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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 discrete-time Fourier transform (DTFT), which is a complex number, 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 (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex number, complex Sine wave, 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 fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Discrete-time Fourier Transform
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. In simpler terms, when you take the DTFT of regularly-spaced samples of a continuous signal, you get repeating (and possibly overlapping) copies of the signal's frequency spectrum, spaced at intervals corresponding to the sampling frequency. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Image Processing
An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a projection on a surface, activation of electronic signals, or digital displays; they can also be reproduced through mechanical means, such as photography, printmaking, or photocopying. Images can also be animated through digital or physical processes. In the context of signal processing, an image is a distributed amplitude of color(s). In optics, the term ''image'' (or ''optical image'') refers specifically to the reproduction of an object formed by light waves coming from the object. A ''volatile image'' exists or is perceived only for a short period. This may be a reflection of an object by a mirror, a projection of a camera obscura, or a scene displayed on a cathode-ray tube. A ''fixed image'', also called a hard copy, is one that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dimension (vector Space)
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is wiktionary:finite, finite, and if its dimension is infinity, infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as [V : F], read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any Field (mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Finite Fourier Transform (other)
__NOTOC__ In mathematics the finite Fourier transform may refer to either *another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., F.J.Harris (pp. 52–53) describes the ''finite Fourier transform'' as a "continuous periodic function" and the discrete Fourier transform (DFT) as "a set of samples of the finite Fourier transform". In actual implementation, that is not two separate steps; the DFT replaces the DTFT. So J.Cooley (pp. 77–78) describes the implementation as ''discrete finite Fourier transform''. or * another name for the Fourier series coefficients. or * another name for one snapshot of a short-time Fourier transform. See also * Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ... Notes Reference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Initialism
An acronym is a type of abbreviation consisting of a phrase whose only pronounced elements are the initial letters or initial sounds of words inside that phrase. Acronyms are often spelled with the initial letter of each word in all caps with no punctuation. For some, an initialism or alphabetism connotes this general meaning, and an ''acronym'' is a subset with a narrower definition; an acronym is pronounced as a word rather than as a sequence of letters. In this sense, ''NASA'' () is an acronym, but '' USA'' () is not. The broader sense of ''acronym'', ignoring pronunciation, is its original meaning and in common use. . Dictionary and style-guide editors dispute whether the term ''acronym'' can be legitimately applied to abbreviations which are not pronounced as words, and they do not agree on acronym spacing, casing, and punctuation. The phrase that the acronym stands for is called its . The of an acronym includes both its expansion and the meaning of its expans ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Digital Circuit
In theoretical computer science, a circuit is a model of computation in which input values proceed through a sequence of gates, each of which computes a function. Circuits of this kind provide a generalization of Boolean circuits and a mathematical model for digital logic circuits. Circuits are defined by the gates they contain and the values the gates can produce. For example, the values in a Boolean circuit are Boolean values, and the circuit includes conjunction, disjunction, and negation gates. The values in an integer circuit are sets of integers and the gates compute set union, set intersection, and set complement, as well as the arithmetic operations addition and multiplication. Formal definition A circuit is a triplet (M, L, G), where * M is a set of values, * L is a set of gate labels, each of which is a function from M^ to M for some non-negative integer i (where i represents the number of inputs to the gate), and * G is a labelled graph, labelled directed acyclic gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Numerical Algorithm
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Computer
A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic sets of operations known as Computer program, ''programs'', which enable computers to perform a wide range of tasks. The term computer system may refer to a nominally complete computer that includes the Computer hardware, hardware, operating system, software, and peripheral equipment needed and used for full operation; or 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, including simple special-purpose devices like microwave ovens and remote controls, and factory devices like industrial robots. Computers are at the core of general-purpose devices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term ''convolution'' refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f*g differs from cross-correlation f \star g only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |