title: Blackberry

text: The usually black fruit is not a berry in the botanical sense of the word. Botanically it is termed an aggregate fruit, composed of small drupelets. It is a widespread and well-known group of over 375 species, many of which are closely related apomictic microspecies native throughout Europe, northwestern Africa, temperate western and central Asia and North and South America.


title: Blackberry

text: Blackberries are perennial plants which typically bear biennial stems ("canes") from the perennial root system.


title: Blackberry

text: In its first year, a new stem, the primocane, grows vigorously to its full length of (in some cases, up to ), arching or trailing along the ground and bearing large palmately compound leaves with five or seven leaflets; it does not produce any flowers. In its second year, the cane becomes a floricane and the stem does not grow longer, but the lateral buds break to produce flowering laterals (which have smaller leaves with three or five leaflets). First- and second-year shoots usually have numerous short-curved, very sharp prickles that are often erroneously called thorns. These prickles can tear through denim with ease and make the plant very difficult to navigate around. Prickle-free cultivars have been developed. The University of Arkansas has developed primocane-fruiting blackberries that grow and flower on first-year growth as much as the primocane-fruiting (also called fall bearing or everbearing) red raspberries do.


title: Blackberry

text: Unmanaged mature plants form a tangle of dense arching stems, the branches rooting from the node tip on many species when they reach the ground. Vigorous and growing rapidly in woods, scrub, hillsides, and hedgerows, blackberry shrubs tolerate poor soils, readily colonizing wasteland, ditches, and vacant lots.


title: Blackberry

text: The flowers are produced in late spring and early summer on short racemes on the tips of the flowering laterals. Each flower is about in diameter, with five white or pale pink petals.


title: Blackberry

text: The drupelets only develop around ovules that are fertilized by the male gamete from a pollen grain. The most likely cause of undeveloped ovules is inadequate pollinator visits. Even a small change in conditions, such as a rainy day or a day too hot for bees to work after early morning, can reduce the number of bee visits to the flower, thus reducing the quality of the fruit. Incomplete drupelet development can also be a symptom of exhausted reserves in the plant's roots or infection with a virus such as raspberry bushy dwarf virus.


title: Blackberry

text: One of the earliest known instances of blackberry consumption comes from the remains of the Haraldskær Woman, the naturally preserved bog body of a Danish woman dating from approximately 2,500 years ago. Forensic evidence found blackberries in her stomach contents, among other foods. The use of blackberries to make wines and cordials was documented in the London Pharmacopoeia in 1696. In the culinary world, blackberries have a long history of use alongside other fruits to make pies, jellies and jams.


title: Blackberry

text: Blackberry plants were used for traditional medicine by Greeks, other European peoples, and aboriginal Americans. A 1771 document described brewing blackberry leaves, stem, and bark for stomach ulcers.


title: Blackberry

text: Blackberry fruit, leaves, and stems have been used to dye fabrics and hair. Native Americans have even been known to use the stems to make rope. The shrubs have also been used for barriers around buildings, crops and livestock. The wild plants have sharp, thick prickles, which offered some protection against enemies and large animals.


title: Blackberry

text: Modern hybridization and cultivar development took place mostly in the United States. In 1880, a hybrid blackberry-raspberry named the loganberry was developed in Santa Cruz, California, by an American judge and horticulturalist, James Harvey Logan. One of the first thornless varieties was developed in 1921, but the berries lost much of their flavor. Common thornless cultivars developed from the 1990s to the early 21st century by the US Department of Agriculture enabled efficient machine-harvesting, higher yields, larger and firmer fruit, and improved flavor, including the "Triple Crown", "Black Diamond", "Black Pearl", and "Nightfall", a marionberry.


title: Blackberry

text: Blackberry leaves are food for certain caterpillars; some grazing mammals, especially deer, are also very fond of the leaves. Caterpillars of the concealer moth "Alabonia geoffrella" have been found feeding inside dead blackberry shoots. When mature, the berries are eaten and their seeds dispersed by mammals, such as the red fox, American black bear and the Eurasian badger, as well as by small birds.


title: Blackberry

text: Blackberries grow wild throughout most of Europe. They are an important element in the ecology of many countries, and harvesting the berries is a popular pastime. However, their rigorous growth and tendency to grow unchecked if not managed correctly means that the plants are also considered a weed, sending down roots from branches that touch the ground, and sending up suckers from the roots. In some parts of the world, such as in Australia, Chile, New Zealand, and the Pacific Northwest of North America, some blackberry species, particularly "Rubus armeniacus" (Himalayan blackberry) and "Rubus laciniatus" (evergreen blackberry), are naturalized and considered an invasive species and a noxious weed.


title: Blackberry

text: Blackberry fruits are red when unripe, leading to an old expression that "blackberries are red when they're green".


title: Blackberry

text: Raw blackberries are 88% water, 10% carbohydrates, 1% protein, and 0.5% fat (table). In a reference amount, raw cultivated blackberries supply 43 calories and rich contents (20% or more of the Daily Value (DV) of dietary fiber, manganese (31% DV), vitamin C (25% DV), and vitamin K (19% DV) (table).


title: Blackberry

text: Blackberries contain numerous large seeds that are not always preferred by consumers. The seeds contain oil rich in omega-3 (alpha-linolenic acid) and omega-6 (linoleic acid) fats as well as protein, dietary fiber, carotenoids, ellagitannins, and ellagic acid.


title: Blackberry

text: The ripe fruit is commonly used in desserts, jams, jelly, wine and liqueurs. It may be mixed with other berries and fruits for pies and crumbles. Blackberries are also used to produce candy.


title: Blackberry

text: Blackberries contain numerous phytochemicals including polyphenols, flavonoids, anthocyanins, salicylic acid, ellagic acid, and fiber. Anthocyanins in blackberries are responsible for their rich dark color. One report placed blackberries at the top of more than 1,000 polyphenol-rich foods consumed in the United States, but this concept of a health benefit from consuming darkly colored foods like blackberries remains scientifically unverified and not accepted for health claims on food labels.


title: Blackberry

text: Worldwide, Mexico is the leading producer of blackberries, with nearly the entire crop being produced for export into the off-season fresh markets in North America and Europe. Until 2018, the Mexican market was almost entirely based on the cultivar 'Tupy' (often spelled 'Tupi', but the EMBRAPA program in Brazil from which it was released prefers the 'Tupy' spelling), but Tupy fell out of favor in some Mexican growing regions. In the US, Oregon is the leading commercial blackberry producer, producing on in 2017.


title: Blackberry

text: Numerous cultivars have been selected for commercial and amateur cultivation in Europe and the United States. Since the many species form hybrids easily, there are numerous cultivars with more than one species in their ancestry.


title: Blackberry

text: 'Marion' (marketed as "marionberry") is an important cultivar that was selected from seedlings from a cross between 'Chehalem' and 'Olallie' (commonly called "Olallieberry") berries. 'Olallie' in turn is a cross between loganberry and youngberry. 'Marion', 'Chehalem' and 'Olallie' are just three of many trailing blackberry cultivars developed by the United States Department of Agriculture Agricultural Research Service (USDA-ARS) blackberry breeding program at Oregon State University in Corvallis, Oregon.


title: Blackberry

text: The most recent cultivars released from this program are the prickle-free cultivars 'Black Diamond', 'Black Pearl', and 'Nightfall' as well as the very early-ripening 'Obsidian' and 'Metolius'. 'Black Diamond' is now the leading cultivar being planted in the Pacific Northwest. Some of the other cultivars from this program are 'Newberry', 'Waldo', 'Siskiyou', 'Black Butte', 'Kotata', 'Pacific', and 'Cascade'.


title: Blackberry

text: Trailing blackberries are vigorous and crown forming, require a trellis for support, and are less cold hardy than the erect or semi-erect blackberries. In addition to the Pacific Northwest, these types do well in similar climates, such as the United Kingdom, New Zealand, Chile, and the Mediterranean countries.


title: Blackberry

text: Semi-erect, prickle-free blackberries were first developed at the John Innes Centre in Norwich, UK, and subsequently by the USDA-ARS in Beltsville, Maryland. These are crown forming and very vigorous and need a trellis for support. Cultivars include 'Black Satin', 'Chester Thornless', 'Dirksen Thornless', 'Hull Thornless', 'Loch Maree', 'Loch Ness', 'Loch Tay', 'Merton Thornless', 'Smoothstem', and 'Triple Crown'. 'Loch Ness' and 'Loch Tay' have gained the RHS's Award of Garden Merit. The cultivar 'Cacanska Bestrna' (also called 'Cacak Thornless') has been developed in Serbia and has been planted on many thousands of hectares there.


title: Blackberry

text: The University of Arkansas has developed cultivars of erect blackberries. These types are less vigorous than the semi-erect types and produce new canes from root initials (therefore they spread underground like raspberries). There are prickly and prickle-free cultivars from this program, including 'Navaho', 'Ouachita', 'Cherokee', 'Apache', 'Arapaho', and 'Kiowa'. They are also responsible for developing the primocane fruiting blackberries such as 'Prime-Jan' and 'Prime-Jim'.


title: Blackberry

text: In raspberries, these types are called primocane fruiting, fall fruiting, or everbearing. 'Prime-Jim' and 'Prime-Jan' were released in 2004 by the University of Arkansas and are the first cultivars of primocane fruiting blackberry. They grow much like the other erect cultivars described above; however, the canes that emerge in the spring will flower in midsummer and fruit in late summer or fall. The fall crop has its highest quality when it ripens in cool mild climate such as in California or the Pacific Northwest.


title: Blackberry

text: 'Illini Hardy', a semi-erect prickly cultivar introduced by the University of Illinois, is cane hardy in zone 5, where blackberry production has traditionally been problematic, since canes often failed to survive the winter.


title: Blackberry

text: Blackberry production in Mexico expanded considerably in the early 21st century. In 2017, Mexico had 97% of the market share for fresh blackberries imported into the United States, while Chile had 61% of the market share for American imports of frozen blackberries.


title: Blackberry

text: While once based on the cultivar 'Brazos', an old erect blackberry cultivar developed in Texas in 1959, the Mexican industry is now dominated by the Brazilian 'Tupy' released in the 1990s. The 'Tupy' has the erect blackberry 'Comanche', and a "wild Uruguayan blackberry" as parents. Since there are no native blackberries in Uruguay, the suspicion is that the widely grown 'Boysenberry' is the male parent. In order to produce these blackberries in regions of Mexico where there is no winter chilling to stimulate flower bud development, chemical defoliation and application of growth regulators are used to bring the plants into bloom.


title: Blackberry

text: Because blackberries belong to the same genus as raspberries, they share the same diseases, including anthracnose, which can cause the berry to have uneven ripening. Sap flow may also be slowed. They also share the same remedies, including the Bordeaux mixture, a combination of lime, water and copper(II) sulfate. The rows between blackberry plants must be free of weeds, blackberry suckers and grasses, which may lead to pests or diseases. Fruit growers are selective when planting blackberry bushes because wild blackberries may be infected, and gardeners are recommended to purchase only certified disease-free plants.


title: Blackberry

text: The spotted-wing drosophila, "Drosophila suzukii", is a serious pest of blackberries. Unlike its vinegar fly relatives, which are primarily attracted to rotting or fermented fruit, "D. suzukii" attacks fresh, ripe fruit by laying eggs under the soft skin. The larvae hatch and grow in the fruit, destroying the fruit's commercial value.


title: Blackberry

text: Another pest is "Amphorophora rubi", known as the blackberry aphid, which eats not just blackberries but raspberries as well.


title: Blackberry

text: "Byturus tomentosus" (raspberry beetle), "Lampronia corticella" (raspberry moth) and "Anthonomus rubi" (strawberry blossom weevil) are also known to infest blackberries.


title: Blackberry

text: The loci controlling the primocane fruiting was mapped in the F Locus, on LG7, whereas thorns/thornlessness was mapped on LG4. Better understanding of the genetics is useful for genetic screening of cross-breds, and for genetic engineering purposes.


title: Blackberry

text: Folklore in the United Kingdom and Ireland tells that blackberries should not be picked after Old Michaelmas Day (11 October) as the devil (or a Púca) has made them unfit to eat by stepping, spitting or fouling on them. There is some value in this legend as autumn's wetter and cooler weather often allows the fruit to become infected by various molds such as "Botryotinia" which give the fruit an unpleasant look and may be toxic. According to some traditions, a blackberry's deep purple color represents Christ's blood and the crown of thorns was made of brambles, although other thorny plants, such as "Crataegus" (hawthorn) and "Euphorbia milii" (crown of thorns plant), have been proposed as the material for the crown.


title: Cheque

text: A cheque, or check (American English; see spelling differences) is a document that orders a bank (or credit union) to pay a specific amount of money from a person's account to the person in whose name the cheque has been issued. The person writing the cheque, known as the "drawer", has a transaction banking account (often called a current, cheque, chequing, checking, or share draft account) where the money is held. The drawer writes various details including the monetary amount, date, and a payee on the cheque, and signs it, ordering their bank, known as the "drawee", to pay the amount of money stated to the payee.


title: Cheque

text: Although forms of cheques have been in use since ancient times and at least since the 9th century, they became a highly popular non-cash method for making payments during the 20th century and usage of cheques peaked. By the second half of the 20th century, as cheque processing became automated, billions of cheques were issued annually; these volumes peaked in or around the early 1990s. Since then cheque usage has fallen, being partly replaced by electronic payment systems, such as debit cards and credit cards. In an increasing number of countries cheques have either become a marginal payment system or have been completely phased out.


title: Cheque

text: A cheque is a negotiable instrument instructing a financial institution to pay a specific amount of a specific currency from a specified transactional account held in the drawer's name with that institution. Both the drawer and payee may be natural persons or legal entities. Cheques are "order instruments", and are not in general payable simply to the bearer as bearer instruments are, but must be paid to the payee. In some countries, such as the US, the payee may endorse the cheque, allowing them to specify a third party to whom it should be paid.


title: Cheque

text: Cheques are a type of bill of exchange that were developed as a way to make payments without the need to carry large amounts of money. Paper money evolved from promissory notes, another form of negotiable instrument similar to cheques in that they were originally a written order to pay the given amount to whoever had it in their possession (the "bearer").


title: Cheque

text: "Check" is the original spelling. The newer spelling, "cheque", is believed to have come into use around 1828, when the switch was made by James William Gilbart in his "Practical Treatise on Banking". The spellings "check", "checque", and "cheque" were used interchangeably from the 17th century until the 20th century. However, since the 19th century, in the Commonwealth and Ireland, the spelling "cheque" (from the French word "chèque") has become standard for the financial instrument, while "check" is used only for other meanings, thus distinguishing the two definitions in writing.


title: Cheque

text: Etymological dictionaries attribute the financial meaning of "check" to come from "a check against forgery", with the use of "check" to mean "control" stemming from a check in chess, a term which came into English through French, Latin, Arabic and ultimately from the Persian word "shah", or "king".


title: Cheque

text: The cheque had its origins in the ancient banking system, in which bankers would issue orders at the request of their customers, to pay money to identified payees. Such an order was referred to as a "bill of exchange". The use of bills of exchange facilitated trade by eliminating the need for merchants to carry large quantities of currency (for example, gold) to purchase goods and services.


title: Cheque

text: There is early evidence of using bill of exchange. In India, during the Maurya Empire (from 321 to 185 BC), a commercial instrument called the adesha was in use, which was an order on a banker desiring him to pay the money of the note to a third person.


title: Cheque

text: The ancient Romans are believed to have used an early form of cheque known as "praescriptiones" in the 1st century BC.


title: Cheque

text: Beginning in the third century AD, banks in Persian territory began to issue letters of credit. These letters were termed "čak", meaning "document" or "contract". The "čak" became the "sakk" later used by traders in the Abbasid Caliphate and other Arab-ruled lands. Transporting a paper "sakk" was more secure than transporting money. In the ninth century, a merchant in one country could cash a "sakk" drawn on his bank in another country. The Persian poet, Ferdowsi, used the term "cheque" several times in his famous book, Shahnameh, when referring to the Sasanid dynasty.


title: Cheque

text: Ibn Hawqal, living in the 10th century, quotes the use of a cheque worth 42,000 dinars in the Ghana Empire of West Africa.


title: Cheque

text: In the 13th century the "bill of exchange" was developed in Venice as a legal device to allow international trade without the need to carry large amounts of gold and silver. Their use subsequently spread to other European countries.


title: Cheque

text: In the early 1500s, to protect large accumulations of cash, people in the Dutch Republic began depositing their money with "cashiers". These cashiers held the money for a fee. Competition drove cashiers to offer additional services including paying money to any person bearing a written order from a depositor to do so. They kept the note as proof of payment. This concept went on to spread to England and elsewhere.


title: Cheque

text: By the 17th century, bills of exchange were being used for domestic payments in England. Cheques, a type of bill of exchange, then began to evolve. Initially, they were called "drawn notes", because they enabled a customer to draw on the funds that he or she had in the account with a bank and required immediate payment. These were handwritten, and one of the earliest known still to be in existence was drawn on Messrs Morris and Clayton, scriveners and bankers based in the City of London, and dated 16 February 1659.


title: Cheque

text: In 1717, the Bank of England pioneered the first use of a pre-printed form. These forms were printed on "cheque paper" to prevent fraud, and customers had to attend in person and obtain a numbered form from the cashier. Once written, the cheque was brought back to the bank for settlement. The suppression of banknotes in eighteenth-century England further promoted the use of cheques.


title: Cheque

text: Until about 1770, an informal exchange of cheques took place between London banks. Clerks of each bank visited all the other banks to exchange cheques while keeping a tally of balances between them until they settled with each other. Daily cheque clearing began around 1770 when the bank clerks met at the Five Bells, a tavern in Lombard Street in the City of London, to exchange all their cheques in one place and settle the balances in cash. This was the first bankers' clearing house.


title: Cheque

text: Provincial clearinghouses were established in major cities throughout the UK to facilitate the clearing of cheques on banks in the same town. Birmingham, Bradford, Bristol, Hull, Leeds, Leicester, Liverpool, Manchester, Newcastle, Nottingham, Sheffield, and Southampton all had their own clearinghouses.


title: Cheque

text: In America, the Bank of New York began issuing cheques after its establishment by Alexander Hamilton in 1784. The oldest surviving example of a complete American chequebook from the 1790s was discovered by a family in New Jersey. The documents are in some ways similar to modern-day cheques, with some data pre-printed on sheets of paper alongside blank spaces for where other information could be hand-written as needed.


title: Cheque

text: It is thought that the Commercial Bank of Scotland was the first bank to personalize its customers' cheques, in 1811, by printing the name of the account holder vertically along the left-hand edge. In 1830 the Bank of England introduced books of 50, 100, and 200 forms and counterparts, bound or stitched. These "cheque books" became a common format for the distribution of cheques to bank customers.


title: Cheque

text: In the late 19th century, several countries formalized laws regarding cheques. The UK passed the Bills of Exchange Act 1882, and India passed the Negotiable Instruments Act, 1881; which both covered cheques.


title: Cheque

text: In 1931, an attempt was made to simplify the international use of cheques by the Geneva Convention on the Unification of the Law Relating to Cheques. Many European and South American states, as well as Japan, joined the convention. However, countries including the US and members of the British Commonwealth did not participate and so it remained very difficult for cheques to be used across country borders.


title: Cheque

text: In 1959 a standard for machine-readable characters (MICR) was agreed upon and patented in the US for use with cheques. This opened the way for the first automated reader/sorting machines for clearing cheques. As automation increased, the following years saw a dramatic change in the way in which cheques were handled and processed. Cheque volumes continued to grow; in the late 20th century, cheques were the most popular non-cash method for making payments, with billions of them processed each year. Most countries saw cheque volumes peak in the late 1980s or early 1990s, after which electronic payment methods became more popular and the use of cheques declined.


title: Cheque

text: In 1969 cheque guarantee cards were introduced in several countries, allowing a retailer to confirm that a cheque would be honored when used at a point of sale. The drawer would sign the cheque in front of the retailer, who would compare the signature to the signature on the card and then write the cheque-guarantee-card number on the back of the cheque. Such cards were generally phased out and replaced by debit cards, starting in the mid-1990s.


title: Cheque

text: From the mid-1990s, many countries enacted laws to allow for cheque truncation, in which a physical cheque is converted into electronic form for transmission to the paying bank or clearing-house. This eliminates the cumbersome physical presentation and saves time and processing costs.


title: Cheque

text: In 2002, the Eurocheque system was phased out and replaced with domestic clearing systems. Old Eurocheques could still be used, but they were now processed by national clearing systems. At that time, several countries took the opportunity to phase out the use of cheques altogether. As of 2010, many countries have either phased out the use of cheques altogether or signaled that they would do so in the future.


title: Cheque

text: As cheque usage increased during the 19th and 20th centuries, additional items were added to increase security or to make processing easier for the financial institution. A signature of the drawer was required to authorize the cheque, and this is the main way to authenticate the cheque. Second, it became customary to write the amount in words as well as in numbers to avoid mistakes and make it harder to fraudulently alter the amount after the cheque had been written. It is not a legal requirement to write the amount in words, although some banks will refuse to accept cheques that do not have the amount in both numbers and words.


title: Cheque

text: An issue date was added, and cheques may become invalid a certain amount of time after issue. In the US and Canada, a cheque is typically valid for six months after the date of issue, after which it is a "stale-dated cheque," but this depends on where the cheque is drawn. In Australia, a cheque is typically valid for fifteen months of the cheque date. A cheque that has an issue date in the future, a post-dated cheque, may not be able to be presented until that date has passed. In some countries writing a post dated cheque may simply be ignored or is illegal. Conversely, an antedated cheque has an issue date in the past.


title: Cheque

text: A cheque number was added and cheque books were issued so that cheque numbers were sequential. This allowed for some basic fraud detection by banks and made sure one cheque was not presented twice.


title: Cheque

text: In some countries, such as the US, cheques may contain a memo line where the purpose of the cheque can be indicated as a convenience without affecting the official parts of the cheque. In the United Kingdom a memo line is not available and such notes may be written on the reverse side of the cheque.


title: Cheque

text: In the US, at the top (when cheque oriented vertically) of the reverse side of the cheque, there are usually one or more blank lines labelled something like "Endorse here".


title: Cheque

text: Starting in the 1960s, machine-readable routing and account information was added to the bottom of cheques in MICR format, which allowed automated sorting and routing of cheques between banks and led to automated central clearing facilities. The information provided at the bottom of the cheque is country-specific and standards are set by each country's cheque clearing system. This means that the payee no longer has to go to the bank that issued the cheque, they can instead deposit it at their own bank or any other bank and the cheque would be routed back to the originating bank, and funds transferred to their own bank account.


title: Cheque

text: In the US, the bottom -inch (16 mm) of the cheque is reserved for MICR characters only. Intrusion into the MICR area can cause problems when the cheque runs through the clearinghouse, requiring someone to print an MICR cheque correction strip and glue it to the cheque. Many new ATMs do not use deposit envelopes and actually scan the cheque at the time it is deposited and will reject cheques due to handwriting incursion which interferes with reading the MICR. This can cause considerable inconvenience as the depositor may have to wait days for the bank to be open and may have difficulty getting to the bank even when they are open; this can delay the availability of the portion of a deposit which their bank makes available immediately as well as the balance of the deposit. Terms of service for many mobile (cell phone camera) deposits also require the MICR section to be readable. Not all of the MICR characters have been printed at the time the cheque is written, as additional characters will be printed later to encode the amount; thus a sloppy signature could obscure characters that will later be printed there. Since MICR characters are no longer necessarily printed in magnetic ink and will be scanned by optical rather than magnetic means, the readers will be unable to distinguish pen ink from pre-printed magnetic ink; these changes allow cheques to be printed on ordinary home and office printers without requiring pre-printed cheque forms, allow ATM deposit capture, allow mobile deposits, and facilitate electronic copies of cheques.


title: Cheque

text: For additional protection, a cheque can be crossed, which restricts the use of the cheque so that the funds must be paid into a bank account. The format and wording varies from country to country, but generally two parallel lines may be placed either vertically across the cheque or in the top left hand corner. In addition the words 'or bearer' must not be used, or if pre-printed on the cheque must be crossed out on the payee line. If the cheque is crossed with the words 'Account Payee' or similar then the cheque can only be paid into the bank account of the person initially named as the payee, thus it cannot be endorsed to a different payee.


title: Cheque

text: Cheques sometimes include additional documents. A page in a chequebook may consist of both the cheque itself and a stub or "" – when the cheque is written, only the cheque itself is detached, and the stub is retained in the chequebook as a record of the cheque. Alternatively, cheques may be recorded with carbon paper behind each cheque, in ledger sheets between cheques or at the back of a chequebook, or in a completely separate transaction register that comes with a chequebook.


title: Cheque

text: When a cheque is mailed, a separate letter or "remittance advice" may be attached to inform the recipient of the purpose of the cheque – formally, which account receivable to credit the funds to. This is frequently done formally using a provided slip when paying a bill, or informally via a letter when sending an ad hoc cheque.


title: Cheque

text: Parties to regular cheques generally include a "drawer", the depositor writing a cheque; a "drawee," the financial institution where the cheque can be presented for payment; and a "payee," the entity to whom the drawer issues the cheque. The drawer "drafts" or "draws" a cheque, which is also called "cutting a cheque", especially in the US. There may also be a "beneficiary"—for example, in depositing a cheque with a custodian of a brokerage account, the payee will be the custodian, but the cheque may be marked "F/B/O" ("for the benefit of") the beneficiary.


title: Cheque

text: Ultimately, there is also at least one "endorsee" which would typically be the financial institution servicing the payee's account, or in some circumstances may be a third party to whom the payee owes or wishes to give money.


title: Cheque

text: A payee that accepts a cheque will typically deposit it in an account at the payee's bank, and have the bank process the cheque. In some cases, the payee will take the cheque to a branch of the drawee bank, and cash the cheque there. If a cheque is refused at the drawee bank (or the drawee bank returns the cheque to the bank that it was deposited at) because there are insufficient funds for the cheque to clear, it is said that the cheque has been "dishonoured". Once a cheque is approved and all appropriate accounts involved have been credited, the cheque is stamped with some kind of cancellation mark, such as a "paid" stamp. The cheque is now a "cancelled cheque". Cancelled cheques are placed in the account holder's file. The account holder can request a copy of a cancelled cheque as proof of a payment. This is known as the cheque clearing cycle.


title: Cheque

text: Cheques can be lost or go astray within the cycle, or be delayed if further verification is needed in the case of suspected fraud. A cheque may thus bounce some time after it has been deposited.


title: Cheque

text: Following concerns about the amount of time it took the Cheque and Credit Clearing Company to clear cheques, the United Kingdom Office of Fair Trading set up a working group in 2006 to look at the cheque clearing cycle. Their report said that clearing times could be improved, but that the costs associated with speeding up the cheque clearing cycle could not be justified considering the use of cheques was declining. However, they concluded the biggest problem was the unlimited time a bank could take to dishonour a cheque. To address this, changes were implemented so that the maximum time after a cheque was deposited that it could be dishonoured was six days, what was known as the "certainty of fate" principle.


title: Cheque

text: An advantage to the drawer of using cheques instead of debit card transactions, is that they know the drawer's bank will not release the money until several days later. Paying with a cheque and making a deposit before it clears the drawer's bank is called "kiting" or "floating" and is generally illegal in the US, but rarely enforced unless the drawer uses multiple chequing accounts with multiple institutions to increase the delay or to steal the funds.


title: Cheque

text: Cheque usage has been declining for some years, both for point of sale transactions (for which credit cards and debit cards are increasingly preferred) and for third party payments (for example, bill payments), where the decline has been accelerated by the emergence of telephone banking, online banking, and mobile banking. Being paper-based, cheques are costly for banks to process in comparison to electronic payments, so banks in many countries now discourage the use of cheques, either by charging for cheques or by making the alternatives more attractive to customers. In particular the handling of money transfer requires more effort and is time-consuming. The cheque has to be handed over in person or sent through mail. The rise of automated teller machines (ATMs) means that small amounts of cash are often easily accessible, so that it is sometimes unnecessary to write a cheque for such amounts instead.


title: Cheque

text: In addition to regular cheques, a number of variations were developed to address specific needs or address issues when using a regular cheque.


title: Cheque

text: Cashier's cheques and banker's drafts, also known as bank cheques, banker's cheques or treasurer's cheques, are cheques issued against the funds of a financial institution rather than an individual account holder. Typically, the term "cashier's check" is used in the US and "banker's draft" is used in the UK and most of the Commonwealth. The mechanism differs slightly from country to country but in general the bank issuing the cheque or draft will allocate the funds at the point the cheque is drawn. This provides a guarantee, save for a failure of the bank, that it will be honoured. Cashier's cheques are perceived to be as good as cash but they are still a cheque, a misconception sometimes exploited by scam artists. A lost or stolen cheque can still be stopped like any other cheque, so payment is not completely guaranteed.


title: Cheque

text: When a certified cheque is drawn, the bank operating the account verifies there are currently sufficient funds in the drawer's account to honour the cheque. Those funds are then set aside in the bank's internal account until the cheque is cashed or returned by the payee. Thus, a certified cheque cannot "bounce", and its liquidity is similar to cash, absent failure of the bank. The bank indicates this fact by making a notation on the face of the cheque (technically called an "acceptance").


title: Cheque

text: A cheque used to pay wages may be referred to as a payroll cheque. Even when the use of cheques for paying wages and salaries became rare, the vocabulary "pay cheque" still remained commonly used to describe the payment of wages and salaries. Payroll cheques issued by the military to soldiers, or by some other government entities to their employees, beneficiants, and creditors, are referred to as warrants.


title: Cheque

text: Warrants look like cheques and clear through the banking system like cheques, but are not drawn against cleared funds in a deposit account. A cheque differs from a warrant in that the warrant is not necessarily payable on demand and may not be negotiable. They are often issued by government entities such as the military to pay wages or suppliers. In this case they are an instruction to the entity's treasurer department to pay the warrant holder on demand or after a specified maturity date.


title: Cheque

text: A traveller's cheque is designed to allow the person signing it to make an unconditional payment to someone else as a result of paying the issuer for that privilege. Traveller's cheques can usually be replaced if lost or stolen, and people frequently used them on holiday instead of cash as many businesses used to accept traveller's cheques as currency. The use of credit or debit cards has begun to replace the traveller's cheque as the standard for vacation money due to their convenience and additional security for the retailer. As a result, many businesses no longer accept traveller's cheques.


title: Cheque

text: A cheque sold by a post office, bank, or merchant such as a grocery store for payment in favour of a third party is referred to as a money order or postal order. These are paid for in advance when the order is drawn and are guaranteed by the institution that issues them and can only be paid to the named third party. This was a common way to send low value payments to third parties, avoiding the risks associated with sending cash by post, prior to the advent of electronic payment methods.


title: Cheque

text: Oversized cheques are often used in public events such as donating money to charity or giving out prizes such as Publishers Clearing House. The cheques are commonly in size; however, according to the "Guinness Book of World Records", the largest ever is . Until recently, regardless of the size, such cheques could still be redeemed for their cash value as long as they would have the same parts as a normal cheque, although usually the oversized cheque is kept as a souvenir and a normal cheque is provided. Any bank could levy additional charges for clearing an oversized cheque. Most banks need to have the machine-readable information on the bottom of cheques read electronically, so only very limited dimensions can be allowed due to standardised equipment.


title: Cheque

text: In the US some public assistance programmes such as the Special Supplemental Nutrition Program for Women, Infants and Children, or Aid to Families with Dependent Children make "vouchers" available to their beneficiaries, which are good up to a certain monetary amount for purchase of grocery items deemed eligible under the particular programme. The voucher can be deposited like any other cheque by a participating supermarket or other approved business.


title: Cheque

text: In many Asian countries cheques were never widely used and generally only used by the wealthy, with cash being used for the majority of payments. Where cheques were used they have been declining rapidly, by 2009 there was negligible consumer cheque usage in Japan, South Korea and Taiwan. This declining trend was accelerated by these developed markets advanced financial services infrastructure. Many of the developing countries in Asia have seen an increasing use of electronic payment systems, 'leap-frogging' the less efficient chequing system altogether.


title: Cheque

text: India is one of the few countries in Asia that did have significant cheque usage. It had a long tradition of using cheques and passed laws formalising cheque usage as early as 1881. In 2009 cheques were still widely used as a means of payment in trade, and also by individuals to pay other individuals or utility bills. One of the reasons was that banks usually provided cheques for free to their individual account holders. However, cheques are now rarely accepted at point of sale in retail stores where cash and cards are payment methods of choice. Electronic payment transfer continued to gain popularity in India and like other countries this caused a subsequent reduction in volumes of cheques issued each year. In 2009 the Reserve Bank of India reported there was a five percent decline in cheque usage compared to the previous year. In 2019, the Reserve Bank of India reported that while cheque usage continued to decline, the decrease was slow. The bank attributed the slow pace of the decline due to the fact that cheque volume had briefly increased after demonetisation in 2016 before continuing to fall, as well as the efficiency of India's cheque clearing system.


title: Cheque

text: Bounced cheques are called . If an account owner bounces two cheques in six months, the bank will suspend the account for two years. If the account belongs to a public company, their stock will also be suspended from trading on the stock exchange, which can lead to bankruptcy.


title: Cheque

text: In Canada, cheque sizes and types, endorsement requirements and MICR tolerances are overseen by Payments Canada.


title: Cheque

text: Canada's usage of cheques is less than that of the US and is declining rapidly at the urging of the Canadian Banking Association. The Government of Canada claims it is 6.5 times more expensive to mail a cheque than to make a direct deposit. The Canadian Payments Association reported that in 2012, cheque use in Canada accounted for only 40% of total financial transactions. The Interac system, which allows instant fund transfers via chip or magnetic strip and PIN, is widely used by merchants to the point that few brick and mortar merchants accept cheques. Many merchants accept Interac debit payments but not credit card payments, even though most Interac terminals can support credit card payments. Financial institutions also facilitate transfers between accounts within different institutions with the Email Money Transfer (EMT) service.


title: Cheque

text: Cheques are still used for government payments, payroll, rent, and utility bill payments, though direct deposits and online or telephone bill payments are more widely and increasingly used.


title: Cheque

text: A void cheque has a special code called a MICR (Magnetic Ink Character Recognition) that is used by the bank to process the payment. The MICR code consists of the cheque number, transit number, and account number. The MICR number on a cheque can be interpreted as follows:


title: Cheque

text: In most European countries, cheques are now rarely used, even for third party payments. In these countries, it is standard practice for businesses to publish their bank details on invoices, to facilitate the receipt of payments by giro. Even before the introduction of online banking, it has been possible in some countries to make payments to third parties using ATMs, which may accurately and rapidly capture invoice amounts, due dates, and payee bank details via a bar code reader to reduce keying. In some countries, entering the bank account number results in the bank revealing the name of the payee as an added safeguard against fraud. In using a cheque, the onus is on the payee to initiate the payment, whereas with a giro transfer, the onus is on the payer to effect the payment (The writer of a paper cheque is pushing on a rope: he cannot force money out of his own account and into the destination's account. By writing the paper cheque, he is handing the far end of the rope to the payee, who will pull in his own good time. In contrast, giro is more akin to wire transfer, in that the payer pushes his money away towards the payee). The process is also procedurally more simple, as no cheques are ever posted, can claim to have been posted, or need banking or clearance.


title: Cheque

text: In Germany, Austria, Switzerland, Liechtenstein, the Netherlands, Belgium, Luxembourg, Norway, Sweden, Finland, Denmark, and Iceland, giro transfers have been standard procedure since the 1950s for regular payments like rent and wages and even mail-order invoices. In the Netherlands, Austria, and Germany, many invoices are accompanied by so-called ' (Netherlands) or ' (German), which are essentially standardized bank transfer order forms preprinted with the payee's account details and the amount payable. The payer fills in his account details and hands the form to a clerk at his bank, which will then transfer the money. It is also very common to allow the payee to automatically withdraw the requested amount from the payer's account ("Lastschrifteinzug" (German) or "Incasso (machtiging)" (Netherlands)). Though similar to paying by cheque, the payee only needs the payer's bank and account number. Since the early 1990s, this method of payment has also been available to merchants. Debit cards are widespread in these countries, since virtually all banks issue debit cards instead of simple ATM cards for use on current accounts. Acceptance of cheques has been further diminished since the late 1990s, because of the abolition of the Eurocheque. Cashing a foreign bank cheque is possible, but usually very expensive.


title: Cheque

text: In Finland, banks stopped issuing personal cheques in about 1993 in favour of giro systems, which are now almost exclusively electronically initiated either via internet banking or payment machines located at banks and shopping malls. All Nordic countries have used an interconnected international giro system since the 1950s, and in Sweden, cheques are now almost totally abandoned; in Denmark, all banks stopped accepting cheques starting on 1 January 2017. Debit cards are now preferred for direct shop payments when not using cash. For large shop payments, such as car purchases, a type of cheque, a money order () is still used.


title: Cheque

text: In Poland cheques were withdrawn from use in 2006, mainly because of lack of popularity due to the widespread adoption of credit and debit cards. Electronic payments across the European Union are now fast and inexpensive—usually free for consumers.


title: Cheque

text: In the United Kingdom, Ireland, France as well as to some extent in Monaco, Italy and San Marino, cheques are still popular, partly because cheques remain free of charge to personal customers; however, bank-to-bank transfers are increasing in popularity. France, however, remains well ahead of its European counterparts in the use of cheque payments, as seen in 2020 where it's estimated that more than 1 billion cheque payments were made, compared to Italy, the country with the next highest number of payments, with under 100 million.


title: Cheque

text: The Cheque was introduced in India by the Bank of Hindustan, the first joint stock bank established in 1770. In 1881, the Negotiable Instruments Act (NI Act) was enacted in India, formalising the usage and characteristics of instruments like the cheque, the bill of exchange, and promissory note. The NI Act provided a legal framework for non-cash paper payment instruments in India. In 1938, the Calcutta Clearing Banks' Association, which was the largest bankers' association at that time, adopted clearing house.


title: Cheque

text: Until 1 April 2012, cheques in India were valid for a period of six months from the date of their issue, before the Reserve Bank of India (RBI) issued a notification reducing their validity to three months from the date of issue.


title: Cheque

text: Beginning in 2010, the RBI along with the National Payments Corporation of India (NPCI) piloted the cheque truncation system (CTS). Under CTS, cheques are no longer physically transported to different clearing houses. They are processed at the bank where they are presented, where an image of the cheque using Magnetic ink character recognition (MICR) is captured and digitally transmitted.


title: Cheque

text: In Australia, following global trends, the use of cheques continues to decline. In 1994 the value of daily cheque transactions was A$25 billion; by 2004 this had dropped to only A$5 billion, and by 2018 this had dropped to only A$1 billion, with almost half of this for B2B transactions. Personal cheque use is practically non-existent thanks to the longstanding use of the EFTPOS system, BPAY, electronic transfers, and debit cards.


title: Cheque

text: In New Zealand, payments by cheque have declined since the mid-1990s in favour of electronic payment methods. In 1993, cheques accounted for over half of transactions through the national banking system, with an annual average of 130 cheques per capita. By 2006 cheques lagged well behind EFTPOS (debit card) transaction and electronic credits, making up only nine per cent of transactions, an annual average of 41 cheque transaction per capita. Kiwibank stopped accepting cheques as payment on 28 February 2020, followed by ANZ on 31 May 2021. Westpac and BNZ stopped accepting cheques on 25 June and 30 June 2021 respectively; ASB was the last major bank to phase out cheques on 27 August 2021.


title: Cheque

text: The Cheques Act 1986 is the body of law governing the issuance of cheques and payment orders in Australia. Procedural and practical issues governing the clearance of cheques and payment orders are handled by Australian Payments Clearing Association (APCA).


title: Cheque

text: In 1999, banks adopted a system to allow faster clearance of cheques by electronically transmitting information about cheques; this brought clearance times down from five to three days. Previously, cheques were required to be physically transported to the paying bank before processing began, and dishonoured cheques were physically returned.


title: Cheque

text: All licensed banks in Australia may issue cheques in their own name. Non-banks are not permitted to issue cheques in their own name but may issue, and have drawn on them, payment orders (which functionally are no different from cheques).


title: Cheque

text: Instrument-specific legislation includes the Cheques Act 1960, part of the Bills of Exchange Act 1908, which codifies aspects related to the cheque payment instrument, notably the procedures for the endorsement, presentment and payment of cheques. A 1995 amendment provided for the electronic presentment of cheques and removed the previous requirement to deliver cheques physically to the paying bank, opening the way for cheque truncation and imaging. Truncation allows for the transmission of an electronic image of all or part of the cheque to the paying bank's branch, instead of cumbersome physical presentment. This reduced the total cheque clearance time and eliminated the costs of physically moving the cheque.


title: Cheque

text: The under supervision of Reserve Bank of New Zealand provide the cheque payment services. Once banked, cheques are processed electronically together with other retail payment instruments. "Homeguard v Kiwi Packaging" is often cited case law regarding the banking of cheques tendered as full settlement of disputed accounts.


title: Cheque

text: In 2020 New Zealand Banks began phasing out of cheques, and they are no longer accepted as payment. All have moved to other types of payment systems


title: Cheque

text: In the UK all cheques must now conform to an industry standard detailing layout and font ("Cheque and Credit Clearing Company (C&CCC) Standard 3"), be printed on a specific weight of paper (CBS1), and contain explicitly defined security features.


title: Cheque

text: Since 1995, all cheque printers must be members of the Cheque Printer Accreditation Scheme (CPAS). The scheme is managed by the Cheque and Credit Clearing Company and requires that all cheques for use in the British clearing process are produced by accredited printers who have adopted stringent security standards.


title: Cheque

text: The rules concerning crossed cheques are set out in Section 1 of the Cheques Act 1992 and prevent cheques being cashed by or paid into the accounts of third parties. On a crossed cheque the words "account payee only" (or similar) are printed between two parallel vertical lines in the centre of the cheque. This makes the cheque non-transferable and is to avoid cheques being endorsed and paid into an account other than that of the named payee. Crossing cheques basically ensures that the money is paid into an account of the intended beneficiary of the cheque.


title: Cheque

text: Following concerns about the amount of time it took banks to clear cheques, the United Kingdom Office of Fair Trading set up a working group in 2006 to look at the cheque clearing cycle. They produced a report recommending maximum times for the cheque clearing which were introduced in UK from November 2007. In the report the date the credit appeared on the recipient's account (usually the day of deposit) was designated "T". At "T + 2" (two business days afterwards) the value would count for calculation of credit interest or overdraft interest on the recipient's account. At "T + 4" clients would be able to withdraw funds on current accounts or at "T + 6" on savings accounts (though this will often happen earlier, at the bank's discretion). "T + 6" is the last day that a cheque can bounce without the recipient's permission—this is known as "certainty of fate". Before the introduction of this standard (also known as 2-4-6 for current accounts and 2-6-6 for savings accounts), the only way to know the "fate" of a cheque has been "Special Presentation", which would normally involve a fee, where the drawee bank contacts the payee bank to see if the payee has that money at that time. "Special Presentation" had been stated at the time of deposit.


title: Cheque

text: Cheque volumes peaked in 1990 when four billion cheque payments were made. Of these, 2.5 billion were cleared through the inter-bank clearing managed by the C&CCC, the remaining 1.5 billion being in-house cheques which were either paid into the branch on which they were drawn or processed intra-bank without going through the clearings. As volumes started to fall, the challenges faced by the clearing banks were then of a different nature: how to benefit from technology improvements in a declining business environment.


title: Cheque

text: Although the UK did not adopt the euro as its national currency when other European countries did in 1999, many banks began offering euro denominated accounts with chequebooks, principally to business customers. The cheques can be used to pay for certain goods and services in the UK. The same year, the C&CCC set up the euro cheque clearing system to process euro denominated cheques separately from sterling cheques in Great Britain.


title: Cheque

text: The UK Payments Council from 30 June 2011 withdrew the existing "Cheque Guarantee Card Scheme" in the UK. This service allowed cheques to be guaranteed at point of sales up to a certain value, normally £50 or £100, when signed in front of the retailer with the additional cheque guarantee card. This was after a long period of decline in their use in favour of debit cards.


title: Cheque

text: The Payments Council proposed to close the centralised cheque clearing altogether in the UK and had set a target date of 31 October 2018. However, on 12 July 2011, the Payments Council announced that after opposition from MPs, charity groups and public opinion, the cheque will remain in use and there would no longer be a reason to seek an alternative paper-initiated payment.


title: Cheque

text: Since 2001, businesses in the United Kingdom have made more electronic payments than cheque payments. Automated payments rose from 753 million in 1995 to 1.1 billion in 2001 and cheques declined in that same period of time from 1.14 to 1.1 billion payments. Most utilities in the United Kingdom charge lower prices to customers who pay by direct debit than for other payment methods, including electronic methods.


title: Cheque

text: The vast majority of retailers in the United Kingdom have not accepted cheques as a means of payment for several years, and cheque guarantee cards are no longer issued. For example, Shell announced in September 2005 that it would no longer accept cheques at their UK petrol stations. This was soon followed by other major fuel retailers, such as Texaco, BP, and Total. Asda announced in April 2006 that it would stop accepting cheques, initially as a trial in the London area, and Boots announced in September 2006 that it would stop accepting cheques, initially as a trial in Sussex and Surrey. Currys (and other stores in the DSGi group) and WH Smith also no longer accept cheques.


title: Cheque

text: Cheques are now widely predicted to become a thing of the past, or at most, a niche product used to pay private individuals or for the very large number of small service providers who are not willing to provide their bank details to customers to allow electronic payments to be made to them or do not wish to be burdened with checking their bank accounts frequently and reconciling them with amounts due (for example, music teachers, driving instructors, children's sports lessons, small shops, schools).


title: Cheque

text: The UK Payments Council announced in December 2009 that cheques would be phased out by October 2018, but only if adequate alternatives were developed. They intended to perform annual checks on the progress of other payments systems and a final review of the decision would have been held in 2016. Concerns were expressed, however, by charities and older people; who are the main users of cheques, and replacement plans were criticised as being open to fraud. It was therefore announced by the UK Payments Council in July 2011 that the cheque would not be eliminated. 432 million inter-bank cheques and credit-items worth £472,000,000,000 were processed in the United Kingdom in 2016 according to Payments UK. In 2017, 405 million cheques worth £356 billion were used for payments and acquiring cash, an average of 1.2 million cheques per day, with more than 10 million being cleared in Northern Ireland alone. The Cheque and Credit Clearing Company noted that cheques continue to be highly valued for paying tradesmen and utility bills, and play a vital role in business, clubs and societies sectors, with nine in 10 business saying that they received or made payment by cheque on a monthly basis.


title: Cheque

text: In June 2014, following a successful trial in the UK by Barclays, the UK government gave the go-ahead for a cheque truncation system, allowing people to pay in a cheque by taking a photo of it, rather than physically depositing the paper cheque at a bank.


title: Cheque

text: In the United States, cheques are referred to as "checks" and are governed by Article 3 of the Uniform Commercial Code, under the rubric of negotiable instruments.


title: Cheque

text: In the US, the terminology for a cheque historically varied with the type of financial institution on which it is drawn. In the case of a savings and loan association it was a "negotiable order of withdrawal" (compare Negotiable Order of Withdrawal account); if a credit union it was a "share draft." "Checks" were associated with chartered commercial banks. However, common usage has increasingly conformed to more recent versions of Article 3, where "check" means any or all of these negotiable instruments. Certain types of cheques drawn on a government agency, especially payroll cheques, may be called a "payroll warrant".


title: Cheque

text: At the bottom of each cheque there is the routing/account number in MICR format. The ABA routing transit number is a nine-digit number in which the first four digits identifies the US Federal Reserve Bank's cheque-processing centre. This is followed by digits 5 through 8, identifying the specific bank served by that cheque-processing centre. Digit 9 is a verification check digit, computed using a complex algorithm of the previous eight digits.


title: Cheque

text: A "draft" in the US Uniform Commercial Code is any bill of exchange, whether payable on demand or at a later date. If payable on demand it is a "demand draft", or if drawn on a financial institution, a cheque.


title: Cheque

text: The electronic cheque or substitute cheque was formally adopted in the US in 2004 with the passing of the "Check Clearing for the 21st Century Act" (or Check 21 Act). This allowed the creation of electronic cheques and translation (truncation) of paper cheques into electronic replacements, reducing cost and processing time.


title: Cheque

text: In 2002 the US still relied heavily on cheques due to the convenience afforded to payers, and due to the absence of a high volume system for low-value electronic payments. In practice, transfers of less than about five dollars are extremely expensive, and transactions of less than 50c impossible (transaction fees swallow the payment or exceed it). The only methods generally available for individuals and small businesses to make payments electronically are electronic funds transfers (EFT) or accepting credit cards. EFT payments require a commercial chequing account (which often has higher fees and minimum balances than individual accounts) and a subscription to EFT service costing anywhere from $10 to $25 a month, plus 10¢ per transaction (making transactions of 10¢ or less impossible, and transactions under $1 very expensive.) Credit card payments cost the recipient (or the payer) 33¢ plus 3% of the transaction, making transactions of 33¢ or less impossible, and transactions of $1 or less have at least a 30% service charge. Generally, payments by cheque (as long as the payer has funds in their account) and the recipient deposits it to their bank account, regardless of amount, have a service charge to both parties of zero.


title: Cheque

text: Since 2002, the decline in cheque usage seen around the world has also started in the US. The cheque, although not as common as it used to be, is still a long way from disappearing completely in the US.


title: Cheque

text: About 70 billion cheques were written annually in the US by 2001, though around 17 million adult Americans have no bank accounts . Certain companies whom a person pays with a cheque will turn it into an Automated Clearing House (ACH) or electronic transaction. Banks try to save time processing cheques by sending them electronically between banks. Cheque clearing is usually done through an electronic cheque broker, such as The Clearing House, Viewpointe LLC or the Federal Reserve Banks. Copies of the cheques are stored at a bank or the broker, for periods up to 99 years, and this is why some cheque archives have grown to 20 petabytes. The access to these archives is now worldwide, as most bank programming is now done offshore. Many utilities and most credit cards will also allow customers to pay by providing bank information and having the payee draw payment from the customer's account (direct debit). Many people in the US pay their bills or transfer money via paper money orders, as these have security advantages over mailing cash and require no bank-account access.


title: Cheque

text: In Turkey, cheques are usually used for commercial transactions only, and using post-dated cheques is legally permissible.


title: Cheque

text: Cheques have been a tempting target for criminals to steal money or goods from the drawer, payee or the banks. A number of measures have been introduced to combat fraud over the years. These range from things like writing a cheque so it is difficult to alter after it is drawn, to mechanisms like crossing a cheque so that it can only be paid into another bank's account providing some traceability. However, the inherent security weaknesses of cheques as a payment method, such as having only the signature as the main authentication method and not knowing if funds will be received until the clearing cycle is complete, have made them vulnerable to a number of different types of fraud.


title: Cheque

text: Taking advantage of the float period (cheque kiting) to delay the notice of non-existent funds. This often involves trying to convince a merchant or other recipient, hoping the recipient will not suspect that the cheque will not clear, giving time for the fraudster to disappear.


title: Cheque

text: Sometimes, forgery is the method of choice in defrauding a bank. One form of forgery involves the use of a victim's legitimate cheques, that have either been stolen and then cashed, or altering a cheque that has been legitimately written to the perpetrator, by adding words or digits to inflate the amount.


title: Cheque

text: Since cheques include significant personal information (name, account number, signature and in some countries driver's license number, the address or phone number of the account holder), they can be used for identity theft. The practice was discontinued as identity theft became widespread.


title: Cheque

text: A dishonoured cheque cannot be redeemed for its value and is worthless; they are also known as an "RDI" (returned deposit item), or "NSF" (non-sufficient funds) cheque. Cheques are usually dishonoured because the drawer's account has been frozen or limited, or because there are insufficient funds in the drawer's account when the cheque was redeemed. A cheque drawn on an account with insufficient funds is said to have "bounced" and may be called a "rubber cheque". Banks will typically charge customers for issuing a dishonoured cheque, and in some jurisdictions such an act is a criminal action. A drawer may also issue a "stop" on a cheque, instructing the financial institution not to honour a particular cheque.


title: Cheque

text: In England and Wales, they are typically returned marked "Refer to Drawer"—an instruction to contact the person issuing the cheque for an explanation as to why the cheque was not honoured. This wording was brought in after a bank was successfully sued for libel after returning a cheque with the phrase "Insufficient Funds" after making an error—the court ruled that as there were sufficient funds the statement was demonstrably false and damaging to the reputation of the person issuing the cheque. Despite the use of this revised phrase, successful libel lawsuits brought against banks by individuals remained for similar errors.


title: Cheque

text: In Scotland, a cheque acts as an assignment of the amount of money to the payee. As such, if a cheque is dishonoured in Scotland, what funds are present in the bank account are "attached" and frozen, until either sufficient funds are credited to the account to pay the cheque, the drawer recovers the cheque and hands it into the bank, or the drawer obtains a letter from the payee stating that they have no further interest in the cheque.


title: Cheque

text: A cheque may also be dishonoured because it is stale or not cashed within a "void after date". Many cheques have an explicit notice printed on the cheque that it is void after some period of days. In the US, banks are not required by the Uniform Commercial Code to honour a , which is a cheque presented six months after it is dated.


title: Cheque

text: In the United States some consumer reporting agencies such as ChexSystems, Early Warning Services, and TeleCheck have been providing cheque verification services that track how people manage their chequing accounts. Banks use the agencies to screen chequing account applicants, and those with low debit scores are denied because banks cannot afford overdrawn accounts.


title: Cheque

text: In the United Kingdom, in common with other items such as Direct Debits or standing orders, dishonoured cheques can be reported on a customer's credit file, although not individually and this does not happen universally amongst banks. Dishonoured payments from current accounts can be marked in the same manner as missed payments on the customer's credit report.


title: Cheque

text: Typically when customers pay bills with cheques (like gas or water bills), the mail will go to a "lock box" at the post office. There a bank will pick up all the mail, sort it, open it, take the cheques and remittance advice out, process it all through electronic machinery, and post the funds to the proper accounts. In modern systems, taking advantage of the Check 21 Act, as in the United States many cheques are transformed into electronic objects and the paper is destroyed.


title: Władysław Szpilman

text: Władysław Szpilman (; 5 December 1911 – 6 July 2000) was a Polish pianist and classical composer of Jewish descent. Szpilman is widely known as the central figure in the 2002 Roman Polanski film "The Pianist", which was based on Szpilman's autobiographical account of how he survived the German occupation of Warsaw and the Holocaust.


title: Władysław Szpilman

text: Szpilman studied piano at music academies in Berlin and Warsaw. He became a popular performer on Polish radio and in concert. Confined within the Warsaw ghetto after the German invasion of Poland, Szpilman spent two years in hiding. Towards the end of his concealment, he was helped by Wilm Hosenfeld, a German officer who detested Nazi policies. After World War II, Szpilman resumed his career on Polish radio. Szpilman was also a prolific composer; his output included hundreds of songs and many orchestral pieces.


title: Władysław Szpilman

text: Szpilman began his study of the piano at the Chopin Academy of Music in Warsaw, Poland, where he studied piano with Aleksander Michałowski and Józef Śmidowicz, first- and second-generation pupils of Franz Liszt. In 1931 he was a student of the prestigious Academy of Arts in Berlin, Germany, where he studied with Artur Schnabel, Franz Schreker, and Leonid Kreutzer. After Adolf Hitler was appointed Chancellor of Germany in 1933, Szpilman returned to Warsaw, where he quickly became a celebrated pianist and composer of both classical and popular music. Primarily a soloist, he was also the chamber music partner of such acclaimed violinists as Roman Totenberg, Ida Haendel and Henryk Szeryng, and in 1934 he toured Poland with U.S. violinist, Bronislav Gimpel.


title: Władysław Szpilman

text: On 5 April 1935 Szpilman joined the Polish Radio, where he worked as a pianist performing classical and jazz music. His compositions at this time included orchestral works, piano pieces, and also music for films, as well as roughly 50 songs, many of which became quite popular in Poland. At the time of the German invasion of Poland in September 1939, he was a celebrity and a featured soloist at the Polskie Radio, which was bombed on 23 September 1939, shortly after broadcasting the last Chopin recital played by Szpilman. The Nazi occupiers established the General Government, and created ghettos in many Polish cities, including Warsaw. Szpilman and his family did not yet need to find a new residence, as their apartment was already in the ghetto area.


title: Władysław Szpilman

text: Władysław Szpilman and his family, along with all other Jews living in Warsaw, were forced to move into a "Jewish quarter" – the Warsaw Ghetto – on 31 October 1940. Once all the Jews were confined within the ghetto, a wall was constructed to separate them from the rest of the Nazi German-occupied city. Szpilman managed to find work as a musician to support his family, which included his mother, father, brother Henryk, and two sisters, Regina and Halina. He first worked at the "Nowoczesna" Cafe, where the patrons sometimes ignored his playing in order to conduct business, as he recalled in the memoir.


title: Władysław Szpilman

text: Szpilman later played in a cafe on Sienna Street and after 1942 in the "Sztuka" Cafe on Leszno Street as well. In these last two cafes he performed chamber music with violinist Zygmunt Lederman, performed in the piano duo with Andrzej Goldfeder, and played with other musicians as well.


title: Władysław Szpilman

text: Everyone in his family was deported in 1942 to Treblinka, an extermination camp within German-occupied Poland roughly northeast of Warsaw. A member of the Jewish Police assisting in deportations, who recognized Szpilman, pulled him from a line of people—including his parents, brother, and two sisters—being loaded onto a train at the transport site (which, as in other ghettos, was called the "Umschlagplatz"). None of Szpilman's family members survived the war. Szpilman stayed in the ghetto as a labourer, and helped smuggle in weapons for the coming Jewish resistance uprising. Szpilman remained in the Warsaw Ghetto until 13 February 1943, shortly before it was abolished after the deportation of most of its inhabitants in April–May 1943.


title: Władysław Szpilman

text: Szpilman found places to hide in Warsaw and survived with the help of his friends from Polish Radio and fellow musicians such as Andrzej Bogucki and his wife Janina, Czesław Lewicki, and Helena Lewicka supported by Edmund Rudnicki, Witold Lutosławski, Eugenia Umińska, Piotr Perkowski, and Irena Sendler. He evaded capture several times. Beginning in August 1944, Szpilman was hiding out in an abandoned building at Aleja Niepodległości Street 223. In November, he was discovered there by the German officer, Captain Wilm Hosenfeld. To Szpilman's surprise, the officer did not arrest or kill him; after discovering that the emaciated Szpilman was a pianist, Hosenfeld asked him to play something on the piano that was on the ground floor. Szpilman played Chopin's Nocturne No. 20 in C♯ minor. After that, the officer brought him bread and jam on numerous occasions. He also offered Szpilman one of his coats to keep warm in the freezing temperatures. Szpilman did not know the name of the German officer until 1951. Despite the efforts of Szpilman and the Poles to rescue him, Hosenfeld died in a Soviet prisoner of war camp in 1952.


title: Władysław Szpilman

text: Szpilman started playing for Polish Radio in 1935 as their house pianist. In 1939, on 23 September, Szpilman was in the middle of broadcasting when Germans opened fire on the studio and he was forced to stop playing. This was the last live music broadcast that was heard until the war's end. When Szpilman resumed his job at Polish Radio in 1945, he did so by carrying on where he left off six years before: poignantly, he opened the first transmission by once again playing Chopin's "Nocturne in C-sharp Minor" ("Lento con gran espressione").


title: Władysław Szpilman

text: From 1945 to 1963, Szpilman was director of the Popular Music Department at Polish Radio. Szpilman performed at the same time as a concert pianist and chamber musician in Poland, as well as throughout Europe, Asia, and America. During this period, he composed several symphonic works and about 500 other compositions that are still popular in Poland today. He also wrote music for radio plays and films and in 1961, he created the International Song Contest in Sopot, Poland, which has been produced every summer for more than 50 years.


title: Władysław Szpilman

text: Szpilman and Bronislav Gimpel founded the Warsaw Piano Quintet in 1963 with which Szpilman performed more than 2000 concerts worldwide until 1986 in such places as Royal Festival Hall in London; Salle Pleyel and Salle Gaveau in Paris; Herkules Saal in Munich; as well as the Salzburger Festspiele, Brahmstage Baden-Baden, Musikhalle Hamburg a.o.


title: Władysław Szpilman

text: From his early Berlin years, Szpilman never gave up the will to write music, even when living in the Warsaw Ghetto. His compositions include orchestral works, concertos, piano pieces, but also significant amounts of music for radio plays and films, as well as around 500 songs. More than 100 of these are very well known as hits and evergreens in Poland. In the 1950s, he wrote about 40 songs for children, for which he received an award from the Polish Composers Union in 1955.


title: Władysław Szpilman

text: His son Andrzej commented in 1998 that Szpilman's works did not reach a larger audience outside Poland, attributing this to the "division of Europe into two halves culturally as well as politically" after the war. His father "shaped the Polish popular music scene over several decades—but the western frontier of Poland constituted a barrier" to music from the Eastern bloc countries. (Andrzej Szpilman's "Foreword" to the 1999 edition of "The Pianist", p. 8)


title: Władysław Szpilman

text: Szpilman's compositions include the suite for piano "Life of the Machines" 1932, Violin Concerto 1933, "Waltzer in the Olden Style" 1937, film soundtracks: "Świt, dzień i noc Palestyny" (1934), "Wrzos" (1938) and "Doctor Murek" (1939), Concertino for Piano and Orchestra (1940), Paraphrase on Own Themes (1948) "Ouverture for Symphonic Orchestra" (1968) and many very popular songs in Poland. His works are now published in printed editions by Boosey & Hawkes/Bote & Bock Music Publishers in New York, Berlin and London.


title: Władysław Szpilman

text: In 1961, he initiated and organized Sopot International Song Festival produced in Poland every summer, now for more than 50 years. He founded the Polish Union of Authors of Popular Music.


title: Władysław Szpilman

text: "The Death of a City" (original "Śmierć miasta") was written by Wladyslaw Szpilman and elaborated by Jerzy Waldorff shortly after the war ended, and first printed in 1946 by publishing house "Wiedza", The book was censored by Stalinist authorities for political reasons. For example, the nationality of benevolent German officer Wilm Hosenfeld was changed to Austrian. As the East German dissident singer-songwriter Wolf Biermann observed in his epilogue for the 1999 English-language edition: "Directly after the war it was impossible to publish a book in Poland which presented a German officer as a brave and helpful man," and an Austrian hero would be "not quite so bad." Biermann added caustically, "In the years of the Cold War Austria and East Germany were linked by a common piece of hypocrisy: both pretended to have been forcibly occupied by Hitler's Germany."


title: Władysław Szpilman

text: In 1998, Szpilman's son Andrzej published new extended edition of his father's memoir, first in German translation from Karin Wolff as "Das wunderbare Überleben" (The Miraculous Survival) by a German publishing house Ullstein Verlag; and then in English translation by Anthea Bell as "The Pianist" with Epilogue by Wolf Biermann. In March 1999 Władysław Szpilman visited London for Jewish Book Week, where he met English readers to mark the publication of the book in Great Britain. It was later published in more than 35 languages, named "Best book of the year" by "Los Angeles Times", "Sunday Times", "Boston Globe", "The Guardian", "The Economist", "Library Journal", won Annual Jewish Quarterly Wingate Prize 2000, "Best book of the year 2001" by magazine "Lire" and "Elle" (Paris) in 2002. New Polish edition, "Pianista : warszawskie wspomnienia 1939–1945" (Kraków: Znak, 2000) became a number 1 on the bestseller list by Polish newspaper "Rzeczpospolita" for 3 years in 2001–2003.


title: Władysław Szpilman

text: As it reached a much larger audience, Szpilman's memoir was widely praised. Britain's "Independent" described it as "a compelling, harrowing masterpiece"; it is "one of the most powerful accounts ever written" of the era declared another leading British daily. The book's description of the famed Warsaw teacher and writer Janusz Korczak has been described as "overwhelmingly powerful and poignant." Korczak declined to save himself from deportation to Treblinka, instead walked with the children of his orphanage to the deportation site and ultimately escorting them "into the next world," as Szpilman related:


title: Władysław Szpilman

text: The 1999 English-language edition also includes excerpts from Wilm Hosenfeld's diary (1942–44). Biermann's Epilogue gives further insight into Hosenfeld's deeds and his character. He aided several other would-be victims in Warsaw; Hosenfeld nonetheless died (in 1952) after seven years in Soviet captivity, despite the efforts of Szpilman to help him.


title: Władysław Szpilman

text: Although it concludes with his survival, Szpilman declined to conclude his memoir on a happy note. In the final paragraphs, he walks the streets of an abandoned and devastated Warsaw: "A stormy wind rattled the scrap-iron in the ruins, whistling and howling through the charred cavities of the windows. Twilight came on. Snow fell from the darkening, leaden sky." As one reviewer noted, "these final sentences distill the style of this astonishing and unforgettable book. Concise yet highly evocative; measured and somewhat detached, yet possessing a poeticism and a consistent spiritual tenor and strength."


title: Władysław Szpilman

text: In 2002, the Polish-French film-maker, Roman Polanski, directed a screen version of the book. The movie won three Academy Awards in 2003 – Oscars for best director; best actor, and best adapted screenplay, the British Academy of Film and Television Arts Best Film Award, and the Palme d'Or at the Cannes Film Festival. Polanski escaped the Kraków Ghetto and survived the Nazi genocides but his mother was killed by the German occupiers. Polanski's film closely follows the book's style and details. Adrien Brody accepting the Oscar for Best Actor in a Leading Role "The Pianist" said – ..."This film would not be possible without the blueprint provided by Wladyslaw Szpilman. This is a tribute to his survival"...


title: Władysław Szpilman

text: Szpilman's son, Andrzej Szpilman, compiled and released a CD with the most popular songs Szpilman had composed under the title "Wendy Lands Sings the Songs of the Pianist" (Universal Music). Other CDs with the works of Szpilman include "Works for Piano and Orchestra by Władysław Szpilman" with Ewa Kupiec (piano), John Axelrod (director), and the Berlin Radio Symphony Orchestra (2004) (Sony classical) and the "Original recordings of The Pianist" and "Władysław Szpilman-Legendary recordings" (Sony classical). In November 1998, Szpilman was honored by the president of Poland with a Commander's Cross with Star of the Order of Polonia Restituta.


title: Władysław Szpilman

text: Szpilman died of natural causes in Warsaw on 6 July 2000, aged 88. He is buried at Powązki Military Cemetery. On 25 September 2011, Polish Radio’s Studio 1 was renamed for Władysław Szpilman. On 4 December 2011, a commemorative plaque to Szpilman, engraved in Polish and English, was unveiled at 223 Niepodległości Avenue in Warsaw, in the presence of his wife Halina Szpilman and son Andrzej, and Wilm Hosenfeld's daughter Jorinde Krejci-Hosenfeld. The next day, on the exact centenary of Szpilman's birth, Polish President Bronisław Komorowski met Szpilman's widow and son, and Krejci-Hosenfeld.


title: Władysław Szpilman

text: Uri Caine, an American classical and jazz pianist and composer, created his own interpretations of Szpilman’s works in a variety of genres. The CD of Caine's concert was released on 24 February 2014.


title: Discrete Fourier transform

text: In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.


title: Discrete Fourier transform

text: The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function). In image processing, the samples can be the values of pixels along a row or column of a raster image. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers.


title: Discrete Fourier transform

text: Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. These implementations usually employ efficient fast Fourier transform (FFT) algorithms; so much so that the terms "FFT" and "DFT" are often used interchangeably. Prior to its current usage, the "FFT" initialism may have also been used for the ambiguous term "finite Fourier transform".


title: Discrete Fourier transform

text: The "discrete Fourier transform" transforms a sequence of "N" complex numbers formula_1 into another sequence of complex numbers, formula_2 which is defined by


title: Discrete Fourier transform

text: The transform is sometimes denoted by the symbol formula_3, as in formula_4 or formula_5 or formula_6.


title: Discrete Fourier transform

text: can also be evaluated outside the domain formula_7, and that extended sequence is formula_8-periodic. Accordingly, other sequences of formula_8 indices are sometimes used, such as formula_10 (if formula_8 is even) and formula_12 (if formula_8 is odd), which amounts to swapping the left and right halves of the result of the transform.


title: Discrete Fourier transform

text: The normalization factor multiplying the DFT and IDFT (here 1 and formula_14) and the signs of the exponents are merely conventions, and differ in some treatments. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be formula_14. A normalization of formula_16 for both the DFT and IDFT, for instance, makes the transforms unitary. A discrete impulse, formula_17 at "n" = 0 and 0 otherwise; might transform to formula_18 for all "k" (use normalization factors 1 for DFT and formula_14 for IDFT). A DC signal, formula_18 at "k" = 0 and 0 otherwise; might inversely transform to formula_21 for all formula_22 (use formula_14 for DFT and 1 for IDFT) which is consistent with viewing DC as the mean average of the signal.


title: Discrete Fourier transform

text: This example demonstrates how to apply the DFT to a sequence of length formula_24 and the input vector


title: Discrete Fourier transform

text: with formula_33 denoting the set of complex numbers. Its inverse is known as Inverse Discrete Fourier Transform (IDFT). In other words, for any formula_34, an "N"-dimensional complex vector has a DFT and an IDFT which are in turn formula_8-dimensional complex vectors.


title: Discrete Fourier transform

text: The DFT is a linear transform, i.e. if formula_36 and formula_37, then for any complex numbers formula_38:


title: Discrete Fourier transform

text: Reversing the time (i.e. replacing formula_22 by formula_41) in formula_42 corresponds to reversing the frequency (i.e. formula_43 by formula_44). Mathematically, if formula_45 represents the vector x then


title: Discrete Fourier transform

text: This table shows some mathematical operations on formula_42 in the time domain and the corresponding effects on its DFT formula_51 in the frequency domain.


title: Discrete Fourier transform

text: where formula_54 is the Kronecker delta. (In the last step, the summation is trivial if formula_55, where it is and otherwise is a geometric series that can be explicitly summed to obtain zero.) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.


title: Discrete Fourier transform

text: If formula_51 and formula_57 are the DFTs of formula_42 and formula_59 respectively then the Parseval's theorem states:


title: Discrete Fourier transform

text: where the star denotes complex conjugation. Plancherel theorem is a special case of the Parseval's theorem and states:


title: Discrete Fourier transform

text: Multiplying formula_42 by a "linear phase" formula_64 for some integer "m" corresponds to a "circular shift" of the output formula_51: formula_51 is replaced by formula_67, where the subscript is interpreted modulo "N" (i.e., periodically). Similarly, a circular shift of the input formula_42 corresponds to multiplying the output formula_51 by a linear phase. Mathematically, if formula_45 represents the vector x then


title: Discrete Fourier transform

text: The convolution theorem for the discrete-time Fourier transform (DTFT) indicates that a convolution of two sequences can be obtained as the inverse transform of the product of the individual transforms. An important simplification occurs when one of sequences is N-periodic, denoted here by formula_74 because formula_75 is non-zero at only discrete frequencies (see ), and therefore so is its product with the continuous function formula_76  That leads to a considerable simplification of the inverse transform.


title: Discrete Fourier transform

text: Customarily, the DFT and inverse DFT summations are taken over the domain formula_81. Defining those DFTs as formula_82 and formula_83, the result is:


title: Discrete Fourier transform

text: In practice, the formula_79 sequence is usually length "N" or less, and formula_86 is a periodic extension of an N-length formula_87-sequence, which can also be expressed as a "circular function":


title: Discrete Fourier transform

text: which gives rise to the interpretation as a "circular" convolution of formula_79 and formula_90 It is often used to efficiently compute their linear convolution. (see Circular convolution, Fast convolution algorithms, and Overlap-save)


title: Discrete Fourier transform

text: It has been shown that any linear transform that turns convolution into pointwise product is the DFT (up to a permutation of coefficients).


title: Discrete Fourier transform

text: where the coefficients "X" are given by the DFT of "x" above, satisfies the interpolation property formula_99 for formula_100.


title: Discrete Fourier transform

text: This interpolation is "not unique": aliasing implies that one could add "N" to any of the complex-sinusoid frequencies (e.g. changing formula_102 to formula_103) without changing the interpolation property, but giving "different" values in between the formula_42 points. The choice above, however, is typical because it has two useful properties. First, it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited. Second, if the formula_42 are real numbers, then formula_106 is real as well.


title: Discrete Fourier transform

text: In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to formula_107 (instead of roughly formula_108 to formula_109 as above), similar to the inverse DFT formula. This interpolation does "not" minimize the slope, and is "not" generally real-valued for real formula_42; its use is a common mistake.


title: Discrete Fourier transform

text: Another way of looking at the DFT is to note that in the above discussion, the DFT can be expressed as the DFT matrix, a Vandermonde matrix,


title: Discrete Fourier transform

text: With unitary normalization constants formula_114, the DFT becomes a unitary transformation, defined by a unitary matrix:


title: Discrete Fourier transform

text: where formula_116 is the determinant function. The determinant is the product of the eigenvalues, which are always formula_117 or formula_118 as described below. In a real vector space, a unitary transformation can be thought of as simply a rigid rotation of the coordinate system, and all of the properties of a rigid rotation can be found in the unitary DFT.


title: Discrete Fourier transform

text: The orthogonality of the DFT is now expressed as an orthonormality condition (which arises in many areas of mathematics as described in root of unity):


title: Discrete Fourier transform

text: If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new coordinate system, then the above is just the statement that the dot product of two vectors is preserved under a unitary DFT transformation. For the special case formula_122, this implies that the length of a vector is preserved as well — this is just Plancherel theorem,


title: Discrete Fourier transform

text: A consequence of the circular convolution theorem is that the DFT matrix diagonalizes any circulant matrix.


title: Discrete Fourier transform

text: A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the (forward) DFT, via several well-known "tricks". (For example, in computations, it is often convenient to only implement a fast Fourier transform corresponding to one transform direction and then to get the other transform direction from the first.)


title: Discrete Fourier transform

text: First, we can compute the inverse DFT by reversing all but one of the inputs (Duhamel "et al.", 1988):


title: Discrete Fourier transform

text: Third, a variant of this conjugation trick, which is sometimes preferable because it requires no modification of the data values, involves swapping real and imaginary parts (which can be done on a computer simply by modifying pointers). Define formula_128 as formula_42 with its real and imaginary parts swapped—that is, if formula_130 then formula_128 is formula_132. Equivalently, formula_128 equals formula_134. Then


title: Discrete Fourier transform

text: That is, the inverse transform is the same as the forward transform with the real and imaginary parts swapped for both input and output, up to a normalization (Duhamel "et al.", 1988).


title: Discrete Fourier transform

text: The conjugation trick can also be used to define a new transform, closely related to the DFT, that is involutory—that is, which is its own inverse. In particular, formula_136 is clearly its own inverse: formula_137. A closely related involutory transformation (by a factor of formula_138) is formula_139, since the formula_140 factors in formula_141 cancel the 2. For real inputs formula_26, the real part of formula_143 is none other than the discrete Hartley transform, which is also involutory.


title: Discrete Fourier transform

text: The eigenvalues of the DFT matrix are simple and well-known, whereas the eigenvectors are complicated, not unique, and are the subject of ongoing research.


title: Discrete Fourier transform

text: This can be seen from the inverse properties above: operating formula_144 twice gives the original data in reverse order, so operating formula_144 four times gives back the original data and is thus the identity matrix. This means that the eigenvalues formula_149 satisfy the equation:


title: Discrete Fourier transform

text: Therefore, the eigenvalues of formula_144 are the fourth roots of unity: formula_149 is +1, −1, +"i", or −"i".


title: Discrete Fourier transform

text: Since there are only four distinct eigenvalues for this formula_153 matrix, they have some multiplicity. The multiplicity gives the number of linearly independent eigenvectors corresponding to each eigenvalue. (There are "N" independent eigenvectors; a unitary matrix is never defective.)


title: Discrete Fourier transform

text: The problem of their multiplicity was solved by McClellan and Parks (1972), although it was later shown to have been equivalent to a problem solved by Gauss (Dickinson and Steiglitz, 1982). The multiplicity depends on the value of "N" modulo 4, and is given by the following table:


title: Discrete Fourier transform

text: No simple analytical formula for general eigenvectors is known. Moreover, the eigenvectors are not unique because any linear combination of eigenvectors for the same eigenvalue is also an eigenvector for that eigenvalue. Various researchers have proposed different choices of eigenvectors, selected to satisfy useful properties like orthogonality and to have "simple" forms (e.g., McClellan and Parks, 1972; Dickinson and Steiglitz, 1982; Grünbaum, 1982; Atakishiyev and Wolf, 1997; Candan "et al.", 2000; Hanna "et al.", 2004; Gurevich and Hadani, 2008).


title: Discrete Fourier transform

text: the discretized and periodically summed Gaussian function yields an eigenvector of the discrete transform:


title: Discrete Fourier transform

text: Two other simple closed-form analytical eigenvectors for special DFT period "N" were found (Kong, 2008):


title: Discrete Fourier transform

text: For DFT period "N" = 2"L" + 1 = 4"K" + 1, where "K" is an integer, the following is an eigenvector of DFT:


title: Discrete Fourier transform

text: The choice of eigenvectors of the DFT matrix has become important in recent years in order to define a discrete analogue of the fractional Fourier transform—the DFT matrix can be taken to fractional powers by exponentiating the eigenvalues (e.g., Rubio and Santhanam, 2005). For the continuous Fourier transform, the natural orthogonal eigenfunctions are the Hermite functions, so various discrete analogues of these have been employed as the eigenvectors of the DFT, such as the Kravchuk polynomials (Atakishiyev and Wolf, 1997). The "best" choice of eigenvectors to define a fractional discrete Fourier transform remains an open question, however.


title: Discrete Fourier transform

text: may be considered to represent a discrete probability mass function of , with an associated probability mass function constructed from the transformed variable,


title: Discrete Fourier transform

text: For the case of continuous functions formula_163 and formula_164, the Heisenberg uncertainty principle states that