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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 are either +1 or −1 and its rows as well as columns are orthogonal, i.e. dot product is zero. 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. The ''naturally ordered'' Hadamard matrix is defined by the recursive formula below, and the ''sequency-ordered'' Hadamard matrix is formed by rearranging the rows so that the number of sign changes in a row is in increasing order. 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'' ∈ '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walsh 16 * Gould's-Morse
Walsh may refer to: People and fictional characters * Walsh (surname), including a list of people and fictional characters with the surname Places * Fort Walsh, one of the first posts of the Royal Canadian Mounted Police * Walsh, Ontario, Norfolk County, Ontario, Canada * Walsh, Colorado, USA * Walsh, Michigan, USA * Walsh, Wisconsin, USA * Walsh County, North Dakota, USA * Walsh, Alberta, a hamlet in Canada * Walsh Lake (Lac-Jacques-Cartier), Canada * Mount Walsh National Park, Australia Schools * Walsh University, North Canton, Ohio * Walsh College, Troy, Michigan * Walsh School of Foreign Service, Washington, D.C. * Walsh Jesuit High School, Cuyahoga Falls, Ohio Ships * USS ''Walsh'' (APD-111), a United States Navy high-speed transport in commission from 1945 to 1946, originally intended to be a destroyer escort Mathematics * Walsh function, an orthogonal basis of the square-integrable functions on the unit interval * Walsh matrix, an orthogonal matrix with several useful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walsh Transform
The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real). The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size . It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the French mathematician Jacques Hadamard (), the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh. Definition The Hadamard transform ''H''''m'' is a 2''m'' × 2''m'' matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2''m'' re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Code-division Multiple Access
Code-division multiple access (CDMA) is a channel access method used by various radio communication technologies. CDMA is an example of multiple access, where several transmitters can send information simultaneously over a single communication channel. This allows several users to share a band of frequencies (see bandwidth). To permit this without undue interference between the users, CDMA employs spread spectrum technology and a special coding scheme (where each transmitter is assigned a code). CDMA optimizes the use of available bandwidth as it transmits over the entire frequency range and does not limit the user's frequency range. It is used as the access method in many mobile phone standards. IS-95, also called "cdmaOne", and its 3G evolution CDMA2000, are often simply referred to as "CDMA", but UMTS, the 3G standard used by GSM carriers, also uses "wideband CDMA", or W-CDMA, as well as TD-CDMA and TD-SCDMA, as its radio technologies. It can be also used as a channel or m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Transform
The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real). The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size . It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the French mathematician Jacques Hadamard (), the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh. Definition The Hadamard transform ''H''''m'' is a 2''m'' × 2''m'' matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2''m'' re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quincunx Matrix
In mathematics, the matrix : \begin 1 & -1 \\ 1 & 1 \end is sometimes called the quincunx matrix. It is a 2×2 Hadamard matrix, and its rows form the basis of a diagonal square lattice consisting of the integer points whose coordinates both have the same parity; this lattice is a two-dimensional analogue of the three-dimensional body-centered cubic lattice.. See also *Quincunx A quincunx () is a geometric pattern consisting of five points arranged in a cross, with four of them forming a square or rectangle and a fifth at its center. The same pattern has other names, including "in saltire" or "in cross" in heraldry (d ... Notes Matrices {{Linear-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Wavelet
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and extensively used as a teaching example. The Haar sequence was proposed in 1909 by Alfréd Haar. Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval , 1 The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1. The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property can, however, be an advantage for the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gray Code
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representation of the decimal value "1" in binary would normally be "" and "2" would be "". In Gray code, these values are represented as "" and "". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two. Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. Motivation and name Many devices indicate position by closing and opening switches. If that device uses natural binary codes, positions 3 and 4 are next to each other but all three bits of the binary representation differ: : The problem with natural binary codes is that physical switches are not ideal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bit-reversal Permutation
In applied mathematics, a bit-reversal permutation is a permutation of a sequence of n items, where n=2^k is a power of two. It is defined by indexing the elements of the sequence by the numbers from 0 to n-1, representing each of these numbers by its binary representation (padded to have length exactly k), and mapping each item to the item whose representation has the same bits in the reversed order. Repeating the same permutation twice returns to the original ordering on the items, so the bit reversal permutation is an involution. This permutation can be applied to any sequence in linear time while performing only simple index calculations. It has applications in the generation of low-discrepancy sequences and in the evaluation of fast Fourier transforms. Example Consider the sequence of eight letters '. Their indexes are the binary numbers 000, 001, 010, 011, 100, 101, 110, and 111, which when reversed become 000, 100, 010, 110, 001, 101, 011, and 111. Thus, the letter ''a'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product. The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the ''Zehfuss matrix'', and the ''Zehfuss product'', after , who in 1858 described this matrix operation, but Kronecker product is currently the most widely used. Definition If A is an matrix and B is a matrix, then the Kr ... [...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] |
Natural And Sequency Ordered Walsh 16
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are part of nature, human activity is often understood as a separate category from other natural phenomena. The word ''nature'' is borrowed from the Old French ''nature'' and is derived from the Latin word ''natura'', or "essential qualities, innate disposition", and in ancient times, literally meant "birth". In ancient philosophy, ''natura'' is mostly used as the Latin translation of the Greek word ''physis'' (φύσις), which originally related to the intrinsic characteristics of plants, animals, and other features of the world to develop of their own accord. The concept of nature as a whole, the physical universe, is one of several expansions of the original notion; it began with certain core applications of the word φύσις by pre-So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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