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Shapiro Polynomials
In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums. In signal processing, the Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. The first few members of the sequence are: : \begin P_1(x) & =1 + x \\ P_2(x) & =1 + x + x^2 - x^3 \\ P_3(x) & =1 + x + x^2 - x^3 + x^4 + x^5 - x^6 + x^7 \\ ... \\ Q_1(x) & =1 - x \\ Q_2(x) & =1 + x - x^2 + x^3 \\ Q_3(x) & =1 + x + x^2 - x^3 - x^4 - x^5 + x^6 - x^7 \\ ... \\ \end where the second sequence, indicated by ''Q'', is said to be ''complementary'' to the first sequence, indicated by ''P''. Construction The Shapiro polynomials ''P''''n''(''z'') may be constructed from the Golay–Rudin–Shapiro sequence ''a''''n'', which equals 1 if the number of pairs of consecutive ones in the binary expansion of ''n'' is even, and −1 otherwise. Thus ''a''0& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics. Examples Some polynomial sequences arise in physics and approximation theory as the solutions of certain ordinary differential equations: * Laguerre polynomials * Chebyshev polynomials * Legendre polynomials * Jacobi polynomials Others come from statistics: * Hermite polynomials Many are studied in algebra and combinatorics: * Monomials * Rising factorials * Falling factorials * All-one polynomials * Abel polynomials * Bell polynomials * Bernoulli polynomials * Cyclotomic polynomials * Dickson polynomials * Fibonacci polynomials * Lagrange polynomials * Lucas polynomials * Spread polynomials * Touchard polynomials * Rook polynomials Classes of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold S
Harold may refer to: People * Harold (given name), including a list of persons and fictional characters with the name * Harold (surname), surname in the English language * András Arató, known in meme culture as "Hide the Pain Harold" Arts and entertainment * Harold (film), ''Harold'' (film), a 2008 comedy film * ''Harold'', an 1876 poem by Alfred, Lord Tennyson * ''Harold, the Last of the Saxons'', an 1848 book by Edward Bulwer-Lytton, 1st Baron Lytton * ''Harold or the Norman Conquest'', an opera by Frederic Cowen * ''Harold'', an 1885 opera by Eduard Nápravník * Harold, a character from the cartoon List of The Grim Adventures of Billy & Mandy characters#Harold, ''The Grim Adventures of Billy & Mandy'' *Harold & Kumar, a US movie; Harold/Harry is the main actor in the show. Places ;In the United States * Alpine, Los Angeles County, California, an erstwhile settlement that was also known as Harold * Harold, Florida, an unincorporated community * Harold, Kentucky, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Sum
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, Data storage, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was published in the Bell System Technical Journal. The paper laid the groundwork for later development of information c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals. Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance. Unit root processes, trend-stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dista ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudin–Shapiro Sequence
In mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite 2-automatic sequence named after Marcel Golay, Walter Rudin, and Harold S. Shapiro, who independently investigated its properties. Definition Each term of the Rudin–Shapiro sequence is either 1 or -1. If the binary expansion of n is given by :n = \sum_ \epsilon_k(n) 2^k, then let :u_n = \sum_ \epsilon_k(n)\epsilon_(n). (So u_n is the number of times the block 11 appears in the binary expansion of n.) The Rudin–Shapiro sequence (r_n)_ is then defined by :r_n = (-1)^. Thus r_n = 1 if u_n is even and r_n = -1 if u_n is odd. The sequence u_n is known as the complete Rudin–Shapiro sequence, and starting at n = 0, its first few terms are: :0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, ... and the corresponding terms r_n of the Rudin–Shapiro sequence are: :+1, +1, +1, −1, +1, +1, −1, +1, +1, +1, +1, −1, −1, −1, +1, −1, ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudin Shapiro 8 Zeros
''Rudin'' (russian: «Рудин», ) is the first novel by Ivan Turgenev, a famous Russian writer best known for his short stories and the novel '' Fathers and Sons''. Turgenev started to work on it in 1855, and it was first published in the literary magazine "Sovremennik" in 1856; several changes were made by Turgenev in subsequent editions. ''Rudin'' was the first of Turgenev's novels, but already in this work the topic of the superfluous man and his inability to act (which became a major theme of Turgenev's literary work) was explored. Similarly to other Turgenev's novels, the main conflict in ''Rudin'' was centred on a love story of the main character and a young, but intellectual and self-conscious woman who is contrasted with the main hero (this type of female character became known in literary criticism as «тургеневская девушка», “Turgenev girl”). __TOC__ Context ''Rudin'' was written by Turgenev in the immediate aftermath of the Crimean War, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: : History The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L2 Norm
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "\,\leq\," in the homogeneity axiom. It can also re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information. Typically, a device that performs data compression is referred to as an encoder, and one that performs the reversal of the process (decompression) as a decoder. The process of reducing the size of a data file is often referred to as data compression. In the context of data transmission, it is called source coding; encoding done at the source of the data before it is stored or transmitted. Source coding should not be confused with channel coding, for error detection and correction or line coding, the means for mapping data onto a signal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golay Pair
: ''For complementary sequences in biology, see complementarity (molecular biology). For integer sequences with complementary sets of members see Lambek–Moser theorem.'' In applied mathematics, complementary sequences (CS) are pairs of sequences with the useful property that their out-of-phase aperiodic autocorrelation coefficients sum to zero. Binary complementary sequences were first introduced by Marcel J. E. Golay in 1949. In 1961–1962 Golay gave several methods for constructing sequences of length 2''N'' and gave examples of complementary sequences of lengths 10 and 26. In 1974 R. J. Turyn gave a method for constructing sequences of length ''mn'' from sequences of lengths ''m'' and ''n'' which allows the construction of sequences of any length of the form 2''N''10''K''26''M''. Later the theory of complementary sequences was generalized by other authors to polyphase complementary sequences, multilevel complementary sequences, and arbitrary complex complementary sequences. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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