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Robert Devereux, 2nd Earl of Essex

Robert Devereux, 2nd Earl of Essex, KG, PC (10 November 1565 – 25 February 1601) was an English nobleman and a favourite of Queen Elizabeth I. Politically ambitious, and a committed general, he was placed under house arrest following a poor campaign in Ireland during the Nine Years' War in 1599. In 1601, he led an abortive coup d'état against the government of Elizabeth I and was executed for treason.

Early life
Devereux was born on 10 November 1565 at Netherwood near Bromyard, in Herefordshire, the son of Walter Devereux, 1st Earl of Essex, and Lettice Knollys. His maternal great-grandmother Mary Boleyn was a sister of Anne Boleyn, the mother of Queen Elizabeth I, making him a first-cousin-twice-removed of the queen.

He was brought up on his father's estates at Chartley Castle, Staffordshire, and at Lamphey, Pembrokeshire, in Wales. He was educated by Thomas Ashton (an influential figure in the evolution of Tudor drama), previously headmaster of Shrewsbury School within his father's household.

His father died in 1576, and the new Earl of Essex became a ward of Lord Burghley of Burghley House. In 1577, he was admitted as a fellow-commoner at Trinity College, Cambridge; in 1579, he matriculated; and in 1581 he graduated as a Master of Arts.

On 21 September 1578, Essex's mother married Robert Dudley, Earl of Leicester, Elizabeth I's long-standing favourite and Robert Devereux's godfather. The following year, Francis Bacon joined the friendship circle comprising Robert, Sir Fulke Greville and Sir Phillip Sidney, Mary Sidney, by then Countess of Pembroke, and Robert's sister Penelope who inspired the "Stella" of Phillip Sidney's Astrophel and Stella sonnet sequence.

Essex performed military service under his stepfather in the Netherlands, before making an impact at court and winning the queen's favour. In 1590, he married Frances Walsingham, daughter of Sir Francis Walsingham and widow of Sir Philip Sidney, by whom he had several children, three of whom survived into adulthood. Elizabeth was against the marriage. Sidney, who was Leicester's nephew, had died from an infected gun wound in 1586, 31 days after his participation in the Battle of Zutphen in which Essex had distinguished himself. In October 1591, Essex's mistress, Elizabeth Southwell, gave birth to their son Walter Devereux (died 1641).

Court and military career

Devereux first came to court in 1584, and by 1587 had become a favourite of the queen, who relished his lively mind and eloquence, as well as his skills as a showman and in courtly love. In June 1587 he replaced the Earl of Leicester as Master of the Horse. After Leicester's death in 1588, the queen transferred the late Earl's royal monopoly on sweet wines to Essex, providing him with revenue from taxes. In 1593, he was made a member of her Privy Council.

It is reported that his friend and confidant Francis Bacon warned him to avoid offending the queen by attempting to gain power and underestimating her ability to rule and wield power.

Essex did underestimate the queen, however, and his later behaviour towards her lacked due respect and showed disdain for the influence of her principal secretary, Robert Cecil. On one occasion during a heated Privy Council debate on the problems in Ireland, the queen reportedly cuffed an insolent Essex round the ear, prompting him to half draw his sword on her.

In 1589, he took part in Francis Drake's English Armada, which sailed to Spain in an unsuccessful attempt to press home the English advantage following the defeat of the Spanish Armada, but the queen had ordered him not to take part. The English Armada was defeated with 40 ships sunk and 15,000 men lost. In 1591, he was given command of a force sent to the assistance of King Henry IV of France. In 1596, he distinguished himself by the capture of Cádiz. During the Islands Voyage expedition to the Azores in 1597, with Walter Raleigh as his second-in-command, he defied the queen's orders, pursuing the Spanish treasure fleet without first defeating the Spanish battle fleet.

When the 3rd Spanish Armada first appeared off the English coast in October 1597, the English fleet was far out to sea, with the coast almost undefended, and panic ensued. This further damaged the relationship between the queen and Essex, even though he was initially given full command of the English fleet when he reached England a few days later. Fortunately, a storm dispersed the Spanish fleet. A number of ships were captured by the English and though there were a few landings, the Spanish withdrew.

Ireland

Essex's greatest failure was as Lord Lieutenant of Ireland, a post which he talked himself into in 1599. The Nine Years' War (1595–1603) was in its middle stages, and no English commander had been successful. More military force was required to defeat the Irish chieftains, led by Hugh O'Neill, the Earl of Tyrone, and supplied from Spain and Scotland.

Essex led the largest expeditionary force ever sent to Ireland—16,000 troops—with orders to put an end to the rebellion. He departed London to the cheers of the queen's subjects, and it was expected the rebellion would be crushed instantly, but the limits of Crown resources and of the Irish campaigning season dictated otherwise.

Essex had declared to the Privy Council that he would confront O'Neill in Ulster. Instead, he led his army into southern Ireland, where he fought a series of inconclusive engagements, wasted his funds, and dispersed his army into garrisons, while the Irish won two important battles in other parts of the country. Rather than face O'Neill in battle, Essex entered a truce that some considered humiliating to the Crown and to the detriment of English authority. The queen told Essex that if she had wished to abandon Ireland it would scarcely have been necessary to send him there.

In all of his campaigns, Essex secured the loyalty of his officers by conferring knighthoods, an honour the queen dispensed sparingly, and by the end of his time in Ireland more than half the knights in England owed their rank to him. The rebels were said to have joked that, "he never drew sword but to make knights", but his practice of conferring knighthoods could in time enable Essex to challenge the powerful factions at Cecil's command.

He was the second Chancellor of the University of Dublin, serving from 1598 to 1601. He was educated at Trinity College Dublin.

First trial
Relying on his general warrant to return to England, given under the great seal, Essex sailed from Ireland on 24 September 1599 and reached London four days later. The queen had expressly forbidden his return and was surprised when he presented himself in her bedchamber one morning at Nonsuch Palace, before she was properly wigged or gowned. On that day, the Privy Council met three times, and it seemed his disobedience might go unpunished, but the queen did confine him to his rooms with the comment that "an unruly beast must be stopped of his provender."

Essex appeared before the full Council on 29 September, when he was compelled to stand before the council during a five-hour interrogation. The Council—his uncle William Knollys, 1st Earl of Banbury included—took a quarter of an hour to compile a report, which declared that his truce with O'Neill was indefensible and his flight from Ireland tantamount to the desertion of duty. He was committed to the custody of Sir Richard Berkeley in his own York House on 1 October, and he blamed Cecil and Raleigh for the queen's hostility. Raleigh advised Cecil to see to it that Essex did not recover power, and Essex appeared to heed advice to retire from public life, despite his popularity with the public.

During his confinement at York House, Essex probably communicated with King James VI of Scotland through Lord Mountjoy, although any plans he may have had at that time to help the Scots king capture the English throne came to nothing. In October, Mountjoy was appointed to replace him in Ireland, and matters seemed to look up for the Earl. In November, the queen was reported to have said that the truce with O'Neill was "so seasonably made... as great good... has grown by it." Others in the council were willing to justify Essex's return from Ireland, on the grounds of the urgent necessity of a briefing by the commander-in-chief.

Cecil kept up the pressure and, on 5 June 1600, Essex was tried before a commission of 18 men. He had to hear the charges and evidence on his knees. Essex was convicted, deprived of public office, and returned to virtual confinement.

Essex's rebellion
In August, his freedom was granted, but the source of his basic income—the sweet wines monopoly—was not renewed. His situation had become desperate, and he shifted "from sorrow and repentance to rage and rebellion." In early 1601, he began to fortify Essex House, his town mansion on the Strand, and gathered his followers.

On the morning of 8 February, he marched out of Essex House with a party of nobles and gentlemen (some later involved in the 1605 Gunpowder Plot) and entered the city of London in an attempt to force an audience with the queen. Cecil immediately had him proclaimed a traitor.

A force under John Leveson placed a barrier across the street at Ludgate Hill. When Essex's men tried to force their way through, Essex's stepfather, Christopher Blount, was injured in the resulting skirmish, and Essex withdrew with his men to Essex House. Essex surrendered after Crown forces besieged Essex House.

Treason trial and death

On 19 February 1601, Essex was tried before his peers on charges of treason. Laura Hanes Cadwallader summarised the indictment:

"The indictment charged Essex with 'conspiring and imagining at London...to depose and slay the queen, and to subvert the Government.' It also stated that Essex had 'endeavoured to raise himself to the Crown of England, and usurp the royal dignity,' and that in order to fulfill these intentions, he and others 'rose and assembled themselves in open rebellion, and moved and persuaded many of the citizens of London to join them in their treason, and endeavoured to get the City of London into their possession and power, and wounded and killed many of the queen's subjects then and there assembled for the purpose of quelling such rebellion.' Essex was charged also with holding the Lord Keeper and the other Privy Councillors in custody 'for four hours and more.'"

Part of the evidence showed that he was in favour of toleration of religious dissent. In his own evidence, he countered the charge of dealing with Catholics, swearing that "papists have been hired and suborned to witness against me." Essex also asserted that Cecil had stated that none in the world but the Infanta of Spain had right to the Crown of England, whereupon Cecil (who had been following the trial at a doorway concealed behind some tapestry) stepped out to make a dramatic denial, going down on his knees to give thanks to God for the opportunity. The witness whom Essex expected to confirm this allegation, his uncle William Knollys, was called and admitted there had once been read in Cecil's presence a book treating such matters. The book may have been either The book of succession supposedly by R. Doleman but probably by Robert Persons or Persons' A Conference about the Next Succession to the Crown of England, works which favoured a Catholic successor friendly to Spain. Knollys denied hearing Cecil make the statement. Thanking God again, Cecil expressed his gratitude that Essex was exposed as a traitor while he was found an honest man.

Essex was found guilty and, on 25 February 1601, was beheaded on Tower Green, the last person to be beheaded in the Tower of London. It was reported to have taken three strokes by the executioner Thomas Derrick to complete the beheading. Previously Thomas Derrick had been convicted of rape but had been pardoned by the Earl of Essex (clearing him of the death penalty) on the condition that he become an executioner at Tyburn. At Sir Walter Raleigh's own execution on 29 October 1618, it was alleged that Raleigh had said to a co-conspirator, "Do not, as my Lord Essex did, take heed of a preacher. By his persuasion, he confessed, and made himself guilty." In that same trial, Raleigh also denied that he had stood at a window during the execution of Essex's sentence, disdainfully puffing out tobacco smoke in sight of the condemned man. Essex in the end shocked many by denouncing his sister Penelope, Lady Rich as his co-conspirator: the queen, who was determined to show as much clemency as possible, ignored the charge.

Some days before the execution, Captain Thomas Lee was apprehended as he kept watch on the door to the queen's chambers. His plan had been to confine her until she signed a warrant for the release of Essex. Captain Lee, who had served in Ireland with the Earl, and who acted as a go-between with the Ulster rebels, was tried and put to death the next day.

Essex's conviction for treason meant that the earldom was forfeit and his son did not inherit the title. However, after the queen's death, King James I of England reinstated the earldom in favour of the disinherited son, Robert Devereux, 3rd Earl of Essex.

The Essex ring
There is a widely repeated romantic legend about a ring given by Elizabeth to Essex. There is a possible reference to the legend by John Webster in his 1623 play The Devil's Law Case suggesting that it was known at this time, but the first printed version of it is in the 1695 romantic novel The Secret History of the most renowned Queen Elizabeth and the Earl of Essex, by a Person of Quality. The version given by David Hume in his History of England says that Elizabeth had given Essex a ring after the expedition to Cádiz that he should send to her if he was in trouble. After his trial, he tried to send the ring to Elizabeth via the Countess of Nottingham, but the countess kept the ring because her husband was an enemy of Essex. As a result, Essex was executed. On her deathbed, the countess is said to have confessed this to Elizabeth, who angrily replied: "May God forgive you, Madam, but I never can". The Queen's Diamond Jubilee Galleries in Westminster Abbey possess a gold ring which is claimed to be this one.

Some historians consider this story of the ring to be a myth, partly because there are no contemporaneous accounts of it. John Lingard in his history of England says the story appears to be fiction. Lytton Strachey states "Such a narrative is appropriate enough to the place where it was first fully elaborated—a sentimental novelette, but it does not belong to history", and Alison Weir calls it a fabrication.

Nevertheless, this version of the story forms the basis of the plot of Gaetano Donizetti's opera Roberto Devereux, with a further twist added to the story, in that Essex is cheating on both the queen and his best friend by having an affair with Lady Nottingham (who in the opera is given the wrong first name of Sarah rather than Catherine): and that this turns out to be (a) the reason why Lord Nottingham turns against his now former friend, when he discovers the ring in question and prevents her sending it, and (b) is the ultimate reason for Queen Elizabeth withdrawing her support for Essex at his trial. The actual question of Devereux's genuine guilt or innocence is sidelined (as is his actual failed rebellion), and the trial is presented as effectively a Parliamentary witch-hunt led by Cecil and Raleigh.

Poetry
Like many other Elizabethan aristocrats Essex was a competent lyric poet, who also participated in court entertainments. He engaged in literary as well as political feuds with his principal enemies, including Walter Raleigh. His poem "Muses no more but mazes" attacks Raleigh's influence over the queen.

Other lyrics were written for masques, including the sonnet "Seated between the old world and the new" in praise of the queen as the moral power linking Europe and America, who supports "the world oppressed" like the mythical Atlas. During his disgrace, he also wrote several bitter and pessimistic verses. His longest poem, "The Passion of a Discontented Mind" (beginning "From silent night..."), is a penitential lament, probably written while imprisoned awaiting execution.

Several of Essex's poems were set to music. English composer John Dowland set a poem called "Can she excuse my wrongs with virtue's cloak?" in his 1597 publication First Booke of Songs: these lyrics have been attributed to Essex, largely on the basis of the dedication of "The Earl of Essex's Galliard", an instrumental version of the same song. Dowland also sets the opening verses of Essex's poem "The Passion of a Discontented Mind" ("From silent night") in his 1612 collection of songs. Orlando Gibbons set lines from the poem in the same year. Settings of Essex's poems "Change thy minde" (set by Richard Martin) and "To plead my faith" (set by Daniel Bacheler) are published in A Musicall Banquet (1610), a collection of songs edited by Robert Dowland.

Portrayals
There have been many portrayals of Essex throughout the years:

Opera

 * Saverio Mercadante's 1833 opera Il Conte d'Essex with libretto by Felice Romani
 * Gaetano Donizetti's 1837 opera Roberto Devereux with libretto by Salvadore Cammarano based mainly on François Ancelot's Elisabeth d'Angleterre.
 * Benjamin Britten's 1953 opera Gloriana is based on Lytton Strachey's Elizabeth and Essex.

Stage

 * In the 1956 essay Hamlet oder Hekuba: der Einbruch der Zeit in das Spiel (Hamlet or Hecuba: the Irruption of Time into the Play), the German legal theorist Carl Schmitt suggests that elements of the Earl's biography, in particular his final days and last words, were incorporated into William Shakespeare's Hamlet at both the level of dialogue and the level of characterisation. Schmitt's overall argument investigates the relationship between history and narrative generally.
 * Essex is briefly alluded to in Shakespeare's Henry V at 5.0.22–34.
 * Essex is said by editor David L. Stevenson to be alluded to in Much Ado About Nothing at 3.1.10–11.
 * Gautier Coste de La Calprenède, Le Comte d'Essex (1639).
 * Thomas Corneille, Le Comte d'Essex (1678).
 * Claude Boyer, Le Comte d'Essex, tragedie. Par Monsieur Boyer de l'Academie françoise (1678).
 * John Banks, The Unhappy Favourite; Or the Earl of Essex, a Tragedy (1682).
 * The night of Essex's execution is dramatised in the Timothy Findley play Elizabeth Rex.
 * Essex is the love interest in La Reine Elizabeth, play by Émile Moreau, 1912, starring Sarah Bernhardt
 * Maxwell Anderson's 1930 play Elizabeth the Queen dramatised the queen's relationship with Devereux, ending with his execution.

Film

 * The 1939 film The Private Lives of Elizabeth and Essex, starring Bette Davis and Errol Flynn, is based on Anderson's play and Lytton Strachey's biographical account Elizabeth and Essex.
 * Their relationship also provided material during the silent era, as in the 1912 film Les Amours de la reine Élisabeth (The Loves of Queen Elizabeth), with Sarah Bernhardt as the queen and Lou Tellegen as Essex.
 * Essex was played by Sam Reid in the 2011 film Anonymous, a fictional biopic that posits that Edward de Vere, the 17th Earl of Oxford, was the true author of William Shakespeare's plays and where both Essex and the Earl of Southampton are the illegitimate sons of Queen Elizabeth (the latter also being de Vere's son who is stated by Robert Cecil to have been Elizabeth's first bastard, making Southampton the product of incest)

TV

 * Charlton Heston portrayed the Earl of Essex opposite Judith Anderson's Elizabeth I in a 1968 television adaption of Maxwell Anderson's Elizabeth the Queen, for the Hallmark Hall of Fame series.
 * The Earl of Essex was portrayed by Robin Ellis in the fifth and sixth episodes of the BBC series Elizabeth R (1971) starring Glenda Jackson as Elizabeth I.
 * The queen's relationship with Essex (played by Hugh Dancy) and his stepfather Leicester (played by Jeremy Irons) was also covered by a 2005 Channel 4/HBO co-production Elizabeth I, starring Helen Mirren.
 * In the 2005 The Virgin Queen, Hans Matheson played the ill-fated Earl of Essex.
 * In the 2017 BBC documentary mini-series Elizabeth I's Secret Agents, the Earl of Essex was portrayed by Joe Wredden.

Video game

 * He is voiced by Rich Keeble in the video game Astrologaster.

Essex in literature
The best known biographical work about Robert Devereux is Lytton Strachey's masterpiece Elizabeth and Essex.

At least two fencing treatises are dedicated to Robert, Earl of Essex. They are as follows:
 * Vincentio Saviolo – His Practice (1595)
 * George Silver – Paradoxes of Defence (1599)

Robert Devereux's death and confession became the subject of two popular 17th-century broadside ballads, set to the English folk tunes Essex Last Goodnight and Welladay. Numerous ballads lamenting his death and praising his military feats were also published throughout the 17th century.

title: Robert Devereux, 2nd Earl of Essex


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Deep Throat

Deep Throat, Deep-throat or Deepthroat may refer to:

Arts and entertainment

 * Deep Throat (film), a 1972 pornographic film
 * Deep Throat (The X-Files), a character in The X-Files
 * "Deep Throat", a 1993 episode of The X-Files
 * Gray Fox (Metal Gear) or Deepthroat, a character in the 1998 Metal Gear Solid
 * Deep Throat (album), a 2002 album by Henry Rollins
 * "Deep Throats", a 2006 episode of Family Guy
 * "Deepthroat" (song), a 2015 song by American rapper Cupcakke from the album Cum Cake

Other uses

 * Deep-throating, a type of fellatio
 * Deep Throat (Watergate), the anonymous source (later revealed to be Mark Felt) in The Washington Post investigation of U.S. President Nixon's 1972 Watergate scandal
 * The Deep Throat Sex Scandal, 2010 American play
 * DeepThroat, the algorithm used by 15.ai to facilitate the cloning of voices with minimal viable data

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

title: Deep Throat


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Wavelet

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing.

For example, a wavelet could be created to have a frequency of middle C and a short duration of roughly one tenth of a second. If this wavelet were to be convolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. Correlation is at the core of many practical wavelet applications.

As a mathematical tool, wavelets can be used to extract information from many kinds of data, including audio signals and images. Sets of wavelets are needed to analyze data fully. "Complementary" wavelets decompose a signal without gaps or overlaps so that the decomposition process is mathematically reversible. Thus, sets of complementary wavelets are useful in wavelet-based compression/decompression algorithms, where it is desirable to recover the original information with minimal loss.

In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of a vector space, for the Hilbert space of square-integrable functions. This is accomplished through coherent states.

In classical physics, the diffraction phenomenon is described by the Huygens–Fresnel principle that treats each point in a propagating wavefront as a collection of individual spherical wavelets. The characteristic bending pattern is most pronounced when a wave from a coherent source (such as a laser) encounters a slit/aperture that is comparable in size to its wavelength. This is due to the addition, or interference, of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of different lengths to the registering surface. Multiple, closely spaced openings (e.g., a diffraction grating), can result in a complex pattern of varying intensity.

Etymology
The word wavelet has been used for decades in digital signal processing and exploration geophysics. The equivalent French word ondelette meaning "small wave" was used by Jean Morlet and Alex Grossmann in the early 1980s.

Wavelet theory
Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis. Discrete wavelet transform (continuous in time) of a discrete-time (sampled) signal by using discrete-time filterbanks of dyadic (octave band) configuration is a wavelet approximation to that signal. The coefficients of such a filter bank are called the shift and scaling coefficients in wavelets nomenclature. These filterbanks may contain either finite impulse response (FIR) or infinite impulse response (IIR) filters. The wavelets forming a continuous wavelet transform (CWT) are subject to the uncertainty principle of Fourier analysis respective sampling theory: given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event. The product of the uncertainties of time and frequency response scale has a lower bound. Thus, in the scaleogram of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.

Wavelet transforms are broadly divided into three classes: continuous, discrete and multiresolution-based.

Continuous wavelet transforms (continuous shift and scale parameters)
In continuous wavelet transforms, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of the Lp function space L2(R) ). For instance the signal may be represented on every frequency band of the form [f, 2f] for all positive frequencies f > 0. Then, the original signal can be reconstructed by a suitable integration over all the resulting frequency components.

The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at scale 1. This subspace in turn is in most situations generated by the shifts of one generating function ψ in L2(R), the mother wavelet. For the example of the scale one frequency band [1, 2] this function is $$\psi(t)=2\,\operatorname{sinc}(2t)-\,\operatorname{sinc}(t)=\frac{\sin(2\pi t)-\sin(\pi t)}{\pi t}$$ with the (normalized) sinc function. That, Meyer's, and two other examples of mother wavelets are:

The subspace of scale a or frequency band [1/a, 2/a] is generated by the functions (sometimes called child wavelets) $$\psi_{a,b} (t) = \frac1{\sqrt a }\psi \left( \frac{t - b}{a} \right),$$ where a is positive and defines the scale and b is any real number and defines the shift. The pair (a, b) defines a point in the right halfplane R+ × R.

The projection of a function x onto the subspace of scale a then has the form $$x_a(t)=\int_\R WT_\psi\{x\}(a,b)\cdot\psi_{a,b}(t)\,db$$ with wavelet coefficients $$WT_\psi\{x\}(a,b)=\langle x,\psi_{a,b}\rangle=\int_\R x(t){\psi_{a,b}(t)}\,dt.$$

For the analysis of the signal x, one can assemble the wavelet coefficients into a scaleogram of the signal.

See a list of some Continuous wavelets.

Discrete wavelet transforms (discrete shift and scale parameters, continuous in time)
It is computationally impossible to analyze a signal using all wavelet coefficients, so one may wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding wavelet coefficients. One such system is the affine system for some real parameters a > 1, b > 0. The corresponding discrete subset of the halfplane consists of all the points (am, nb am) with m, n in Z. The corresponding child wavelets are now given as $$\psi_{m,n}(t) = \frac1{\sqrt{a^m}}\psi\left(\frac{t - nba^m}{a^m}\right).$$

A sufficient condition for the reconstruction of any signal x of finite energy by the formula $$ x(t)=\sum_{m\in\Z}\sum_{n\in\Z}\langle x,\,\psi_{m,n}\rangle\cdot\psi_{m,n}(t)$$ is that the functions $$\{\psi_{m,n}:m,n\in\Z\}$$ form an orthonormal basis of L2(R).

Multiresolution based discrete wavelet transforms (continuous in time)
In any discretised wavelet transform, there are only a finite number of wavelet coefficients for each bounded rectangular region in the upper halfplane. Still, each coefficient requires the evaluation of an integral. In special situations this numerical complexity can be avoided if the scaled and shifted wavelets form a multiresolution analysis. This means that there has to exist an auxiliary function, the father wavelet φ in L2(R), and that a is an integer. A typical choice is a = 2 and b = 1. The most famous pair of father and mother wavelets is the Daubechies 4-tap wavelet. Note that not every orthonormal discrete wavelet basis can be associated to a multiresolution analysis; for example, the Journe wavelet admits no multiresolution analysis.

From the mother and father wavelets one constructs the subspaces $$V_m=\operatorname{span}(\phi_{m,n}:n\in\Z),\text{ where }\phi_{m,n}(t)=2^{-m/2}\phi(2^{-m}t-n)$$ $$W_m=\operatorname{span}(\psi_{m,n}:n\in\Z),\text{ where }\psi_{m,n}(t)=2^{-m/2}\psi(2^{-m}t-n).$$ The father wavelet $$V_{i}$$ keeps the time domain properties, while the mother wavelets $$W_{i}$$ keeps the frequency domain properties.

From these it is required that the sequence $$\{0\}\subset\dots\subset V_{1}\subset V_{0}\subset V_{-1}\subset V_{-2}\subset\dots\subset L^2(\R)$$ forms a multiresolution analysis of L2 and that the subspaces $$\dots,W_1,W_0,W_{-1},\dots$$ are the orthogonal "differences" of the above sequence, that is, Wm is the orthogonal complement of Vm inside the subspace Vm−1, $$V_m\oplus W_m=V_{m-1}.$$

In analogy to the sampling theorem one may conclude that the space Vm with sampling distance 2m more or less covers the frequency baseband from 0 to 1/2m-1. As orthogonal complement, Wm roughly covers the band [1/2m−1, 1/2m].

From those inclusions and orthogonality relations, especially $$V_0\oplus W_0=V_{-1}$$, follows the existence of sequences $$h=\{h_n\}_{n\in\Z}$$ and $$g=\{g_n\}_{n\in\Z}$$ that satisfy the identities $$g_n=\langle\phi_{0,0},\,\phi_{-1,n}\rangle$$ so that $\phi(t)=\sqrt2 \sum_{n\in\Z} g_n\phi(2t-n),$ and $$h_n=\langle\psi_{0,0},\,\phi_{-1,n}\rangle$$ so that $\psi(t)=\sqrt2 \sum_{n\in\Z} h_n\phi(2t-n).$ The second identity of the first pair is a refinement equation for the father wavelet φ. Both pairs of identities form the basis for the algorithm of the fast wavelet transform.

From the multiresolution analysis derives the orthogonal decomposition of the space L2 as $$L^2 = V_{j_0} \oplus W_{j_0} \oplus W_{j_0-1} \oplus W_{j_0-2} \oplus W_{j_0-3} \oplus \cdots$$ For any signal or function $$S\in L^2$$ this gives a representation in basis functions of the corresponding subspaces as $$S = \sum_{k} c_{j_0,k}\phi_{j_0,k} + \sum_{j\le j_0}\sum_{k} d_{j,k}\psi_{j,k}$$ where the coefficients are $$c_{j_0,k} = \langle S,\phi_{j_0,k}\rangle $$ and $$d_{j,k} = \langle S,\psi_{j,k}\rangle. $$

Time-causal wavelets
For processing temporal signals in real time, it is essential that the wavelet filters do not access signal values from the future as well as that minimal temporal latencies can be obtained. Time-causal wavelets representations have been developed by Szu et al and Lindeberg, with the latter method also involving a memory-efficient time-recursive implementation.

Mother wavelet
For practical applications, and for efficiency reasons, one prefers continuously differentiable functions with compact support as mother (prototype) wavelet (functions). However, to satisfy analytical requirements (in the continuous WT) and in general for theoretical reasons, one chooses the wavelet functions from a subspace of the space $$L^1(\R)\cap L^2(\R).$$ This is the space of Lebesgue measurable functions that are both absolutely integrable and square integrable in the sense that $$\int_{-\infty}^{\infty} |\psi (t)| \, dt <\infty$$ and $$\int_{-\infty}^{\infty} |\psi (t)|^2 \, dt < \infty.$$

Being in this space ensures that one can formulate the conditions of zero mean and square norm one: $$\int_{-\infty}^{\infty} \psi (t) \, dt = 0$$ is the condition for zero mean, and $$\int_{-\infty}^{\infty} |\psi (t)|^2\, dt = 1$$ is the condition for square norm one.

For ψ to be a wavelet for the continuous wavelet transform (see there for exact statement), the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform.

For the discrete wavelet transform, one needs at least the condition that the wavelet series is a representation of the identity in the space L2(R). Most constructions of discrete WT make use of the multiresolution analysis, which defines the wavelet by a scaling function. This scaling function itself is a solution to a functional equation.

In most situations it is useful to restrict ψ to be a continuous function with a higher number M of vanishing moments, i.e. for all integer m < M $$\int_{-\infty}^{\infty} t^m\,\psi (t)\, dt = 0.$$

The mother wavelet is scaled (or dilated) by a factor of a and translated (or shifted) by a factor of b to give (under Morlet's original formulation):

$$\psi _{a,b} (t) = {1 \over \sqrt a}\psi \left( {t - b \over a} \right).$$

For the continuous WT, the pair (a,b) varies over the full half-plane R+ × R; for the discrete WT this pair varies over a discrete subset of it, which is also called affine group.

These functions are often incorrectly referred to as the basis functions of the (continuous) transform. In fact, as in the continuous Fourier transform, there is no basis in the continuous wavelet transform. Time-frequency interpretation uses a subtly different formulation (after Delprat).

Restriction：
 * 1) $$ \frac{1}{\sqrt{a}} \int_{-\infty}^\infty \varphi_{a1,b1}(t)\varphi\left(\frac{t-b}{a}\right) \, dt$$ when $a_{1} = a$ and $b_{1} = b$,
 * 2) $$\Psi (t)$$ has a finite time interval

Comparisons with Fourier transform (continuous-time)
The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. In fact, the Fourier transform can be viewed as a special case of the continuous wavelet transform with the choice of the mother wavelet $$\psi (t) = e^{-2 \pi i t}$$. The main difference in general is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The short-time Fourier transform (STFT) is similar to the wavelet transform, in that it is also time and frequency localized, but there are issues with the frequency/time resolution trade-off.

In particular, assuming a rectangular window region, one may think of the STFT as a transform with a slightly different kernel $$\psi (t) = g(t-u) e^{-2 \pi i t}$$ where $$g(t-u)$$ can often be written as $\operatorname{rect}\left(\frac{t-u}{\Delta_t}\right)$, where $$\Delta_t$$ and u respectively denote the length and temporal offset of the windowing function. Using Parseval's theorem, one may define the wavelet's energy as $$E = \int_{-\infty}^{\infty} |\psi (t)|^2\, dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |\hat{\psi}(\omega)|^2 \, d\omega$$ From this, the square of the temporal support of the window offset by time u is given by $$\sigma_u^2 = \frac{1}{E}\int |t-u|^2 |\psi(t)|^2 \, dt$$

and the square of the spectral support of the window acting on a frequency $$\xi$$ $$\hat{\sigma}_\xi^2 =\frac{1}{2\pi E} \int |\omega-\xi|^2|\hat{\psi}(\omega)|^2 \, d\omega$$

Multiplication with a rectangular window in the time domain corresponds to convolution with a $$\operatorname{sinc}(\Delta_t\omega)$$ function in the frequency domain, resulting in spurious ringing artifacts for short/localized temporal windows. With the continuous-time Fourier transform, $$\Delta_t \to \infty$$ and this convolution is with a delta function in Fourier space, resulting in the true Fourier transform of the signal $$x(t)$$. The window function may be some other apodizing filter, such as a Gaussian. The choice of windowing function will affect the approximation error relative to the true Fourier transform.

A given resolution cell's time-bandwidth product may not be exceeded with the STFT. All STFT basis elements maintain a uniform spectral and temporal support for all temporal shifts or offsets, thereby attaining an equal resolution in time for lower and higher frequencies. The resolution is purely determined by the sampling width.

In contrast, the wavelet transform's multiresolutional properties enables large temporal supports for lower frequencies while maintaining short temporal widths for higher frequencies by the scaling properties of the wavelet transform. This property extends conventional time-frequency analysis into time-scale analysis.

The discrete wavelet transform is less computationally complex, taking O(N) time as compared to O(N log N) for the fast Fourier transform (FFT). This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT which uses the same basis functions as the discrete Fourier transform (DFT). This complexity only applies when the filter size has no relation to the signal size. A wavelet without compact support such as the Shannon wavelet would require O(N2). (For instance, a logarithmic Fourier Transform also exists with O(N) complexity, but the original signal must be sampled logarithmically in time, which is only useful for certain types of signals. )

Definition of a wavelet
A wavelet (or a wavelet family) can be defined in various ways:

Scaling filter
An orthogonal wavelet is entirely defined by the scaling filter – a low-pass finite impulse response (FIR) filter of length 2N and sum 1. In biorthogonal wavelets, separate decomposition and reconstruction filters are defined.

For analysis with orthogonal wavelets the high pass filter is calculated as the quadrature mirror filter of the low pass, and reconstruction filters are the time reverse of the decomposition filters.

Daubechies and Symlet wavelets can be defined by the scaling filter.

Scaling function
Wavelets are defined by the wavelet function ψ(t) (i.e. the mother wavelet) and scaling function φ(t) (also called father wavelet) in the time domain.

The wavelet function is in effect a band-pass filter and scaling that for each level halves its bandwidth. This creates the problem that in order to cover the entire spectrum, an infinite number of levels would be required. The scaling function filters the lowest level of the transform and ensures all the spectrum is covered. See for a detailed explanation.

For a wavelet with compact support, φ(t) can be considered finite in length and is equivalent to the scaling filter g.

Meyer wavelets can be defined by scaling functions

Wavelet function
The wavelet only has a time domain representation as the wavelet function ψ(t).

For instance, Mexican hat wavelets can be defined by a wavelet function. See a list of a few continuous wavelets.

History
The development of wavelets can be linked to several separate trains of thought, starting with Alfréd Haar's work in the early 20th century. Later work by Dennis Gabor yielded Gabor atoms (1946), which are constructed similarly to wavelets, and applied to similar purposes.

Notable contributions to wavelet theory since then can be attributed to George Zweig’s discovery of the continuous wavelet transform (CWT) in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound), Pierre Goupillaud, Alex Grossmann and Jean Morlet's formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work on discrete wavelets (1983), the Le Gall–Tabatabai (LGT) 5/3-taps non-orthogonal filter bank with linear phase (1988),  Ingrid Daubechies' orthogonal wavelets with compact support (1988), Stephanie Mallat's non-orthogonal multiresolution framework (1989), Ali Akansu's binomial QMF (1990), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland's harmonic wavelet transform (1993), and set partitioning in hierarchical trees (SPIHT) developed by Amir Said with William A. Pearlman in 1996.

The JPEG 2000 standard was developed from 1997 to 2000 by a Joint Photographic Experts Group (JPEG) committee chaired by Touradj Ebrahimi (later the JPEG president). In contrast to the DCT algorithm used by the original JPEG format, JPEG 2000 instead uses discrete wavelet transform (DWT) algorithms. It uses the CDF 9/7 wavelet transform (developed by Ingrid Daubechies in 1992) for its lossy compression algorithm, and the Le Gall–Tabatabai (LGT) 5/3 discrete-time filter bank (developed by Didier Le Gall and Ali J. Tabatabai in 1988) for its lossless compression algorithm. JPEG 2000 technology, which includes the Motion JPEG 2000 extension, was selected as the video coding standard for digital cinema in 2004.

Timeline

 * First wavelet (Haar's wavelet) by Alfréd Haar (1909)
 * Since the 1970s: George Zweig, Jean Morlet, Alex Grossmann
 * Since the 1980s: Yves Meyer, Didier Le Gall, Ali J. Tabatabai, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Ali Akansu, Victor Wickerhauser
 * Since the 1990s: Nathalie Delprat, Newland, Amir Said, William A. Pearlman, Touradj Ebrahimi, JPEG 2000

Wavelet transforms
A wavelet is a mathematical function used to divide a given function or continuous-time signal into different scale components. Usually one can assign a frequency range to each scale component. Each scale component can then be studied with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.

Wavelet transforms are classified into discrete wavelet transforms (DWTs) and continuous wavelet transforms (CWTs). Note that both DWT and CWT are continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid.

There are a large number of wavelet transforms each suitable for different applications. For a full list see list of wavelet-related transforms but the common ones are listed below:
 * Continuous wavelet transform (CWT)
 * Discrete wavelet transform (DWT)
 * Fast wavelet transform (FWT)
 * Lifting scheme and generalized lifting scheme
 * Wavelet packet decomposition (WPD)
 * Stationary wavelet transform (SWT)
 * Fractional Fourier transform (FRFT)
 * Fractional wavelet transform (FRWT)

Generalized transforms
There are a number of generalized transforms of which the wavelet transform is a special case. For example, Yosef Joseph introduced scale into the Heisenberg group, giving rise to a continuous transform space that is a function of time, scale, and frequency. The CWT is a two-dimensional slice through the resulting 3d time-scale-frequency volume.

Another example of a generalized transform is the chirplet transform in which the CWT is also a two dimensional slice through the chirplet transform.

An important application area for generalized transforms involves systems in which high frequency resolution is crucial. For example, darkfield electron optical transforms intermediate between direct and reciprocal space have been widely used in the harmonic analysis of atom clustering, i.e. in the study of crystals and crystal defects. Now that transmission electron microscopes are capable of providing digital images with picometer-scale information on atomic periodicity in nanostructure of all sorts, the range of pattern recognition and strain /metrology applications for intermediate transforms with high frequency resolution (like brushlets and ridgelets ) is growing rapidly.

Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform in the fractional Fourier transform domains. This transform is capable of providing the time- and fractional-domain information simultaneously and representing signals in the time-fractional-frequency plane.

Applications
Generally, an approximation to DWT is used for data compression if a signal is already sampled, and the CWT for signal analysis. Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific research.

Like some other transforms, wavelet transforms can be used to transform data, then encode the transformed data, resulting in effective compression. For example, JPEG 2000 is an image compression standard that uses biorthogonal wavelets. This means that although the frame is overcomplete, it is a tight frame (see types of frames of a vector space), and the same frame functions (except for conjugation in the case of complex wavelets) are used for both analysis and synthesis, i.e., in both the forward and inverse transform. For details see wavelet compression.

A related use is for smoothing/denoising data based on wavelet coefficient thresholding, also called wavelet shrinkage. By adaptively thresholding the wavelet coefficients that correspond to undesired frequency components smoothing and/or denoising operations can be performed.

Wavelet transforms are also starting to be used for communication applications. Wavelet OFDM is the basic modulation scheme used in HD-PLC (a power line communications technology developed by Panasonic), and in one of the optional modes included in the IEEE 1901 standard. Wavelet OFDM can achieve deeper notches than traditional FFT OFDM, and wavelet OFDM does not require a guard interval (which usually represents significant overhead in FFT OFDM systems).

As a representation of a signal
Often, signals can be represented well as a sum of sinusoids. However, consider a non-continuous signal with an abrupt discontinuity; this signal can still be represented as a sum of sinusoids, but requires an infinite number, which is an observation known as Gibbs phenomenon. This, then, requires an infinite number of Fourier coefficients, which is not practical for many applications, such as compression. Wavelets are more useful for describing these signals with discontinuities because of their time-localized behavior (both Fourier and wavelet transforms are frequency-localized, but wavelets have an additional time-localization property). Because of this, many types of signals in practice may be non-sparse in the Fourier domain, but very sparse in the wavelet domain. This is particularly useful in signal reconstruction, especially in the recently popular field of compressed sensing. (Note that the short-time Fourier transform (STFT) is also localized in time and frequency, but there are often problems with the frequency-time resolution trade-off. Wavelets are better signal representations because of multiresolution analysis.)

This motivates why wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional Fourier transform. Many areas of physics have seen this paradigm shift, including molecular dynamics, chaos theory, ab initio calculations, astrophysics, gravitational wave transient data analysis, density-matrix localisation, seismology, optics, turbulence and quantum mechanics. This change has also occurred in image processing, EEG, EMG, ECG analyses, brain rhythms, DNA analysis, protein analysis, climatology, human sexual response analysis, general signal processing, speech recognition, acoustics, vibration signals, computer graphics, multifractal analysis, and sparse coding. In computer vision and image processing, the notion of scale space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.

Wavelet denoising

Suppose we measure a noisy signal $$x = s + v $$, where $$s$$ represents the signal and $$v$$ represents the noise. Assume $$s$$ has a sparse representation in a certain wavelet basis, and $$v \ \sim\ \mathcal{N}(0,\,\sigma^2I)$$

Let the wavelet transform of $$x$$ be $$y = W^T x = W^T s + W^T v = p + z$$, where $$p = W^T s$$ is the wavelet transform of the signal component and $$z = W^T v$$ is the wavelet transform of the noise component.

Most elements in $$p$$ are 0 or close to 0, and $$z \ \sim\ \ \mathcal{N}(0,\,\sigma^2I)$$

Since $$W$$ is orthogonal, the estimation problem amounts to recovery of a signal in iid Gaussian noise. As $$p$$ is sparse, one method is to apply a Gaussian mixture model for $$p$$.

Assume a prior $$p \ \sim\ a\mathcal{N}(0,\,\sigma_1^2) +(1- a)\mathcal{N}(0,\,\sigma_2^2)$$, where $$\sigma_1^2$$ is the variance of "significant" coefficients and $$\sigma_2^2$$ is the variance of "insignificant" coefficients.

Then $$\tilde p = E(p/y) = \tau(y) y$$, $$\tau(y)$$ is called the shrinkage factor, which depends on the prior variances $$\sigma_1^2$$ and $$\sigma_2^2$$. By setting coefficients that fall below a shrinkage threshold to zero, once the inverse transform is applied, an expectedly small amount of signal is lost due to the sparsity assumption. The larger coefficients are expected to primarily represent signal due to sparsity, and statistically very little of the signal, albeit the majority of the noise, is expected to be represented in such lower magnitude coefficients... therefore the zeroing-out operation is expected to remove most of the noise and not much signal. Typically, the above-threshold coefficients are not modified during this process. Some algorithms for wavelet-based denoising may attenuate larger coefficients as well, based on a statistical estimate of the amount of noise expected to be removed by such an attenuation.

At last, apply the inverse wavelet transform to obtain $$ \tilde s = W \tilde p$$

Multiscale climate network
Agarwal et al. proposed wavelet based advanced linear and nonlinear methods to construct and investigate Climate as complex networks at different timescales. Climate networks constructed using SST datasets at different timescale averred that wavelet based multi-scale analysis of climatic processes holds the promise of better understanding the system dynamics that may be missed when processes are analyzed at one timescale only

Discrete wavelets

 * Beylkin (18)
 * Moore Wavelet
 * Biorthogonal nearly coiflet (BNC) wavelets
 * Coiflet (6, 12, 18, 24, 30)
 * Cohen-Daubechies-Feauveau wavelet (Sometimes referred to as CDF N/P or Daubechies biorthogonal wavelets)
 * Daubechies wavelet (2, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.)
 * Binomial QMF (Also referred to as Daubechies wavelet)
 * Haar wavelet
 * Mathieu wavelet
 * Legendre wavelet
 * Villasenor wavelet
 * Symlet

Real-valued

 * Beta wavelet
 * Hermitian wavelet
 * Meyer wavelet
 * Mexican hat wavelet
 * Poisson wavelet
 * Shannon wavelet
 * Spline wavelet
 * Strömberg wavelet

Complex-valued

 * Complex Mexican hat wavelet
 * fbsp wavelet
 * Morlet wavelet
 * Shannon wavelet
 * Modified Morlet wavelet

title: Wavelet


text:
Haar wavelet

In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example.

The Haar sequence was proposed in 1909 by Alfréd Haar. Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval [0, 1]. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1.

The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property can, however, be an advantage for the analysis of signals with sudden transitions (discrete signals), such as monitoring of tool failure in machines.

The Haar wavelet's mother wavelet function $$\psi(t)$$ can be described as
 * $$\psi(t) = \begin{cases}

1 \quad & 0 \leq t < \frac{1}{2},\\ -1 & \frac{1}{2} \leq t < 1,\\ 0 &\mbox{otherwise.} \end{cases}$$

Its scaling function $$\varphi(t)$$ can be described as
 * $$\varphi(t) = \begin{cases}1 \quad & 0 \leq t < 1,\\0 &\mbox{otherwise.}\end{cases}$$

Haar functions and Haar system
For every pair n, k of integers in $$\mathbb{Z}$$, the Haar function ψn,k is defined on the real line $$\mathbb{R}$$ by the formula
 * $$ \psi_{n,k}(t) = 2^{n / 2} \psi(2^n t-k), \quad t \in \mathbb{R}.$$

This function is supported on the right-open interval In,k = [ k2−n, (k+1)2−n), i.e., it vanishes outside that interval. It has integral 0 and norm 1 in the Hilbert space L2($\mathbb{R}$),
 * $$ \int_{\mathbb{R}} \psi_{n, k}(t) \, d t = 0, \quad \|\psi_{n, k}\|^2_{L^2(\mathbb{R})} = \int_{\mathbb{R}} \psi_{n, k}(t)^2 \, d t = 1.$$

The Haar functions are pairwise orthogonal,
 * $$ \int_{\mathbb{R}} \psi_{n_1, k_1}(t) \psi_{n_2, k_2}(t) \, d t = \delta_{n_1n_2} \delta_{k_1k_2}, $$

where $$\delta_{ij}$$ represents the Kronecker delta. Here is the reason for orthogonality: when the two supporting intervals $$I_{n_1, k_1}$$ and $$I_{n_2, k_2}$$ are not equal, then they are either disjoint, or else the smaller of the two supports, say $$I_{n_1, k_1}$$, is contained in the lower or in the upper half of the other interval, on which the function $$\psi_{n_2, k_2}$$ remains constant. It follows in this case that the product of these two Haar functions is a multiple of the first Haar function, hence the product has integral 0.

The Haar system on the real line is the set of functions
 * $$ \{ \psi_{n,k}(t) \; : \; n \in \mathbb{Z}, \; k \in \mathbb{Z} \}.$$

It is complete in L2($$\mathbb{R}$$): The Haar system on the line is an orthonormal basis in L2($$\mathbb{R}$$).

Haar wavelet properties
The Haar wavelet has several notable properties:

1. Any continuous real function with compact support can be approximated uniformly by linear combinations of $\varphi(t),\varphi(2t),\varphi(4t),\dots,\varphi(2^n t),\dots$ and their shifted functions. This extends to those function spaces where any function therein can be approximated by continuous functions.

2. Any continuous real function on [0, 1] can be approximated uniformly on [0, 1] by linear combinations of the constant function 1, $\psi(t),\psi(2t),\psi(4t),\dots,\psi(2^n t),\dots$ and their shifted functions.

3. Orthogonality in the form

\int_{-\infty}^{\infty}2^{(n+n_1)/2}\psi(2^n t-k)\psi(2^{n_1} t - k_1)\, dt = \delta_{nn_1}\delta_{kk_1}. $ Here, $\delta_{ij}$ represents the Kronecker delta. The dual function of ψ(t) is ψ(t) itself.

4. Wavelet/scaling functions with different scale n have a functional relationship: since

\begin{align} \varphi(t) &= \varphi(2t)+\varphi(2t-1)\\[.2em] \psi(t) &= \varphi(2t)-\varphi(2t-1), \end{align}$ it follows that coefficients of scale n can be calculated by coefficients of scale n+1: If $ \chi_w(k, n)= 2^{n/2}\int_{-\infty}^\infty x(t)\varphi(2^nt-k)\, dt$ and $ \Chi_w(k, n)= 2^{n/2}\int_{-\infty}^\infty x(t)\psi(2^nt-k)\, dt$ then
 * $ \chi_w(k,n)= 2^{-1/2} \bigl( \chi_w(2k,n+1)+\chi_w(2k+1,n+1) \bigr)$
 * $ \Chi_w(k,n)= 2^{-1/2} \bigl( \chi_w(2k,n+1)-\chi_w(2k+1,n+1) \bigr).$

Haar system on the unit interval and related systems
In this section, the discussion is restricted to the unit interval [0, 1] and to the Haar functions that are supported on [0, 1]. The system of functions considered by Haar in 1910, called the Haar system on [0, 1] in this article, consists of the subset of Haar wavelets defined as
 * $$\{ t \in [0, 1] \mapsto \psi_{n,k}(t) \; : \; n, k \in \N \cup \{0\}, \; 0 \leq k < 2^n\},$$

with the addition of the constant function 1 on [0, 1].

In Hilbert space terms, this Haar system on [0, 1] is a complete orthonormal system, i.e., an orthonormal basis, for the space L2([0, 1]) of square integrable functions on the unit interval.

The Haar system on [0, 1] —with the constant function 1 as first element, followed with the Haar functions ordered according to the lexicographic ordering of couples (n, k)— is further a monotone Schauder basis for the space Lp([0, 1]) when 1 ≤ p < ∞. This basis is unconditional when 1 < p < ∞.

There is a related Rademacher system consisting of sums of Haar functions,
 * $$r_n(t) = 2^{-n/2} \sum_{k=0}^{2^n - 1} \psi_{n, k}(t), \quad t \in [0, 1], \ n \ge 0.$$

Notice that |rn(t)| = 1 on [0, 1). This is an orthonormal system but it is not complete. In the language of probability theory, the Rademacher sequence is an instance of a sequence of independent Bernoulli random variables with mean 0. The Khintchine inequality expresses the fact that in all the spaces Lp([0, 1]), 1 ≤ p < ∞, the Rademacher sequence is equivalent to the unit vector basis in ℓ2. In particular, the closed linear span of the Rademacher sequence in Lp([0, 1]), 1 ≤ p < ∞, is isomorphic to ℓ2.

The Faber–Schauder system
The Faber–Schauder system is the family of continuous functions on [0, 1] consisting of the constant function 1, and of multiples of indefinite integrals of the functions in the Haar system on [0, 1], chosen to have norm 1 in the maximum norm. This system begins with s0 = 1, then is the indefinite integral vanishing at 0 of the function 1, first element of the Haar system on [0, 1]. Next, for every integer n ≥ 0, functions sn,k are defined by the formula

s_{n, k}(t) = 2^{1 + n/2} \int_0^t \psi_{n, k}(u) \, d u, \quad t \in [0, 1], \ 0 \le k < 2^n.$$ These functions sn,k are continuous, piecewise linear, supported by the interval In,k that also supports ψn,k. The function sn,k is equal to 1 at the midpoint xn,k of the interval In,k, linear on both halves of that interval. It takes values between 0 and 1 everywhere.

The Faber–Schauder system is a Schauder basis for the space C([0, 1]) of continuous functions on [0, 1]. For every f in C([0, 1]), the partial sum
 * $$ f_{n+1} = a_0 s_0 + a_1 s_1 + \sum_{m = 0}^{n-1} \Bigl( \sum_{k=0}^{2^m - 1} a_{m,k} s_{m, k} \Bigr) \in C([0, 1])$$

of the series expansion of f in the Faber–Schauder system is the continuous piecewise linear function that agrees with f at the 2n + 1 points k2−n, where 0 ≤ k ≤ 2n. Next, the formula
 * $$ f_{n+2} - f_{n+1} = \sum_{k=0}^{2^n - 1} \bigl( f(x_{n,k}) - f_{n+1}(x_{n, k}) \bigr) s_{n, k} = \sum_{k=0}^{2^n - 1} a_{n, k} s_{n, k} $$

gives a way to compute the expansion of f step by step. Since f is uniformly continuous, the sequence {fn} converges uniformly to f. It follows that the Faber–Schauder series expansion of f converges in C([0, 1]), and the sum of this series is equal to f.

The Franklin system
The Franklin system is obtained from the Faber–Schauder system by the Gram–Schmidt orthonormalization procedure. Since the Franklin system has the same linear span as that of the Faber–Schauder system, this span is dense in C([0, 1]), hence in L2([0, 1]). The Franklin system is therefore an orthonormal basis for L2([0, 1]), consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis for C([0, 1]). The Franklin system is also an unconditional Schauder basis for the space Lp([0, 1]) when 1 < p < ∞. The Franklin system provides a Schauder basis in the disk algebra A(D). This was proved in 1974 by Bočkarev, after the existence of a basis for the disk algebra had remained open for more than forty years.

Bočkarev's construction of a Schauder basis in A(D) goes as follows: let f be a complex valued Lipschitz function on [0, π]; then f is the sum of a cosine series with absolutely summable coefficients. Let T(f) be the element of A(D) defined by the complex power series with the same coefficients,

 * $$ \left\{ f : x \in [0, \pi] \rightarrow \sum_{n=0}^\infty a_n \cos(n x) \right\} \longrightarrow \left\{ T(f) : z \rightarrow \sum_{n=0}^\infty a_n z^n, \quad |z| \le 1 \right\}.$$

Bočkarev's basis for A(D) is formed by the images under T of the functions in the Franklin system on [0, π]. Bočkarev's equivalent description for the mapping T starts by extending f to an even Lipschitz function g1 on [−π, π], identified with a Lipschitz function on the unit circle T. Next, let g2 be the conjugate function of g1, and define T(f) to be the function in A(D) whose value on the boundary T of D is equal to g1 + ig2.

When dealing with 1-periodic continuous functions, or rather with continuous functions f on [0, 1] such that, one removes the function from the Faber–Schauder system, in order to obtain the periodic Faber–Schauder system. The periodic Franklin system is obtained by orthonormalization from the periodic Faber–-Schauder system. One can prove Bočkarev's result on A(D) by proving that the periodic Franklin system on [0, 2π] is a basis for a Banach space Ar isomorphic to A(D). The space Ar consists of complex continuous functions on the unit circle T whose conjugate function is also continuous.

Haar matrix
The 2×2 Haar matrix that is associated with the Haar wavelet is
 * $$ H_2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}.$$

Using the discrete wavelet transform, one can transform any sequence $$(a_0,a_1,\dots,a_{2n},a_{2n+1})$$ of even length into a sequence of two-component-vectors $$ \left(\left(a_0,a_1\right),\left(a_2,a_3\right),\dots,\left(a_{2n},a_{2n+1}\right)\right) $$. If one right-multiplies each vector with the matrix $$ H_2 $$, one gets the result $$\left(\left(s_0,d_0\right),\dots,\left(s_n,d_n\right)\right)$$ of one stage of the fast Haar-wavelet transform. Usually one separates the sequences s and d and continues with transforming the sequence s. Sequence s is often referred to as the averages part, whereas d is known as the details part.

If one has a sequence of length a multiple of four, one can build blocks of 4 elements and transform them in a similar manner with the 4×4 Haar matrix
 * $$ H_4 = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 0 & 0\\ 0 & 0 & 1 & -1 \end{bmatrix},$$

which combines two stages of the fast Haar-wavelet transform.

Compare with a Walsh matrix, which is a non-localized 1/–1 matrix.

Generally, the 2N×2N Haar matrix can be derived by the following equation.

 * $$ H_{2N} = \begin{bmatrix} H_{N} \otimes [1, 1] \\ I_{N} \otimes [1, -1] \end{bmatrix}$$
 * where $$I_{N} = \begin{bmatrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 \end{bmatrix}$$ and $$\otimes$$ is the Kronecker product.

The Kronecker product of $$A \otimes B$$, where $$A$$ is an m×n matrix and $$B$$ is a p×q matrix, is expressed as

 * $$A \otimes B = \begin{bmatrix} a_{11}B & \dots & a_{1n}B \\ \vdots & \ddots & \vdots \\ a_{m1}B & \dots & a_{mn}B\end{bmatrix}.$$

An un-normalized 8-point Haar matrix $$H_8$$ is shown below

 * $$H_{8} = \begin{bmatrix} 1&1&1&1&1&1&1&1 \\ 1&1&1&1&-1&-1&-1&-1 \\ 1&1&-1&-1&0&0&0&0& \\ 0&0&0&0&1&1&-1&-1 \\ 1&-1&0&0&0&0&0&0& \\ 0&0&1&-1&0&0&0&0 \\ 0&0&0&0&1&-1&0&0& \\ 0&0&0&0&0&0&1&-1 \end{bmatrix}.$$

Note that, the above matrix is an un-normalized Haar matrix. The Haar matrix required by the Haar transform should be normalized.

From the definition of the Haar matrix $$H$$, one can observe that, unlike the Fourier transform, $$H$$ has only real elements (i.e., 1, -1 or 0) and is non-symmetric.

Take the 8-point Haar matrix $$H_8$$ as an example. The first row of $$H_8$$ measures the average value, and the second row of $$H_8$$ measures a low frequency component of the input vector. The next two rows are sensitive to the first and second half of the input vector respectively, which corresponds to moderate frequency components. The remaining four rows are sensitive to the four section of the input vector, which corresponds to high frequency components.

Haar transform
The Haar transform is the simplest of the wavelet transforms. This transform cross-multiplies a function against the Haar wavelet with various shifts and stretches, like the Fourier transform cross-multiplies a function against a sine wave with two phases and many stretches.

Introduction
The Haar transform is one of the oldest transform functions, proposed in 1910 by the Hungarian mathematician Alfréd Haar. It is found effective in applications such as signal and image compression in electrical and computer engineering as it provides a simple and computationally efficient approach for analysing the local aspects of a signal.

The Haar transform is derived from the Haar matrix. An example of a 4×4 Haar transformation matrix is shown below.

 * $$H_4 = \frac{1}{2}

\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & -\sqrt{2}\end{bmatrix} $$

The Haar transform can be thought of as a sampling process in which rows of the transformation matrix act as samples of finer and finer resolution.

Compare with the Walsh transform, which is also 1/–1, but is non-localized.

Property
The Haar transform has the following properties

 * 1) No need for multiplications. It requires only additions and there are many elements with zero value in the Haar matrix, so the computation time is short. It is faster than Walsh transform, whose matrix is composed of +1 and −1.
 * 2) Input and output length are the same. However, the length should be a power of 2, i.e. $$N = 2^k,  k\in \mathbb{N}$$.
 * 3) It can be used to analyse the localized feature of signals. Due to the orthogonal property of the Haar function, the frequency components of input signal can be analyzed.

Haar transform and Inverse Haar transform
The Haar transform yn of an n-input function xn is

 * $$ y_n = H_n x_n$$

The Haar transform matrix is real and orthogonal. Thus, the inverse Haar transform can be derived by the following equations.

 * $$ H = H^*, H^{-1} = H^T, \text{ i.e. } HH^T = I $$

 * where $$I$$ is the identity matrix. For example, when n = 4

 * $$ H_4^{T}H_4 = \frac{1}{2}\begin{bmatrix} 1&1&\sqrt{2}&0 \\ 1&1&-\sqrt{2}&0 \\ 1&-1&0&\sqrt{2} \\ 1&-1&0&-\sqrt{2}\end{bmatrix}

\cdot\; \frac{1}{2}\begin{bmatrix} 1&1&1&1 \\ 1&1&-1&-1 \\ \sqrt{2}&-\sqrt{2}&0&0 \\ 0&0&\sqrt{2}&-\sqrt{2}\end{bmatrix} = \begin{bmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} $$

Thus, the inverse Haar transform is

 * $$ x_{n} = H^{T}y_{n}$$

Example
The Haar transform coefficients of a n=4-point signal $$x_{4} = [1,2,3,4]^{T}$$ can be found as

 * $$ y_{4} = H_4 x_4 =

\frac{1}{2}\begin{bmatrix} 1&1&1&1 \\ 1&1&-1&-1 \\ \sqrt{2}&-\sqrt{2}&0&0 \\ 0&0&\sqrt{2}&-\sqrt{2}\end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4\end{bmatrix} = \begin{bmatrix} 5 \\ -2 \\ -1/\sqrt{2} \\ -1/\sqrt{2}\end{bmatrix} $$

The input signal can then be perfectly reconstructed by the inverse Haar transform

 * $$ \hat{x_{4}} = H_{4}^{T}y_{4} =

\frac{1}{2}\begin{bmatrix} 1&1&\sqrt{2}&0 \\ 1&1&-\sqrt{2}&0 \\ 1&-1&0&\sqrt{2} \\ 1&-1&0&-\sqrt{2}\end{bmatrix} \begin{bmatrix} 5 \\ -2 \\ -1/\sqrt{2} \\ -1/\sqrt{2}\end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} $$

title: Haar wavelet


text:
List of Fourier-related transforms

This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component.

Continuous transforms
Applied to functions of continuous arguments, Fourier-related transforms include:
 * Two-sided Laplace transform
 * Mellin transform, another closely related integral transform
 * Laplace transform
 * Fourier transform, with special cases:
 * Fourier series
 * When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called Fourier series coefficients.  The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients.
 * When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed.  The same discrete set is obtained by treating the duration of the segment as one period of a periodic function and computing the Fourier series coefficients.
 * Sine and cosine transforms: When the input function has odd or even symmetry around the origin, the Fourier transform reduces to a sine or cosine transform.
 * Hartley transform
 * Short-time Fourier transform (or short-term Fourier transform) (STFT)
 * Rectangular mask short-time Fourier transform
 * Chirplet transform
 * Fractional Fourier transform (FRFT)
 * Hankel transform: related to the Fourier Transform of radial functions.
 * Fourier–Bros–Iagolnitzer transform
 * Linear canonical transform

Discrete transforms
For usage on computers, number theory and algebra, discrete arguments (e.g. functions of a series of discrete samples) are often more appropriate, and are handled by the transforms (analogous to the continuous cases above):

 * Discrete-time Fourier transform (DTFT): Equivalent to the Fourier transform of a "continuous" function that is constructed from the discrete input function by using the sample values to modulate a Dirac comb. When the sample values are derived by sampling a function on the real line, ƒ(x), the DTFT is equivalent to a periodic summation of the Fourier transform of ƒ.  The DTFT output is always periodic (cyclic).  An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded (or finite), the length of one cycle.
 * discrete Fourier transform (DFT):
 * When the input sequence is periodic, the DTFT output is also a Dirac comb function, modulated by the coefficients of a Fourier series which can be computed as a DFT of one cycle of the input sequence. The number of discrete values in one cycle of the DFT is the same as in one cycle of the input sequence.
 * When the non-zero portion of the input sequence has finite duration, the DTFT is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed.  The same discrete set is obtained by treating the duration of the segment as one cycle of a periodic function and computing the DFT.
 * Discrete sine and cosine transforms: When the input sequence has odd or even symmetry around the origin, the DTFT reduces to a discrete sine transform (DST) or discrete cosine transform (DCT).
 * Regressive discrete Fourier series, in which the period is determined by the data rather than fixed in advance.
 * Discrete Chebyshev transforms (on the 'roots' grid and the 'extrema' grid of the Chebyshev polynomials of the first kind). This transform is of much importance in the field of spectral methods for solving differential equations because it can be used to swiftly and efficient go from grid point values to Chebyshev series coefficients.
 * Generalized DFT (GDFT), a generalization of the DFT and constant modulus transforms where phase functions might be of linear with integer and real valued slopes, or even non-linear phase bringing flexibilities for optimal designs of various metrics, e.g. auto- and cross-correlations.
 * Discrete-space Fourier transform (DSFT) is the generalization of the DTFT from 1D signals to 2D signals. It is called "discrete-space" rather than "discrete-time" because the most prevalent application is to imaging and image processing where the input function arguments are equally spaced samples of spatial coordinates $$(x,y)$$. The DSFT output is periodic in both variables.
 * Z-transform, a generalization of the DTFT to the entire complex plane
 * Modified discrete cosine transform (MDCT)
 * Discrete Hartley transform (DHT)
 * Also the discretized STFT (see above).
 * Hadamard transform (Walsh function).
 * Fourier transform on finite groups.
 * Discrete Fourier transform (general).

The use of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a fast Fourier transform (FFT). The Nyquist–Shannon sampling theorem is critical for understanding the output of such discrete transforms.

title: List of Fourier-related transforms