle, now Columbus Circle) and 155th Streets would be paved and widened, becoming an avenue with landscaped medians. It was called "Western Boulevard" or "The Boulevard". An 1897 official map of the city shows a segment of what is now Broadway as "Kingsbridge Road" in the vicinity of Washington Heights.

On February 14, 1899, the name "Broadway" was extended to the entire Broadway / Bloomingdale / Boulevard / Kingsbridge complex.

20th century
In the 20th century, a 30-block stretch of Broadway, extending mainly between Times Square at 42nd Street and Sherman Square at 72nd Street, formed part of Manhattan's "Automobile Row". Before the first decade of the 20th century, the area was occupied mostly by equestrian industries and was "thoroughly lifeless", but by 1907, The New York Times characterized this section of Broadway as having "almost a solid line of motor vehicle signs all the way from Times Square to Sherman Square". In the late 1900s and early 1910s, several large automobile showrooms, stores, and garages were built on Broadway, including the U.S. Rubber Company Building at 58th Street, the B.F. Goodrich showroom at 1780 Broadway (between 58th and 57th Streets), the Fisk Building at 250 West 57th Street, and the Demarest and Peerless Buildings at 224 West 57th Street.

Broadway once was a two-way street for its entire length. The present status, in which it runs one-way southbound south of Columbus Circle (59th Street), came about in several stages. On June 6, 1954, Seventh Avenue became southbound and Eighth Avenue became northbound south of Broadway. None of Broadway became one-way, but the increased southbound traffic between Columbus Circle (Eighth Avenue) and Times Square (Seventh Avenue) caused the city to re-stripe that section of Broadway for four southbound and two northbound lanes. Broadway became one-way from Columbus Circle south to Herald Square (34th Street) on March 10, 1957, in conjunction with Sixth Avenue becoming one-way from Herald Square north to 59th Street and Seventh Avenue becoming one-way from 59th Street south to Times Square (where it crosses Broadway). On June 3, 1962, Broadway became one-way south of Canal Street, with Trinity Place and Church Street carrying northbound traffic.

Another change was made on November 10, 1963, when Broadway became one-way southbound from Herald Square to Madison Square (23rd Street) and Union Square (14th Street) to Canal Street, and two routes – Sixth Avenue south of Herald Square and Centre Street, Lafayette Street, and Fourth Avenue south of Union Square – became one-way northbound. Finally, at the same time as Madison Avenue became one-way northbound and Fifth Avenue became one-way southbound, Broadway was made one-way southbound between Madison Square (where Fifth Avenue crosses) and Union Square on January 14, 1966, completing its conversion south of Columbus Circle.

21st century
In 2001, a one-block section of Broadway between 72nd Street and 73rd Street at Verdi Square was reconfigured. Its easternmost lanes, which formerly hosted northbound traffic, were turned into a public park when a new subway entrance for the 72nd Street station was built in the exact location of these lanes. Northbound traffic on Broadway is now channeled onto Amsterdam Avenue to 73rd Street, makes a left turn on the three-lane 73rd Street, and then a right turn on Broadway shortly afterward.

In August 2008, two traffic lanes from 42nd to 35th Streets were taken out of service and converted to public plazas. Bike lanes were added on Broadway from 42nd Street to Union Square.

Since May 2009, the portions of Broadway through Duffy Square, Times Square, and Herald Square have been closed entirely to automobile traffic, except for cross traffic on the Streets and Avenues, as part of a traffic and pedestrianization experiment, with the pavement reserved exclusively for walkers, cyclists, and those lounging in temporary seating placed by the city. The city decided that the experiment was successful, and decided to make the change permanent in February 2010. Though the anticipated benefits to traffic flow were not as large as hoped, pedestrian injuries dropped dramatically and foot traffic increased in the designated areas; the project was popular with both residents and businesses. The current portions converted into pedestrian plazas are between West 47th and 42nd Streets within Times and Duffy Squares, and between West 35th and 33rd Streets in the Herald Square area. Additionally, portions of Broadway in Madison Square and Union Square have been dramatically narrowed, allowing ample pedestrian plazas to exist along the side of the road.

2010s
A terrorist attempted to set off a bomb on Broadway in Times Square on May 1, 2010. The attempted bomber was sentenced to life in prison.

In May 2013, the NYCDOT decided to redesign Broadway between 35th and 42nd Streets for the second time in five years, owing to poor connections between pedestrian plazas and decreased vehicular traffic. With the new redesign, the bike lane is now on the right side of the street; it was formerly on the left side adjacent to the pedestrian plazas, causing conflicts between pedestrian and bicycle traffic.

In spring 2017, as part of a capital reconstruction of Worth Square, Broadway between 24th and 25th Streets was converted to a shared street, where through vehicles are banned and delivery vehicles are restricted to 5 mph. Delivery vehicles go northbound from Fifth Avenue to 25th Street for that one block, reversing the direction of traffic and preventing vehicles from going south on Broadway south of 25th Street. The capital project expands on a 2008 initiative where part of the intersection of Broadway and Fifth Avenue was repurposed into a public plaza, simplifying that intersection. As part of the 2017 project, Worth Square was expanded, converting the adjoining block of Broadway into a "shared street".

In September 2019, the pedestrian space in the Herald Square area was expanded between 33rd and 32nd Streets alongside Greeley Square. Five blocks of Broadway—from 50th to 48th, 39th to 39th, and 23rd to 21st Street—were converted into shared streets in late 2021. The block between 40th and 39th Streets, known as Golda Meir Square, was closed to vehicular traffic at that time.

2020s
During 2020, the section from 31st to 25th Street was converted to a temporary pedestrian-only street called NoMad Piazza as part of the New York City Department of Transportation's Open Streets program. Following the success of the pedestrian-only street, the Flatiron/23rd Street Partnership BID closed the section between 25th and 27th Streets to vehicular traffic again during 2021 and 2022.

City officials announced in March 2023 that the section of Broadway between 32nd and 21st Streets would be redesigned as part of a project called Broadway Vision. The section between 32nd and 25th Streets would receive a bidirectional bike lane and would be converted to a shared street. Cars would be banned permanently from 27th to 25th Street. That work was finished the same July. In March 2024, the DOT announced plans to convert the section between 17th and 21st Streets into a shared street.

Route description
Broadway runs the length of Manhattan Island, roughly parallel to the North River (the portion of the Hudson River bordering Manhattan), from Bowling Green at the south to Inwood at the northern tip of the island. South of Columbus Circle, it is a one-way southbound street. Since 2009, vehicular traffic has been banned at Times Square between 47th and 42nd Streets, and at Herald Square between 35th and 33rd Streets as part of a pilot program; the right-of-way is intact and reserved for cyclists and pedestrians. From the northern shore of Manhattan, Broadway crosses Spuyten Duyvil Creek via the Broadway Bridge and continues through Marble Hill (a discontiguous portion of the borough of Manhattan) and the Bronx into Westchester County. U.S. 9 continues to be known as Broadway until its junction with NY 117.

Lower Manhattan
The section of lower Broadway from its origin at Bowling Green to City Hall Park is the historical location for the city's ticker-tape parades, and is sometimes called the "Canyon of Heroes" during such events. West of Broadway, as far as Canal Street, was the city's fashionable residential area until c. 1825; landfill has more than tripled the area, and the Hudson River shore now lies far to the west, beyond Tribeca and Battery Park City.

Broadway marks the boundary between Greenwich Village to the west and the East Village to the east, passing Astor Place. It is a short walk from there to New York University near Washington Square Park, which is at the foot of Fifth Avenue. A bend in front of Grace Church allegedly avoids an earlier tavern; from 10th Street it begins its long diagonal course across Manhattan, headed almost due north.

Midtown Manhattan
Because Broadway preceded the grid that the Commissioners' Plan of 1811 imposed on the island, Broadway crosses midtown Manhattan diagonally, intersecting with both the east–west streets and north–south avenues. Broadway's intersections with avenues, marked by "squares" (some merely triangular slivers of open space), have induced some interesting architecture, such as the Flatiron Building.

At Union Square, Broadway crosses 14th Street, merges with Fourth Avenue, and continues its diagonal uptown course from the Square's northwest corner; Union Square is the only location wherein the physical section of Broadway is discontinuous in Manhattan (other portions of Broadway in Manhattan are pedestrian-only plazas). At Madison Square, the location of the Flatiron Building, Broadway crosses Fifth Avenue at 23rd Street, thereby moving from the east side of Manhattan to the west, and is discontinuous to vehicles for a one-block stretch between 24th and 25th Streets. At Greeley Square (West 32nd Street), Broadway crosses Sixth Avenue (Avenue of the Americas), and is discontinuous to vehicles until West 35th Street. Macy's Herald Square department store, one block north of the vehicular discontinuity, is located on the northwest corner of Broadway and West 34th Street and southwest corner of Broadway and West 35th Street; it is one of the largest department stores in the world.

One famous stretch near Times Square, where Broadway crosses Seventh Avenue in midtown Manhattan, is the home of many Broadway theatres, housing an ever-changing array of commercial, large-scale plays, particularly musicals. This area of Manhattan is often called the Theater District or the Great White Way, a nickname originating in the headline "Found on the Great White Way" in the February 3, 1902, edition of the New York Evening Telegram. The journalistic nickname was inspired by the millions of lights on theater marquees and billboard advertisements that illuminate the area. After becoming the city's de facto red-light district in the 1960s and 1970s (as can be seen in the films Taxi Driver and Midnight Cowboy), since the late 1980s Times Square has emerged as a family tourist center, in effect being Disneyfied following the company's purchase and renovation of the New Amsterdam Theatre on 42nd Street in 1993.

The New York Times, from which the Square gets its name, was published at offices at 239 West 43rd Street; the paper stopped printing papers there on June 15, 2007.

Upper West Side
At the southwest corner of Central Park, Broadway crosses Eighth Avenue (called Central Park West north of 59th Street) at West 59th Street and Columbus Circle; on the site of the former New York Coliseum convention center is the new shopping center at the foot of the Time Warner Center, headquarters of Time Warner. From Columbus Circle northward, Broadway becomes a wide boulevard to 169th Street; it retains landscaped center islands that separate northbound from southbound traffic. The medians are a vestige of the central mall of "The Boulevard" that had become the spine of the Upper West Side, and many of these contain public seating.

Broadway intersects with Columbus Avenue (known as Ninth Avenue south of West 59th Street) at West 65th and 66th Streets where the Juilliard School and Lincoln Center, both well-known performing arts landmarks, as well as the Manhattan New York Temple of the Church of Jesus Christ of Latter-day Saints are located.

Between West 70th and 73rd Streets, Broadway intersects with Amsterdam Avenue (known as 10th Avenue south of West 59th Street). The wide intersection of the two thoroughfares has historically been the site of numerous traffic accidents and pedestrian casualties, partly due to the long crosswalks. Two small triangular plots of land were created at points where Broadway slices through Amsterdam Avenue. One is a tiny fenced-in patch of shrubbery and plants at West 70th Street called Sherman Square (although it and the surrounding intersection have also been known collectively as Sherman Square), and the other triangle is a lush tree-filled garden bordering Amsterdam Avenue from just above West 72nd Street to West 73rd Street. Named Verdi Square in 1921 for its monument to Italian composer Giuseppe Verdi, which was erected in 1909, this triangular sliver of public space was designated a Scenic Landmark by the Landmarks Preservation Commission in 1974, one of nine city parks that have received the designation. In the 1960s and 1970s, the area surrounding both Verdi Square and Sherman Square was known by local drug users and dealers as "Needle Park", and was featured prominently in the gritty 1971 dramatic film The Panic in Needle Park, directed by Jerry Schatzberg and starring Al Pacino in his second onscreen role.

The original brick and stone shelter leading to the entrance of the 72nd Street subway station, one of the first 28 subway stations in Manhattan, remains located on one of the wide islands in the center of Broadway, on the south side of West 72nd Street. For many years, all traffic on Broadway flowed on either side of this median and its subway entrance, and its uptown lanes went past it along the western edge of triangular Verdi Square. In 2001 and 2002, renovation of the historic 72nd Street station and the addition of a second subway control house and passenger shelter on an adjacent center median just north of 72nd Street, across from the original building, resulted in the creation of a public plaza with stone pavers and extensive seating, connecting the newer building with Verdi Square, and making it necessary to divert northbound traffic to Amsterdam Avenue for one block. While Broadway's southbound lanes at this intersection were unaffected by the new construction, its northbound lanes are no longer contiguous at this intersection. Drivers can either continue along Amsterdam Avenue to head uptown or turn left on West 73rd Street to resume traveling on Broadway.

Several notable apartment buildings are in close proximity to this intersection, including The Ansonia, its ornate architecture dominating the cityscape here. After the Ansonia first opened as a hotel, live seals were kept in indoor fountains inside its lobby. Later, it was home to the infamous Plato's Retreat nightclub. Immediately north of Verdi Square is the Apple Bank Building, formerly the Central Savings Bank, which was built in 1926 and designed to resemble the Federal Reserve Bank of New York. Broadway is also home to the Beacon Theatre at West 74th Street, designated a national landmark in 1979 and still in operation as a concert venue after its establishment in 1929 as a vaudeville and music hall, and "sister" venue to Radio City Music Hall.

At its intersection with West 78th Street, Broadway shifts direction and continues directly uptown and aligned approximately with the Commissioners' grid. Past the bend are the historic Apthorp apartment building, built in 1908, and the First Baptist Church in the City of New York, incorporated in New York in 1762, its current building on Broadway erected in 1891. The road heads north and passes historically important apartment houses such as the Belnord, the Astor Court Building, and the Art Nouveau Cornwall.

At Broadway and 95th Street is Symphony Space, established in 1978 as home to avant-garde and classical music and dance performances in the former Symphony Theatre, which was originally built in 1918 as a premier "music and motion-picture house". At 99th Street, Broadway passes between the controversial skyscrapers of the Ariel East and West.

At 107th Street, Broadway merges with West End Avenue, with the intersection forming Straus Park with its Titanic Memorial by Augustus Lukeman.

Northern Manhattan and the Bronx
Broadway then passes the campus of Columbia University at 116th Street in Morningside Heights, in part on the tract that housed the Bloomingdale Insane Asylum from 1808 until it moved to Westchester County in 1894. Still in Morningside Heights, Broadway passes the park-like campus of Barnard College. Next, the Gothic quadrangle of Union Theological Seminary, and the brick buildings of the Jewish Theological Seminary of America with their landscaped interior courtyards, face one another across Broadway. On the next block is the Manhattan School of Music.

Broadway then runs past the Manhattanville campus of Columbia University, and the main campus of CUNY–City College near 135th Street; the Gothic buildings of the original City College campus are out of sight, a block to the east. Also to the east are the brownstones of Hamilton Heights. Hamilton Place is a surviving section of Bloomingdale Road, and originally the address of Alexander Hamilton's house, The Grange, which has been moved.

Broadway achieves a verdant, park-like effect, particularly in the spring, when it runs between the uptown Trinity Church Cemetery and the former Trinity Chapel, now the Church of the Intercession near 155th Street.

NewYork–Presbyterian Hospital lies on Broadway near 166th, 167th, and 168th Streets in Washington Heights. The intersection with St. Nicholas Avenue at 167th Street forms Mitchell Square Park. At 178th Street, US 9 becomes concurrent with Broadway.

Broadway crosses the Harlem River on the Broadway Bridge to Marble Hill. Afterward, it then enters the Bronx, where it is the eastern border of Riverdale and the western border of Van Cortlandt Park. At 253rd Street, NY 9A joins with US 9 and Broadway. (NY 9A splits off Broadway at Ashburton Avenue in Yonkers.)

Westchester County
The northwestern corner of the park marks the New York City limit and Broadway enters Westchester County in Yonkers, where it is now known as South Broadway. It trends ever westward, closer to the Hudson River, remaining a busy urban commercial street. In downtown Yonkers, it drops close to the river, becomes North Broadway and 9A leaves via Ashburton Avenue. Broadway climbs to the nearby ridgetop runs parallel to the river and the railroad, a few blocks east of both as it passes St. John's Riverside Hospital. The neighborhoods become more residential and the road gently undulates along the ridgetop. In Yonkers, Broadway passes the historic Philipse Manor house, which dates back to colonial times.

It remains Broadway as it leaves Yonkers for Hastings-on-Hudson, where it splits into separate north and south routes for 0.6 mi. The trees become taller and the houses, many separated from the road by stone fences, become larger. Another National Historic Landmark, the John William Draper House, was the site of the first astrophotograph of the Moon.

In the next village, Dobbs Ferry, Broadway has various views of the Hudson River while passing through the residential section. Broadway passes by the Old Croton Aqueduct and nearby the shopping district of the village. After intersecting with Ashford Avenue, Broadway passes Mercy College, then turns left again at the center of town just past South Presbyterian Church, headed for equally comfortable Ardsley-on-Hudson and Irvington. Villa Lewaro, the home of Madam C. J. Walker, the first African-American millionaire, is along the highway here. At the north end of the village of Irvington, a memorial to writer Washington Irving, after whom the village was renamed, marks the turnoff to his home at Sunnyside. Entering into the southern portion of Tarrytown, Broadway passes by historic Lyndhurst mansion, a massive mansion built along the Hudson River built in the early 1800s.

North of here, at the Kraft Foods technical center, the Tappan Zee Bridge becomes visible. After crossing under the Thruway and I-87 again, here concurrent with I-287, and then intersecting with the four-lane NY 119, where 119 splits off to the east, Broadway becomes the busy main street of Tarrytown. Christ Episcopal Church, where Irving worshiped, is along the street. Many high-quality restaurants and shops are along this main road. This downtown ends at the eastern terminus of NY 448, where Broadway slopes off to the left, downhill, and four signs indicate that Broadway turns left, passing the Old Dutch Church of Sleepy Hollow, another NHL. The road then enters Sleepy Hollow (formerly North Tarrytown), passing the visitors' center for Kykuit, the National Historic Landmark that was (and partially still is) the Rockefeller family's estate. Broadway then passes the historic Sleepy Hollow Cemetery, which includes the resting place of Washington Irving and the setting for "The Legend of Sleepy Hollow".

Broadway expands to four lanes at the trumpet intersection with NY 117, where it finally ends and U.S. 9 becomes Albany Post Road (and Highland Avenue) at the northern border of Sleepy Hollow, New York.

Canyon of Heroes
Canyon of Heroes is occasionally used to refer to the section of lower Broadway in the Financial District that is the location of the city's ticker-tape parades. The traditional route of the parade is northward from Bowling Green to City Hall Park. Most of the route is lined with tall office buildings along both sides, affording a view of the parade for thousands of office workers who create the snowstorm-like jettison of shredded paper products that characterize the parade.

While typical sports championship parades have been showered with some 50 tons of confetti and shredded paper, the V-J Day parade on August 14–15, 1945 – marking the end of World War II – was covered with 5,438 tons of paper, based on estimates provided by the New York City Department of Sanitation.

More than 200 black granite strips embedded in the sidewalks along the Canyon of Heroes list honorees of past ticker-tape parades.

Great White Way
"The Great White Way" is a nickname for a section of Broadway in Midtown Manhattan, specifically the portion that encompasses the Theater District, between 42nd and 53rd Streets, and encompassing Times Square.

In 1880, a stretch of Broadway between Union Square and Madison Square was illuminated by Brush arc lamps, making it among the first electrically lighted streets in the United States. By the 1890s, the portion from 23rd Street to 34th Street was so brightly illuminated by electrical advertising signs, that people began calling it "The Great White Way". When the theater district moved uptown, the name was transferred to the Times Square area.

The phrase "Great White Way" has been attributed to Shep Friedman, columnist for the New York Morning Telegraph in 1901, who lifted the term from the title of a book about the Arctic by Albert Paine. The headline "Found on the Great White Way" appeared in the February 3, 1902, edition of the New York Evening Telegram.

A portrait of Broadway in the early part of the 20th century and "The Great White Way" late at night appeared in "Artist In Manhattan" (1940) written by the artist-historian Jerome Myers:

"Early morn on Broadway, the same light that tips the mountain tops of the Colorado canyons gradually discloses the quiet anatomy, the bare skeletons of the huge iron signs that trellis the sky, now denuded of the attractions of the volcanic night. Almost lifeless, the tired entertainers of the night clubs and their friends straggle to their rooms, taximen compare notes and earnings, the vast street scene has had its curtain call, the play is over.

Dear old Broadway, for many years have I dwelt on your borders. I have known the quiet note of your dawn. Even earlier I would take my coffee at Martin's, at 54th Street–now, alas, vanished–where I would see creatures of the night life before they disappeared with the dawn.

One night a celebrated female impersonator came to the restaurant in all his regalia, directly from a club across the street. Several taximen began to poke fun at him. Unable any longer to bear their taunts, he got up and knocked all the taximen out cold. Then he went back to the club, only to lament under his bitter tears, "See how they've ruined my dress!"

Gone are the old-time Broadway oyster bars and chop houses that were the survivors of a tradition of their sporting patrons, the bon vivants of Manhattan. Gone are the days when the Hoffman House flourished on Madison Square, with its famous nudes by Bouguereau; when barrooms were palaces, on nearly every corner throughout the city; when Steve Brodie, jumping from Brooklyn Bridge, splashed the entire country with publicity; when Bowery concert halls dispensed schooners of beer for a nickel, with a stage show thrown in; when Theis's Music Hall still resounded on 14th Street with its great mechanical organ, the wonder of its day, a place of beauty, with fine paintings and free company and the frankest of female life. Across the street was Tammany Hall, and next to it Tony Pastor's, where stars of the stage were born. Tony himself, in dress clothes and top hat, sang his ballads, a gallant trouper introducing Lillian Russell and others to fame through his audience."

Transportation
From south to north, Broadway at one point or another runs over or under various New York City Subway lines, including the IRT Lexington Avenue Line, the BMT Broadway Line, IRT Broadway–Seventh Avenue Line, and IND Eighth Avenue Line (the IND Sixth Avenue Line is the only north–south trunk line in Manhattan that does not run along Broadway).
 * The IRT Lexington Avenue Line runs under Broadway from Bowling Green to Fulton Street.
 * The BMT Broadway Line runs under it from City Hall to Times Square–42nd Street.
 * The IRT Broadway–Seventh Avenue Line runs under and over Broadway from Times Square to 168th Street, and again from 218th Street to its terminal in the Bronx at Van Cortlandt Park–242nd Street.
 * The northern portion of the IND Eighth Avenue Line runs under Broadway from Dyckman Street to Inwood–207th Street.

Early street railways on Broadway included the Broadway and Seventh Avenue Railroad's Broadway and University Place Line (1864?) between Union Square (14th Street) and Times Square (42nd Street), the Ninth Avenue Railroad's Ninth and Amsterdam Avenues Line (1884) between 65th Street and 71st Street, the Forty-second Street, Manhattanville and St. Nicholas Avenue Railway's Broadway Branch Line (1885?) between Times Square and 125th Street, and the Kingsbridge Railway's Kingsbridge Line north of 169th Street. The Broadway Surface Railroad's Broadway Line, a cable car line, opened on lower Broadway (below Times Square) in 1893, and soon became the core of the Metropolitan Street Railway, with two cable branches: the Broadway and Lexington Avenue Line and Broadway and Columbus Avenue Line.

These streetcar lines were replaced with bus routes in the 1930s and 1940s. Before Broadway became one-way, the main bus routes along it were the New York City Omnibus Company's (NYCO) 6 (Broadway below Times Square), 7 (Broadway and Columbus Avenue), and 11 (Ninth and Amsterdam Avenues), and the Surface Transportation Corporation's M100 (Kingsbridge) and M104 (Broadway Branch). Additionally, the Fifth Avenue Coach Company's (FACCo) 4 and 5 used Broadway from 135th Street north to Washington Heights, and their 5 and 6 used Broadway between 57th Street and 72nd Street. With the implementation of one-way traffic, the northbound 6 and 7 were moved to Sixth Avenue.

, Broadway is served by the M4 (ex-FACCo 4), M7 (ex-NYCO 7), M55, M100, and M104. Other routes that use part of Broadway include the M5 (ex-FACCo 5), M10, M20, M60 Select Bus Service, Bx7, Bx9, and Bx20.

Bee-Line buses also serves Broadway within Riverdale and Westchester County. Routes 1, 2, 3, 4, 6, 13, and several others run on a portion of Broadway.

Notable buildings

Broadway is lined with many famous and otherwise noted and historic buildings, such as:
 * 2 Broadway
 * 280 Broadway (also known as the Marble Palace, the A.T. Stewart Company Store, or The Sun Building)
 * Alexander Hamilton U.S. Custom House (1 Bowling Green, between the two legs of the southern end of Broadway)
 * American Surety Building (100 Broadway)
 * Ansonia Hotel (2109 Broadway)
 * Bowling Green Fence and Park (between 25 and 26 Broadway)
 * Bowling Green Offices Building (11 Broadway)
 * Brill Building (1619 Broadway)
 * Corbin Building (196 Broadway)
 * Cunard Building (25 Broadway)
 * Dyckman House (4881 Broadway)
 * Equitable Building (120 Broadway)
 * Flatiron Building (Fifth Avenue and Broadway at 23rd Street)
 * Gilsey House (1200 Broadway)
 * Gorham Manufacturing Company Building (889-91 Broadway)
 * Home Life Building (253, 256 Broadway)
 * International Mercantile Marine Company Building (1 Broadway)
 * Morgan Stanley Building (1585 Broadway)
 * One Times Square (1475 Broadway)
 * Paramount Building (1501 Broadway)
 * Standard Oil Building (26 Broadway)
 * Trinity Church (79 Broadway)
 * Union Theological Seminary (3041 Broadway)
 * United Palace (4140 Broadway)
 * United States Lines-Panama Pacific Lines Building (1 Broadway)
 * Winter Garden Theatre (1634 Broadway)
 * Woolworth Building (233 Broadway)

Historic buildings on Broadway that are now demolished include:
 * Appleton Building
 * Alexander Macomb House
 * Barnum's American Museum
 * Equitable Life Building
 * Grand Central Hotel (673 Broadway)
 * Mechanics' Hall
 * Metropolitan Opera House, from 1883 to 1966, between 39th and 40th Streets
 * Singer Tower (Liberty Street and Broadway)
 * St. Nicholas Hotel

title: Broadway (Manhattan)


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Bilinear transform

The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.

The bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often used for converting a transfer function $$ H_a(s) $$ of a linear, time-invariant (LTI) filter in the continuous-time domain (often named an analog filter) to a transfer function $$H_d(z)$$ of a linear, shift-invariant filter in the discrete-time domain (often named a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the $$ j \omega $$ axis, $$ \mathrm{Re}[s]=0 $$, in the s-plane to the unit circle, $$ |z| = 1 $$, in the z-plane. Other bilinear transforms can be used for warping the frequency response of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays $$ \left( z^{-1} \right) $$ with first order all-pass filters.

The transform preserves stability and maps every point of the frequency response of the continuous-time filter, $$ H_a(j \omega_a) $$ to a corresponding point in the frequency response of the discrete-time filter, $$ H_d(e^{j \omega_d T}) $$ although to a somewhat different frequency, as shown in the Frequency warping section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency. The change in frequency is barely noticeable at low frequencies but is quite evident at frequencies close to the Nyquist frequency.

Discrete-time approximation
The bilinear transform is a first-order Padé approximant of the natural logarithm function that is an exact mapping of the z-plane to the s-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the Z transform of the discrete-time sequence with the substitution of

\begin{align} z &= e^{sT}  \\ &= \frac{e^{sT/2}}{e^{-sT/2}} \\ &\approx \frac{1 + s T / 2}{1 - s T / 2} \end{align} $$

where $$ T $$ is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation; or, in other words, the sampling period. The above bilinear approximation can be solved for $$ s $$ or a similar approximation for $$ s = (1/T) \ln(z) $$ can be performed.

The inverse of this mapping (and its first-order bilinear approximation) is

\begin{align} s &= \frac{1}{T} \ln(z) \\ &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5  + \frac{1}{7} \left( \frac{z-1}{z+1} \right)^7 + \cdots \right] \\ &\approx \frac{2}{T} \frac{z - 1}{z + 1} \\ &= \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}} \end{align} $$

The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, $$ H_a(s) $$

 * $$s \leftarrow \frac{2}{T} \frac{z - 1}{z + 1}.$$

That is

 * $$H_d(z) = H_a(s) \bigg|_{s = \frac{2}{T} \frac{z - 1}{z + 1}}= H_a \left( \frac{2}{T} \frac{z-1}{z+1} \right). \ $$

Stability and minimum-phase property preserved
A continuous-time causal filter is stable if the poles of its transfer function fall in the left half of the complex s-plane. A discrete-time causal filter is stable if the poles of its transfer function fall inside the unit circle in the complex z-plane. The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane. Thus, filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.

Likewise, a continuous-time filter is minimum-phase if the zeros of its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.

Transformation of a General LTI System
A general LTI system has the transfer function $$   H_a(s) = \frac{b_0 + b_1s + b_2s^2 + \cdots + b_Qs^Q}{a_0 + a_1s + a_2s^2 + \cdots + a_Ps^P} $$ The order of the transfer function $N$ is the greater of $P$ and $Q$ (in practice this is most likely $P$ as the transfer function must be proper for the system to be stable). Applying the bilinear transform $$   s = K\frac{z - 1}{z + 1} $$ where $K$ is defined as either $2/T$ or otherwise if using frequency warping, gives $$   H_d(z) = \frac{b_0 + b_1\left(K\frac{z - 1}{z + 1}\right) + b_2\left(K\frac{z - 1}{z + 1}\right)^2 + \cdots + b_Q\left(K\frac{z - 1}{z + 1}\right)^Q} {a_0 + a_1\left(K\frac{z - 1}{z + 1}\right) + a_2\left(K\frac{z - 1}{z + 1}\right)^2 + \cdots + b_P\left(K\frac{z - 1}{z + 1}\right)^P} $$ Multiplying the numerator and denominator by the largest power of $(z + 1)^{−1}$ present, $(z + 1)^{-N}$, gives $$  H_d(z) = \frac{b_0(z+1)^N + b_1K(z-1)(z+1)^{N-1} + b_2K^2(z-1)^2(z+1)^{N-2} + \cdots + b_QK^Q(z-1)^Q(z+1)^{N-Q}} {a_0(z+1)^N + a_1K(z-1)(z+1)^{N-1} + a_2K^2(z-1)^2(z+1)^{N-2} + \cdots + a_PK^P(z-1)^P(z+1)^{N-P}} $$ It can be seen here that after the transformation, the degree of the numerator and denominator are both $N$.

Consider then the pole-zero form of the continuous-time transfer function $$   H_a(s) = \frac{(s - \xi_1)(s - \xi_2) \cdots (s - \xi_Q)}{(s - p_1)(s - p_2) \cdots (s - p_P)} $$ The roots of the numerator and denominator polynomials, $ξ_{i}$ and $p_{i}$, are the zeros and poles of the system. The bilinear transform is a one-to-one mapping, hence these can be transformed to the z-domain using $$   z = \frac{K + s}{K - s} $$ yielding some of the discretized transfer function's zeros and poles $ξ'_{i}$ and $p'_{i}$ $$   \begin{aligned} \xi'_i &= \frac{K + \xi_i}{K - \xi_i} \quad 1 \leq i \leq Q \\ p'_i &= \frac{K + p_i}{K - p_i}    \quad 1 \leq i \leq P    \end{aligned} $$ As described above, the degree of the numerator and denominator are now both $N$, in other words there is now an equal number of zeros and poles. The multiplication by $(z + 1)^{-N}$ means the additional zeros or poles are $$   \begin{aligned} \xi'_i &= -1 \quad Q < i \leq N \\ p'_i &= -1 \quad P < i \leq N   \end{aligned} $$ Given the full set of zeros and poles, the z-domain transfer function is then $$   H_d(z) = \frac{(z - \xi'_1)(z - \xi'_2) \cdots (z - \xi'_N)} {(z - p'_1)(z - p'_2) \cdots (z - p'_N)} $$

Example
As an example take a simple low-pass RC filter. This continuous-time filter has a transfer function

 * $$\begin{align}

H_a(s) &= \frac{1/sC}{R+1/sC} \\ &= \frac{1}{1 + RC s}. \end{align}$$

If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for $$s$$ the formula above; after some reworking, we get the following filter representation:

 * $$H_d(z) \ $$
 * $$ =H_a \left( \frac{2}{T} \frac{z-1}{z+1}\right) \ $$
 * $$= \frac{1}{1 + RC \left( \frac{2}{T} \frac{z-1}{z+1}\right)} \ $$
 * $$= \frac{1 + z}{(1 - 2 RC / T) + (1 + 2RC / T) z} \ $$
 * $$= \frac{1 + z^{-1}}{(1 + 2RC / T) + (1 - 2RC / T) z^{-1}}. \ $$
 * }
 * $$= \frac{1 + z}{(1 - 2 RC / T) + (1 + 2RC / T) z} \ $$
 * $$= \frac{1 + z^{-1}}{(1 + 2RC / T) + (1 - 2RC / T) z^{-1}}. \ $$
 * }
 * $$= \frac{1 + z^{-1}}{(1 + 2RC / T) + (1 - 2RC / T) z^{-1}}. \ $$
 * }
 * $$= \frac{1 + z^{-1}}{(1 + 2RC / T) + (1 - 2RC / T) z^{-1}}. \ $$
 * }

The coefficients of the denominator are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients used for implementing a real-time digital filter.

Transformation for a general first-order continuous-time filter
It is possible to relate the coefficients of a continuous-time, analog filter with those of a similar discrete-time digital filter created through the bilinear transform process. Transforming a general, first-order continuous-time filter with the given transfer function

 * $$H_a(s) = \frac{b_0 s + b_1}{a_0 s + a_1} = \frac{b_0 + b_1 s^{-1}}{a_0 + a_1 s^{-1}}$$

using the bilinear transform (without prewarping any frequency specification) requires the substitution of

 * $$s \leftarrow K \frac{1 - z^{-1}}{1 + z^{-1}}$$

where

 * $$K \triangleq \frac{2}{T} $$.

However, if the frequency warping compensation as described below is used in the bilinear transform, so that both analog and digital filter gain and phase agree at frequency $$\omega_0$$, then

 * $$K \triangleq \frac{\omega_0}{\tan\left(\frac{\omega_0 T}{2}\right)} $$.

This results in a discrete-time digital filter with coefficients expressed in terms of the coefficients of the original continuous time filter:
 * $$H_d(z)=\frac{(b_0 K + b_1) + (-b_0 K + b_1)z^{-1}}{(a_0 K + a_1) + (-a_0 K + a_1)z^{-1}}$$

Normally the constant term in the denominator must be normalized to 1 before deriving the corresponding difference equation. This results in

 * $$H_d(z)=\frac{\frac{b_0 K + b_1}{a_0 K + a_1} + \frac{-b_0 K + b_1}{a_0 K + a_1}z^{-1}}{1 + \frac{-a_0 K + a_1}{a_0 K + a_1}z^{-1}}. $$

The difference equation (using the Direct form I) is

y[n] = \frac{b_0 K + b_1}{a_0 K + a_1} \cdot x[n] + \frac{-b_0 K + b_1}{a_0 K + a_1} \cdot x[n-1] - \frac{-a_0 K + a_1}{a_0 K + a_1} \cdot y[n-1] \. $$

General second-order biquad transformation
A similar process can be used for a general second-order filter with the given transfer function

 * $$H_a(s) = \frac{b_0 s^2 + b_1 s + b_2}{a_0 s^2 + a_1 s + a_2} = \frac{b_0 + b_1 s^{-1} + b_2 s^{-2}}{a_0 + a_1 s^{-1} + a_2 s^{-2}} \ . $$

This results in a discrete-time digital biquad filter with coefficients expressed in terms of the coefficients of the original continuous time filter:
 * $$H_d(z)=\frac{(b_0 K^2 + b_1 K + b_2) + (2b_2 - 2b_0 K^2)z^{-1} + (b_0 K^2 - b_1 K + b_2)z^{-2}}{(a_0 K^2 + a_1 K + a_2) + (2a_2 - 2a_0 K^2)z^{-1} + (a_0 K^2 - a_1 K + a_2)z^{-2}}$$

Again, the constant term in the denominator is generally normalized to 1 before deriving the corresponding difference equation. This results in

 * $$H_d(z)=\frac{\frac{b_0 K^2 + b_1 K + b_2}{a_0 K^2 + a_1 K + a_2} + \frac{2b_2 - 2b_0 K^2}{a_0 K^2 + a_1 K + a_2}z^{-1} + \frac{b_0 K^2 - b_1 K + b_2}{a_0 K^2 + a_1 K + a_2}z^{-2}}{1 + \frac{2a_2 - 2a_0 K^2}{a_0 K^2 + a_1 K + a_2}z^{-1} + \frac{a_0 K^2 - a_1 K + a_2}{a_0 K^2 + a_1 K + a_2}z^{-2}}. $$

The difference equation (using the Direct form I) is

y[n] = \frac{b_0 K^2 + b_1 K + b_2}{a_0 K^2 + a_1 K + a_2} \cdot x[n] + \frac{2b_2 - 2b_0 K^2}{a_0 K^2 + a_1 K + a_2} \cdot x[n-1] + \frac{b_0 K^2 - b_1 K + b_2}{a_0 K^2 + a_1 K + a_2} \cdot x[n-2] - \frac{2a_2 - 2a_0 K^2}{a_0 K^2 + a_1 K + a_2} \cdot y[n-1] - \frac{a_0 K^2 - a_1 K + a_2}{a_0 K^2 + a_1 K + a_2} \cdot y[n-2] \. $$

Frequency warping
To determine the frequency response of a continuous-time filter, the transfer function $$ H_a(s) $$ is evaluated at $$s = j \omega_a $$ which is on the $$ j \omega $$ axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function $$ H_d(z) $$ is evaluated at $$z = e^{ j \omega_d T} $$ which is on the unit circle, $$ |z| = 1 $$. The bilinear transform maps the $$ j \omega $$ axis of the s-plane (which is the domain of $$ H_a(s) $$) to the unit circle of the z-plane, $$ |z| = 1 $$ (which is the domain of $$ H_d(z) $$), but it is not the same mapping $$ z = e^{sT} $$ which also maps the $$ j \omega $$ axis to the unit circle. When the actual frequency of $$ \omega_d $$ is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency, $$ \omega_a $$, for the continuous-time filter that this $$ \omega_d $$ is mapped to.

 * $$H_d(z) = H_a \left( \frac{2}{T} \frac{z-1}{z+1}\right) $$

 * $$H_d(e^{ j \omega_d T}) $$
 * $$= H_a \left( \frac{2}{T} \frac{e^{ j \omega_d T} - 1}{e^{ j \omega_d T} + 1}\right) $$
 * $$= H_a \left( \frac{2}{T} \cdot \frac{e^{j \omega_d T/2} \left(e^{j \omega_d T/2} - e^{-j \omega_d T/2}\right)}{e^{j \omega_d T/2} \left(e^{j \omega_d T/2} + e^{-j \omega_d T/2 }\right)}\right) $$
 * $$= H_a \left( \frac{2}{T} \cdot \frac{\left(e^{j \omega_d T/2} - e^{-j \omega_d T/2}\right)}{\left(e^{j \omega_d T/2} + e^{-j \omega_d T/2 }\right)}\right) $$
 * $$= H_a \left(j \frac{2}{T} \cdot \frac{ \left(e^{j \omega_d T/2} - e^{-j \omega_d T/2}\right) /(2j)}{\left(e^{j \omega_d T/2} + e^{-j \omega_d T/2 }\right) / 2}\right) $$
 * $$= H_a \left(j \frac{2}{T} \cdot \frac{ \sin(\omega_d T/2) }{ \cos(\omega_d T/2) }\right) $$
 * $$= H_a \left(j \frac{2}{T} \cdot \tan \left( \omega_d T/2 \right) \right) $$
 * }
 * $$= H_a \left( \frac{2}{T} \cdot \frac{\left(e^{j \omega_d T/2} - e^{-j \omega_d T/2}\right)}{\left(e^{j \omega_d T/2} + e^{-j \omega_d T/2 }\right)}\right) $$
 * $$= H_a \left(j \frac{2}{T} \cdot \frac{ \left(e^{j \omega_d T/2} - e^{-j \omega_d T/2}\right) /(2j)}{\left(e^{j \omega_d T/2} + e^{-j \omega_d T/2 }\right) / 2}\right) $$
 * $$= H_a \left(j \frac{2}{T} \cdot \frac{ \sin(\omega_d T/2) }{ \cos(\omega_d T/2) }\right) $$
 * $$= H_a \left(j \frac{2}{T} \cdot \tan \left( \omega_d T/2 \right) \right) $$
 * }
 * $$= H_a \left(j \frac{2}{T} \cdot \frac{ \sin(\omega_d T/2) }{ \cos(\omega_d T/2) }\right) $$
 * $$= H_a \left(j \frac{2}{T} \cdot \tan \left( \omega_d T/2 \right) \right) $$
 * }
 * $$= H_a \left(j \frac{2}{T} \cdot \tan \left( \omega_d T/2 \right) \right) $$
 * }
 * }

This shows that every point on the unit circle in the discrete-time filter z-plane, $$z = e^{ j \omega_d T}$$ is mapped to a point on the $$j \omega$$ axis on the continuous-time filter s-plane, $$s = j \omega_a$$. That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is

 * $$ \omega_a = \frac{2}{T} \tan \left( \omega_d \frac{T}{2} \right) $$

and the inverse mapping is

 * $$ \omega_d = \frac{2}{T} \arctan \left( \omega_a \frac{T}{2} \right). $$

The discrete-time filter behaves at frequency $$\omega_d$$ the same way that the continuous-time filter behaves at frequency $$ (2/T) \tan(\omega_d T/2) $$. Specifically, the gain and phase shift that the discrete-time filter has at frequency $$\omega_d$$ is the same gain and phase shift that the continuous-time filter has at frequency $$(2/T) \tan(\omega_d T/2)$$. This means that every feature, every "bump" that is visible in the frequency response of the continuous-time filter is also visible in the discrete-time filter, but at a different frequency. For low frequencies (that is, when $$\omega_d \ll 2/T$$ or $$\omega_a \ll 2/T$$), then the features are mapped to a slightly different frequency; $$\omega_d \approx \omega_a $$.

One can see that the entire continuous frequency range

 * $$ -\infty < \omega_a < +\infty $$

is mapped onto the fundamental frequency interval

 * $$ -\frac{\pi}{T} < \omega_d < +\frac{\pi}{T}. $$

The continuous-time filter frequency $$ \omega_a = 0 $$ corresponds to the discrete-time filter frequency $$ \omega_d = 0 $$ and the continuous-time filter frequency $$ \omega_a = \pm \infty $$ correspond to the discrete-time filter frequency $$ \omega_d = \pm \pi / T. $$

One can also see that there is a nonlinear relationship between $$ \omega_a $$ and $$ \omega_d.$$ This effect of the bilinear transform is called frequency warping. The continuous-time filter can be designed to compensate for this frequency warping by setting $$ \omega_a = \frac{2}{T} \tan \left( \omega_d \frac{T}{2} \right) $$ for every frequency specification that the designer has control over (such as corner frequency or center frequency). This is called pre-warping the filter design.

It is possible, however, to compensate for the frequency warping by pre-warping a frequency specification $$ \omega_0 $$ (usually a resonant frequency or the frequency of the most significant feature of the frequency response) of the continuous-time system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system. When designing a digital filter as an approximation of a continuous time filter, the frequency response (both amplitude and phase) of the digital filter can be made to match the frequency response of the continuous filter at a specified frequency $$ \omega_0 $$, as well as matching at DC, if the following transform is substituted into the continuous filter transfer function. This is a modified version of Tustin's transform shown above.

 * $$s \leftarrow \frac{\omega_0}{\tan\left(\frac{\omega_0 T}{2}\right)} \frac{z - 1}{z + 1}.$$

However, note that this transform becomes the original transform

 * $$s \leftarrow \frac{2}{T} \frac{z - 1}{z + 1}$$

as $$ \omega_0 \to 0 $$.

The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with Impulse invariance.

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

title: Bilinear transform


text:
Talk:Bilinear transform

Question

 * s = (2/T) * ((1 - z) / (1 + z))

I take it that z is the "transfer function in the analog domain"? What is T? Is z a complex function? s is the transformed transfer function? What are the arguments of these functions? --AxelBoldt

Do the changes answer your questions? -- Ap

Yes, it is a lot clearer (I made some cosmetic changes now that I understand what is going on). Articles about "stable filter" and "transfer function" would really be useful for context, though. --AxelBoldt

Formula
What is the correct formula for this transform? The current formula does not seem to agree with the "Background" paragraph. --AxelBoldt

I found two essentially equivalent definitions:

 * s = (2/T) * ((1 - z-1) / (1 + z-1))

and

 * s = (2/T) * ((z - 1) / (z + 1)).

The first one is from Feedback control of dynamic systems by Franklin et al. The second is from the help text of the bilinear operation in Octave.

What exactly do you mean by does not seem to agree? -- Ap

In the Background paragraph of the main article, it is explained that the transform maps complex numbers from the interior of the unit circle to the negative half plane. But I think the above transform maps complex numbers from the interior of the unit circle to the positive half plane. Or am I confused? --AxelBoldt

Oops, yes I am confused indeed. The above is correct, and gives values in the negative half plane. --AxelBoldt

 * True, both equations are the same, but I find it easier to convert the first form (above) into executable code. The second one requires a little algebra to make it useful in the real world, IMHO. Am I wrong? Madhu 23:57, 4 July 2006 (UTC)

Multiply the first equation by $$\frac{z}{z}$$ and you get the second equation. They are the same. Cburnett 05:31, 28 Nov 2004 (UTC)

"When the Laplace transform of a discrete-time signal with each element of the discrete-time sequence attached to an appropriately delayed unit impulse, the result is precisely the Z transform of the discrete-time sequence with the substitution......"

-There is something wrong grammatically here. Is there a verb missing?

 * "is" is the verb ("the result" is the subject), but it is poorly worded. for some reason i think it's my fault. i'll work on fixing it. r b-j 23:59, 8 December 2005 (UTC)

I understand how your first approximation z=((1+sT/2)/(1-sT/2) arises and leads to the bilinear transform, but the infinite series you show for ln(z) really puzzles me. I can't concieve of any Taylor series expansion to produce it.  Can you shed some light on where it comes from?  Thanks.  [SW]  155.104.37.17 17:19, 30 January 2007 (UTC)

 * See Logarithm. Oli Filth 12:54, 15 April 2007 (UTC)

Shouldn't there be a short definition for T somewhere, perhaps a link to http://en.wikipedia.org/wiki/Sampling_(signal_processing)#Sampling_interval ? It took me an extremely long time to understand this page because I had assumed that T stood for the sampling frequency, not the sampling period (reciprocal of sampling frequency). DavidCreswick 05:32, 20 October 2007 (UTC)

 * T is already defined in the article as the sample time. Oli Filth(talk) 11:56, 20 October 2007 (UTC)

Question
In the Frequency wrapping section, the T is left out in the formula's (i.e. exp(jwT)-> exp(jw)), is that ok?? Arthurdenhaan (talk) 13:55, 1 November 2011 (UTC)

--> I think so. the 'T' should be inserted at appropriate locations in sentences and equation images. So, I corrected them.

http://en.wikipedia.org/wiki/Discretization — Preceding unsigned comment added by 128.39.12.150 (talk) 12:03, 27 June 2011 (UTC)

Reversion of 952723881
I have reverted 952723881 as the maths in the revision was incorrect. I believe the edit was in good faith. Frankplow (talk) 10:55, 14 August 2022 (UTC)

I have rewritten the section as, although much of the working was incorrect, the results were partially correct, useful and not shown elsewhere on the page. Frankplow (talk) 11:11, 16 August 2022 (UTC)

Derivation of frequency warping
The warping equation in Digital signal processing, John G Proakis, $$\Omega = \frac{2}{T}\tan\left(\frac{\omega}{2}\right)$$ is different than the derived one in the article. Any insights into this? Spartie19 (talk) 12:50, 17 June 2024 (UTC)

 * The units don't work in that formula; it is probably wrong. Constant314 (talk) 13:27, 17 June 2024 (UTC)
 * Hi Constant314. Always good to have your input. That's concerning since I have a module currently teaching it with that formula. From what differences I can spot it seems they used $$z = re^{j\omega}$$ instead of $$z = re^{j\omega T}$$ in the book. Also some courses did use the formula in the book from what I could find online. Which implies that either I'm missing something or there is a bunch of institutions teaching the wrong formula. Spartie19 (talk) 13:49, 17 June 2024 (UTC)
 * I find this lot in academics where units are simply ignored as an unnecessary distraction. Also, T=1 is sometimes assumed and then forgotten. You see this especially in earlier works on digital signal processing.  Once it is in the computer, everything is just a number.  ω is assumed to be normallized tto range between +/- 0.5. Constant314 (talk) 15:47, 17 June 2024 (UTC)
 * That definitely seems to be the case. Thank you for clearing up my confusion and for the heads up on normalizing $$\omega$$, it's something I would have probably overlooked. Spartie19 (talk) 16:37, 17 June 2024 (UTC)

title: Talk:Bilinear transform


text:
Brian Boitano

Brian Anthony Boitano (born October 22, 1963) is an American figure skater from Sunnyvale, California. He is the 1988 Olympic champion, the 1986 and 1988 World Champion, and the 1985–1988 U.S. National Champion.

Boitano turned professional following the 1988 season. Under new rules by the ISU, he returned to competition in 1993 and competed at the 1994 Winter Olympics, where he placed sixth. In 1996, he was inducted into the World Figure Skating Hall of Fame and the United States Figure Skating Hall of Fame.

Early life
Brian Boitano was born in 1963 and raised in Mountain View, California. Boitano is a graduate of Marian A. Peterson High School in Sunnyvale, California. He is of Italian American descent, with family from northern Italy. As an adult, he has lived in San Francisco.

Early career
Beginning skating as a child, Brian Boitano first made his mark on the international scene when he won the bronze medal at the 1978 World Junior Figure Skating Championships, beating future rival Brian Orser for that medal.

Early in his career, Boitano was known primarily for his jumping. He, along with several other skaters, helped push the technical envelope of men's skating. In 1982, Boitano became the first American to land a triple Axel. In 1987, he introduced his signature jump, the 'Boitano triple Lutz', in which the skater raises his left arm above his head. He attempted a quadruple jump throughout the 1986–87 season and at the 1988 World Figure Skating Championships, but did not cleanly land the jump; he double-footed the landing on two occasions.

At the 1983 World Championships, he became the first skater to ever land all six triple jumps in competition. He would eventually include and successfully land eight triple jumps in his free skate program, the maximum number possible (see Zayak rule). He would jump two flip jumps and two triple Axels to compete with his rival, Brian Orser, who jumped one triple flip and one triple Axel. It was not until failing to defend his World title in 1987 that Boitano focused specifically on improving his artistry. Toward this end, he worked with renowned choreographer Sandra Bezic.

Boitano placed second at the 1984 United States Figure Skating Championships, earning a place in the 1984 Winter Olympics. He placed 5th at the Olympics, setting the stage for his success over the next four years.

World Champion
Following the 1984 Olympics, several skaters emerged as likely medal hopes following the retirement of Scott Hamilton.

Boitano won the 1985 United States Figure Skating Championships, the first of his four titles. At the first World Championships of the post-Hamilton era in 1985, Alexander Fadeev won, with Brian Orser finishing 2nd and Boitano 3rd. He had injured tendons in his right ankle a few weeks before the 1986 U.S. Championships but went on to win his second national title. At the 1986 World Championships, Boitano took the title, while Fadeev had a disastrous free skate despite having been in an excellent position to win; Orser finished 2nd once again.

During the 1986–87 season, Boitano had introduced three new elements to his programs: the 'Tano triple lutz and a quadruple toe loop, as well as wearing a blindfold, although he never succeeded in landing a clean quadruple jump in competition. The 1987 World Championships were held in Cincinnati, giving the defending world champion a home-field advantage. The outcome of the event would set the tone for the 1988 Olympics. He fell on his quadruple toe loop attempt and placed second.

After losing the world title to Orser at home, Boitano and his coach Linda Leaver decided that some changes needed to be made if he was to become the Olympic champion. He had always been good at the technical requirements ("The first mark"), but he was weak on the artistic ("the second mark"). He was a self-described "jumping robot." In order to help his growth as an artist, he hired choreographer Sandra Bezic to choreograph his programs for the 1987–1988 Olympic season.

Bezic choreographed two programs that featured clean lines and accentuated the skating abilities of the 5' 11" Boitano. The short program was based on Giacomo Meyerbeer's ballet Les Patineurs, in which he plays a cocky young man showing off his tricks, using movements dating to the 19th century. In one moment, he wipes ice shavings, also called snow, off his skate blade and tosses it over his shoulder after landing a triple Axel combination. The free skating program was based on the film score, Napoleon, detailing various phases of a soldier's life.

Boitano debuted his new programs at 1987 Skate Canada, held in the Saddledome in Calgary, Alberta, Canada. This was where he would compete against Brian Orser for the Olympic title three months later. His new programs were received with standing ovations by the audience. Although Orser won the competition, Boitano skated clean, landing seven triple jumps, including a footwork section into a jump, but popped his planned second triple Axel. The team was so confident about the strength of his new programs that they omitted the quadruple toe loop which, if landed, could have put him a shoulder above Orser in technical merit.

The short program at the 1988 United States Figure Skating Championships proved to be a highlight. Boitano received marks of 6.0 from eight of the nine judges for presentation, the second mark. His free skate was flawed. Due to delays, he did not skate until after midnight. Still, he won the competition, and went into the Olympics as the national champion (U.S.), as did Orser (representing Canada).

1988 Olympics: Battle of the Brians
Going into the Olympics, Boitano and Brian Orser each had won a world title and each had an excellent, balanced repertoire. Boitano was known as the slightly better technician and Orser as the better artist. Adding to the rivalry, Boitano and Orser were both performing military-themed programs. Boitano's free skate was set to music from Napoleon and Josephine, the television miniseries. For his free skate, Boitano wore a blue stretch suit with red braids and epaulets, and used military gestures and postures as much as his music allowed.

The Battle of the Brians at the 1988 Winter Olympics was the highlight of Boitano's amateur career. Boitano and Orser were effectively tied going into the free skating portion of the event and whoever won that portion would win the event. Alexander Fadeev had won the compulsory figures section of the competition, with Boitano second and Orser third. In the short program, Orser placed first and Boitano second. The free skating was, at the time, worth 50% of the score, and so Boitano's lead would not be enough to hold him in first place if he lost the free skate.

Boitano skated a clean, technically excellent long program, with eight triple jumps, including two axels, and a triple flip-triple toe loop combination. Landing his second triple axel jump cleanly was probably a critical factor in the battle. Orser made one small mistake on a jump and omitted his planned second triple axel. Boitano won the battle in a 5–4 split. It was later discovered that the Canadian Figure Skating Association had engaged in "vote trading" with several countries on the judging panel, particularly East Germany and the USSR. This ultimately backfired, as the Soviet judge refused to follow this agreement and voting "with his conscience," placing Boitano first. Had he followed his federation's directive, Boitano would have lost the gold medal. The judge was promptly suspended by his federation. Experts questioned why the scores were so close between the two skaters because Boitano had 2 triple axels, 2 triple flips and a triple triple combination, elements that were not included in Orser's program.

With his win, Boitano became the first Olympic champion to land the full complement of six types of triple jumps. Boitano won the gold medal, wearing skates with American flag appliqués. These are now part of the collections of the National Museum of American History at the Smithsonian Institution.

Following the Olympics, both Orser and Boitano went to the World Championships, which Boitano won. Boitano turned professional soon after.

Professional career and return to amateur standing
Following the Olympics, Boitano went on to dominate competitions in the professional ranks, winning ten straight professional competitions, including five consecutive World Professional Championship titles and four consecutive wins at the Challenge of Champions. Boitano also appeared in Carmen on Ice, for which he won an Emmy. He wanted to return to amateur competition and make another run at the Olympics.

In June 1993, the International Skating Union (ISU) introduced a clause, commonly known as the "Boitano rule," which allowed professionals to reinstate as "amateur" or "eligible" skaters. Many others joined Boitano, including Ukrainian Viktor Petrenko, 1988 bronze medalist and 1992 gold medalist. The ISU decision was the result of Boitano's active involvement during the early 1990s, when the International Olympic Committee lifted the remaining limits on athletes' remuneration. Previously, the Committee had been accused of rejecting Western professionals, while allowing Eastern Bloc state-sponsored "amateurs" to compete. Boitano reinstated as an amateur to compete in the 1994 Winter Olympics in Lillehammer, Norway.

Boitano competed at the 1994 United States Figure Skating Championships, led after the short program, but lost to Scott Davis in the long program in a 6–3 split decision. Boitano was named to the Olympic team. Going into the Olympics as a medal favorite in a strong field, Boitano missed his triple Axel combination during the short program for the first time in his career. This mistake proved extremely costly, and knocked Boitano out of medal contention. He skated a good long program and finished 6th.

Boitano returned to the professional ranks afterward. In 1996 he was inducted into the World Figure Skating Hall of Fame and the United States Figure Skating Hall of Fame.

Personal life
In December 2013, Boitano was named to the United States delegation to the 2014 Winter Olympics in Sochi, Russia. In conjunction with that appointment, Boitano publicly came out as gay.

The Sochi games and Russia were the targets of criticism and LGBT activism because of a Russian anti-gay "propaganda" law passed in June 2013. In January 2014, Boitano told the Associated Press that he had never wanted to come out until he was named to the delegation.

Boitano's older brother, Mark Boitano, is a real estate agent and former politician. He served as a member of the New Mexico Senate from 1997 to 2013.

South Park song
A caricature of Boitano as a superhero appears as a semi-recurring character in the cartoon series South Park. The film South Park: Bigger, Longer & Uncut (1999) features a musical number titled "What Would Brian Boitano Do?". He was also featured in Jesus vs. Santa.

Food Network show
On August 23, 2009, Food Network debuted a new series entitled What Would Brian Boitano Make?, which borrows both its name and opening musical theme from the South Park: Bigger, Longer & Uncut song. The show features Boitano preparing meals for his friends. The series was picked up for a ten-episode second season.

Other television and film appearances

 * Boitano starred, along with Katarina Witt and archrival Brian Orser, in the 1990 German dance film Carmen on Ice; Boitano played Don Jose. All three won an Emmy Award for "Outstanding performance in classical music/dance programming".
 * He was featured in the Super Bowl XXVI halftime show "Winter Magic", along with Gloria Estefan and Dorothy Hamill.
 * Boitano had a cameo in the 2007 film Blades of Glory as a world skating federation judge.
 * He and fellow figure skater Michelle Kwan had a cameo as themselves in the 2005 Disney film, Ice Princess, appearing as commentators during the Sectionals competition.
 * He appeared on Giada at Home for one episode.
 * He appeared as a guest judge on Top Chef Masters, Season 4 episode 3. The episode is titled "What would Brian Boitano Make?"
 * He appeared on Fox Business Network's Stossel, episode "Spontaneous Order" (February 10, 2011).
 * He appeared in an episode of Check, Please! Bay Area, a restaurant review program which airs on KQED-TV in San Francisco.
 * He hosted a series on HGTV, called The Brian Boitano Project, which premiered January 16, 2014, in which he purchased a near derelict ancestral home in Northern Italy, home to many Boitanos. During the series he gives the home in Favale di Malvaro a sympathetic restoration/renovation and shops flea markets with two nieces to find decor and furnishings. Local artisans, carpenters, masons and painters create a gem where he can live part-time and host Boitanos from afar.
 * Boitano appeared as a guest on Season 18 of the reality series Hell's Kitchen, where he along with Gordon Ramsay and Traci Des Jardins co-judged the team challenge in the episode "Hell Freezes Over".

title: Brian Boitano


text:
List of political scandals in the United Kingdom

This is a list of political scandals in the United Kingdom in chronological order. Scandals implicating political figures or governments of the UK, often reported in the mass media, have long had repercussions for their popularity. Issues in political scandals have included alleged or proven financial and sexual matters, or various other allegations or actions taken by politicians that led to controversy. In British media and political discourse, such scandals have sometimes been referred to as political sleaze since the 1990s. Notable scandals include the Marconi scandal, Profumo affair and the 2009 expenses scandal.

1890s

 * Liberator Building Society scandal, in which the Liberal Party MP Jabez Balfour was exposed as running several fraudulent companies to conceal financial losses. Balfour fled to Argentina, but was eventually arrested and imprisoned.

1910s

 * Marconi scandal of insider trading by Liberal Party Ministers including:
 * Rufus Isaacs, 1st Marquess of Reading, the Attorney General
 * The Master of Elibank, Lord Murray, the Treasurer of the Liberal Party,
 * David Lloyd George, 1st Earl Lloyd-George of Dwyfor, the Chancellor of the Exchequer
 * Herbert Samuel, 1st Viscount Samuel, Postmaster General; was falsely implicated. (1912)
 * Shell Crisis of 1915, which led to the fall of H. H. Asquith's Liberal Party government during World War I.

1920s

 * Lloyd George and the honours scandal. Honours sold for large campaign contributions (1922)
 * Zinoviev Letter (1924)

1930s

 * Jimmy Thomas budget leak (1936)

1940s

 * Hugh Dalton budget leak (1947)
 * John Belcher corruptly influenced – led to Lynskey Tribunal

1950s

 * Crichel Down and the resignation of Thomas Dugdale (1954)
 * Suez Crisis (1956)

1960s

 * Vassall affair (1963): civil servant John Vassall, working for Minister Tam Galbraith, was revealed to be a spy for the Soviet Union and was arrested. The affair was investigated in the Vassall tribunal.
 * Profumo affair (1963): Secretary of State for War John Profumo had an affair with Christine Keeler (to whom he had been introduced by artist Stephen Ward) who was having an affair with a Soviet spy at the same time.
 * The Robert Boothby (Tory), Tom Driberg (Labour), Kray brothers affair and consequent cover-up involving senior politicians of both parties. The Daily Mirror published some details of the matter and was falsely sued for libel.

1970s

 * Corrupt architect John Poulson and links to Conservative Home Secretary Reginald Maudling, Labour council leader T. Dan Smith and others (1972–1974): Maudling resigned, Smith sentenced to imprisonment.
 * Earl Jellicoe and Lord Lambton sex scandal (1973): Conservatives, junior defence minister Lambton is arrested for using prostitutes and Cabinet minister Jellicoe also confesses.
 * Labour MP John Stonehouse's faked suicide (1974)
 * Harold Wilson's Prime Minister's Resignation Honours (known satirically as the "Lavender List") gives honours to a number of wealthy businessmen whose principles were considered antipathetic to those held by the Labour Party (May 1976)
 * Peter Jay's appointment as British Ambassador to the US by his father in law, the then Labour Prime Minister James Callaghan. At the time Jay was a journalist with little diplomatic experience. (1976)
 * "Rinkagate": the Thorpe affair. Liberal Party leader Jeremy Thorpe was arrested and tried for allegedly paying a hitman to murder his lover, model Norman Scott, while walking his dog on Exmoor; the hitman only shot the dog, Rinka. Thorpe was forced to resign due to his clandestine gay affairs, but was acquitted of conspiracy to murder.

1980s

 * Joseph Kagan, Baron Kagan, earlier ennobled by the Labour Prime Minister Harold Wilson's notorious Lavender List (1976), was convicted of fraud (1980)
 * Cecil Parkinson affair with secretary Sara Keays resulting in their child, Flora Keays (1983)
 * Al Yamamah contract alleged to have been obtained by bribery (1985)
 * Westland affair (1986): The Defence Secretary, Michael Heseltine resigned from his Cabinet job in a disagreement with Prime Minister Margaret Thatcher over the Westland affair. Heseltine walked out of a meeting at Number 10 as his views on the future of the Westland helicopter company were being ignored at the time.
 * Jeffrey Archer and the prostitute allegations (1986), and his subsequent conviction for perjury (2001)
 * Westminster cemeteries scandal (1987)
 * Edwina Currie resigns as a junior Health minister after claiming that millions of British eggs were infected with salmonella, stating that "most of [British] egg production" was infected (1988)
 * "Homes for votes" gerrymandering scandal (1987–1989)

1990s

 * Arms-to-Iraq and the closely connected Iraqi Supergun affair (1990)
 * David Mellor resignation after press disclosure of his affair with Antonia de Sancha and gratis holiday from a daughter of a PLO official (1992)
 * Michael Mates gift of a watch ("Don't let the bastards grind you down") to Asil Nadir (1993)
 * Monklandsgate dominated the 1994 Monklands East by-election. It mainly consisted of allegations of sectarian spending discrepancies between Protestant Airdrie and Catholic Coatbridge, fuelled by the fact that all 17 of the ruling Labour group were Roman Catholics. (1994)
 * Back to Basics, a government policy slogan portrayed by opponents and the press as a morality campaign to compare it with a contemporaneous succession of sex scandals in John Major's government which led to the resignation of Tim Yeo and the Earl of Caithness, among others (1994)
 * Cash-for-questions affair involving Neil Hamilton, Tim Smith and Mohamed Al-Fayed (1994)
 * Jonathan Aitken and the Paris Ritz Hotel bill allegations, and his subsequent conviction for perjury after his failed libel action against The Guardian, resulting in Aitken being only the third person to have to resign from the Privy Council in the 20th century. (1995)
 * Conservative MP Jerry Hayes was "outed" as a homosexual by the News of the World with the headline "TORY MP 2-TIMED WIFE WITH UNDER-AGE GAY LOVER". Hayes had met Young Conservative Paul Stone at the 1991 Conservative conference and that same evening, "committed a lewd act which was in breach of the law at the time". Stone had been 18 at the time, whilst the legal age for homosexual sex in 1991 was 21. He had previously supported Section 28 and other anti-gay legislation. (1997)
 * Bernie Ecclestone was involved in a political scandal when it transpired he had given the Labour Party a million pound donation – which raised eyebrows when the incoming Labour government changed its policy to allow Formula One to continue being sponsored by tobacco manufacturers. The Labour Party returned the donation when the scandal came to light. (1997)
 * Peter Mandelson, Trade and Industry Secretary, resigned after failing to disclose £373,000 loan from Paymaster General Geoffrey Robinson. (1998)
 * Ron Davies resigned from the cabinet after being robbed by a man he met at Clapham Common (a well-known gay cruising ground) and then lying about it (1998)

2000s

 * Officegate (2001). Henry McLeish, Labour First Minister of Scotland, failed to refund the House of Commons for income he had received from the sub-let of his constituency office in Glenrothes while still a Westminster MP.
 * Keith Vaz, Peter Mandelson and the Hinduja brothers. Mandelson forced to resign for a second time due to misleading statements. (2001)
 * Jo Moore, within an hour of the September 11 attacks, sent an email to the press office of her department suggesting: "It's now a very good day to get out anything we want to bury. Councillors' expenses?" Although prior to the catastrophic collapse of the towers, the phrase "a good day to bury bad news" (not actually used by Moore) has since been used to refer to other instances of attempting to hide one item of news behind a more publicised issue.
 * Betsygate (2002), which revolved around the level of pay that Iain Duncan Smith's wife Elisabeth received as his diary secretary.
 * In 2002, Edwina Currie revealed that she had had an affair, beginning in 1984, with John Major before he became Prime Minister of the United Kingdom. This was criticised as Major had frequently pushed his Back To Basics agenda (see above), which was taken by the media as a form of moral absolutism.
 * The Burrell affair – allegations about the behaviour of the British royal family and their servants with possible constitutional implications. (2002)
 * Ron Davies stood down from the Welsh assembly following accusations of illicit gay sex. Davies had claimed he had been badger-watching in the area. (2003)
 * The apparent suicide of Dr. David Kelly and the Hutton Inquiry. On 17 July 2003, Kelly, an employee of the Ministry of Defence, apparently committed suicide after being misquoted by BBC journalist Andrew Gilligan as saying that Tony Blair's Labour government had knowingly "sexed up" the "September Dossier", a report into Iraq and weapons of mass destruction. The government was cleared of wrongdoing, while the BBC was strongly criticised by the subsequent inquiry, leading to the resignation of the BBC's chairman and director-general.
 * In April 2004, Beverly Hughes was forced to resign as minister for Immigration, Citizenship and Counter Terrorism when it was shown that she had been informed of procedural improprieties concerning the granting of visas to certain categories of workers from Eastern Europe. She had earlier told the House of Commons that if she had been aware of such facts she would have done something about it.
 * In 2005, David McLetchie, leader of the Scottish Conservatives, was forced to resign after claiming the highest taxi expenses of any MSP. These included personal journeys, journeys related solely with his second job as a solicitor, and Conservative Party business, for example travel to Conservative conferences. Conservative backbench MSP Brian Monteith had the whip withdrawn for briefing against his leader to the Scotland on Sunday newspaper.
 * Liberal Democrats Home Affairs spokesman Mark Oaten resigned after it was revealed by the News of the World that he paid rentboys to perform sexual acts on him.
 * David Mills financial allegations (2006). Tessa Jowell, Labour cabinet minister, was embroiled in a scandal about a property remortgage allegedly arranged to enable her husband, David Mills, to realise £350,000 from an off-shore hedge fund, money he allegedly received as a gift following testimony he had provided for Silvio Berlusconi in the 1990s. Nicknamed by the press as "Jowellgate".
 * Cash for Honours (2006). In March 2006 it emerged that the Labour Party had borrowed millions of pounds in 2005 to help fund their general election campaign. While not illegal, on 15 March the Treasurer of the party, Jack Dromey stated publicly that he had neither knowledge of nor involvement in these loans and had only become aware when he read about it in the newspapers. A story was running at the time that Dr Chai Patel and others had been recommended for life peerages after lending the Labour party money. He called on the Electoral Commission to investigate the issue of political parties taking out loans from non-commercial sources.
 * Following revelations about Dr Chai Patel and others who were recommended for peerages after lending the Labour party money, the Treasurer of the party, Jack Dromey said he had not been involved and did not know the party had secretly borrowed millions of pounds in 2005. He called on the Electoral Commission to investigate the issue of political parties taking out loans from non-commercial sources.
 * Angus McNeil (2007). The married SNP MP who made the initial police complaint over the cash for honours scandal was forced to make an apology after it was revealed that in 2005 he had a "heavy petting" session with two teenage girls aged 17 and 18 in a hotel room at the same time his wife was pregnant with their third child.
 * News of the World royal phone hacking scandal
 * In November 2007, it emerged that more than £400,000 had been accepted by the Labour Party from one person through a series of third parties, causing the Electoral Commission to seek an explanation. Peter Watt resigned as the General Secretary of the party the day after the story broke and was quoted as saying that he knew about the arrangement but had not appreciated that he had failed to comply with the reporting requirements.
 * On 24 January 2008, Peter Hain resigned his two cabinet posts (Secretary of State for Work and Pensions and Secretary of State for Wales) after the Electoral Commission referred donations to his Deputy Leadership campaign to the police.
 * Derek Conway (2008). The Conservative Party MP was found to have reclaimed salaries he had paid to his two sons who had in fact not carried out the work to the extent claimed. He was ordered to repay £16,918, suspended from the House of Commons for 10 days and removed from the party whip.
 * Cash for Influence (2009). Details of covertly recorded discussions with four Labour Party peers which their ability to influence legislation and the consultancy fees that they charged (including retainer payments of up to £120,000) were published by The Sunday Times.
 * United Kingdom parliamentary expenses scandal (2009). Widespread actual and alleged misuse of the permitted allowances and expenses claimed by Members of Parliament and attempts by MPs and peers to exempt themselves from Freedom of Information legislation.

2010

 * The Iris Robinson scandal in which First Minister of Northern Ireland Peter Robinson stepped aside for six weeks in January 2010 following revelations of his wife's involvement in an extramarital affair, her attempted suicide, and allegations that he had failed to properly declare details of loans she had procured for her lover to develop a business venture.
 * Red Sky scandal, involving contracts given to company Red Sky by the Northern Ireland Housing Executive.
 * The 2010 cash for influence scandal, in which undercover reporters for the Dispatches television series posed as political lobbyists offering to pay Members of Parliament to influence policy.
 * On 29 May 2010 Chief Secretary to the Treasury David Laws resigned from the Cabinet and was referred to the Parliamentary Commissioner for Standards after The Daily Telegraph newspaper published details of Laws claiming around £40,000 in expenses on a second home owned by a secret partner between 2004 and 2009, whilst House of Commons rules have prevented MPs from claiming second home expenses on properties owned by a partner since 2006. By resigning Laws became the shortest serving Minister in modern British political history with less than 18 days' service as a Cabinet Minister.

2011

 * On 14 October 2011 Secretary of State for Defence Liam Fox resigned from the Cabinet after he "mistakenly allowed the distinction between [his] personal interest and [his] government activities to become blurred" over his friendship with Adam Werritty. (He again served as a cabinet minister under Theresa May.)
 * News International phone hacking scandal
 * The Ed Balls document leak was exposed by the Daily Telegraph and showed that shadow chancellor Ed Balls was involved in a supposed plot known as 'Project Volvo' to oust Tony Blair as leader and replace him with Gordon Brown shortly after the 2005 election.

2012

 * Conservative Party 'cash for access' scandal involving Peter Cruddas and Sarah Southern, March 2012.
 * In February 2012 Liberal Democrat MP Chris Huhne resigned from the Cabinet when he was charged with perverting the course of justice over a 2003 speeding case. His wife Vicky Pryce had claimed that she was driving the car, and accepted the licence penalty points on his behalf so that he could avoid being banned from driving. Huhne pleaded guilty at his trial, resigned as a member of parliament, and he and Pryce were sentenced to eight months in prison for perverting the course of justice.
 * In October 2012, Andrew Mitchell resigned from his post as Chief Whip following allegations made about his conduct during an altercation with police at Downing Street on 19 September, the incident becoming known as "plebgate".

2013

 * In the 2013 Labour Party Falkirk candidate selection, which began following the announcement that the incumbent MP Eric Joyce was to step down at the 2015 general election, allegations were made on the significant infiltration of the selection process by the Unite trade union, the Labour Party's largest financial backer.

2014

 * In April 2014 Maria Miller, the Culture Secretary, resigned following pressure relating to the results of an investigation into her past expenses claims.
 * On 20 November 2014 Emily Thornberry resigned her shadow cabinet position shortly after polls closed in the Rochester and Strood by-election. Earlier in the day, she had received criticism after tweeting a photograph of a house in the constituency adorned with three flags of St. George and the owner's white van parked outside on the driveway, under the caption "Image from #Rochester", provoking accusations of snobbery. She was criticised by fellow Labour Party MPs, including leader Ed Miliband who said her tweet conveyed a "sense of disrespect".
 * Namagate, involving allegations that First Minister of Northern Ireland Peter Robinson may have financially benefitted from a deal with National Asset Management Agency (NAMA).

2015

 * In September 2015, Lord Ashcroft published a biography of David Cameron, which suggested that the then Prime Minister took drugs regularly and performed an "outrageous initiation ceremony" which involved inserting "a private part of his anatomy" into the mouth of a dead pig during his time in university. This became known as "piggate". It also led to questions about the Prime Minister's honesty with party donors' known tax statuses as Lord Ashcroft suggested he had openly discussed his non-domiciled status with him in 2009, earlier than previously thought.

2017

 * In 2017 the contaminated blood scandal, in which many haemophiliacs died from infected Factor medicine, hit the headlines and Parliament with allegations of an "industrial scale" criminal cover-up. MP Ken Clarke retracted remarks from his autobiography relating to the scandal and a public inquiry is now underway.
 * The Renewable Heat Incentive scandal in Northern Ireland, in which Arlene Foster set up a green energy scheme but failed to introduce cost controls, creating perverse incentives which eventually led to a £480m bill to the Northern Ireland budget. There were allegations that members of the Democratic Unionist Party attempted to postpone the closure of the scheme, which gave way to a spike in applications and causing the public purse millions of pounds. In January 2017, the scandal caused the resignation of the deputy First Minister, Martin McGuinness, after Foster refused to stand aside as First Minister pending an investigation, collapsing the Executive Office and triggering an early election of the Northern Ireland Assembly. The resulting political rifts meant the Assembly did not meet again until 2020.

2018

 * The 2018 Windrush scandal, involving members of the Windrush generation being wrongly detained, deported, or threatened with deportation which caused the resignation of then Home Secretary, Amber Rudd.

2020

 * The Dominic Cummings scandal, where Dominic Cummings, chief strategist of prime minister Boris Johnson, broke COVID-19 pandemic restrictions during the UK's first nationwide lockdown while experiencing symptoms of the disease. Cummings and Johnson rejected calls for the former to resign. It was suggested that the scandal undermined the public's compliance with pandemic restrictions.
 * The Alex Salmond scandal concerned how the Scottish Government, led by incumbent first minister of Scotland Nicola Sturgeon, breached its own guidelines in its investigation into the harassment claims against Sturgeon's predecessor as first minister, Alex Salmond. The Scottish Government lost a judicial review into their actions and had to pay over £500,000 to Salmond for legal expenses. Salmond claimed that senior figures in Sturgeon's government and the ruling Scottish National Party (SNP) conspired against him for political reasons. Critics accused Sturgeon of breaking the ministerial code which resulted in calls for her resignation. Sturgeon disputed the allegations, arguing that while mistakes had been made, her government acted appropriately.

2021

 * The Greensill scandal, in which former prime minister David Cameron approached a number of government ministers on behalf of Greensill Capital to lobby for the company to receive Covid Corporate Financing Facility loans.
 * The Sun published pictures and then video of leaked CCTV footage from inside the Department of Health of health secretary Matt Hancock and Gina Coladangelo kissing in a breach of COVID-19 social distancing guidance. Boris Johnson accepted Matt Hancock's apology and stated that he "considers the matter closed", but Hancock resigned as health secretary the following day.
 * In November 2021, the Parliamentary Commissioner for Standards found that Conservative MP Owen Paterson had broken paid advocacy rules, but instead of approving his suspension, the government enforced a three-line whip on Conservative MPs to pass a motion that the investigation was "clearly flawed". After an outcry from opposition parties, the government made a U-turn and Paterson resigned.
 * Partygate, involving social gatherings by Downing Street and Conservative Party staff during COVID-19 restrictions in late 2020.

2022

 * Neil Parish, Conservative MP for Tiverton and Honiton, was forced to resign in April after it was discovered that he had watched pornography in the House of Commons on at least two occasions.
 * Chris Pincher scandal: Chris Pincher, the deputy chief whip of the Conservative Party, resigned on 30 June following allegations about him groping two men. Further allegations of harassment emerged against Pincher, along with claims that prime minister Boris Johnson had already been informed of his behaviour. The incremental effect of this and other recent controversies led to the resignation of 59 Conservative politicians, most notably Rishi Sunak as chancellor and Sajid Javid as health secretary. This in turn led to Boris Johnson committing to resign as leader of the Conservative Party, and thus as prime minister when his replacement as leader had been chosen by his party.

2024

 * In March 2024, The Guardian reported that Frank Hester, the largest ever donor to the Conservative Party, had made comments in a 2019 company meeting about the MP Diane Abbott. The paper reported that he said that looking at Abbott makes you "want to hate all black women" and that she "should be shot", as well as making comments about a female executive from another organisation, saying "it would be much better if she died", and about his own Asian female employees, saying "we take the piss out of the fact that all our Chinese girls sit together in Asian corner".
 * Election betting scandal: Following the scheduling of the 2024 United Kingdom general election for 4 July, it was discovered that Craig Williams, Parliamentary Private Secretary to Prime Minister Rishi Sunak, had placed a £100 bet on the election being in July. Further investigation uncovered multiple similar bets made by Conservative Party members and MPs, including cabinet minister Alister Jack, as well as police officers on Sunak's protection detail. Both the Gambling Commission and the police are investigating whether any criminal offences were committed by people involved in the scandal.