Arthur Koestler

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Arthur Koestler, CBE (UK: /ˈkɜːstlər/, US: /ˈkɛst-/; German: [ˈkœstlɐ]; Hungarian: Kösztler Artúr; 5 September 1905 – 1 March 1983) was a Hungarian-born author and journalist. Koestler was born in Budapest and, apart from his early school years, was educated in Austria. In 1931, Koestler joined the Communist Party of Germany, but he resigned in 1938 after becoming disillusioned with Stalinism.

Having moved to Britain in 1940, he published his novel Darkness at Noon, an anti-totalitarian work that gained him international fame. Over the next 43 years, Koestler espoused many political causes and wrote novels, memoirs, biographies, and numerous essays. In 1949, Koestler began secretly working with a British Cold War anti-communist propaganda department known as the Information Research Department (IRD), which would republish and distribute many of his works, and also fund his activities. In 1968, he was awarded the Sonning Prize "for [his] outstanding contribution to European culture". In 1972 he was made a Commander of the Order of the British Empire (CBE).

In 1976, he was diagnosed with Parkinson's disease and in 1979 with terminal leukaemia. On March 1, 1983, Koestler and his wife Cynthia took their lives at their London home by swallowing lethal quantities of barbiturate-based Tuinal capsules.

Life

[Koestler] began his education in the twilight of the Austro-Hungarian Empire, at an experimental kindergarten in Budapest. His mother was briefly a patient of Sigmund Freud. In interwar Vienna he wound up as the personal secretary of Vladimir Jabotinsky, one of the early leaders of the Zionist movement. Travelling in Soviet Turkmenistan as a young and ardent Communist, he ran into Langston Hughes. While reporting on the Spanish Civil War, he met W. H. Auden at a "crazy party" in Valencia before winding up in one of Franco's prisons. In Weimar Berlin he fell into the circle of the Comintern agent Willi Münzenberg, through whom he met the leading German Communists [and fellow-travellers] of the era, including Johannes Becher, Hanns Eisler and Bertolt Brecht. Afraid of being caught by the Gestapo while fleeing France, he borrowed suicide pills from Walter Benjamin. He took them several weeks later when it seemed he would be unable to get out of Lisbon, but he did not die. Along the way he had lunch with Thomas Mann, got drunk with Dylan Thomas, made friends with George Orwell, flirted with Mary McCarthy and lived in Cyril Connolly's London flat. In 1940 Koestler was released from a French detention camp, partly thanks to the intervention of Harold Nicolson and Noël Coward. In the 1950s he helped to found the Congress for Cultural Freedom, together with Melvin Lasky and Sidney Hook. In the 1960s he took LSD with Timothy Leary. In the 1970s he was still giving lectures that impressed, among others, the young Salman Rushdie.

Anne Applebaum, reviewing Michael Scammell: Koestler: The Literary and Political Odyssey of a Twentieth-Century Skeptic

Origins and early life

Koestler was born in Budapest to Jewish parents Henrik and Adele Koestler (née Jeiteles). He was an only child. His father Henrik Koestler was born on 18 August 1869 in the town of Miskolc in northeastern Hungary. His paternal grandfather Lipót Koestler, was a soldier in the Austro-Hungarian Army. In 1861 he married Karolina Schon, the daughter of a prosperous timber merchant. Their son Henrik was born several years later. Henrik left school at age 16 and took a job as an errand boy with a firm of drapers. He taught himself English, German and French, and eventually became a partner in the firm. He set up his own business importing textiles into Hungary.

Arthur's mother, Adele Jeiteles, was born on 25 June 1871 into a prominent Jewish family in Prague. Among her ancestors was Jonas Mischel Loeb Jeitteles, a prominent 18th-century physician and essayist, whose son Juda Jeitteles became a well-known poet. Beethoven set some of his poems to music. Adele's father, Jacob Jeiteles, moved the family to Vienna, where she grew up in relative prosperity until about 1890. Faced with financial difficulties, her father abandoned his wife and daughter, and emigrated to the United States.

Adele and her mother moved from Vienna to Budapest to stay with Adele's older married sister. Although Koestler "liked later to claim his family had flared up from nothing into sudden wealth and then vanished just as fast into exile or the gas chambers... It wasn't true: his mother was from one of the richest Jewish families in Austro-Hungary."

Henrik Koestler met Adele in 1898 and married her in 1900. Arthur, their only child, was born on 5 September 1905. The Koestlers lived in spacious, well-furnished, rented apartments in various predominantly Jewish districts of Budapest. During Arthur's early years, they employed a cook/housekeeper, as well as a foreign governess. His primary school education started at an experimental private kindergarten founded by Laura Striker (née Polányi). Her daughter Eva Striker later became Koestler's lover, and they remained friends all his life.

The outbreak of World War I in 1914 deprived Koestler's father of foreign suppliers and his business collapsed. Facing destitution, the family moved temporarily to a boarding house in Vienna. When the war ended, the family returned to Budapest.

As noted in Koestler's autobiography, he and his family were sympathetic to the short-lived Hungarian Bolshevik Revolution of 1919. Though the small soap factory owned at the time by Koestler's father was nationalised, the elder Koestler was appointed its director by the revolutionary government and was well-paid. Even though the autobiography was published in 1953, after Koestler had become an outspoken anti-Communist, he wrote favourably of the Hungarian Communists and their leader Béla Kun. He fondly recalled the hopes for a better future he had felt as a teenager in revolutionary Budapest.

Later the Koestlers witnessed the temporary occupation of Budapest by the Romanian Army and then the White Terror under the right-wing regime of Admiral Horthy. In 1920 the family returned to Vienna, where Henrik set up a successful new import business.

In September 1922 Arthur enrolled in the University of Vienna to study engineering, and joined a Zionist duelling student fraternity, 'Unitas.' . When Henrik's latest business failed, Koestler stopped attending lectures, and was expelled for non-payment of fees. In March 1926 he wrote a letter to his parents telling them that he was going to Mandate Palestine for a year to work as an assistant engineer in a factory, in order to gain experience to help him obtain a job in Austria. On 1 April 1926 he left Vienna for Palestine.

Palestine, Paris, Berlin and polar flight, 1926–1931

For a few weeks Koestler lived in a kibbutz, but his application to join the collective (Kvutzat Heftziba) was rejected by its members. For the next twelve months he supported himself with menial jobs in Haifa, Tel Aviv, and Jerusalem. Frequently penniless and starving, he often depended on friends and acquaintances for survival. He occasionally wrote or edited broadsheets and other publications, mostly in German. In early 1927 he left Palestine briefly for Berlin, where he ran the Secretariat of Ze'ev Jabotinsky's Revisionist Party.

Later that year, through a friend, Koestler obtained the position of Middle East correspondent for the prestigious Berlin-based Ullstein-Verlag group of newspapers. He returned to Jerusalem, where for the next two years he produced detailed political essays, as well as some lighter reportage, for his principal employer and for other newspapers. He was resident at this time at 29 Rehov Hanevi'im, in Jerusalem. He travelled extensively, interviewed heads of state, kings, presidents and prime ministers, and greatly enhanced his reputation as a journalist. As noted in his autobiography, he came to realise that he would never really fit into Palestine's Zionist Jewish community, the Yishuv, and particularly that he would not be able to have a journalistic career in Hebrew.

In June 1929, while on leave in Berlin, Koestler successfully lobbied at Ullstein for a transfer away from Palestine. In September he was sent to Paris to fill a vacancy in the bureau of the Ullstein News Service. In 1931, he was called to Berlin and appointed science editor of the Vossische Zeitung and science adviser to the Ullstein newspaper empire. In July 1931, he was Ullstein's choice to represent the paper on board the Graf Zeppelin's week-long polar flight, which carried a team of scientists and the polar aviator Lincoln Ellsworth to 82 degrees North and back. Koestler was the only journalist on board: his live wireless broadcasts, and subsequent articles and lecture tours throughout Europe, brought him further kudos. Soon afterwards he was appointed foreign editor and assistant editor-in-chief of the mass-circulation Berliner Zeitung am Mittag.

In 1931, Koestler, encouraged by Eva Striker, and impressed by the achievements of the Soviet Union, became a supporter of Marxism-Leninism. On 31 December 1931, he applied for membership of the Communist Party of Germany. As noted in his biography, he was disappointed in the conduct of the Vossische Zeitung, "The Flagship of German Liberalism", which adapted to changing times by firing Jewish journalists, hiring writers with marked German Nationalist views, and dropping its longstanding campaign against capital punishment. Koestler concluded that Liberals and moderate Democrats could not stand up against the rising Nazi tide and that the Communists were the only real counter-force.

1930s

Koestler wrote a book on the Soviet Five-Year Plan, but it did not meet with the approval of the Soviet authorities and was never published in Russian. Only the German version, extensively censored, was published in an edition for German-speaking Soviet citizens.

In 1932 Koestler travelled in Turkmenistan and Central Asia, where he met and traveled with Langston Hughes. In September 1933 he returned to Paris and for the next two years was active in anti-Fascist movements. He wrote propaganda under the direction of Willi Münzenberg, the Comintern's chief propaganda director in the West.

In 1935 Koestler married Dorothy Ascher, a fellow Communist activist. They separated amicably in 1937.

In 1936, during the Spanish Civil War, he undertook a visit to General Francisco Franco's headquarters in Seville on behalf of the Comintern, pretending to be a Franco sympathiser and using credentials from the London daily News Chronicle as cover. He collected evidence of the direct involvement of Fascist Italy and Nazi Germany on Franco's side, which at that time the Nationalist rebels were still trying to conceal. He had to escape after he was recognised and denounced as a Communist by a German former colleague. Back in France he wrote L'Espagne Ensanglantée, which was later incorporated into his book Spanish Testament.

In 1937 he returned to Loyalist Spain as a war correspondent for the News Chronicle, and was in Málaga when it fell to Mussolini's troops, who were fighting on the side of the Nationalists. He took refuge in the house of retired zoologist Sir Peter Chalmers Mitchell, and they were both arrested by Franco's chief propagandist, Luis Bolín, who had sworn that if he ever got his hands on Koestler, he would "shoot him like a dog". From February until June, Koestler was imprisoned in Seville under sentence of death. He was eventually exchanged for a "high value" Nationalist prisoner held by the Loyalists, the wife of one of Franco's ace fighter pilots. Koestler was one of the few authors to have been sentenced to death, an experience he wrote about in Dialogue with Death. As he noted in his autobiography, his estranged wife Dorothy Ascher had greatly contributed to saving his life by intensive, months-long lobbying on his behalf in Britain. When he went to Britain after his release, the couple tried to resume their marriage, but Koestler's gratitude to her proved an insufficient foundation for a daily life together.

Koestler returned to France, where he agreed to write a sex encyclopaedia to earn money to live on. It was published to great success under the title The Encyclopœdia of Sexual Knowledge, under the pseudonyms of "Drs A. Costler, A. Willy, and Others".

In July 1938 Koestler finished work on his novel The Gladiators. Later that year he resigned from the Communist Party and started work on a new novel, which was published in London under the title Darkness at Noon (1941). Also in 1938 he became editor of Die Zukunft (The Future), a German-language weekly published in Paris. Koestler's breaking with the Communist Party may have been influenced by the similar step taken by his fellow activist Willi Münzenberg.

In 1939 Koestler met and formed an attachment to the British sculptor Daphne Hardy. They lived together in Paris, and she translated the manuscript of Darkness at Noon from German into English in early 1940. She smuggled it out of France when they left ahead of the German occupation and arranged for its publication after reaching London that year.

War years

After the outbreak of World War II, Koestler returned from the South of France to Paris. He attempted to turn himself in to the authorities as a foreign national several times and was finally arrested on 2 October 1939. The French government first detained Koestler at Stade Roland Garros until he was moved to Le Vernet Internment Camp among other "undesirable aliens", most of them refugees. He was released in early 1940 in response to strong British pressure.

Milicent Bagot, an intelligence officer at MI5, recommended his release from Camp Vernet, but said that he should not be granted a British visa. (John le Carré used Bagot as a model for Connie Sachs in his spy novels featuring "George Smiley". Bagot was the first to warn that Kim Philby of MI6 was probably spying for the USSR.) Koestler describes the period 1939 to 1940 and his incarceration in Le Vernet in his memoir Scum of the Earth.

Shortly before the German invasion of France, Koestler joined the French Foreign Legion in order to get out of the country. He deserted in North Africa and tried to return to England. He heard a false report that the ship on which Hardy was travelling had sunk, and that she and his manuscript were lost. He attempted suicide, but survived.

Arriving in the UK without an entry permit, Koestler was imprisoned pending examination of his case. He was still in prison when Daphne Hardy's English translation of his book Darkness at Noon was published in early 1941.

Immediately after Koestler was released, he volunteered for Army service. While awaiting his call-up papers, between January and March 1941, he wrote his memoir Scum of the Earth, the first book he wrote in English. For the next twelve months he served in the Pioneer Corps.

January 1945, Kibbutz Ein HaShofet, Koestler is 5th from the right

In March 1942 Koestler was assigned to the Ministry of Information, where he worked as a scriptwriter for propaganda broadcasts and films. In his spare time he wrote Arrival and Departure, the third in his trilogy of novels that included Darkness at Noon. He also wrote several essays, which were subsequently collected and published in The Yogi and the Commissar. One of the essays, titled "On Disbelieving Atrocities" (originally published in The New York Times), was about the Nazi atrocities against the Jews.

Daphne Hardy, who had been doing war work in Oxford, joined Koestler in London in 1943, but they parted company a few months later. They remained good friends until Koestler's death.

In December 1944 Koestler travelled to Palestine with accreditation from The Times. There he had a clandestine meeting with Menachem Begin, the head of the Irgun paramilitary organisation, who was wanted by the British and had a 500-pound bounty on his head. Koestler tried to persuade him to abandon militant attacks and accept a two-state solution for Palestine, but failed. Many years later Koestler wrote in his memoirs: "When the meeting was over, I realised how naïve I had been to imagine that my arguments would have even the slightest influence."

Staying in Palestine until August 1945, Koestler collected material for his next novel, Thieves in the Night. When he returned to England, Mamaine Paget, whom he had started to see before going out to Palestine, was waiting for him. In August 1945 the couple moved to the cottage of Bwlch Ocyn, a secluded farmhouse that belonged to Clough Williams-Ellis, in the Vale of Ffestiniog. Over the next three years, Koestler became a close friend of writer George Orwell. The region had its own intellectual circle, which would have been sympathetic to Koestler: Williams-Ellis' wife, Amabel, a niece of Lytton Strachey, was also a former communist; other associates included Rupert Crawshay-Williams, Michael Polanyi, Storm Jameson and, most significantly, Bertrand Russell, who lived just a few miles from the Koestler cottage.

Post-war years

In 1948, when war broke out between the newly declared State of Israel and the neighbouring Arab states, Koestler was accredited by several newspapers, American, British and French, and travelled to Israel. Mamaine Paget went with him. They arrived in Israel on 4 June and stayed there until October. Later that year they decided to leave the UK for a while and move to France. News that his long-pending application for British nationality had been granted reached him in France in late December; early in 1949 he returned to London to swear the oath of allegiance to the British Crown.

In January 1949 Koestler and Paget moved to a house he had bought in France. There he wrote a contribution to The God That Failed and finished work on Promise and Fulfilment: Palestine 1917-1949. The latter book received poor reviews in both the U.S. and the UK. In 1949 he also published the non-fiction Insight and Outlook. This too received lukewarm reviews. In July Koestler began work on Arrow in the Blue, the first volume of his autobiography. He hired a new part-time secretary, Cynthia Jefferies, who replaced Daphne Woodward. Cynthia and Koestler eventually married. In the autumn he started work on The Age of Longing, on which he continued to work until mid-1950.

Koestler had reached agreement with his first wife, Dorothy, on an amicable divorce, and their marriage was dissolved on 15 December 1949. This cleared the way for his marriage to Mamaine Paget, which took place on 15 April 1950 at the British Consulate in Paris.

In June Koestler delivered a major anti-Communist speech in Berlin under the auspices of the Congress for Cultural Freedom, an organisation funded (though he did not know this) by the Central Intelligence Agency (CIA) of the United States. In the autumn he went to the United States on a lecture tour, during which he lobbied for permanent resident status in the U.S. At the end of October, on impulse, he bought Island Farm, a small island with a house on it on the Delaware River near New Hope, Pennsylvania. He intended to live there at least for part of each year.

In January 1951 a dramatised version of Darkness at Noon, by Sidney Kingsley, opened in New York. It won the New York Drama Critics Award. Koestler donated all his royalties from the play to a fund he had set up to help struggling authors, the Fund for Intellectual Freedom (FIF). In June a bill was introduced in the U.S. Senate to grant Koestler permanent residence in the U.S. Koestler sent tickets for the play to his House sponsor Richard Nixon and his Senate sponsor Owen Brewster, a close confidant of Joseph McCarthy. The bill became law on 23 August 1951 as Private Law 221 Chapter 343 "AN ACT For the relief of Arthur Koestler".

In 1951 the last of Koestler's political works, The Age of Longing, was published. In it he examined the political landscape of post-war Europe and the problems facing the continent.

In August 1952 his marriage to Mamaine collapsed. They separated, but remained close until her sudden and unexpected death in June 1954. The book Living with Koestler: Mamaine Koestler's Letters 1945–51, edited by Mamaine's twin sister Celia Goodman, gives insight into their lives together.

Koestler decided to make his permanent home in Britain. In May 1953 he bought a three-storey Georgian town house on Montpelier Square in London, and sold his houses in France and the United States.

The first two volumes of his autobiography, Arrow in the Blue, which covers his life up to December 1931 when he joined the German Communist Party, and The Invisible Writing, which covers the years 1932 to 1940, were published in 1952 and 1954, respectively. A collection of essays, The Trail of the Dinosaur and Other Essays, on the perils he saw facing western civilisation, was published in 1955.

On 13 April 1955 Janine Graetz, with whom Koestler had an on-off relationship over a period of years, gave birth to his daughter Cristina. Despite repeated attempts by Janine to persuade Koestler to show some interest in her, Koestler had almost no contact with Cristina throughout his life. Early in 1956 he arranged for Cynthia Jeffries to have an abortion when she became pregnant; it was then illegal. Koestler's main political activity during 1955 was his campaign for the abolition of capital punishment (which in the UK was by hanging). In July he started work on Reflections on Hanging.

Later life, 1956–1975

Although Koestler resumed work on a biography of Kepler in 1955, it was not published until 1959. In the interim it was entitled The Sleepwalkers. The emphasis of the book had changed and broadened to "A History of Man's Changing Vision of the Universe", which also became the book's subtitle. Copernicus and Galileo were added to Kepler as the major subjects of the book.

Later in 1956, as a consequence of the Hungarian Uprising, Koestler became busy organising anti-Soviet meetings and protests. In June 1957 Koestler gave a lecture at a symposium in Alpbach, Austria, and fell in love with the village. He bought land there, had a house built, and for the next twelve years used it as a place for summer vacations and for organising symposia.

In May 1958 he had a hernia operation. In December he left for India and Japan, and was away until early 1959. Based on his travels, he wrote the book The Lotus and the Robot.

In early 1960, on his way back from a conference in San Francisco, Koestler interrupted his journey at the University of Michigan, Ann Arbor, where some experimental research was going on with hallucinogens. He tried psilocybin and had a "bad trip". Later, when he arrived at Harvard to see Timothy Leary, he experimented with more drugs, but was not enthusiastic about that experience either.

In November 1960 he was elected to a Fellowship of The Royal Society of Literature.

In 1962, along with his agent, A D Peters and the editor of The Observer, David Astor, Koestler set up a scheme to encourage prison inmates to engage in arts activities and to reward their efforts. Koestler Arts supports over 7,000 entrants from UK prisons each year and awards prizes in fifty different artforms. In September each year, Koestler Arts run an exhibition at London's Southbank Centre.

Koestler's book The Act of Creation was published in May 1964. In November he undertook a lecture tour of various universities in California. In 1965 he married Cynthia in New York; they moved to California, where he participated in a series of seminars at the Center for Advanced Study in the Behavioral Sciences at Stanford University.

Koestler spent most of 1966 and the early months of 1967 working on The Ghost in the Machine. In his article "Return Trip to Nirvana", published in 1967 in the Sunday Telegraph, Koestler wrote about the drug culture and his own experiences with hallucinogens. The article also challenged the defence of drugs in Aldous Huxley's The Doors of Perception.

In April 1968 Koestler was awarded the Sonning Prize "for [his] outstanding contribution to European culture". The Ghost in the Machine was published in August of same year and in the autumn he received an honorary doctorate from Queen's University, Kingston, Canada. In the later part of November the Koestlers flew to Australia for a number of television appearances and press interviews.

The first half of the 1970s saw the publication of four more books by Koestler: The Case of the Midwife Toad (1971), The Roots of Coincidence and The Call-Girls (both 1972), and The Heel of Achilles: Essays 1968–1973 (1974). In the New Year Honours List for 1972 he was made a Commander of the Order of the British Empire (CBE).

Final years, 1976–1983

Early in 1976 Koestler was diagnosed with Parkinson's disease. The trembling of his hand made writing progressively more difficult. He cut back on overseas trips and spent the summer months at a farmhouse in Denston, Suffolk, which he had bought in 1971. That same year saw the publication of The Thirteenth Tribe, which presents his Khazar hypothesis of Ashkenazi ancestry.

In 1978 Koestler published Janus: A Summing Up. In 1980 he was diagnosed with chronic lymphocytic leukaemia. His book Bricks to Babel was published that year. His final book, Kaleidoscope, containing essays from Drinkers of Infinity and The Heel of Achilles: Essays 1968–1973, with some later pieces and stories, was published in 1981.

During the final years of his life, Koestler, Brian Inglis and Tony Bloomfield established the KIB Society (named from the initials of their surnames) to sponsor research "outside the scientific orthodoxies". After his death it was renamed The Koestler Foundation.

In his capacity as Vice-President of the Voluntary Euthanasia Society, later renamed Exit, Koestler wrote a pamphlet on suicide, outlining the case both for and against, with a section dealing specifically with how best to do it.

Koestler and Cynthia killed themselves on the evening of 1 March 1983 at their London home, 8 Montpelier Square, with overdoses of the barbiturate Tuinal taken with alcohol. Their bodies were discovered on the morning of 3 March, by which time they had been dead for 36 hours.

Koestler had stated more than once that he was afraid, not of being dead, but of the process of dying. His suicide was not unexpected among his close friends. Shortly before his suicide his doctor had discovered a swelling in the groin which indicated a metastasis of the cancer.

Koestler's suicide note:

To whom it may concern.

The purpose of this note is to make it unmistakably clear that I intend to commit suicide by taking an overdose of drugs without the knowledge or aid of any other person. The drugs have been legally obtained and hoarded over a considerable period.

Trying to commit suicide is a gamble the outcome of which will be known to the gambler only if the attempt fails, but not if it succeeds. Should this attempt fail and I survive it in a physically or mentally impaired state, in which I can no longer control what is done to me, or communicate my wishes, I hereby request that I be allowed to die in my own home and not be resuscitated or kept alive by artificial means. I further request that my wife, or a physician, or any friend present, should invoke habeas corpus against any attempt to remove me forcibly from my house to hospital.

My reasons for deciding to put an end to my life are simple and compelling: Parkinson's disease and the slow-killing variety of leukaemia (CCI). I kept the latter a secret even from intimate friends to save them distress. After a more or less steady physical decline over the last years, the process has now reached an acute state with added complications which make it advisable to seek self-deliverance now, before I become incapable of making the necessary arrangements.

I wish my friends to know that I am leaving their company in a peaceful frame of mind, with some timid hopes for a de-personalised after-life beyond due confines of space, time and matter and beyond the limits of our comprehension. This "oceanic feeling" has often sustained me at difficult moments, and does so now, while I am writing this.

What makes it nevertheless hard to take this final step is the reflection of the pain it is bound to inflict on my surviving friends, above all my wife Cynthia. It is to her that I owe the relative peace and happiness that I enjoyed in the last period of my life – and never before.

The note was dated June 1982. Below it appeared the following:

Since the above was written in June 1982, my wife decided that after thirty-four years of working together she could not face life after my death.

Further down the page appeared Cynthia's own farewell note:

I fear both death and the act of dying that lies ahead of us. I should have liked to finish my account of working for Arthur – a story which began when our paths happened to cross in 1949. However, I cannot live without Arthur, despite certain inner resources.

Double suicide has never appealed to me, but now Arthur's incurable diseases have reached a stage where there is nothing else to do.

The funeral was held at the Mortlake Crematorium in South London on 11 March 1983.

Controversy arose over why Koestler allowed, consented to, or (according to some critics) compelled his wife's simultaneous suicide. She was only 55 years old and was believed to be in good health. In a typewritten addition to her husband's suicide note, Cynthia wrote that she could not live without her husband. Reportedly, few of the Koestlers' friends were surprised by this admission, apparently perceiving that Cynthia lived her life through her husband and that she had no "life of her own". Her absolute devotion to Koestler can be seen clearly in her partially completed memoirs. Yet according to a profile of Koestler by Peter Kurth:

All their friends were troubled by what Julian Barnes calls "the unmentionable, half-spoken question" of Koestler's responsibility for Cynthia's actions. "Did he bully her into it?" asks Barnes. And "if he didn't bully her into it, why didn't he bully her out of it?" Because, with hindsight, the evidence that Cynthia's life had been ebbing with her husband's was all too apparent.

With the exception of some minor bequests, Koestler left the residue of his estate, about £1 million (worth about £3.59 million in 2021), to the promotion of research into the paranormal through the founding of a chair in parapsychology at a university in Britain. The trustees of the estate had great difficulty finding a university willing to establish such a chair. Oxford, Cambridge, King's College London and University College London were approached, and all refused. Eventually the trustees reached agreement with the University of Edinburgh to set up a chair in accordance with Koestler's request.

Personal life and allegations

Koestler's relations with women have been a source of controversy. David Cesarani alleged in his biography of Koestler, published in 1998, that Koestler had been a serial rapist, citing the case of the British feminist writer Jill Craigie who said that she had been his victim in 1951. Feminist protesters forced the removal of his bust from Edinburgh University. In his biography, Koestler: The Indispensable Intellectual (2009), Michael Scammell countered that Craigie was the only woman to go on record that she had been raped by Koestler, and had done so at a dinner party more than fifty years after the event. Claims that Koestler had been violent were added by Craigie later, although Scammell concedes that Koestler could be rough and sexually aggressive.

Some critics believed that Cesarani's claims of Koestler having been a 'serial rapist' were unfounded; in his review of Cesarani's biography in The New York Times, the historian Mark Mazower observed: "Even those who applaud Cesarani for bringing the rape issue forward may wonder whether his approach is not too one-sided to make for a convincing portrait. Koestler was a domineering man. But he attracted women and many remained close friends after they had slept with him. It is implausible to write them all off as masochists, as Cesarani effectively does. Some broke with him; but then so did many other friends and acquaintances. "Similarly, John Banville, in the London Review of Books, wrote:

Koestler himself, and at least one Hungarian friend, saw nothing odd in (Koestler's) bed-hopping. "In Central Europe", George Mikes wrote in defence of Koestler, "every woman was regarded as fair game. She could always say 'no' and ... her no would be taken for an answer, even if grudgingly." Cesarani will have none of this political incorrectness, and stoutly declares: "There is evidence that as well as his consistent violence against women Koestler was a serial rapist." The evidence that Cesarani adduces in support of this accusation is an account of a particular encounter between Koestler and Jill Craigie, the wife of Michael Foot.

Cesarani and others claim that Koestler had misogynistic tendencies. He engaged in numerous sexual affairs and generally treated the women in his life badly. In his autobiography, The Invisible Writing, Koestler admits to having denounced Nadezhda Smirnova, with whom he was having a relationship, to the Soviet secret police.

Influence and legacy

It is difficult to think of a single important twentieth-century intellectual who did not cross paths with Arthur Koestler, or a single important twentieth-century intellectual movement that Koestler did not either join or oppose. From progressive education and Freudian psychoanalysis through Zionism, communism, and existentialism to psychedelic drugs, parapsychology, and euthanasia, Koestler was fascinated by every philosophical fad, serious and unserious, political and apolitical, of his era.

– Anne Applebaum, The New York Review Of Books

Koestler wrote several major novels, two volumes of autobiographical works, two volumes of reportage, a major work on the history of science, several volumes of essays, and a considerable body of other writing and articles on subjects as varied as genetics, euthanasia, Eastern mysticism, neurology, chess, evolution, psychology, the paranormal and more.

Darkness at Noon was one of the most influential anti-Soviet books ever written. Its influence in Europe on Communists and sympathisers and, indirectly, on the outcomes of elections in Europe, was substantial. Geoffrey Wheatcroft believes that Koestler's most important books were the five completed before he was 40: his first memoirs and the trilogy of anti-totalitarian novels that included Darkness at Noon.

Politics and causes

Koestler embraced a multitude of political as well as non-political issues. Zionism, communism, anti-communism, voluntary euthanasia, abolition of capital punishment, particularly hanging, and the abolition of quarantine for dogs being reimported into the United Kingdom are examples.

Science

In his book The Case of the Midwife Toad (1971) Koestler defended the biologist Paul Kammerer, who claimed to have found experimental support for Lamarckian inheritance. According to Koestler, Kammerer's experiments on the midwife toad may have been tampered with by a Nazi sympathiser at the University of Vienna. Koestler came to the conclusion that a kind of modified "Mini-Lamarckism" may explain some rare evolutionary phenomena.

Koestler criticised neo-Darwinism in a number of his books, but he was not opposed to the theory of evolution in general terms. Biology professor Harry Gershenowitz described Koestler as a "populariser" of science despite his views not being accepted by the "orthodox academic community". According to an article in the Skeptical Inquirer, Koestler was an "advocate of Lamarckian evolution – and a critic of Darwinian natural selection as well as a believer in psychic phenomena".

In addition to his specific critiques of neo-Darwinism, Koestler was opposed to what he saw as dangerous scientific reductionism more generally, including the behaviourism school of psychology, promoted in particular by B. F. Skinner during the 1930s. Koestler assembled a group of high-profile antireductionist scientists, including C. H. Waddington, W. H. Thorpe and Ludwig von Bertalanffy, for a meeting at his retreat in Alpbach in 1968. This was one of many attempts which Koestler made to gain acceptance within the mainstream of science, a strategy which brought him into conflict with individuals such as Peter Medawar who saw themselves as defending the integrity of science from outsiders. Although he never gained significant credibility as a scientist, Koestler published a number of works at the border between science and philosophy, such as Insight and Outlook, The Act of Creation and The Ghost in the Machine.

Paranormal

Mysticism and a fascination with the paranormal imbued much of Koestler's later work and he discussed paranormal phenomena, such as extrasensory perception, psychokinesis and telepathy. In his book The Roots of Coincidence (1972) he claims that such phenomena can never be explained by theoretical physics. According to Koestler, distinct types of coincidence could be classified, such as "the library angel", in which information (typically in libraries) becomes accessible through serendipity, chance or coincidence, rather than through the use of a catalogue search. The book mentions yet another line of unconventional research by Paul Kammerer, the theory of coincidence or seriality. He also presents critically the related concepts of Carl Jung. More controversial were Koestler's studies and experiments on levitation and telepathy.

Judaism

Koestler was Jewish by birth, but he did not practise the religion. In an interview published in the British newspaper The Jewish Chronicle in 1950 he argued that Jews should either emigrate to Israel or assimilate completely into the majority cultures they lived in.

In The Thirteenth Tribe (1976) Koestler advanced a theory that Ashkenazi Jews are descended, not from the Israelites of antiquity, but from the Khazars, a Turkic people in the Caucasus that converted to Judaism in the 8th century and was later forced westwards. Koestler argued that a proof that Ashkenazi Jews have no biological connection to biblical Jews would remove the racial basis of European anti-Semitism.

In reference to the Balfour Declaration Koestler stated, "one nation solemnly promised to a second nation the country of a third."

Collaboration with the Information Research Department

Much of Arthur Koestler's work was funded and distributed secretly by a covert propaganda wing of the UK Foreign Office, known as the Information Research Department (IRD). Koestler enjoyed strong personal relationships with IRD agents from 1949 onwards, and was supportive of the department's anti-communist goals. Koestler's relationship with the British government was so strong that he had become a de facto advisor to British propagandists, urging them to create a popular series of anti-communist left-wing literature to rival the success of the Left Book Club.

Languages

Koestler first learned Hungarian, but later his family spoke mostly German at home. From his early years he became fluent in both languages. It is likely that he picked up some Yiddish too, through contact with his grandfather. By his teens he was fluent in Hungarian, German, French and English.

During his years in Palestine Koestler became sufficiently fluent in Hebrew to write stories in that language, as well as to create what is believed to have been the world's first Hebrew crossword puzzle. During his years in the Soviet Union (1932–33), although he arrived with a vocabulary of only 1,000 words of Russian, and no grammar, he picked up enough colloquial Russian to speak the language.

Koestler wrote his books in German up to 1940, but then wrote only in English. (L'Espagne ensanglantée was translated into French from German.)

Koestler is said to have coined the word mimophant to describe Bobby Fischer.

Quotes

"Liking a writer and then meeting the writer is like liking goose liver and then meeting the goose".

In August 1945 Koestler was in Palestine where he read in the Palestine Post about the dropping of the atomic bomb on Hiroshima. "That's the end of the world war", he said to a friend — "and it is also the beginning of the end of the world."

Published works

Fiction (novels)

• 1934 (2013). Die Erlebnisse des Genossen Piepvogel in der Emigration
• 1939. The Gladiators (about the revolt of Spartacus)
• 1940. Darkness at Noon
• 1943. Arrival and Departure
• 1946. Thieves in the Night
• 1951. The Age of Longing, ISBN 978-0-09-104520-3.
• 1972. The Call-Girls: A Tragicomedy with a Prologue and Epilogue. A novel about scholars making a living on the international seminar-conference circuit. ISBN 978-0-09-112550-9

Drama

• 1945. Twilight Bar.

Autobiographical writings

• 1937. Spanish Testament.
• 1941. Scum of the Earth.
• 1942. Dialogue with Death.
• 1952. Arrow in the Blue: The First Volume of an Autobiography, 1905–31, 2005 reprint, ISBN 978-0-09-949067-8
• 1954. The Invisible Writing: The Second Volume of an Autobiography, 1932–40, 1984 reprint, ISBN 978-0-8128-6218-8
• 1984. Stranger on the Square co-written with Cynthia Koestler, published posthumously, edited and with an Introduction and Epilogue by Harold Harris, London: Hutchinson, 1984, ISBN 978-0-09-154330-3.

NB The books The Lotus and the Robot, The God that Failed, and Von weissen Nächten und roten Tagen, as well as his numerous essays, all may contain further autobiographical information.

Other non-fiction

• 1934. Von weissen Nächten und roten Tagen. About Koestler's travels in the USSR. In his The Invisible Writing, Koestler calls the book Red Days and White Nights, or, more usually, Red Days. Of the five foreign language editions − Russian, German, Ukrainian, Georgian, Armenian − planned, only the German version was eventually published in Kharkov, Ukrainian S.S.R. The edition is very rare.
• 1937. L'Espagne ensanglantée.
• 1942 (summer) Le yogi et le commissaire.
• 1945. The Yogi and the Commissar and Other Essays.
• 1948. "What the Modern World is Doing to the Soul of Man". Essay in The Challenge of Our Time. 1948.
• 1949. Promise and Fulfilment: Palestine 1917–1949.
• 1949. Insight and Outlook. 1949.
• 1952. The Trail of the Dinosaur. Google Books
• 1955. The Trail of the Dinosaur and Other Essays.
• 1955. The Anatomy of Snobbery in The Anchor review No.1
• 1956. Reflections on Hanging.
• 1959. The Sleepwalkers: A History of Man's Changing Vision of the Universe. ISBN 978-0-14-019246-9   An account of changing scientific paradigms.
• 1960. The Watershed: A Biography of Johannes Kepler. (excerpted from The Sleepwalkers.) ISBN 978-0-385-09576-1
• 1960. The Lotus and the Robot, ISBN 978-0-09-059891-5. Koestler's journey to India and Japan, and his assessment of East and West.
• 1961. Control of the Mind.
• 1961. Hanged by the Neck. Reuses some material from Reflections on Hanging.
• 1963. Suicide of a Nation.
• 1964. The Act of Creation.
• 1967. The Ghost in the Machine. Penguin reprint 1990: ISBN 978-0-14-019192-9.
• 1968. Drinkers of Infinity: Essays 1955–1967.
• 1971. The Case of the Midwife Toad, ISBN 978-0-394-71823-1. An account of Paul Kammerer's research on Lamarckian evolution and what he called "serial coincidences".
• 1972. The Roots of Coincidence, ISBN 978-0-394-71934-4. Sequel to The Case of the Midwife Toad.
• 1973. The Lion and the Ostrich.
• 1974. The Heel of Achilles: Essays 1968–1973, ISBN 978-0-09-119400-0.
• 1976. The Thirteenth Tribe: The Khazar Empire and Its Heritage, ISBN 978-0-394-40284-0.
• 1976. Astride the Two Cultures: Arthur Koestler at 70, ISBN 978-0-394-40063-1.
• 1977. Twentieth Century Views: A Collection of Critical Essays, ISBN 978-0-13-049213-5.
• 1978. Janus: A Summing Up, ISBN 978-0-394-50052-2. Sequel to The Ghost in the Machine
• 1980. Bricks to Babel. Random House, ISBN 978-0-394-51897-8. This 1980 anthology of passages from many of his books, described as "A selection from 50 years of his writings, chosen and with new commentary by the author", is a comprehensive introduction to Koestler's writing and thought.
• 1981. Kaleidoscope. Essays from Drinkers of Infinity and The Heel of Achilles, plus later pieces and stories.

Writings as a contributor

• The Encyclopœdia of Sexual Knowledge (1934) (In his autobiography The Invisible Writing, Koestler uses the ligature œ in the spelling of the word "Encyclopaedia".)
• Foreign Correspondent (1940) uncredited contributor to Alfred Hitchcock film produced by Walter Wanger
• The God That Failed (1950) (collection of testimonies by ex-Communists)
• Attila, the Poet (1954) (Encounter; 1954.2 (5)). On loan at the UCL library of the School of Slavonic & Eastern European Studies.
• University College London
• Beyond Reductionism: The Alpbach Symposium. New Perspectives in the Life Sciences (co-editor with J. R. Smythies, 1969), ISBN 978-0-8070-1535-3
• The Challenge of Chance: A Mass Experiment in Telepathy and Its Unexpected Outcome (1973)
• The Concept of Creativity in Science and Art (1976)
• Life After Death, (co-editor, 1976)
• Humour and Wit. I: Encyclopædia Britannica. 15th ed. vol. 9.(1983)
• humour, Encyclopædia Britannica

Spectral density

The spectral density of a fluorescent light as a function of optical wavelength shows peaks at atomic transitions, indicated by the numbered arrows.

 

The voice waveform over time (left) has a broad audio power spectrum (right).

The power spectrum ${\displaystyle S_{xx}(f)}$ of a time series ${\displaystyle x(t)}$ describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.

When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over all time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating ${\displaystyle x^2(t)}$ over the time domain, as dictated by Parseval's theorem.

The spectrum of a physical process ${\displaystyle x(t)}$ often contains essential information about the nature of ${\displaystyle x}$. For instance, the pitch and timbre of a musical instrument are immediately determined from a spectral analysis. The color of a light source is determined by the spectrum of the electromagnetic wave's electric field ${\displaystyle E(t)}$ as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the Fourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a dispersive prism is used to obtain a spectrum of light in a spectrograph, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency.

However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer). The power spectrum is important in statistical signal processing and in the statistical study of stochastic processes, as well as in many other branches of physics and engineering. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency.

Units

See also: Fourier transform § Units

In physics, the signal might be a wave, such as an electromagnetic wave, an acoustic wave, or the vibration of a mechanism. The power spectral density (PSD) of the signal describes the power present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in watts per hertz (W/Hz).

When a signal is defined in terms only of a voltage, for instance, there is no unique power associated with the stated amplitude. In this case "power" is simply reckoned in terms of the square of the signal, as this would always be proportional to the actual power delivered by that signal into a given impedance. So one might use units of V² Hz⁻¹ for the PSD. Energy spectral density (ESD) would have units would be V² s Hz⁻¹, since energy has units of power multiplied by time (e.g., watt-hour).

In the general case, the units of PSD will be the ratio of units of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD in units of meters squared per hertz, m²/Hz. In the analysis of random vibrations, units of g² Hz⁻¹ are frequently used for the PSD of acceleration, where g denotes the g-force.

Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of x(t) will remain unspecified, but the independent variable will be assumed to be that of time.

Definition

Energy spectral density

Energy spectral density describes how the energy of a signal or a time series is distributed with frequency. Here, the term energy is used in the generalized sense of signal processing; that is, the energy ${\displaystyle E}$ of a signal ${\displaystyle x(t)}$ is:

${\displaystyle E\triangleq {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|x(t)\right|}^{2}dt.}$

The energy spectral density is most suitable for transients—that is, pulse-like signals—having a finite total energy. Finite or not, Parseval's theorem (or Plancherel's theorem) gives us an alternate expression for the energy of the signal:

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x(t){|}^{2}dt={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|\hat{x}(f)\right|}^{2}df,}$

where:

${\displaystyle \hat{x}(f)\triangleq {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i2\pi ft}x(t)dt}$

is the value of the Fourier transform of ${\displaystyle x(t)}$ at frequency ${\displaystyle f}$ (in Hz). The theorem also holds true in the discrete-time cases. Since the integral on the right-hand side is the energy of the signal, the value of${\displaystyle {\left|\hat{x}(f)\right|}^{2}df}$ can be interpreted as a density function multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency ${\displaystyle f}$ in the frequency interval ${\displaystyle f+df}$.

Therefore, the energy spectral density of ${\displaystyle x(t)}$ is defined as:

${\displaystyle {\overline{S}}_{xx}(f)\triangleq {\left|\hat{x}(f)\right|}^2}$

(Eq.1)

The function ${\displaystyle {\overline{S}}_{xx}(f)}$ and the autocorrelation of ${\displaystyle x(t)}$ form a Fourier transform pair, a result also known as the Wiener–Khinchin theorem (see also Periodogram).

As a physical example of how one might measure the energy spectral density of a signal, suppose ${\displaystyle V(t)}$ represents the potential (in volts) of an electrical pulse propagating along a transmission line of impedance ${\displaystyle Z}$, and suppose the line is terminated with a matched resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By Ohm's law, the power delivered to the resistor at time ${\displaystyle t}$ is equal to ${\displaystyle V(t{)}^2/Z}$, so the total energy is found by integrating ${\displaystyle V(t{)}^2/Z}$ with respect to time over the duration of the pulse. To find the value of the energy spectral density ${\displaystyle {\overline{S}}_{xx}(f)}$ at frequency ${\displaystyle f}$, one could insert between the transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies (${\displaystyle \mathrm{\Delta }f}$, say) near the frequency of interest and then measure the total energy ${\displaystyle E(f)}$ dissipated across the resistor. The value of the energy spectral density at ${\displaystyle f}$ is then estimated to be ${\displaystyle E(f)/\mathrm{\Delta }f}$. In this example, since the power ${\displaystyle V(t{)}^2/Z}$ has units of V² Ω⁻¹, the energy ${\displaystyle E(f)}$ has units of V² s Ω⁻¹ = J, and hence the estimate ${\displaystyle E(f)/\mathrm{\Delta }f}$ of the energy spectral density has units of J Hz⁻¹, as required. In many situations, it is common to forget the step of dividing by ${\displaystyle Z}$ so that the energy spectral density instead has units of V² Hz⁻².

This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values ${\displaystyle x_n}$ such as a signal sampled at discrete times ${\displaystyle t_n=t_0+(n\mathrm{\Delta }t)}$:

${\displaystyle {\overline{S}}_{xx}(f)=\underset{N\to \mathrm{\infty }}{lim}(\mathrm{\Delta }t{)}^2\underset{{\left|{\hat{x}}_d(f)\right|}^2}{\underbrace{{\left|\sum _{n=-N}^N{x}_n{e}^{-i2\pi fn\mathrm{\Delta }t}\right|}^2}},}$

where ${\displaystyle {\hat{x}}_d(f)}$ is the discrete-time Fourier transform of ${\displaystyle x_n.}$  The sampling interval ${\displaystyle \mathrm{\Delta }t}$ is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit ${\displaystyle \mathrm{\Delta }t\to 0.}$  But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see normalized frequency)

Power spectral density

Not to be confused with spectral power distribution.

 

The power spectrum of the measured cosmic microwave background radiation temperature anisotropy in terms of the angular scale. The solid line is a theoretical model, for comparison.

The above definition of energy spectral density is suitable for transients (pulse-like signals) whose energy is concentrated around one time window; then the Fourier transforms of the signals generally exist. For continuous signals over all time, one must rather define the power spectral density (PSD) which exists for stationary processes; this describes how the power of a signal or time series is distributed over frequency, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, is simply identified with the squared value of the signal. For example, statisticians study the variance of a function over time ${\displaystyle x(t)}$ (or over another independent variable), and using an analogy with electrical signals (among other physical processes), it is customary to refer to it as the power spectrum even when there is no physical power involved. If one were to create a physical voltage source which followed ${\displaystyle x(t)}$ and applied it to the terminals of a one ohm resistor, then indeed the instantaneous power dissipated in that resistor would be given by ${\displaystyle x(t{)}^2}$ watts.

The average power ${\displaystyle P}$ of a signal ${\displaystyle x(t)}$ over all time is therefore given by the following time average, where the period ${\displaystyle T}$ is centered about some arbitrary time ${\displaystyle t=t_0}$:

${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{t_0-T/2}^{t_0+T/2}|x(t){|}^{2}dt}$

However, for the sake of dealing with the math that follows, it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral. As such, we have an alternative representation of the average power, where ${\displaystyle x_T(t)=x(t)w_T(t)}$ and ${\displaystyle w_T(t)}$ is unity within the arbitrary period and zero elsewhere.

${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x_T(t){|}^{2}dt}$

Clearly in cases where the above expression for P is non-zero (even as T grows without bound) the integral itself must also grow without bound. That is the reason that we cannot use the energy spectral density itself, which is that diverging integral, in such cases.

In analyzing the frequency content of the signal ${\displaystyle x(t)}$, one might like to compute the ordinary Fourier transform ${\displaystyle \hat{x}(f)}$; however, for many signals of interest the Fourier transform does not formally exist. Regardless, Parseval's theorem tells us that we can re-write the average power as follows.

${\displaystyle P=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|{\hat{x}}_T(f){|}^{2}df}$

Then the power spectral density is simply defined as the integrand above.

${\displaystyle S_{xx}(f)=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}|{\hat{x}}_T(f){|}^2}$

(Eq.2)

From here, we can also view ${\displaystyle |{\hat{x}}_T(f){|}^2}$ as the Fourier transform of the time convolution of ${\displaystyle x_T^{\ast }(-t)}$ and ${\displaystyle x_T(t)}$

${\displaystyle {\left|{\hat{x}}_T(f)\right|}^2=\mathcal{F}\left\{x_T^{\ast }(-t)\mathbf{\ast }{x}_T(t)\right\}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau }$

Now, if we divide the time convolution above by the period ${\displaystyle T}$ and take the limit as ${\displaystyle T\to \mathrm{\infty }}$, it becomes the autocorrelation function of the non-windowed signal ${\displaystyle x(t)}$, which is denoted as ${\displaystyle R_{xx}(\tau )}$, provided that ${\displaystyle x(t)}$ is ergodic, which is true in most, but not all, practical cases.

${\displaystyle \underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\left|{\hat{x}}_T(f)\right|}^2={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xx}(\tau )e^{-i2\pi f\tau }d\tau }$

From here we see, again assuming the ergodicity of ${\displaystyle x(t)}$, that the power spectral density can be found as the Fourier transform of the autocorrelation function (Wiener–Khinchin theorem).

${\displaystyle S_{xx}(f)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xx}(\tau )e^{-i2\pi f\tau }d\tau ={\hat{R}}_{xx}(f)}$

(Eq.3)

Many authors use this equality to actually define the power spectral density.

The power of the signal in a given frequency band ${\displaystyle [f_1,f_2]}$, where ${\displaystyle 0<f_1<f_2}$, can be calculated by integrating over frequency. Since ${\displaystyle S_{xx}(-f)=S_{xx}(f)}$, an equal amount of power can be attributed to positive and negative frequency bands, which accounts for the factor of 2 in the following form (such trivial factors depend on the conventions used):

${\displaystyle P_{\mathsf{\text{bandlimited}}}=2{\int }_{f_1}^{f_2}{S}_{xx}(f)df}$

More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval ${\displaystyle T}$ is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than ${\displaystyle 1/T}$ are not sampled, and results at frequencies which are not an integer multiple of ${\displaystyle 1/T}$ are not independent. Just using a single such time series, the estimated power spectrum will be very "noisy"; however this can be alleviated if it is possible to evaluate the expected value (in the above equation) using a large (or infinite) number of short-term spectra corresponding to statistical ensembles of realizations of ${\displaystyle x(t)}$ evaluated over the specified time window.

Just as with the energy spectral density, the definition of the power spectral density can be generalized to discrete time variables ${\displaystyle x_n}$. As before, we can consider a window of ${\displaystyle -N\le n\le N}$ with the signal sampled at discrete times ${\displaystyle x_n=x_0+(n\mathrm{\Delta }t)}$ for a total measurement period ${\displaystyle T=(2N+1)\mathrm{\Delta }t}$.

${\displaystyle S_{xx}(f)=\underset{N\to \mathrm{\infty }}{lim}\frac{(\mathrm{\Delta }t{)}^2}{T}{\left|\sum _{n=-N}^N{x}_n{e}^{-i2\pi fn\mathrm{\Delta }t}\right|}^2}$

Note that a single estimate of the PSD can be obtained through a finite number of samplings. As before, the actual PSD is achieved when ${\displaystyle N}$ (and thus ${\displaystyle T}$) approaches infinity and the expected value is formally applied. In a real-world application, one would typically average a finite-measurement PSD over many trials to obtain a more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called a periodogram. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval ${\displaystyle T}$ approach infinity (Brown & Hwang).

If two signals both possess power spectral densities, then the cross-spectral density can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the cross-correlation.

Properties of the power spectral density

Some properties of the PSD include:

• The power spectrum is always real and non-negative, and the spectrum of a real valued process is also an even function of frequency: ${\displaystyle S_{xx}(-f)=S_{xx}(f)}$.
• For a continuous stochastic process x(t), the autocorrelation function R_xx(t) can be reconstructed from its power spectrum S_xx(f) by using the inverse Fourier transform
• Using Parseval's theorem, one can compute the variance (average power) of a process by integrating the power spectrum over all frequency:

  ${\displaystyle P=\mathrm{Var}(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{S}_{xx}(f)df}$

• For a real process x(t) with power spectral density ${\displaystyle S_{xx}(f)}$, one can compute the integrated spectrum or power spectral distribution ${\displaystyle F(f)}$, which specifies the average bandlimited power contained in frequencies from DC to f using:
  ${\displaystyle F(f)=2{\int }_0^f{S}_{xx}(f^{\prime })d{f}^{\prime }.}$
  Note that the previous expression for total power (signal variance) is a special case where ƒ → ∞.

Cross power spectral density

See also: Coherence (signal processing)

Given two signals ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$, each of which possess power spectral densities ${\displaystyle S_{xx}(f)}$ and ${\displaystyle S_{yy}(f)}$, it is possible to define a cross power spectral density (CPSD) or cross spectral density (CSD). To begin, let us consider the average power of such a combined signal.

${\displaystyle \begin{array}{rl}P& =\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left[x_T(t)+y_T(t)\right]}^{\ast }\left[x_T(t)+y_T(t)\right]dt\\ & =\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|x_T(t){|}^2+x_T^{\ast }(t)y_T(t)+y_T^{\ast }(t)x_T(t)+|y_T(t){|}^{2}dt\end{array}}$

Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain

${\displaystyle \begin{array}{rlrl}{S}_{xy}(f)& =\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}\left[{\hat{x}}_T^{\ast }(f){\hat{y}}_T(f)\right]& S_{yx}(f)& =\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}\left[{\hat{y}}_T^{\ast }(f){\hat{x}}_T(f)\right]\end{array}}$

where, again, the contributions of ${\displaystyle S_{xx}(f)}$ and ${\displaystyle S_{yy}(f)}$ are already understood. Note that ${\displaystyle S_{xy}^{\ast }(f)=S_{yx}(f)}$, so the full contribution to the cross power is, generally, from twice the real part of either individual CPSD. Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit ${\displaystyle T\to \mathrm{\infty }}$ becomes the Fourier transform of a cross-correlation function.

${\displaystyle \begin{array}{rl}{S}_{xy}(f)& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}_T^{\ast }(t-\tau )y_T(t)dt\right]e^{-i2\pi f\tau }d\tau ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\ S_{yx}(f)& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[\underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{y}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{yx}(\tau )e^{-i2\pi f\tau }d\tau \end{array}}$

where ${\displaystyle R_{xy}(\tau )}$ is the cross-correlation of ${\displaystyle x(t)}$ with ${\displaystyle y(t)}$ and ${\displaystyle R_{yx}(\tau )}$ is the cross-correlation of ${\displaystyle y(t)}$ with ${\displaystyle x(t)}$. In light of this, the PSD is seen to be a special case of the CSD for ${\displaystyle x(t)=y(t)}$. If ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$ are real signals (e.g. voltage or current), their Fourier transforms ${\displaystyle \hat{x}(f)}$ and ${\displaystyle \hat{y}(f)}$ are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full CPSD is just one of the CPSDs scaled by a factor of two.

${\displaystyle {\mathrm{CPSD}}_{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}$

For discrete signals x_n and y_n, the relationship between the cross-spectral density and the cross-covariance is

${\displaystyle S_{xy}(f)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{R}_{xy}({\tau }_n)e^{-i2\pi f{\tau }_n}\mathrm{\Delta }\tau }$

Estimation

Main article: Spectral density estimation

The goal of spectral density estimation is to estimate the spectral density of a random signal from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve parametric or non-parametric approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an autoregressive model. A common non-parametric technique is the periodogram.

The spectral density is usually estimated using Fourier transform methods (such as the Welch method), but other techniques such as the maximum entropy method can also be used.

Related concepts

Not to be confused with spectral density (physical science).

• The spectral centroid of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.
• The spectral edge frequency (SEF), usually expressed as "SEF x", represents the frequency below which x percent of the total power of a given signal are located; typically, x is in the range 75 to 95. It is more particularly a popular measure used in EEG monitoring, in which case SEF has variously been used to estimate the depth of anesthesia and stages of sleep.
• A spectral envelope is the envelope curve of the spectrum density. It describes one point in time (one window, to be precise). For example, in remote sensing using a spectrometer, the spectral envelope of a feature is the boundary of its spectral properties, as defined by the range of brightness levels in each of the spectral bands of interest.
• The spectral density is a function of frequency, not a function of time. However, the spectral density of a small window of a longer signal may be calculated, and plotted versus time associated with the window. Such a graph is called a spectrogram. This is the basis of a number of spectral analysis techniques such as the short-time Fourier transform and wavelets.
• A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. For transfer functions (e.g., Bode plot, chirp) the complete frequency response may be graphed in two parts: power versus frequency and phase versus frequency—the phase spectral density, phase spectrum, or spectral phase. Less commonly, the two parts may be the real and imaginary parts of the transfer function. This is not to be confused with the frequency response of a transfer function, which also includes a phase (or equivalently, a real and imaginary part) as a function of frequency. The time-domain impulse response ${\displaystyle h(t)}$ cannot generally be uniquely recovered from the power spectral density alone without the phase part. Although these are also Fourier transform pairs, there is no symmetry (as there is for the autocorrelation) forcing the Fourier transform to be real-valued. See Ultrashort pulse#Spectral phase, phase noise, group delay.
• Sometimes one encounters an amplitude spectral density (ASD), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz^−1/2. This is useful when the shape of the spectrum is rather constant, since variations in the ASD will then be proportional to variations in the signal's voltage level itself. But it is mathematically preferred to use the PSD, since only in that case is the area under the curve meaningful in terms of actual power over all frequency or over a specified bandwidth.

Applications

Further information: Spectrum

Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as visible light (perceived as color), musical notes (perceived as pitch), radio/TV (specified by their frequency, or sometimes wavelength) and even the regular rotation of the earth. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed. In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component. And additionally there may be peaks corresponding to harmonics of a fundamental peak, indicating a periodic signal which is not simply sinusoidal. Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a notch filter.

Electrical engineering

Spectrogram of an FM radio signal with frequency on the horizontal axis and time increasing upwards on the vertical axis.

The concept and use of the power spectrum of a signal is fundamental in electrical engineering, especially in electronic communication systems, including radio communications, radars, and related systems, plus passive remote sensing technology. Electronic instruments called spectrum analyzers are used to observe and measure the power spectra of signals.

The spectrum analyzer measures the magnitude of the short-time Fourier transform (STFT) of an input signal. If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral density.

Cosmology

Primordial fluctuations, density variations in the early universe, are quantified by a power spectrum which gives the power of the variations as a function of spatial scale.

Climate Science

Power spectral-analysis have been used to examine the spatial structures for climate research. These results suggests atmospheric turbulence link climate change to more local regional volatility in weather conditions.