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Fast Walsh–Hadamard Transform
In computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHT''h'') is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive implementation of the WHT of order n = 2^m would have a computational complexity of O(n^2). The FWHT''h'' requires only n \log n additions or subtractions. The FWHT''h'' is a divide-and-conquer algorithm that recursively breaks down a WHT of size n into two smaller WHTs of size n/2. This implementation follows the recursive definition of the 2^m \times 2^m Hadamard matrix H_m: :H_m = \frac \begin H_ & H_ \\ H_ & -H_ \end. The 1/\sqrt2 normalization factors for each stage may be grouped together or even omitted. The sequency-ordered, also known as Walsh-ordered, fast Walsh–Hadamard transform, FWHT''w'', is obtained by computing the FWHT''h'' as above, and then rearranging the outputs. A simple fast nonrecursive implementation of the Walsh–Hadamard transform follows from decomposition of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Walsh Hadamard Transform 8
Fast or FAST may refer to: Arts and entertainment * "Fast" (Juice Wrld song), 2019 * "Fast" (Luke Bryan song), 2016 * "Fast" (Sueco song), 2019 * "Fast" (GloToven song), 2019 * ''Fast'', an album by Custom, 2002 * ''Fast'', a 2010 short film starring Charlyne Yi * " Fast (Motion)", 2021 song by Saweetie * ''Fast & Furious'', an action franchise Computing and software * FAST protocol, an adaptation of the FIX protocol, optimized for streaming * FAST TCP, a TCP congestion avoidance algorithm * Facilitated Application Specification Techniques, a team-oriented approach for requirement gathering * Fatigue Avoidance Scheduling Tool, software to develop work schedules * Features from accelerated segment test, computer vision method for corner detection * Feedback arc set in Tournaments, a computational problem in graph theory Government * Faʻatuatua i le Atua Samoa ua Tasi, a political party in Samoa * Fixing America's Surface Transportation Act, passed by the United St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1010 0110 Walsh Spectrum (fast WHT)
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. In mathematics The number 1 is the first natural number after 0. Each natural number, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divide-and-conquer Algorithm
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g., the Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform ( FFT). Designing efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable to a recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by mathematical induction, and its computational cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. 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 (mathematics), 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 can occur. A process that exhibits recursion is ''recursive''. Video feedback displays recursive images, as does an infinity mirror. 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 reduce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Matrix
In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometry, geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vector (mathematics and physics), vectors, while in combinatorics, combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The ''n''-dimensional Parallelepiped#Parallelotope, parallelotope spanned by the rows of an ''n'' × ''n'' Hadamard matrix has the maximum possible volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrix (mathematics), matrices with entrie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walsh Matrix
In mathematics, a Walsh matrix is a specific square matrix of dimensions 2, where ''n'' is some particular natural number. The entries of the matrix (mathematics), matrix are either +1 or −1 and its rows as well as columns are orthogonal. The Walsh matrix was proposed by Joseph L. Walsh in 1923. Each row of a Walsh matrix corresponds to a Walsh function. The Walsh matrices are a special case of ''Hadamard matrices'' where the rows are rearranged so that the number of sign changes in a row is in increasing order. In short, a Hadamard matrix is defined by the Recursion, recursive formula below and is ''naturally ordered'', whereas a Walsh matrix is ''sequency-ordered''. Confusingly, different sources refer to either matrix as the Walsh matrix. The Walsh matrix (and Walsh functions) are used in computing the Walsh transform and have applications in the efficient implementation of certain signal processing operations. Formula The Hadamard matrices of dimension 2^k for k \in \ ... [...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] |
Digital Signal Processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent Sampling (signal processing), samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor. Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include Audio signal processing, audio and speech processing, sonar, radar and other sensor array processing, spectral density estimation, statistical signal processing, digital image processing, data compression, video coding, audio coding, image compression, signal processing for telecommunications, control systems, biomedical engineering, and seismology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
