Fast Fourier transform and Native American identity in the United States: Difference between pages

Revision as of 03:43, 9 October 2003

Revision as of 18:49, 15 July 2007

Line 1:

[[Image:Amerikanska folk, Nordisk familjebok.jpg|thumb|250 px|right|In the United States, there are over 560 federally recognized tribes, and over 1.8 million Native Americans.]]

[[nl:FFT]] [[de:FFT]] [[pl:FFT]] [[fr:Transformée de Fourier rapide]]

'''Native American identity in the United States''' is an issue which seeks to define "[[Native American]]" or "(American) Indian" both for people who consider themselves Native American and for people who do not. An [[Identity (social science)|identity]] is sought which will provide for a stable definition for legal, social, and personal purposes. There are a number of different factors which have been used to define "Indianness," and the source and potential use of the definition play a role in what definition is used. Facets which characterize "Indianness" include [[culture]], [[society]], [[genes]]/[[biology]], [[law]], and [[self-identity]].<ref>Garroutte (2003), Paredes (1995)</ref> An important question is whether the definition should be dynamic and changeable across time and situation, or whether it is possible to define "Indianness" in a static way.<ref>Peroff (1997) p487</ref> The dynamic definitions may be based in how Indians adapt and adjust to dominant society, which may be called an "oppositional process" by which the boundaries between Indians and the dominant groups are maintained. Another reason for dynamic definitions is the process of "[[ethnogenesis]]," which is the process by which the ethnic identity of the group is developed and renewed as social organizations and cultures evolve.<ref>Peroff (1997) p487</ref> The question of identity, especially [[Indigenous peoples|aboriginal]] identity is common in many societies worldwide.<ref>Peroff (1997) p487</ref>

[[ja:&#39640;&#36895;&#12501;&#12540;&#12522;&#12456;&#22793;&#25563;]]

A '''fast Fourier transform''' ('''FFT''') is an efficient [[algorithm]] to compute the [[discrete Fourier transform]] (DFT) and its inverse. It is of great importance to a wide variety of applications, from [[digital signal processing]] to solving [[partial differential equation]]s to algorithms for quickly [[multiplication algorithm|multiplying large integers]]. This article describes the algorithms; see [[discrete Fourier transform]] for properties and applications of the transform.

The future of their identity is extremely important to [[Native American]]s. Activist [[Russell Means]] bemoans the crumbling Indian way of life, the loss of traditions, languages, and sacred places. He remarks that there may soon be no more Native Americans, only "Native American-Americans, like [[Polish-American]]s and [[Italian American]]s." As the number of Indians has grown (ten times as many today as in 1890), the number who carry on tribal traditions shrinks (one fifth as many as in 1890): "we might speak our language, we might look like Indians and sound like Indians, but we won’t be Indians."<ref>Peroff (1997) p492</ref>

Let ''x''₀, ...., ''x''_''n''-1 be [[complex number|complex numbers]]. The DFT is defined by the formula

==Definitions==

:<math> f_j = \sum_{k=0}^{n-1} x_k e^{-\frac{2\pi i}{n} jk }

\qquad

j = 0,\dots,n-1. </math>

===Blood quantum===

Evaluating these sums directly would take O(''n''²) arithmetical operations (see [[Big O notation]]). An FFT is an algorithm to compute the same result in only O(''n'' log ''n'') operations.

{{main|Blood quantum}}

A common source of definition for an individual being Indian is based on their blood quantum (often one-fourth) or documented Indian heritage. Almost two thirds of all Indian federally recognized Indian tribes in the United States require a certain blood quantum for membership.<ref>Garroutte (2003) p16</ref> Indian heritage is a requirement for membership in most American Indian Tribes.<ref>Peroff (1997) p487</ref> The [[Indian Reorganization Act]] of 1934 used three criteria: tribal membership, ancestral descent, and blood quantum (one half). This was very influential in using blood quantum to restrict the definition of Indian.<ref>Brownell (2001) p284</ref> The use of blood quantum is problematic as Indians [[interracial marriage|interracially marry]] at a higher rater than any other [[demography of the United States|United States ethnic or racial category]], which ultimately could lead to absorption into the rest of American society.<ref>Peroff (1997) p487 gives the rate of interracial marriage for Native Americans as 75%, whites as 5% and blacks as 8%</ref>

===Traditional===

[[Image:teepee-logcabin.jpg|thumb|Reservation life has often been a blend of the traditional and the contemporary. In 1877, this Lakota family living at South Dakota's Rose Bud Agency had both teepees and log cabins.]]

Traditional definitions of "Indianness" are also important. There is a sense of "peoplehood" which links Indianness to sacred traditions, places, and shared history as indigenous people.<ref>Peroff (1997) p487</ref> This definition transcends academic and legal terminology.<ref>Peroff (1997) p487</ref> Language is also seen as an important part of identity, and learning [[Indigenous languages of the Americas|Native languages]], especially for youth in a community, is an important part in tribal survival.<ref>Etheridge (2007)</ref>

Some Indian artists find traditional definitions especially important. [[Crow Nation|Crow]] poet [[Henry Real Bird]] offers his own definition, "An Indian is one who offers tobacco to the ground, feeds the water, and prays to the four winds in his own language," Pulitzer Prize winning [[Kiowa]] author [[N. Scott Momaday]] gives a definition that is less supernatural but still based in the traditions and experience of a person and their family, "An Indian is someone who thinks of themselves as an Indian. But that's not so easy to do and one has to earn the entitlement somehow. You have to have a certain experience of the world in order to formulate this idea. I consider myself an Indian; I've had the experience of an Indian. I know how my father saw the world, and his father before him."<ref>Bordewich (1996) p67</ref>

Since the inverse DFT is the same as the DFT, but with the sign of the exponent flipped and a 1/''n'' factor, any FFT algorithm can easily be adapted for it as well.

====Residence on tribal lands====

== The Cooley-Tukey algorithm ==

[[Image:Bia-map-indian-reservations-usa.png|left|thumb|300px|BIA map of reservations in the United States]]

Related to the remembrance and practice of traditions is the residence on tribal lands and [[Indian reservation]]s. Peroff (2002) emphasizes the role proximity to other Native Americans (and ultimately proximity to tribal lands) plays in ones identity as a Native American.<ref>Peroff (2002) uses complexity theory methods to model the maintenance of traditions and self-identity based on proximity</ref>

===Unknown===

The most common FFT is the '''Cooley-Tukey''' algorithm. In its most basic form, this method first computes the Fourier transforms of the even-indexed numbers ''x''₀, ''x''₂, ..., ''x''_''n''-2, and of the odd-indexed numbers ''x''₁, ''x''₃, ..., ''x''_''n''-1, and then combines those two results to produce the Fourier transform of the whole sequence. This idea can then be performed [[recursion|recursively]] to reduce the overall runtime to O(''n'' log ''n''). This simplified form assumes that ''n'' is a power of two; since the number of sample points ''n'' can usually be chosen freely by the application, this is often not an important restriction.

American Indians were perhaps clearly identifiable at the turn of the 20th century, but today the concept is contested. Malcolm Margolin, co-editor of News From Native California muses, "I don’t know what an Indian is... [but] Some people are clearly Indian, and some are clearly not."<ref>Peroff (1997) p489 quoting Fost (1991) p. 28</ref>

Further, it is difficult to know what might be meant by any Native American ''racial'' identity. Race is a disputed term, but is often said to be a social (or political) rather than biological construct. The issue of Native American racial identity is discussed in Russell (2002, p68), "American Indians have always had the theoretical option of removing themselves from a tribal community and becoming legally white. American law has made it easy for Indians to disappear because that disappearance has always been necessary to the '[[manifest destiny]]' that the United States span the continent that was, after all, occupied." Russell contrasts this with the reminder that Native Americans are "members of communities before members of a race"<ref>Russell (2002) p68 is quoting López (1994) p55</ref>.

We write ''n''<nowiki>'</nowiki> = ''n''/2 and denote the DFT of the even-indexed numbers ''x''<nowiki>'</nowiki>₀ = ''x''₀, ''x''<nowiki>'</nowiki>₁ = ''x''₂, ..., ''x''<nowiki>'</nowiki>_{''n''<nowiki>'</nowiki>-1} = ''x''_''n''-2 by ''f''₀', ..., ''f''&nbsp;<nowiki>'</nowiki>_{''n''<nowiki>'</nowiki>-1}; and the DFT of the odd-indexed numbers ''x''<nowiki>''</nowiki>₀ = ''x''₁, ''x''<nowiki>''</nowiki>₁ = ''x''₃, ..., ''x''<nowiki>''</nowiki>_{''n''<nowiki>'</nowiki>-1} = ''x''_''n''-1 by ''f''₀<nowiki>''</nowiki>, ..., ''f''&nbsp;<nowiki>''</nowiki>_{''n''<nowiki>'</nowiki>-1}. Then it follows:

Some social scientists relate this uncertainty to the theory of the constructed nature of identity. Many social scientists discuss the construction of identity. [[Benidect Anderson]]'s "[[Imagined Communities]]" are an example. However, some see construction of identity as being a part of how a group remembers its past, tells its stories, and interprets its [[myths]]. Thus [[cultural identity]] are made within the discourses of history and culture. Thus identity may not be a fact based in the essence of a person, but a positioning, based in politics and social situations.<ref>Hall (1997) p53</ref>

:<math> \begin{matrix}

===Construction by others===

f_j & = & \sum_{k=0}^{\frac{n}{2}-1} x_{2k} e^{-\frac{2\pi i}{n} j(2k)}

+ \sum_{k=0}^{\frac{n}{2}-1} x_{2k+1} e^{-\frac{2\pi i}{n} j(2k+1)} \\ \\

[[Image:Bismarck Indian School.jpg|thumb|left|Students at the Bismarck Indian School in the early twentieth century.]]

& = & \sum_{k=0}^{n'-1} x'_{k} e^{-\frac{2\pi i}{n'} jk}

European conceptions of "Indianness" are notable both for how they influence how American Indians see themselves and for how they have persisted as stereotypes which may negatively affect treatment of Indians. The [[noble savage]] stereotype is famous, but [[European colonization of the Americas|American colonists]] held other stereotypes as well. For example, some colonists imagined Indians as living in a state similar to their own ancestors, for example the [[Pict]]s, [[Gaul]]s, and [[Briton]]s before "[[Julius Caesar]] with his [[Roman legion]]s (or some other) had ... laid the ground to make us tame and civil"<ref>quoted from Robert Johnson, promoter for the fledgling Virginia Colony in Dyar (2003) p819</ref>.

+ e^{-\frac{2\pi i}{n}j} \sum_{k=0}^{n'-1} x''_k e^{-\frac{2\pi i}{n'} jk} \\ \\

In the nineteenth and twentieth century, particularly until [[John Collier (reformer)|John Collier]]'s tenure as Commissioner of Indian Affairs began in 1933, various policies of the United States federal and state governments have amounted to what some consider an attack on Indian cultural identity. These policies have included the banning of traditional religious ceremonies, forced cutting of Indian boys' hair, forced "conversion" to [[Christianity]] by withholding rations, forcing Indian children to go to boarding schools, boarding schools where the use of [[Native American languages]] was not permitted, freedom of speech restrictions, and restricted allowances of travel between reservations<ref>Russell (2002) p66-67</ref>. This was also true in the Southwest sections of the U.S. then under Spanish control (until [[the Treaty of Guadalupe-Hildago]] in 1848), where the majority (80%) of inhabitants were Indigenous<ref>Russell (2002) p67</ref>.

& = & \left\{ \begin{matrix}

f'_j + e^{-\frac{2\pi i}{n}j} f''_j & \mbox{if } j<n' \\ \\

===United States government definitions===

f'_{j-n'} - e^{-\frac{2\pi i}{n}(j-n')} f''_{j-n'} & \mbox{if } j \geq n' \end{matrix} \right.

{{main|Native American recognition in the United States}}

[[Image:IndianAct2.jpg|right|frame|President Coolidge stands with four [[Osage Nation|Osage Indians]] at a White House ceremony]]

Some authors have pointed to a connection between social identity of Native Americans and their political status as members of a tribe<ref>Ray (2007) p399</ref>. There are 561 [[List of Native American Tribal Entities|federally recognized tribal governments]] in the United States, and in the eyes of the U.S. federal government, these tribes possess the right to establish the legal requirements for membership<ref>This right was upheld by the US Supreme Court in Santa Clara Pueblo v. Martinez in 1978, which is discussed in Ray (2007) p403, see also {{cite web |url=http://usinfo.state.gov/eur/Archive/2005/Jan/28-691277.html |work=usinfo.state.gov |title=The U.S. Relationship To American Indian and Alaska Native Tribes |accessdate=February 08 |accessyear=2006}}.</ref>. In recent times, preference is given to "political" definitions where legislation has identified Indians based on their membership in federally recognized tribes<ref>Most often given is the two-part definition: an "Indian" is someone who is a member of an Indian tribe and an "Indian tribe" as any tribe, band, nation, or organized Indian community recognized by the United States</ref>. The government and many tribes prefer this definition because it allows the tribes to determine the meaning of "Indianness" in their own membership criteria. However, some still criticize this as the federal government's historic role in setting certain conditions on the nature of membership criteria means that this definition does not transcend federal government influence<ref>Brownell (2001) p299</ref>. Thus in some sense, one has greater claim to a Native American identity if one belongs to a federally recognized tribe, recognition that many who claim Indian identity do not have<ref>Nagel remarks that 1,878,285 people marked Native American as their ethnicity on the 1990 US Census, while the number of members of federally recognized tribes is much smaller, Nagel (1995) p948</ref>. Holly Reckord, an anthropologist who heads the BIA Branch of Acknowledgment and Recognition discusses the most common outcome for those who seek membership: "We check and find that they haven't a trace of Indian ancestry, yet they are still totally convinced that they are Indians. even if you have a trace of Indian blood, why do you want to select that for your identity, and not your Irish or Italian? It's not clear why, but at this point in time, a lot of people want to be Indian"<ref>Bordewich (1996) 66</ref>.

[[Image:Navajo blanket.jpg|thumb|right|A [[Navajo rug|Navajo blanket]] from the early 20th century.]]

Under the [[Arts and Crafts Act]] of 1990, having the status of a state recognized Indian tribe is discussd, as well as having tribal recognition as an "Indian artisan" independent of tribal membership. In certain circumstances, This allows people who identify as Indian to legally label their products as "Indian made," even when they aren't members of a federally recognized tribe<ref>Brownell (2001) p313</ref>.

Using federal laws to define "Indian" signals to some a continued government control over Indians, even as the government seeks to establish a sense of deference. Thus Indianness becomes a rigid legal term defined by the BIA, rather than an expression of tradition, history, and culture. Many groups which claim descendants from tribes that predate European contact not federally recognized. According to Rennard Strickland, an Indian Law scholar, the federal government uses the process of recognizing groups to "divide and conquer Indians: "the question of who is 'more' or 'most' Indian may draw people away from common concerns."<ref>Brownell (2001) p302</ref>

===Self-identification===

In some cases, one's opinion about ones self is sufficient to define one as Indian. One can often choose to identify as Indian without outside verification when filling out a census form, a college application, or writing a letter to the editor of a newspaper<ref>Peroff (1997) p487</ref>. A "self-identified Indian" is a person who may not satisfy the legal requirements which define a Native American according to the United States government but who understand and express their own identity as Native American.<ref>Garroutte (2003) p82</ref> However, many people who do not satisfy these requirements identify themselves as Native American - whether due to biology, culture, or some other reason. The [[United States census]] allows citizens to check any ethnicity without requirements of validation. Thus, the census allows individuals to self-identify as Indian, merely by checking the racial category, "Native American/Alaska Native," <ref>Brownell (2001) p276-277 notes that much of the $180 billion dollars a year in federal for the benefit of Indians are apportioned on the basis of this census population</ref>. In 1990, about 60 percent of the over 1.8 million identifying themselves in the census as American Indian were actually enrolled in a federally recognized tribe<ref>Thornton 1997, page 38</ref>. Using self identification allows both uniformity and includes many different ideas of "Indianness."<ref>Brownell (2001) p315</ref> It also avoids marginalizing the nearly half a million who receive no benefits because they are unenrolled members of a federally recognized tribe, or full members of tribes that have never been recognized or whose recognition was terminated by the government during programs in the 1950s and 1960s.<ref>Brownell (2001) p299</ref> Identity is in some way a personal issue; based on the way one feels about oneself and one's experiences. Horse (2001) describes five influences on self-identity as Indian:

*"The extent to which one is grounded in one’s Native American language and culture, one’s cultural identity"

*"The validity of one’s American Indian genealogy"

*"The extent to which one holds a traditional American Indian general philosophy or worldview (emphasizing balance and harmony and drawing on Indian spirituality)"

*"One’s self-concept as an American Indian"

*"One’s enrollment (or lack of it) in a tribe"<ref>Horse (2005) p65</ref>

[[University of Kansas]] Sociologist, Joane Nagel traces the tripling in the number of Americans reporting American Indian as their race in the U.S. Census from 1960 to 1990 (from 523,591 to 1,878,285) to federal Indian policy, American ethnic politics, and American Indian political Activism. Much of the population growth was due to "[[ethnic switching]]," where people who previously marked one group, later mark another. This is made possible by our increasing stress on social constructions role in determining ethnicity<ref>Nagel (1995) p948</ref>.

\end{matrix}

</math>

Some Native American groups embrace the idea of self-identification as a primal indicator of Indian identity, even more than cultural identity<ref>Garroutte (2003) p83</ref>. Garroutte identifies some practical problems with self-identification as a policy, quoting the struggles of Indian service providers who deal with many people who had ancestors, some steps removed, who were Indian. She quotes a social worker, "Hell, if all that was real, there are more Cherokees in the world than there are [[Chinese people|Chinese]]." The use of self identification in US censuses has changed since 2000 as now people are allowed to check multiple categories.<ref>Russell 149</ref>

A generalization of this trick (see below), including its recursive application, was already known around [[1805]] to [[Carl Friedrich Gauss]], who used it to interpolate the trajectories of the [[asteroid|asteroids]] [[2 Pallas|Pallas]] and [[3 Juno|Juno]], but his work was not widely recognized (being published only posthumously and in Latin); Gauss did not analyze the asymptotic complexity, however. Various limited forms were also rediscovered several times throughout the 19th and early 20th centuries. FFTs became popular after J. W. Cooley of [[International Business Machines|IBM]] and J. W. Tukey of [[Princeton University|Princeton]] published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer (including how to arrange for the output to be produced in the natural ordering).

Many Individuals seek broader definitions of Indian for their own reasons. In legislative hearings related to the Arts and Crafts Act of 1990, One Indian artist whose mother is not Indian, but whose father is [[Seneca]] and who was raised on the [[Seneca reservation]] said, "I do not question the rights of the tribes to set whatever criteria they want for enrollment eligibility; but in my view, that is the extent of their rights, to say who is an enrolled Seneca or [[Mohawk]] or [[Navajo]] or [[Cheyenne]] or any other tribe. Since there are mixed bloods with enrollment numbers and some of those with very small percentages of genetic Indian ancestry, I don't feel they have the right to say to those of us without enrollment numbers that we are not of Indian heritage, only that we are not enrolled... To say that I am not [Indian] and to prosecute me for telling people of my Indian heritage is to deny me some of my [[civil liberties]]...and constitutes racial discrimination."<ref>Brownell (2001) p314</ref>

This process is an example of the general technique of [[divide and conquer (computer science)|divide and conquer]] algorithms; in many traditional implementations, however, the explicit recursion is avoided, and instead one traverses the computational tree in [[breadth-first search|breadth-first]] fashion.

Native American Literature professor Becca Gercken-Hawkins writes about the trouble of recognition for those who do not look Indian; "I self-identify as Cherokee and [[Irish American]], and even though I do not look especially Indian with my dark curly hair and light skin, I easily meet my tribe's blood quantum standards. My family has been working for years to get the documentation that will allow us to be enrolled members of the [[Eastern Band of Cherokee Indians]]. Because of my appearance and my lack of enrollment status, I expect questions regarding my identity, but even so, I was surprised when a fellow graduate student advised me—in all seriousness—to straighten my hair and work on a tan before any interviews. Thinking she was joking, I asked if I should put a feather in my hair, and she replied with a straight face that a feather might be a bit much, but I should at least wear traditional Native jewelry."<ref>Gercken-Hawkins (2003) p200</ref>

The above re-expression of a size-''n'' DFT as two size-''n''/2 DFTs is sometimes called the '''Danielson-Lanczos''' lemma, since the identity was noted by those two authors in 1942 (influenced by [[Carl David Tolm&eacute; Runge|Runge's]] 1903 work). They applied their lemma in a "backwards" recursive fashion, repeatedly ''doubling'' the DFT size until the transform spectrum converged (although they apparently didn't realize the logarithmic asymptotic complexity they had achieved). The Danielson-Lanczos work predated widespread availability of computing machines and required hand calculation; they reported a computation time of 140 minutes for a size-64 DFT operating on real inputs (see below) to 3-5 significant digits. Cooley and Tukey's 1965 paper reported a running time of 0.02 minutes for a size-2048 complex DFT on an [[IBM 7094]] (probably in 36-bit [[floating point|single precision]], ~8 digits). Rescaling the time by ''n'' log ''n'', this corresponds roughly to a speedup factor of around 800,000. The more modern FFT library FFTW, on a 2GHz [[Pentium 4]] in 64-bit double precision (~16 digits), can compute a size-64 real-input DFT in 1&mu;s and a size-2048 complex DFT in 100&mu;s, speedups of about 8,000,000,000 and 10,000 over Danielson &amp; Lanczos and Cooley &amp; Tukey, respectively, not even including the considerable improvements in accuracy.

Cynthia Hunt, who self-identifies as a member of the non-federally recognized Lumbee tribe, says: "I feel as if I'm not a real Indian until I've got that BIA stamp of approval .... You're told all your life that you're Indian, but sometimes you want to be that kind of Indian that everybody else accepts as Indian."<ref>Brownell (2001) p275</ref>

(140 minutes for size 64 may sound like a long time, but it corresponds to an average of at most 16 seconds per floating-point operation, around 20% of which are multiplications...this is a fairly impressive rate for a human being to sustain for almost two and a half hours, especially when you consider the bookkeeping overhead.)

Cherokee/Choktaw author [[Louis Owens]] discusses his feelings about his status of not being a real Indian because he's not enrolled. "Because growing up in different times, I naively thought that Indian was something we were, not something we did or had or were required to prove on demand. Listening to my mother's stories about [[Oklahoma]], about brutally hard lives and dreams that cut across the fabric of every experience, I thought I was Indian."<ref>Eva Marie Garroutte opens with these lines in her book, ''Real Indians: identity and the survival of Native America'' (2003)</ref>

==== Variations on the Cooley-Tukey algorithm ====

Yet, self-identification is problematic on many levels. It is sometimes said, in fun, that the largest tribe in the United States may be the "Wantabes."<ref>Brownell (2001) p315</ref>

More generally, Cooley-Tukey algorithms recursively re-express a DFT of a composite size ''n'' = ''n''₁''n''₂ as ''n''₁ DFTs of size ''n''₂, followed by multiplication by complex [[roots of unity]] called '''twiddle factors''', followed by ''n''₂ DFTs of size ''n''₁. Typically, either ''n''₁ or ''n''₂ is a small factor, called the '''radix'''. If ''n''₁ is the radix, it is called a '''decimation in time''' (DIT) algorithm, whereas if ''n''₂ is the radix, it is '''decimation in frequency''' (DIF, also called the Sande-Tukey algorithm). The version presented above is a radix-2 DIT algorithm; in the final expression, the phase multiplying the odd transform is the twiddle factor, and the +/- combination (''butterfly'') of the even and odd transforms is a size-2 DFT. (The radix's small DFT is sometimes known as a '''butterfly''', so-called because of the shape of the data-flow diagram for the radix-2 case.)

==Historic struggles==

There are many other variations on the Cooley-Tukey algorithm. '''Mixed-radix''' implementations handle composite sizes with a variety of (typically small) factors in addition to two, usually (but not always) employing the O(''n''²) algorithm for the prime base cases of the recursion. '''Split-radix''' merges radices 2 and 4, exploiting the fact that the first transform of radix 2 requires no twiddle factor, in order to achieve a minimal operation count for power-of-two sizes. (On present-day computers, performance is determined more by [[cache]] and [[CPU pipeline]] considerations than by strict operation counts; well optimized FFT implementations often employ larger radices and/or hard-coded base-case transforms of significant size.) Another way of looking at the Cooley-Tukey algorithm is that it re-expresses a size ''n'' one-dimensional DFT as an ''n''₁ by ''n''₂ two-dimensional DFT (plus twiddles), where the output matrix is [[Transpose|transposed]]. The net result of all of these transpositions, for a radix-2 algorithm, corresponds to a bit reversal of the input (DIF) or output (DIT) indices. If, instead of using a small radix, one employs a radix of roughly &radic;''n'' and explicit input/output matrix transpositions, it is called a '''four-step''' algorithm (or ''six-step'', depending on the number of transpositions), initially proposed for memory locality (cache) optimization (Gentleman and Sande, 1966; also Bailey, 1990).

[[Florida State]] anthropologist J. Anthony Paredes considers the question of Indianness that may be asked about pre-ceramic peoples (what modern archaeologists call the [[Archaic period in the Americas|"Early" and "Middle Archaic" period]]), pre-maize burial mound cultures, etc. Paredes asks, "Would any <nowiki>[</nowiki>[[Mississippian culture|Mississippian]] high priest<nowiki>]</nowiki> have been any less awed than ourselves to come upon a so-called [[Paleo-Indian]] hunter hurling a spear at a woolly [[mastodon]]?" His question reflects the point that indigenous cultures are themselves the products of millennia of history and change.<ref>Paredes (1995) p346</ref>

The question of "Indianness" was different in colonial times. Integration into Indian tribes was not difficult, as Indians typically accepted persons based not on ethnic or racial characteristics, but on learnable and acquirable designators such as "language, culturally appropriate behavior, social affiliation, and loyalty."<ref>Dyar (2003) p823</ref> Non-Indian captives were often adopted into society, including, famously, Mary Jemison. As a side note, the "[[Running the gauntlet|gauntlet]]" was a ceremony that was often misunderstood as a form of torture, but was seen as a way for the captives to leave their European society and become a tribal member.<ref>Dyar (2003) p823</ref>

The general Cooley-Tukey factorization rewrites the indices ''j'' and ''k'' as ''j'' = ''n''₂ ''j''₁ + ''j''₂ and ''k'' = ''n''₁ ''k''₂ + ''k''₁, respectively, where the indices ''j''_a and ''k''_a run from 0..''n''_a-1 (for ''a'' of 1 or 2). That is, it re-indexes the input (''k'') and output (''j'') as ''n''₁ by ''n''₂ two-dimensional arrays in column-major and row-major order, respectively. When this reindexing is substituted into

the DFT formula for ''jk'', the ''n''₂''j''₁''n''₁''k''₂ cross term vanishes (its exponential is unity), and the remaining terms give

[[Image:JohnRossC.jpg|thumb|left|Cherokee chief John Ross]]

:<math>f_{n_2 j_1 + j_2} =

But since the mid 19th century, things have become more controversial. In the early 1860s, Novelist [[John Rollin Ridge]] led a group of delegates to [[Washington D.C.]] in an attempt to gain federal recognition for a "Southern Cherokee Nation" which was a faction that was opposed to the leadership of rival faction leader and Cherokee Chief [[John Ross (Cherokee chief)|John Ross]].<ref>Christiensen 1992</ref> In the 1920s, a famous case was set to investigate the true ethnic identity of a woman known as "[[Princess Chinquilla]]" and her associate [[Red Fox James]] (aka Skiuhushu).<ref>Carpenter (2005) p139</ref>

\sum_{k_1=0}^{n_1-1}

\left[ e^{-\frac{2\pi i}{n} j_2 k_1 } \right]

\left( \sum_{k_2=0}^{n_2-1} x_{n_1 k_2 + k_1}

e^{-\frac{2\pi i}{n_2} j_2 k_2 } \right)

e^{-\frac{2\pi i}{n_1} j_1 k_1 }

Chinquilla was a New York woman who claimed to have been separated from her Cheyenne parents at birth. She and James created a fraternal club which was to counter existing groups "founded by white people to help the red race" in that it was founded by Indians. The club's opening received much praise for supporting this purpose, and was seen as very authentic; it involved a Council Fire, the peace pipe, and speeches by [[Robert Ely]], [[White Horse Eagle]], and [[American Indian Defense Association]] President Haven Emerson. In the 1920s, fraternal clubs were fairly common in New York, and titles such as "princess" and "chief" were bestowed by the club to Natives and non-Natives.<ref>Carpenter (2005) p143</ref> This allowed non-Natives to "try on" Indian identities. In 1911, the [[Society of American Indians]] was founded by [[Arthur C. Parker]], [[Carlos Montezuma]], and others as the first national association founded and run primarily by Native Americans. The group campaigned for full citizenship for Indians, and other reforms, goals similar to other groups and fraternal clubs, which led to blurred distinctions between the different groups and their members.<ref>Carpenter (2005) p141</ref> With different groups and people of different ethnicities involved in parallel and often competing groups, accusations that one was not a real Indian was a painful accusation for those involved. In 1918, [[Arapaho]] Cleaver Warden testified in hearings related to Indian religious ceremonies, "We only ask a fair and impartial trial by reasonable white people, not half breeds who do not know a bit of their ancestors or kindred. A true Indian is one who helps for a race and not that secretary of the [[Society of American Indians]]."

</math>

Just as the struggle for recognition is not new, Indian entrepreneurship based on that recognition is not new. An example is a stipulation of the [[Creek Treaty of 1805]] which gave [[Creeks]] the exclusive right to operate certain ferries and "houses of entertainment" along a federal road from [[Ocmulgee, Georgia]] to [[Mobile, Alabama]], and which went over parts of Creek Nation land purchased as an easement.<ref>Paredes (1995) p357</ref>

where the inner sum is a DFT of size ''n''₂, the outer sum is a DFT of size ''n''₁, and the [...] bracketed term is the twiddle factor.

==Unity and nationalism==

The 1965 Cooley-Tukey paper noted that one can employ an arbitrary radix ''r'' (as well as mixed radices), but failed to realize that the radix butterfly is itself a DFT that can use FFT algorithms. Hence, they reckoned the complexity to be O(''r''² ''n''/''r'' log_''r''''n''), and erroneously concluded that the optimal radix was 3 (the closest integer to ''[[E (mathematical constant)|e]]''). Gauss also derived the algorithm for arbitrary radices, and gave explicit examples of both radix-3 and radix-6 steps.

[[Image:TecumsehColor.JPG|right|thumb|[[1848]] drawing of Tecumseh was based on a sketch done from life in [[1808]].]]

== Other FFT algorithms ==

The issue of Indianness had somewhat expanded meaning in the 1960s with Indian nationalist movements such as the [[American Indian Movement]]. The American Indian Movement unified nationalist identity was in contrast to the "brotherhood of tribes" nationalism of groups like the [[National Indian Youth Council]] and the [[National Congress of American Indians]].<ref>Bonney (1977) p210</ref> This unified Indian identity has been cited to the teachings of 19th century [[Shawnee]] leader [[Tecumseh]] to unify all Indians against "white oppression."<ref>Particularly cited is Tecumseh's concern with the alienation of Indian lands and his 1812 statement about Indian unity as discussed in Bonney (1977) p229</ref> The movements of the 1960s changed dramatically how Indians see their identity, both as separate from Anglos, as a member of a tribe, and as a member of a unified category encompassing all Indians.<ref>Schulz (1998)</ref>

==Examples==

There are other FFT algorithms distinct from Cooley-Tukey. For [[relatively prime]] ''n''₁ and ''n''₂, one can use the [[Prime-factor FFT algorithm|Prime-Factor]] (Good-Thomas) algorithm (PFA), based on the [[Chinese Remainder Theorem]], to factorize the DFT similarly to Cooley-Tukey but without the twiddle factors. The [[Rader-Brenner FFT algorithm|Rader-Brenner algorithm]] is a Cooley-Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability. Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and [[QFT algoritm|QFT]] algorithms. (The Rader-Brenner and QFT algorithms were proposed for power-of-two sizes, but it is possible that they could be adapted to general composite ''n''. Bruun's algorithm applies to arbitrary even composite sizes.) [[Bruun's FFT algorithm|Bruun's algorithm]], in particular, is based on interpreting the FFT as a recursive factorization of the [[polynomial]] ''z''^''n'' - 1, here into real-coefficient polynomials of the form ''z^m'' - 1 and

Different tribes have unique cultures, histories, and situations that have made particular the question of identity in each tribe. Tribal membership may be based on descent, blood quantum, and/or reservation habitation.

''z^2m'' + ''az^m'' + 1. Another polynomial viewpoint is exploited by the [[Winograd FFT algorithm|Winograd]] algorithm, which factorizes ''zⁿ'' - 1 into [[Cyclotomic polynomial|cyclotomic polynomials]]&mdash;these often have coefficients of 1, 0, or -1, and therefore require few (if any) multiplications, so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors. In particular, Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of ''prime'' sizes. [[Rader's FFT algorithm|Rader's algorithm]], exploiting the existence of a [[generating set of a group|generator]] for the multiplicative [[group (mathematics)|group]] modulo prime ''n'', expresses a DFT of prime size ''n'' as a cyclic [[convolution]] of (composite) size ''n'' - 1, which can then be computed by a pair of ordinary FFTs via the [[convolution theorem]] (although Winograd uses other convolution methods). Another prime-size FFT is due to L. I. Bluestein, and is sometimes called the [[Bluestein's FFT algorithm|chirp-z algorithm]]; it also re-expresses a DFT as a convolution, but this time of the ''same'' size (which can be zero-padded to a power of two and evaluated by radix-2 Cooley-Tukey FFTs, for example), via the identity ''jk'' = -(''j''-''k'')²/2 + ''j''²/2 + ''k''²/2.

===Cherokee===

''See also: [[Prime-factor FFT algorithm]], [[Bruun's FFT algorithm]], [[Rader's FFT algorithm]], [[Bluestein's FFT algorithm]].''

Historically, race was not a factor in the acceptance of individuals into [[Cherokee society]], since historically, the [[Cherokee people]] viewed their self-identity as a political rather than racial distinction.<ref> AIRFA Federal Precedence Applied in State Court http://www.nativeamericanchurch.net/stott.html</ref> Going far back into antiquity based upon existing social and historical evidence as well as oral traditions among the [[Cherokee]] themselves, the Cherokee society was best described as an Indian Republic. Theda Perdue (2000) recounts a story from "before the American Revolution" where a black slave named Molly is accepted as a Cherokee as a "replacement" for a woman who was beaten to death by her white husband. According to Cherokee tradition, vengeance for the woman's death was required for her soul to find peace, and the husband was able to prevent his own execution by fleeing to the town of [[Chota]] (where according to Cherokee Law he was safe) and purchasing Molly as an exchange. When the wives family accepted Molly, later known as "Chickaw," she became a part of their clan (the [[Deer Clan]]), and thus Cherokee.<ref>Perdue (2000)</ref>

== FFT algorithms specialized for real and/or symmetric data ==

Inheritance was largely matrilineal, and kinship and clan membership was of primary importance until around 1810, when the seven [[Cherokee clans]] began the abolition of blood vengeance by giving the sacred duty to the new Cherokee National government. Clans formally relinquished judicial responsibilities by the 1820s when the Cherokee Supreme Court was established. When in 1825, the National Council extended citizenship to biracial children of Cherokee men, the matrilineal definition of clans was broken and clan membership no longer defined Cherokee citizenship. These ideas were largely incorporated into the 1827 Cherokee constitution.<ref>Perdue (2000) p564</ref> The constitution did, state that "No person who is of negro or mulatlo [sic]parentage, either by the father or mother side, shall be eligible to hold any office of profit, honor or trust under this Government," with an exception for, "negroes and descendants of white and Indian men by negro women who may have been set free."<ref>Perdue (2000) p564-565</ref> Although by this time, some Cherokee considered clans to be anachronistic, this feeling may have been more widely held among the elite than the general population.<ref>Perdue (2000) p566</ref> Thus even in the initial constitution, the Cherokee reserved the right to define who was and was not Cherokee as a political rather than racial distinction. Novelist [[John Rollin Ridge]] led a group of delegates to Washington D.C. as early as the 1860s in an attempt to gain federal recognition for a "Southern Cherokee Nation" which was a faction that was opposed to the leadership of rival faction leader and Cherokee Chief [[John Ross (Cherokee chief)|John Ross]].<ref>Christiensen 1992</ref>

In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry

:<math>f_{n-j} = f_j^*,</math>

and efficient FFT algorithms have been designed for this situation. One approach consists of taking the ordinary Cooley-Tukey algorithm and removing the redundant parts of the computation, saving roughly a factor of two in time and memory. Alternatively, it is possible to express an ''even''-length real-input DFT as a complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real data), followed by O(''n'') post-processing operations.

===Navajo===

It was once believed that real-input DFTs could be more efficiently computed by means of the [[Discrete Hartley transform]] (DHT), but this was subsequently disproved: a specialized real-input DFT algorithm (FFT) can typically be found that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs.

Most of the 158,633 [[Navajo]]s enumerated in the 1980 census and the 219,198 Navajos enumerated in the 1990 census were enrolled in the Navajo Nation, which is the nation with the largest number of enrolled citizens. It is notable that their is such a small number of people who identify as Navajo who are not registered.<ref>Thornton 2004</ref>

===Lumbee===

There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can gain another factor of ~2 in time/space and the DFT becomes the discrete cosine/sine transform(s) ([[discrete cosine transform|DCT]]/[[discrete sine transform|DST]]). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed via FFTs of real data combined with O(''n'') pre/post processing.

When the [[Lumbee]] of [[North Carolina]] petitioned for recognition in 1974, many federally recognized tribes adamantly opposed them. These tribes made no secret of their fear that passage of the legislation would dilute services to historically recognized tribes.<ref>Brownell (2001) p304</ref> The Lumbee were at one point known by the state as the Cherokee tribe of Robeson County and applied for federal benefits under that name in the early 20th century.<ref>Barrett (2007)</ref> The [[Eastern Band of Cherokee Indians]] has been at the forefront of the opposition of the Lumbee. It is sometimes noted that if granted full federal recognition, the designation would bring tens of millions of dollars in federal benefits, and also the chance to open a casino along Interstate 95 (which would compete with a nearby Eastern Cherokee Nation casino).<ref>Barrett (2007)</ref>

==Footnotes==

== Accuracy and approximations ==

{{reflist|3}}

==Bibliography==

All of the FFT algorithms discussed so far compute the DFT exactly (in exact arithmetic, i.e. neglecting [[floating-point]] errors). A few "FFT" algorithms have been proposed, however, that compute the DFT ''approximately'', with an error that can be made arbitrarily small at the expense of increased computations. Such algorithms trade the approximation error for increased speed or other properties. For example, an approximate FFT algorithm by Edelman et al. (1999) achieves lower communication requirements for [[parallel computing]] with the help of a fast-multipole method. A [[wavelet]]-based approximate FFT by Guo and Burrus (1996) takes sparse inputs/outputs (time/frequency localization) into account more efficiently than is possible with an exact FFT. Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al. (1995). Only the Edelman algorithm works equally well for sparse and non-sparse data, however, since it is based on the compressibility (rank deficiency) of the Fourier matrix itself rather than the compressibility (sparsity) of the data.

<div class="references-small">

* Barrett, Barbara. (2007) "Two N.C. tribes fight for identity; Delegation split on Lumbee recognition" The News & Observer (Raleigh, North Carolina) April 19, 2007

* Bonney, Rachel A. (1977) "The Role of AIM Leaders in Indian Nationalism." American Indian Quarterly, Vol. 3, No. 3. (Autumn, 1977), pp. 209-224.

Even the "exact" FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small; most FFT algorithms, e.g. Cooley-Tukey, have excellent numerical properties. The upper bound on the relative error for the Cooley-Tukey algorithm is O(&epsilon; log ''n''), compared to O(&epsilon; ''n''^3/2) for the naive DFT formula (Gentleman and Sande, 1966), where &epsilon; is the machine floating-point relative precision. In fact, the average errors are much better than these upper bounds, being only O(&epsilon &radic;log ''n'') for Cooley-Tukey and O(&epsilon; &radic;''n'') for the naive DFT (Schatzman, 1996). These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT (i.e. the [[trigonometric function]] values), and it is not unusual for incautious FFT implementations to have much worse accuracy, e.g. if they use inaccurate [[generating trigonometric tables|trigonometric recurrence]] formulas. Some FFTs other than Cooley-Tukey, such as the Rader-Brenner algorithm, are intrinsically less stable.

----

* Bordewich, Fergus M. (1996) ''Killing the White Man's Indian: Reinventing Native Americans at the End of the Twentieth Century''. First Anchor Books, ISBN 0385420366

==References==

* James W. Cooley and John W. Tukey, "An algorithm for the machine calculation of complex Fourier series," ''Math. Comput.'' '''19''', 297&ndash;301 (1965).

* Brownell, Margo S. (2001) "Who is an Indian? Searching for an Answer to the Question at the Core

* Carl Friedrich Gauss, "Nachlass: Theoria interpolationis methodo nova tractata," ''Werke'' band '''3''', 265&ndash;327 (K&ouml;nigliche Gesellschaft der Wissenschaften, G&ouml;ttingen, 1866). See also M. T. Heideman, D. H. Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier transform," ''IEEE ASSP Magazine'' '''1''' (4), 14&ndash;21 (1984).

of Federal Indian Law." Michigan Journal of Law Reform 34(1-2):275-320.

* G. C. Danielson and C. Lanczos, "Some improvements in practical Fourier analysis and their application to X-ray scattering from liquids," ''J. Franklin Inst.'' '''233''', 365&ndash;380 and 435&ndash;452 (1942).

* P. Duhamel and M. Vetterli, "Fast Fourier transforms: a tutorial review and a state of the art," ''Signal Processing'' '''19''', 259&ndash;299 (1990).

* Carpenter, Cari. (2005) "Detecting Indianness: Gertrude Bonnin's Investigation of Native American Identity." Wicazo Sa Review - Volume 20, Number 1, Spring 2005, pp. 139-159

* W. M. Gentleman and G. Sande, "Fast Fourier transforms&mdash;for fun and profit," ''Proc. AFIPS'' '''29''', 563&ndash;578 (1966).

* David H. Bailey, "FFTs in External or Hierarchical Memory," ''J. Supercomputing'' '''4''' (1), 23&ndash;35 (1990).

* Carter, Kent. (1988) "Wantabes and Outalucks: Searching for Indian Ancestors in Federal Records". Chronicles of Oklahoma 66 (Spring 1988): 99-104 (Accessed June 30, 2007 [http://www.archives.gov/genealogy/heritage/native-american/ancestor-search.html here])

* H. Guo, G. A. Sitton, and C. S. Burrus, "The Quick Discrete Fourier Transform," ''Proc. IEEE Conf. Acoust. Speech and Sig. Processing (ICASSP)'' '''3''', 445&ndash;448 (1994).

* H. V. Sorensen, D. L. Jones, M. T. Heideman, and C. S. Burrus, "Real-valued fast Fourier transform algorithms," ''IEEE Trans. Acoust. Speech Sig. Processing'' '''ASSP-35''', 849&ndash;863 (1987).

* Cohen, F. (1982) ''Handbook of Federal Indian law''. Charlottesville: Bobbs-Merrill, ISBN 0872154130

* A. Edelman, P. McCorquodale, and S. Toledo, "The future fast Fourier transform?" ''SIAM J. Sci. Computing'' '''20''', 1094&ndash;1114 (1999).

* H. Guo and C. S. Burrus, "Fast approximate Fourier transform via wavelets transform," ''Proc. SPIE Intl. Soc. Opt. Eng.'' '''2825''', 250&ndash;259 (1996).

* Dyar, Jennifer. (2003) "Fatal Attraction: The White Obsession with Indianness." The Historian. June 2003. Vol 65 Issue 4 pages 817–836

* O. V. Shentov, S. K. Mitra, U. Heute, and A. N. Hossen, "Subband DFT. I. Definition, interpretations and extensions," ''Signal Processing'' '''41''' (3), 261&ndash;277 (1995).

* James C. Schatzman, "Accuracy of the discrete Fourier transform and the fast Fourier transform," ''SIAM J. Sci. Comput.'' '''17''' (5), 1150&ndash;1166 (1996).

* Etheridge, Tiara. (2007) "Displacement, loss still blur American Indian identities" April 25, 2007 Wednesday, Oklahoma Daily, University of Oklahoma

* Matteo Frigo and Steven G. Johnson: ''FFTW'', http://www.fftw.org/. A free ([[GPL]]) C library for computing discrete Fourier transforms in one or more dimensions, of arbitrary size, using Fast Fourier Transforms. Also M. Frigo and S. G. Johnson, "FFTW: An adaptive software architecture for the FFT," ''Proc. ICASSP'' '''3''', 1381&ndash;1384 (1998).

* Field, W. Les (with the Muwekma Ohlone Tribe). (2003) "Unacknowledged Tribes, Dangerous Knowledge, The Muwekma Ohlone and How Indian Identities Are 'Known.'" Wicazo Sa Review 18.2 pages 79-94

* Garroutte, Eva Marie. (2003) ''Real Indians: identity and the survival of Native America''. University of California Press, ISBN 0520229770

* Gercken-Hawkins, Becca (2003) "'Maybe you only look white:' Ethnic Authority and Indian Authenticity in Academia." The American Indian Quarterly 27.1&2 pages 200-202

* Hall, Stuart. (1997) "The work of representation." In ''Representation: Cultural Representations and Signifying Practices'', ed. Stuart Hall, 15-75. London: Sage Publications, ISBN 0761954325

* Horse, Perry G. (2005) "Native American identity" New Directions for Student Services. Volume 2005, Issue 109 , Pages 61 - 68

* Lawrence, Bonita. (2003) "Gender, Race, and the Regulation of Native Identity in Canada and the United States: An Overview" Hypatia 18.2 pages 3-31

* Morello, Carol. (2001) "Native American Roots, Once Hidden, Now Embraced". Washington Post, April 7, 2001

* Nagel, J. (1995) "Politics and the Resurgence of American Indian Ethnic Identity," American

Sociological Review 60: 947–965.

* Paredes, J. Anthony. (1995) "Paradoxes of Modernism and Indianness in the Southeast." American Indian Quarterly, Vol. 19, No. 3. (Summer, 1995), pp. 341-360.

* Peroff, Nicholas C. (1997) "Indian Identity" The Social Science Journal, Volume 34, Number 4, pages 485-494.

* Peroff, N.C. (2002) "Who is an American Indian?" Social Science Journal. Volume 39, Number 3, pages 349

* Pierpoint, Mary. (2000) "Unrecognized Cherokee claims cause problems for nation". Indian Country Today. August 16, 2000 (Accessed May 16, 2007 [http://www.indiancountry.com/content.cfm?id=703 here])

* Porter, F.W. III (ed.) (1983). "Nonrecognized American Indian tribes: An historical and legal perspective." Occasional Paper Series No. 7. Chicago, IL: D’Arcy McNickle Center for the History of the American Indian, The Newberry Library.

* Ray, S. Allan. ''A Race or a Nation? Cherokee National Identity and the Status of Freedmen's Descendents''. Michigan Journal of Race and Law. Vol. 12, 2007 (Accessible as a working paper as of July 12, 2007 [http://law.bepress.com/cgi/viewcontent.cgi?article=7231&context=expresso here]).

* Russell, Steve. (2004) "Review of Real Indians: Identity and the Survival of Native America" PoLAR: Political and Legal Anthropology Review. May 2004, Vol. 27, No. 1, pp. 147-153

* Russell, Steve (2002). "Apples are the Color of Blood". Critical Sociology Vol. 28, 1, 2002, p. 65

* Schulz, Amy J. (1998) "Navajo Women and the Politics of Identity." Social Problems, Vol. 45, No. 3. (Aug., 1998), pp. 336-355.

* Sturm, Circe. (1998) "Blood Politics, Racial Classification, and Cherokee National Identity: The Trials and Tribulations of the Cherokee Freedmen". American Indian Quarterly, Winter/Spring 1998, Vol 22. No 1&2 pgs 230-258

* Thornton, Russell. (1992) ''The Cherokees: A Population History''. University of Nebraska Press, ISBN 0803294107

* Thornton, Russell. (1997) "Tribal Membership Requirements and the Demography of 'Old' and 'New' Native Americans". Population Research and Policy Review Vol. 16, Issue 1, p. 33 ISBN 0803244169

* "Census 2000 PHC-T-18. American Indian and Alaska Native Tribes in the United States: 2000" United States Census Bureau, Census 2000, Special Tabulation (Accessed May 27, 2007 [http://www.census.gov/population/cen2000/phc-t18/tab001.pdf here])

</div>