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Split-radix FFT
The split-radix FFT is a fast Fourier transform (FFT) algorithm for computing the discrete Fourier transform (DFT), and was first described in an initially little-appreciated paper by R. Yavne (1968) and subsequently rediscovered simultaneously by various authors in 1984. (The name "split radix" was coined by two of these reinventors, P. Duhamel and H. Hollmann.) In particular, split radix is a variant of the Cooley–Tukey FFT algorithm that uses a blend of radices 2 and 4: it recursively expresses a DFT of length ''N'' in terms of one smaller DFT of length ''N''/2 and two smaller DFTs of length ''N''/4. The split-radix FFT, along with its variations, long had the distinction of achieving the lowest published arithmetic operation count (total exact number of required real additions and multiplications) to compute a DFT of power-of-two sizes ''N''. The arithmetic count of the original split-radix algorithm was improved upon in 2004 (with the initial gains made in unpublished wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Fourier Transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O\left(N^2\right), which arises if one simply applies the definition of DFT, to O(N \log N), where N is the data size. The difference in speed can be enormous, especially for long data sets where ''N'' may be in the thousands or millions. In the presence of round-off error, many FFT algorithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
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. The DFT is the most important discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Duhamel
Pierre is a masculine given name. It is a French form of the name Peter. Pierre originally meant "rock" or "stone" in French (derived from the Greek word πέτρος (''petros'') meaning "stone, rock", via Latin "petra"). It is a translation of Aramaic כיפא (''Kefa),'' the nickname Jesus gave to apostle Simon Bar-Jona, referred in English as Saint Peter. Pierre is also found as a surname. People with the given name * Abbé Pierre, Henri Marie Joseph Grouès (1912–2007), French Catholic priest who founded the Emmaus Movement * Monsieur Pierre, Pierre Jean Philippe Zurcher-Margolle (c. 1890–1963), French ballroom dancer and dance teacher * Pierre (footballer), Lucas Pierre Santos Oliveira (born 1982), Brazilian footballer * Pierre, Baron of Beauvau (c. 1380–1453) * Pierre, Duke of Penthièvre (1845–1919) * Pierre, marquis de Fayet (died 1737), French naval commander and Governor General of Saint-Domingue * Prince Pierre, Duke of Valentinois (1895–1964), father ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henk D
Henk is a Dutch male given name, originally a short form of Hendrik. It influenced "Hank" which is used in English-speaking countries (mainly in the US) as a form of "Henry". People named "Henk" include: Academics *Henk Aertsen (born 1943), Dutch Anglo-Saxon linguist *Henk Barendregt (born 1947), Dutch logician *Henk Jaap Beentje (born 1951), Dutch botanist *Henk Blezer (born 1961), Dutch Tibetologist, Indologist, and scholar of Buddhist studies *Henk Bodewitz (born 1939), Dutch Sanskrit scholar *Henk J. M. Bos (born 1940), Dutch historian of mathematics *Henk Braakhuis (born 1939), Dutch historian of philosophy *Henk Buck (born 1930), Dutch organic chemist *Henk van Dongen (1936–2011), Dutch organizational theorist and policy advisor *Henk Dorgelo (1894–1961), Dutch physicist and academic *Henk van der Flier (born 1945), Dutch psychologist *Henk A. M. J. ten Have (born 1951), Dutch medical ethicist *Henk van de Hulst (1918–2000), Dutch astronomer and mathematician *Henk Lom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cooley–Tukey FFT Algorithm
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N_1N_2 in terms of ''N''1 smaller DFTs of sizes ''N''2, recursively, to reduce the computation time to O(''N'' log ''N'') for highly composite ''N'' (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below. Because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley–Tukey, or the prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors. The algorithm, along with its recursive application, was invented by Carl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's ances ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Of Two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system. Computer science Two to the exponent of , written as , is the number of ways the bits in a binary word of length can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 () to () inclusively. Corresponding signed integer values can be positive, negative and zero; see signed n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as Computer program, programs. These programs enable computers to perform a wide range of tasks. A computer system is a nominally complete computer that includes the Computer hardware, hardware, operating system (main software), and peripheral equipment needed and used for full operation. This term may also refer to a group of computers that are linked and function together, such as a computer network or computer cluster. A broad range of Programmable logic controller, industrial and Consumer electronics, consumer products use computers as control systems. Simple special-purpose devices like microwave ovens and remote controls are included, as are factory devices like industrial robots and computer-aided design, as well as general-purpose devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field (mathematics), field. If the characteristic of a field, characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, converse (logic), conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Numbers
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulo Operation
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is the remainder of the Euclidean division of by , where is the dividend and is the divisor. For example, the expression "5 mod 2" would evaluate to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. Although typically performed with and both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of is 0 to inclusive ( mod 1 is always 0; is undefined, possibly resulting in a division by zero error in some programming languages). See Modular arithmetic for an older and related c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
