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Butterfly (FFT Algorithm)
In the context of fast Fourier transform algorithms, a butterfly is a portion of the computation that combines the results of smaller discrete Fourier transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into subtransforms). The name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as described below.Alan V. Oppenheim, Ronald W. Schafer, and John R. Buck, ''Discrete-Time Signal Processing'', 2nd edition (Upper Saddle River, NJ: Prentice Hall, 1989) The earliest occurrence in print of the term is thought to be in a 1969 MIT technical report. The same structure can also be found in the Viterbi algorithm, used for finding the most likely sequence of hidden states. Most commonly, the term "butterfly" appears in the context of the Cooley–Tukey FFT algorithm, which recursively breaks down a DFT of composite size ''n'' = ''rm'' into ''r'' smaller transforms of size ''m'' where ''r'' is the "radix" of the transform. Thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twiddle Factor
A twiddle factor, in fast Fourier transform (FFT) algorithms, is any of the trigonometric constant coefficients that are multiplied by the data in the course of the algorithm. This term was apparently coined by Gentleman & Sande in 1966, and has since become widespread in thousands of papers of the FFT literature. More specifically, "twiddle factors" originally referred to the root-of-unity complex multiplicative constants in the butterfly operations of the Cooley–Tukey FFT algorithm, used to recursively combine smaller discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ...s. This remains the term's most common meaning, but it may also be used for any data-independent multiplicative constant in an FFT. The prime-factor FFT algorithm is one unusual ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal-flow Graph
A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized Flow graph (mathematics), flow graph, a directed graph in which nodes represent system variables, and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes. Thus, signal-flow graph theory builds on that of directed graphs (also called Digraph (mathematics), digraphs), which includes as well that of Orientation (graph theory)#Oriented graphs, oriented graphs. This mathematical theory of digraphs exists, of course, quite apart from its applications. i SFGs are most commonly used to represent signal flow in a physical systems, physical system and its controller(s), forming a cyber-physical system. Among their other uses are the representation of signal flow in various electronic networks and amplifiers, digital filters, state-variable filters and some other types of analog f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zassenhaus Lemma
In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Zassenhaus, is a technical result on the lattice of subgroups of a group (mathematics), group or the lattice of submodules of a module (mathematics), module, or more generally for any modular lattice. :Lemma. Suppose G is a group with subgroups A and C. Suppose B\triangleleft A and D\triangleleft C are normal subgroups. Then there is an group isomorphism, isomorphism of quotient groups: ::\frac \cong \frac. This can be generalized to the case of a group with operators (G, \Omega) with stable subgroups A and C, the above statement being the case of \Omega=G group action, acting on itself by conjugation (group theory), conjugation. Zassenhaus mathematical proof, proved this lemma specifically to give the most direct proof of the Schreier refinement theorem. The 'butterfly' becomes apparent when trying to draw the Hasse diagram of the various groups involved. Zassenhaus' lemma for groups can be derived from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Diagram
Mathematical diagrams, such as charts and graphs, are mainly designed to convey mathematical relationships—for example, comparisons over time. Specific types of mathematical diagrams Argand diagram A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram The complex plane is sometimes called the ''Argand plane'' because it is used in ''Argand diagrams''. These are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates — the magnitude or ''modulus'' of the product is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterfly
Butterflies are winged insects from the lepidopteran superfamily Papilionoidea, characterized by large, often brightly coloured wings that often fold together when at rest, and a conspicuous, fluttering flight. The oldest butterfly fossils have been dated to the Paleocene, about 56 million years ago, though molecular evidence suggests that they likely originated in the Cretaceous. Butterflies have a four-stage life cycle, and like other holometabolous insects they undergo complete metamorphosis. Winged adults lay eggs on the food plant on which their larvae, known as caterpillars, will feed. The caterpillars grow, sometimes very rapidly, and when fully developed, pupate in a chrysalis. When metamorphosis is complete, the pupal skin splits, the adult insect climbs out, expands its wings to dry, and flies off. Some butterflies, especially in the tropics, have several generations in a year, while others have a single generation, and a few in cold locations may take s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity
In mathematics, a root of unity 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. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. 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 nu ... [...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). A Fourier transform 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 Matrix decomposition, factorizing the DFT matrix into a product of Sparse matrix, sparse (mostly zero) factors. As a result, it manages to reduce the Computational complexity theory, complexity of computing the DFT from O(n^2), which arises if one simply applies the definition of DFT, to O(n \log n), where is the data size. The difference in speed can be enormous, especially for long data sets where may be in the thousands or millions. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime number, prime, or the Unit (ring theory), unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
